Knightian Robustness of the Vickrey Mechanism
Knightian Robustness of the Vickrey Mechanism
arXiv:1403.6413v2 [cs.GT] 1 Apr 2014
Alessandro Chiesa
Silvio Micali
Zeyuan Allen Zhu
MIT April 1, 2014 Abstract We investigate the resilience of some classical mechanisms to alternative specifications of preferences and information structures. Specifically, we analyze the Vickrey mechanism for auctions of multiple identical goods when the only information a player i has about the profile of true valuations, θ∗ , consists of a set of distributions, from one of which θi∗ has been drawn. In this setting, the players no longer have complete preferences, and the Vickrey mechanism is no longer dominant-strategy. However, we prove that its efficiency performance is excellent, and essentially optimal, in undominated strategies.
Keywords: Knightian uncertainty, implementation in undominated strategies, incomplete preferences
1
Introduction
We prove that some classical mechanisms are more robust than previously thought: namely, that they guarantee desirable outcomes even when the players have no beliefs about their opponents, and very limited knowledge about themselves. In particular, we prove that the Vickrey mechanism enjoys such robustness. The Vickrey mechanism efficiently allocates multiple identical goods by ensuring that it is a dominant strategy for each player i to report his true valuation θi∗ . Being dominant-strategy truthful, this mechanism of course (1) works even when the players have no beliefs about their opponents, but also (2) assumes that each player knows his own true valuation precisely. In real life, however, a player i may be uncertain about his own θi∗ , as it may depend on variables that are not directly observable by him. For example, in an auction for the exclusive right to a newly discovered oil well, a player’s valuation may depend on the exact quality of the crude, on the likelihood that other oil wells will be discovered, on the political stability of the country where the well resides, etc. Knightian Valuation Uncertainty. A simple way to capture a player i’s uncertainty about his own valuation is the ‘single-distribution’ model, where i does not know θi∗ , but only the true distribution Di∗ , over the set Θi of all his possible valuations, from which θi∗ has been drawn. We instead investigate a more general form of self uncertainty. Namely, i only knows a set of distributions, from one of which θi∗ has been drawn. We refer to this model as Knightian valuation uncertainty, as it is a special case of the uncertainty model envisaged by Frank H. Knight almost a century ago [Kni21], and later formalized by Truman F. Bewley [Bew02]. Various reasons sometimes make this model more realistic than the single-distribution one. First of all, before the auction starts, a player i may not have the time necessary to compute the true probability distribution Di∗ for θi∗ , but may have sufficient time to distill a set of possible distributions for θi∗ , Ki , which is guaranteed to include Di∗ .1 For instance, assume that, in the time available, i has been able to determine that Di∗ is a Gaussian distribution, but has not been able to learn its mean µ and its standard deviation σ beyond the fact that the first is between 9 and 10, and the second between 1 and 2. Then, Ki = {N (µ, σ) : µ ∈ [9, 11], σ ∈ [1, 3]}. 1
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Secondly, sometimes no single distribution for θi∗ may be very meaningful. Consider an auction of m BitCoins. Bitcoin is a recent, hard to exchange, and wildly fluctuating, cryptocurrency, whose security depends on an unproven mathematical assumption, M A.2 This is an auction of identical goods, and in it a player i’s valuation depends not only on hard-to-observe variables, such as the future legality of BitCoins and the likelihood of the rise of new crypto-currencies, but also on whether M A is true or false. Since the correctness of a mathematical statement is not a probabilistic event, i may find it hard to form Bayesian beliefs about M A. Accordingly, even if all other variables of BitCoins can be precisely captured by a single distribution, i’s valuation is best modeled as being drawn from one of two distinct distributions, Dtrue if M A is true, and Dfalse if M A is false. Thirdly, Knightian valuation uncertainty may also arise from conflicting expert opinions. Consider an auction of a novel technology, where a firm i’s valuation depends on how much the technology might increase i’s productivity. Due to the novelty of the technology, firm i does not know how useful it will prove, and hires k (properly incentivized) independent experts to figure it out, trusting that at least one of them will be right. If each of them reports a different distribution for θi∗ , either because time was limited or because some of the experts made an error, then i is faced with a set of distributions, and believes that θi∗ has been drawn from one of them. Findings. The Vickrey mechanism can certainly be used in multi-unit auctions with Knightian valuation uncertainty, but it would no longer be dominant-strategy. We prove, however, that this is no ‘big loss’. Although dominant-strategy mechanisms for multi-unit auctions do exist,3 we prove that all of them (including those that allow a player to report a set of valuation distributions rather than a single valuation) perform poorly in terms of social-welfare. For the special case of single-good auctions, Theorem 1 points out that no such mechanism, as well as no ex-post Nash mecha2
BitCoins are ‘minted” via a given easy-to-compute function h, and are unforgeable only under the assumption that it is computationally infeasible to find two different strings x and y such that h(x) = h(y). 3 At least the ‘degenerate’ mechanism, which assigns each copy of the good at random, is one of them. In a sense, we prove that all dominant-strategy mechanisms must be de facto degenerate in the Knightian valuation model, even for single-good auctions.
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nism, whether deterministic or randomized, can guarantee higher efficiency than that obtainable by allocating the good at random. Dominant-strategy mechanisms provide designers with the best guarantee that the players will choose the desired strategies. However, one may also be quite confident that a player will not choose a strategy outside his undominated set, and we prove that, no matter which undominated strategies the players might choose, the Vickrey mechanism is guaranteed to return an allocation of very high social welfare. More precisely, Theorem 2 guarantees that, in the Knightian valuation model, the social-welfare performance of the Vickrey mechanism gracefully degrades with the uncertainty level of the players (naturally measured) and the number of copies of the good. Let us emphasize that this performance guarantee is proved without ‘inertia’.4 Of course, the fact that the Vickrey mechanism performs well in undominated strategies does not exclude that different mechanisms may perform even better. Theorem 3, however, shows that the performance of the Vickrey mechanism is essentially optimal among all finite undominated-strategy mechanisms, probabilistic or not. Here, a mechanism is ‘finite’ if it assigns a finite set of strategies to each player. Together, Theorem 1, Theorem 2 and Theorem 3 show that the Vickrey mechanism is very robust to alternative specifications of preferences and information structures. We believe that such robustness is an important property of a mechanism. In our last section, we point out that this robustness is also enjoyed by additional, but not all, classical mechanisms. 4 As originally put forward by Bewley, the inertia assumption states that a player with incomparable options will chose a ‘reference point’ unless a strictly better alternative exists. (Lopomo, Rigotti, and Shannon rely on the inertia assumption when putting forward maximal incentive compatibility.) As we shall argue in our technical sections, in the Vickrey mechanism, a player i with k possible distributions for θi∗ , D1 , . . . , Dk , has k natural reference points to report: namely, E[D1 ], . . . , E[Dk ], where each expectation E[Dj ] is an m-dimensional vector if there are m copies of the good. Indeed, reporting each of these valuations is undominated, and when all the players report these reference points, the Vickrey mechanism performs well. However, these are not the only undominated strategies, and Theorem 2 guarantees that the Vickrey mechanism continues to perform well no matter which undominated strategy each player may report, without any behavioral assumptions.
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2
Related Work
Knightian players have received much attention in decision theory. Aumann [Aum62], Dubra, Maccheroni and Ok [DMO04], Ok [Ok02], and Nascimento [Nas11] investigate decision with incomplete orders of preferences. Various criteria for selecting a single distribution out of a set of distributions have been studied by Danan [Dan10], Schmeidler [Sch89], and Gilboa and Schmeidler [GS89]. (In fact, Bose, Ozdenoren and Pape [BOP06] and Bodoh-Creed [Bod12] use the model from [GS89] to study auctions.) General equilibrium models with incompletely ordered preferences have been considered by Mas-Colell [Mas74], Gale and Mas-Colell [GM75], Shafer and Sonnenschein [SS75], and Fon and Otani [FO79]. More recently, Rigotti and Shannon [RS05] have characterized the set of equilibria in a financial market problem.5 Knightian mechanisms were first considered by Lopomo, Rigotti, and Shannon [LRS09], for the rental extraction problem, in a model different from ours. In their model, there is a single player, whose Knightian uncertainty arises from a variable that he cannot observe, but is exactly known to the mechanism. More precisely, the player’s utility for a given outcome depends on (a) his own type, t ∈ T , which the player knows exactly, and (b) the state of the world, s ∈ S, which only the mechanism knows exactly, while the
player only knows that it is drawn from a distribution in some set Π(t) ⊆ ∆(S).
They proposed two notions of implementation in their Knightian model: optimal incentive compatibility and maximal incentive compatibility. Their first notion corresponds to our Knightian dominant-strategy truthfulness, but does not coincide with it, due to the difference between theirs and our model.6 Their second notion 5
A strategy profile is an equilibrium if no player can deviate and strictly benefit no matter which distribution is picked from his set. Notice that such an equilibrium is not a notion of dominance. 6 Optimal incentive compatibility applies to mechanisms φ : T × S → O mapping the reported type of the player t ∈ T and the true state of the world s ∈ S to an outcome φ(t, s) ∈ O. Formally (see [LRS09, Definition 3]), such a mechanism φ is optimal incentive for all types t and compatible if all mixed strategies σ ∈ ∆(T ): Es∼π U (φ(t, s), t, s) ≥ Es∼π Eθ∼σ U (φ(θ, s), t, s) ∀π ∈ Π(t). By contrast, we consider auctions with multiple players, so that to express Knightian dominance the strategy subprofiles of a player’s opponents must be universally quantified. Moreover, and more importantly, our mechanisms have no information about the players’ true valuations.
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corresponds to a weaker version of our implementation of Knightian undominated strategies, and does not apply to the Vickrey mechanism in our model.7 Lopomo, Rigotti, and Shannon also studied variants of their notions in a principal-agent model with Knightian uncertainty [LRS11]. Implementation in (traditional) undominated strategies was originally proposed by Jackson [Jac92, JPS94]. An example of of such an implementation in the exactvaluation model is given by the mechanism of Babaioff et al. [BLP06] for efficiency in single-value multi-minded auctions.
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Notation and Model
3.1
Notation for Multi-Unit Auctions
In a multi-unit auction there are identical copies of the same good. We denote by n the number of players, and by m the number of copies of the good. The set of all P def possible allocations is A = {A ∈ Zn≥0 : ni=0 ai = m}. In an allocation A, A0 is the
number of unallocated copies and Ai the number of copies allocated to player i.
The set of all possible valuation profiles is Θ = Θ1 × · · · × Θn . Throughout this def
paper, for each player i, Θi = {θi : [m] → R≥0 | θi (1) ≥ · · · ≥ θi (m) ≥ 0}. That is, we
never assume any restrictions on the players’ possible valuations, except for marginal
valuations being non-increasing, the standard assumption envisaged by Vickrey.8 The profile of the players’ true valuations is θ∗ = (θ1∗ , . . . , θn∗ ) ∈ Θ. def
The set of possible outcomes is Ω = A × Rn≥0 . If (A, P ) ∈ Ω, we refer Pi as the
7 In maximal incentive compatibility, truthful reporting is not strictly dominated, there may be additional undominated strategies, for which the mechanism may fail to produce a desired outcome. By contrast, in implementation in Knightian undominated strategies, a mechanism must produce a desired outcome no matter which undominated strategy each player may choose. Moreover, their notion applies only to ‘Knightian direct’ mechanisms, but not to the Vickrey mechanism in our model. Indeed, the Vickrey mechanism allows a player to report only a single valuation, while in our Knightian model a player has a set of possible valuations. Thus, he could not ‘truthfully’ report his candidate set even if he wanted. 8 Multi-unit auctions are sometimes referred to as Vickrey auctions (in particular, see Ausubel and Milgrom [AM06]). Indeed, Vickrey was the first one to provide an efficient mechanism for such auctions [Vic61], and his mechanism was later generalized to become the VCG mechanism. If a player i has a valuation θi ∈ Θi , then θi (j) is his marginal value of receiving the j-th copy of the good. Non-decreasing marginal valuations in multi-unit auctions were indeed envisaged by Vickrey.
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price charged to player i. The utility of a player i, with valuation θi , for an outcome ω = (A, P ) is def
Ui (θi , ω) =
PA i
j=0 θi (j)
− Pi .
If ω is a distribution over outcomes, then Ui (θi , ω) is the expected utility of player i. The social welfare of an outcome ω = (A, P ), and that of an allocation A, where A = (a0 , a1 , . . . , an ), relative to a valuation profile θ, is SW(θ, ω) = SW(θ, A) = P Pai i j=1 θi (j). The maximum social welfare relative to θ, MSW(θ), is maxA∈A SW(θ, A).
The maximum social welfare is MSW = MSW(θ∗ ).
A mechanism M specifies, for each player i, a set Si . We interchangeably refer to each member of Si as a pure strategy/action/report of i, and similarly, a member of ∆(Si ) as a mixed strategy/action/report of i.9 After each player i, simultaneously with his opponents, reports a strategy si in Si , M maps the reported strategy profile s to an outcome M (s) ∈ Ω. If M is probabilistic, then M (s) ∈ ∆(Ω).
3.2
Knightian Valuation Uncertainty
In our model, a player i’s sole information about θ∗ consists of Ki , a set of distributions
over Θi , from one of which θi∗ has been drawn. (The true valuations are uncorrelated.) That is, Ki is i’s sole (and private) information about his own true valuation θi∗ .
Furthermore, for every opponent j, i has no information (or beliefs) about θj∗ or Kj .
For auctions, because each Θi is convex, this model has an equivalent, but non-
distributional formulation. Definition 3.1 (Knightian valuation model). For each player i, i’s sole information about θ∗ is a set Ki , the candidate (valuation) set of i, such that θi∗ ∈ Ki ⊂ Θi . We refer to an element of Ki as a candidate valuation.
def
The set of all possible candidate sets of a player i is Ki = {Ki ⊆ Θi : Ki 6= ∅}. 9
Often, in pre-Bayesian settings, the notion of a strategy and that of an action are distinct. Indeed, a strategy si of a player i maps the set of all possible types of i to the set of i’ possible actions/reports. But since strategies are universally quantified in all relevant definitions of this paper, we have no need to separate (and for simplicity refrain from separating) the notions of strategies and actions.
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The reason behind the above equivalence is that, in auctions, given that all a player i cares about is his expected (quasi-linear) utility, he may ‘collapse’ each distribution Di ∈ Ki to its expectation Eθi ∼Di [θi ].10
In the Knightian valuation model, a mechanism’s performance will of course de-
pend on the inaccuracy of the players’ candidate sets, which we measure as follows. Definition 3.2. For all players i, candidate set Ki , and copies j ∈ [m], we let def
Ki (j) = {θi (j) | θi ∈ Ki },
def
Ki⊥ (j) = inf Ki (j),
and
def
Ki> (j) = sup Ki (j).
A candidate set Ki is (at most) δ-approximate if Ki> (j) − Ki⊥ (j) ≤ δ for all j ∈ [m].
An auction is (at most) δ-approximate if the set of all possible candidate sets of a def
player i is Kδi = {Ki ∈ Ki : Ki is δ-approximate}. Remarks 1. Non-Convexity. Note that a candidate set Ki may not be convex. This fact may be puzzling, because in a typical single-good auction we expect that a player having a and b as two possible valuations must also have
a+b 2
as another possible
valuation. Let us stress that ‘the possibility of holes’ in Ki is not a restriction, nor a speciousness of our model. To the contrary, it is the natural sub-product of the generality of the Knightian setting. To make our results stronger, when proving that a mechanism performs well, we consider all possible candidate sets, including non convex ones. When proving that a mechanism performs poorly, we consider candidate sets that are convex. 10
For example, in a 2-unit auction, a player’s valuation consists of a pair (v(1), v(2)) where v(1) ≥ v(2) ≥ 0. Suppose a player i knows that θi∗ is drawn from a distribution Di∗ over such pairs. In def addition, let (e1 , e2 ) = Eθi ∼Di θi (1), θi (2) ∈ Θi , and let ω be an outcome in which i wins exactly one copy of the good. Then, i’s expected utility for ω is Eθi ∼Di [θi (1)] − p = e1 − p, as if his θi∗ were (e1 , e2 ). Since this is true for every (even probabilistically chosen) outcome, i may very well act as if his true valuation were exactly θi∗ = (e1 , e2 ). Let us now be more abstract. The desired equivalence could be formalized in three steps. Consider a player i only knowing a set of distributions Ki from one of which θi∗ has been drawn. In a first step, i can be identified with a player only knowing a multiset of valuations Ki0 , containing his true valuation, if there exists a bijection φ : Ki ↔ Ki0 such that, for all outcome ω and Di ∈ Ki , Eθi ∼Di [Ui (θi , ω)] = Ui (φ(Di ), ω). In a second step, the latter player can be identified with a player only knowing a set of valuations Ki , containing his true valuation and coinciding with Ki0 after removing redundant valuations. In a third step, we choose φ to be the function mapping a distribution D to its expectation. In the above proof sketch we rely on the fact that φ(Di ) = E[Di ] exists in Θi for all Di ∈ ∆(Θi ), which is indeed the case because Θi is convex. The above equivalence thus holds not only for multi-unit auctions, but also for any auction where each Θi is a convex set.
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2. Beliefs. Assume that Ki = {a, b}. Then, despite our prior remark, it may still
be hard to accept that player i may know his true valuation to be either a or b, without forming some (even partial) beliefs. For instance, without exactly knowing the probabilities of a and b, i may believe that his true valuation is more probable to be a than b. Note, however, that this belief corresponds to the player having a set of distributions Ki0 = {Dp : p ∈ [0.5, 1]} where each Dp is a distribution taking value a with probability p, and value b with probability 1 − p. Again, i may as well collapse each distribution Dp to its expected value, so as to end up, de facto, with a set of candidate valuations: namely, Ki0 = {pa + (1 − p)b : p ∈ [0.5, 1]} ⊆ Θi . In other words, candidate sets may be very expressive. When we say that the candidate set is Ki , we assume that all (partial) beliefs that player i may have about his own valuation θi∗ have already been taken into account.
˙ 3. Additive vs. Multiplicative Inaccuracy. We could have measured Knightian uncertainty via a multiplicative, rather than additive, parameter δ. Although all results in this paper can be restated for a multiplicative δ (see Appendix D), the additive version provides the simplest proofs and is more meaningful.11 4. Richness of Candidate Sets. In a δ-approximate auction, the mechanism could face all possible δ-approximate candidate sets Ki : for instance, {2, 2 + δ},
[100, 100 + δ/2], or [200, 200 + δ] \ {200 + δ/i : i = 2, 3, . . .}.
This richness of candidate sets is important, because the Vickrey mechanism performs well in the classical model for all possible valuations, that is, for Θi as defined in section 3.1. 11 For simplicity, consider a single-copy (i.e., single-good) auction. Here, the multiplicative version of the inaccuracy can be defined as follows. Let δ be in [0, 1]. A candidate set Ki is δ-approximate if inf Ki ≥ (1 − δ) · sup Ki . The auction is δ-approximate if each Ki is δ-approximate. Suppose now that K1 = [1000, 1010] and K2 = [10, 20]. Then, the maximum social welfare of this auction is at least 1000, no matter what the true valuations of the two players might be. Using our additive definition, this auction is only 10-approximate, where 10 is small relative to the maximum social welfare. By contrast, using the multiplicative definition, the auction is 50%-approximate. That is, it has the same inaccuracy as when K1 = [1000, 2000] and K2 = [10, 20].
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The richness of candidate sets makes the robustness of the Vickrey mechanism —that is Theorem 2— stronger. Moreover, as we shall see, it is crucial for our negative results —that is, for Theorems 1 and 3. 5. Social welfare. Note that the true social welfare of an outcome ω = (A, P ), that of an allocation A, and the true maximum social welfare continue to be SW(θ∗ , ω), SW(θ∗ , A), and MSW(θ∗ ), whether or not every player i knows θi∗ exactly.
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First Theorem
It is easy to see (but anyway proved by Lopomo, Rigotti and Shannon [LRS09] for their model, and in Appendix A for our model) that, in line with the revelation principle [Gib73, DHM79, Mye79], if a social choice correspondence f is implementable in Knightian dominant strategies or ex-post Nash equilibrium, then f is also implementable by a Knightian dominant-strategy-truthful mechanism. We thus state our first theorem in terms of Knightian DST mechanisms, which we define below. Definition 4.1. A mechanism is Knightian direct if Si = Ki for each player i. Such a mechanism M is Knightian dominant-strategy-truthful (Knightian DST) if ∀Ki , Ki0 ∈ Ki ∀K−i ∈ K−i ∀θi ∈ Ki Ui θi , M (Ki , K−i ) ≥ Ui θi , M (Ki0 , K−i ) .
Theorem 1. For all (possibly probabilistic) Knightian DST mechanisms M for single-good auctions, and for all δ > 0, there exist profiles K of δ-approximate candidate sets such that MSW(θ) +δ . E SW(θ, M (K)) ≤ n Above, the expectation is over the possible random choices of the mechanism M . ∀θ ∈ K1 × · · · × Kn
The proof of Theorem 1 can be found in Appendix B.12 12
Note that our proof of Theorem 1 continues to hold even when M is allowed to know δ in advance. In other words, the proof holds when M knows that every Ki comes from Kδi . (The Knightian DST requirement can be easily weakened by a δ-DST requirement, where the mechanism only takes as input δ-approximate candidate sets.)
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Here, we just wish to make three remarks. 1. Visualization of Bad Cases. To appreciate how intrinsically inadequate DST (and ex-post Nash) mechanisms are in single-good Knightian auctions, consider the following example. Let n = 3, δ = 10, let M be any dominant-strategy (or ex-post Nash) mechanism, and let the players have the following δ-approximate candidate sets: Case 1: K1 = [9, 10]
K2 = [9, 10]
Case 2: K1 = [9, 10]
K2 = [1000, 1010] K3 = [9, 10]
Case 3: K1 = [1000, 1010] K2 = [9, 10]
K3 = [1000, 1010] K3 = [9, 10].
Then, when the players choose a profile of dominant (or ex-post Nash equilibrium) strategies, Claim B.2 shows that (a) if M is deterministic, then it must assign the good to a ‘low-valuation player’ for at least one of the three cases above; and (b) if M is probabilistic, then it must assign the good to the ‘high-valuation player’ with probability ≥ 1/3 for at least one of the three cases above. 2. Richness of Candidate Sets. In our model, where candidate sets are unconstrained, it is natural to require that a good mechanism must perform well for all possible candidate sets of the players.13 Only for analyzing a mechanism’s performance may we restrict our attention to δ-approximate candidate sets. This richness of candidate sets is crucially relied upon by our proof of Theorem 1, but is not a limitation of our proof techniques. Indeed, if the players’ possible candidate sets were sufficiently restricted, then dominant-strategy mechanisms could actually guarantee maximum social welfare.14 13
In the model of [LRS09] this is unnecessary. Indeed, the optimal incentive compatible mechanism of their Theorem 4 achieves full rent extraction if, in our language, each possible candidate set of the single player is disjoint from the convex hull of the union of all other candidate sets he may have. 14 For instance, assume that, in an n-player auction of a single good, for some δ < 1/3, the only possible candidate sets for player i are of the form [kn + i, kn + i + δ] where k is an arbitrary positive integer. Let M be the mechanism such that (1) Si = {kn + i : k ∈ Z+ } for each player i, (2) for all s ∈ S1 × · · · × Sn , M (s) = 2P(s), where 2P is the second-price mechanism. Then it is clear that (a) for player i whose candidate set is [kn + i, kn + i + δ], reporting kn + i is a dominant strategy, and (b) when dominant strategies are played, M produces an outcome with maximum social welfare.
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3. Ubiquity of Bad Cases. The proof of Theorem 1 not only shows the existence of bad cases for every possible dominant-strategy truthful mechanism M for single-good auctions, but also that such bad cases are ‘extremely abundant’. In particular, if M is deterministic, Claim B.1 implies the following. For all players i and strategy subprofiles K−i of i’s opponents, if there exists a candidate set Ki of cardinality > 1 such that i does not get the good in M (Ki , K−i ), then, for all candidate sets Ki0 of cardinality > 1, i does not get the good in M (Ki0 , K−i ).
4.1
Intuition for Proving Theorem 1
Denote by MiA (Ki , K−i ) the probability that player i wins the good in mechanism M when players report the profile (Ki , K−i ). The key point in our proof is to show that, for every K−i , MiA (Ki , K−i ) = MiA (Ki0 , K−i ) whenever |Ki ∩ Ki0 | ≥ 2. Fixing the choice of K−i , let us view all the candidate sets of player i as nodes in a graph, and connect two nodes Ki and Ki0 whenever they share two candidate valuations. If the player’s candidate sets are rich enough so that this graph is connected, then the probability MiA ( · , K−i ) must be constant regardless of player i’s report. From this, it immediately follows that M cannot guarantee non-trivial social welfare.
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Second Theorem
We wish to prove that the social-welfare performance of the Vickrey mechanism gracefully degrades with δ, no matter which undominated strategies the players may choose. Recall that the Vickrey mechanism, denoted by Vickrey, is a direct mechanism (i.e., satisfies Si = Θi ) and maps a profile of valuations θ ∈ Θ1 × · · · × Θn , to an outcome P Pk (A, P ); where A ∈ arg maxA∈A SW(θ, A), Pi = MSW(θ−i ) − k6=i aj=1 θk (j), and
possible ties are broken lexicographically.15 15
More precisely, on a reported valuation profile θ, the Vickrey mechanism sorts all the values {θi (j) : i ∈ [n], j ∈ [m]} in a non-increasing order, and then chooses the m largest entries to assign the m copies of the good. If ties occur in this ordering, that is, θi (j) = θi0 (j 0 ), then θi (j) precedes θi0 (j 0 ) if and only if either (1) i < i0 or (2) i = i0 and j < j 0 . This technical assumption can be replaced with any other order that only resorts to the players’ names to break ties.
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For the Knightian valuation model, in this section, we prove in Theorem 2 that Vickrey delivers high social welfare in (Knightian) undominated strategies, which we define below. Definition 5.1. In a mechanism M , a pure strategy si ∈ Si of a player i is (weakly) dominated by another possibly mixed strategy σi ∈ ∆(Si ) of i with respect to his Ki , in symbols si ≺(i,Ki ) σi , if
(1) ∀θi ∈ Ki ∀s−i ∈ S−i (2) ∃θi ∈ Ki ∃s−i ∈ S−i
Ui θi , M (σi , s−i ) ≥ Ui θi , M (si , s−i ) , and Ui θi , M (σi , s−i ) > Ui θi , M (si , s−i ) .16
A strategy si ∈ Si is (weakly Knightian) undominated, if there exists no σi ∈ ∆(Si )
such that si ≺(i,Ki ) σi . We denote the set of such undominated strategies by UDi (Ki ). If K is a product or a profile of candidate sets, that is, if K = (K1 , . . . , Kn ) or def
K = K1 × · · · × Kn , then UD(K) = UD1 (K1 ) × · · · × UDn (Kn ). Our notion of an undominated strategy intends to capture the ‘weakest condition’ for which si should be discarded in favor of σi . Note that it is a natural extension of its classical counterpart. In particular, we allow for the possibility of a pure strategy si to be dominated by a mixed one σi .17 Of course, other extensions are also possible. However, they either (1) fail to capture our intended ‘weakest condition’, (2) fail to coincide with the classical notion when players know their valuations exactly, or (3) are equivalent to ours under mild 16
This notion is thus different from strong dominance, where inequality (1) is always strict. For strong dominance in the the exact-valuation case, see, for instance, [FT91, LS08]. 17 We allow for this possibility because, also in the Knightian setting, a pure strategy si may not be dominated by any other pure strategy ti , while being dominated by some mixed strategy σi . Let us construct a simple example, outside auctions, by modifying a classical example for strategic-form games. Consider two players, where the row player has two actions, Up and Down, the column player has three actions, Left, Middle and Right, and the column player’s utilities 1 3 0 are as follows: . Then, for the column player, Left is not dominated by Middle or 1 0 3
Right, but is dominated by half Middle plus half Right. This phenomenon generalizes to that when the column player has Knightian uncertainties, so that his utilities are now intervals as follows: [0.8, 1.2] [2.7, 3.3] [0, 0.3] . In this game, Left is still dominated by ‘1/2 Middle + 1/2 Right’, [0.8, 1.2] [0, 0.3] [2.7, 3.3] but not by Middle or Right alone.
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conditions.18 Finally, the difficulty of defining undominated strategies pointed out by Jackson [Jac92] does not apply to our paper.19 We are now ready to state our second theorem. Theorem 2. In a m-unit Knightian auction, for all δ, all products K of δ-approximate candidate sets, all profiles v ∈ UD(K), and all θ ∈ K SW θ, Vickrey(v) ≥ MSW(θ) − 2mδ .
5.1
Intuition for Proving Theorem 2
The good social-welfare performance of the Vickrey mechanism follows from a structural lemma, Lemma 5.2, guaranteeing that, for all players i, all candidate sets Ki , and all copies j: If a strategy vi ∈ UD(Ki ), then vi (j) ∈ Ki⊥ (j), Ki> (j) .
It is immediate to see that this condition holds in the simpler case of a single-copy (i.e., single-good) auction, where the Vickrey mechanism coincides with the secondprice one. Indeed, assume that for some player i, Ki⊥ = a and Ki> = b. Then, it is easy to check that reporting any value greater than b (respectively, smaller than a) is (weakly) dominated by reporting precisely b (respectively, a). 18 Our extension desires to capture the ‘weakest condition’ for which si should be discarded in favor of σi . To express condition (2) in Definition 5.1, we must quantify the true valuation θi ∈ Ki and the pure strategy subprofile of i’s opponents s−i ∈ S−i . There are three alternative possibilities to consider. Namely, (a) ∀θi ∀s−i , (b) ∃θi ∀s−i , and (c) ∀θi ∃s−i . Alternatives (a) and (b) do not yield the classical notion of (weak) dominance when Ki is a singleton. Alternative (c) fails to capture our desideratum. (Indeed, since σi is already no worse than si , for player i to discard strategy si in favor of σi , it should suffice that si be strictly worse than σi for a single possible valuation θi ∈ Ki .) Alternatively, one may also consider defining an undominated strategy in the spirit of an equilibrium notion in a pre-Bayesian game. Here, a strategy of player i is a function fi mapping i’s candidate set Ki to an action. Accordingly, a profile of strategies (f1 , . . . , fn ) is ‘undominated’ if, for each player i, all K, and all θ ∈ K , there is no σ such that U θ , M (σ , f (K ) ≥ i i i i i i −i −i Ui θi , M (fi (Ki ), f−i (K−i ) , where at least one inequality is strict. Note, however, that this notion coincides with ours if each strategy fi is onto. That is, if for every possible action si , there exists some Ki such that si = fi (Ki ). Also note that, in the case of the Vickrey mechanism, fi should be onto. Indeed, when Ki is a singleton, Ki = {si }, player i should only want to play si , and thus fi ({si }) should coincide with si . 19 As pointed out by Jackson [Jac92] in the exact-valuation case, the general notion of an undominated strategy is more complex. However, for bounded mechanisms, the simpler notion above coincides with the general notion, even in the Knightian setting. Since this class of mechanisms includes the Vickrey mechanism and all finite ones, we adopt this simpler notion for this paper.
13
Consider now the Vickrey mechanism in a multi-unit auction, and let aj = Ki⊥ (j) and bj = Ki> (j). Here, it is again intuitively clear that reporting a valuation vi0 such that vi0 (j) < aj (resp. vi0 (j) > bj ) for all j ∈ [m] is a dominated strategy for player
i. However, i may consider some strategy vi00 that bids above bj for some copy j and below ak for some other copy k. Such a strategy vi00 is, in general, not dominated by “bidding bj for all j ∈ [m]” or “bidding aj for all j ∈ [m]”. However, one can still carefully construct a strategy vi∗ that dominates vi00 .
More specifically, we construct vi∗ from vi00 by changing only one, but carefully chosen, coordinate j ∗ ∈ [m]. This allows us to argue that, for all v−i ∈ Θ−i , the
outcomes Vickrey(vi∗ , v−i ) and Vickrey(vi00 , v−i ) differ minimally. That is, if player i receives a different number of copies in these two outcomes, then it must be that i has received precisely j ∗ copies in the first outcome, and precisely j ∗ − 1 (or j ∗ + 1) in
the second. By taking into account the Vickrey prices, it becomes easy to conclude that vi∗ dominates vi00 .
5.2
A Structural Lemma
Let us now formally state and prove our structural lemma. Lemma 5.2. In Vickrey, for each player i, candidate set Ki , and copy j vi ∈ UDi (Ki ) ⇒ vi (j) ∈ [Ki⊥ (j), Ki> (j)]. Proof of Lemma 5.2. Recall that the Vickrey mechanism is direct, that is, Si = Θi for all players i, and that multi-unit auctions have non-increasing marginal valuations, that is, θi (1) ≥ θi (2) ≥ · · · ≥ θi (m) for each θi ∈ Θi . ⊥
⊥
>
This implies
>
that Ki (1), . . . , Ki (m) and Ki (1), . . . , Ki (m) are non-decreasing sequences. Thus, K > , K ⊥ ∈ Θ. That is, both Ki> and Ki⊥ are valid strategies in the Vickrey mechanism. We start by proving, by contradiction, that
vi ∈ UDi (Ki ) =⇒ vi (j) ≥ Ki⊥ (j) for all j ∈ [m].
(5.1)
Assume that implication (5.1) is false; let j ∗ ∈ [m] be the first coordinate j such that
14
vi (j) < Ki⊥ (j); and define the function vi∗ : [m] → R≥0 as follows: v (j), if j 6= j ∗ ; i ∗ vi (j) = K ⊥ (j), if j = j ∗ . i
Since vi and K ⊥ are monotonically non-increasing, so is vi∗ . Indeed, • if j ∗ > 1, then vi∗ (j ∗ − 1) = vi (j ∗ − 1) ≥ Ki⊥ (j ∗ − 1) ≥ Ki⊥ (j ∗ ) = vi∗ (j ∗ ) • if j ∗ < m, then vi∗ (j ∗ ) = Ki⊥ (j ∗ ) > vi (j ∗ ) ≥ vi (j ∗ + 1) = v ∗ (j ∗ + 1) . Thus also vi∗ is a valid valuation in Θi . We now reach a contradiction by showing that vi∗ weakly dominates vi , that is, ∀ θi ∈ Ki , ∀ v−i , 0 ∃ θi0 ∈ Ki , ∃ v−i ,
Ui θi , Vickrey(vi∗ , v−i ) ≥ Ui θi , Vickrey(vi , v−i ) , 0 0 Ui θi0 , Vickrey(vi∗ , v−i ) > Ui θi0 , Vickrey(vi , v−i ) .
(5.2) (5.3)
To show (5.2), choose arbitrarily v−i ∈ Θ−i , and consider the following two cases: (1) In Vickrey(vi∗ , v−i ) and Vickrey(vi , v−i ), i receives the same number of copies. In this case, inequality (5.2) holds because its two sides are equal for all θ. (2) In Vickrey(vi∗ , v−i ) and Vickrey(vi , v−i ), i receives a different number of copies. In this case, one can carefully verify that player i wins j ∗ copies of the good in Vickrey(vi∗ , v−i ) and only j ∗ − 1 copies in Vickrey(vi , v−i ).20 Thus, (5.2) holds
because of the following two reasons.
(a) i’s price for his extra j ∗ -th copy of the good is ≤ K ⊥ (j ∗ ).
Indeed, being dominant-strategy in the classical setting, Vickrey guar-
antees that i pays for his j ∗ -th copy at most the value he reports for it. 20
Recall that, when each player reports a valuation vi , the Vickrey mechanism orders the nm values {vi (j) : i ∈ [n], j ∈ [m]} (and breaking ties lexicographically), and allocates the m copies of the good by looking at the first m values in this order. Since the only difference between vi∗ and vi is that vi∗ (j ∗ ) > vi (j ∗ ), the ordering of the reported nm values is minimally affected. That is, if player i receives different numbers of copies in outcome (vi , v−i ) and outcome (vi∗ , v−i ), then it must be that vi (j ∗ ) is outside the largest m sorted numbers under (vi , v−i ), but vi∗ (j ∗ ) is within the largest m entries under (vi∗ , v−i ). This implies that i wins j ∗ − 1 copies in Vickrey(vi , v−i ) but j ∗ copies in Vickrey(vi∗ , v−i ).
15
(b) i’s value for this j ∗ -th copy is ≥ K ⊥ (j ∗ ).
Indeed, for any possible choice of θi in Ki , θi (j ∗ ) ≥ K ⊥ (j ∗ ).
Therefore, inequality (5.2) holds. Let us now show that also inequality (5.3) holds. To do so, we need to construct a ‘witness’ candidate valuation θi0 ∈ Ki and a ‘witness’
0 0 strategy sub-profile v−i . In fact, we will construct some v−i so that (5.3) holds for all 0 θi0 . Let v−i be such that for any player k 6= i,
vi (j ∗ ) + Ki⊥ (j ∗ ) < Ki⊥ (j ∗ ) ∀j ∈ [m] =x= 2 ∗ 0 Then, player i wins exactly j copies in Vickrey(vi∗ , v−i ) and exactly j ∗ − 1 copies vk0 (j)
def
def
0 ). Indeed, there are exactly j ∗ − 1 numbers greater than x in vi , in Vickrey(vi , v−i
0 while exactly j ∗ of such in v ∗ . Moreover, i’s price in Vickrey(vi∗ , v−i ) is < Ki⊥ (j ∗ ). In
fact, his price for the j ∗ -th copy of the good coincides with the maximum reported
valuation of another player for such an extra copy. Thus, (5.3) holds for all possible 0 θi0 ∈ Ki and the so constructed v−i .
Since both (5.2) and (5.3) hold, valuation vi∗ (weakly) dominates vi , contradicting
the hypothesis that vi ∈ UDi (Ki ). The contradiction proves (5.1). An absolutely symmetrical argument shows that21
vi ∈ UDi (Ki ) =⇒ vi (j) ≤ Ki> (j) for all j ∈ [m]. Together, statements (5.1) and (5.4) imply that Lemma 5.2 holds.
5.3
(5.4)
Proof of Theorem 2
Let v ∈ UD(K) be any profile of undominated strategies, and a0 , a1 , . . . , an represent the allocation in the outcome Vickrey(v), where each player i receives ai
copies of the goods, and a0 is the number of unallocated copies. For any θ ∈ K,
let b0 , b1 , . . . , bn represent the allocation that maximizes social welfare under θ, i.e., In this symmetrical case, one needs to define j ∗ ∈ [m] to be the last coordinate such that vi (j ) > Ki> (j). 21
∗
16
MSW(θ) =
Pn Pbi i=1
`=1 θi (`). ai n X X
SW(θ, Vickrey(v)) =
i=1 `=1
Then, (1)
θi (`) ≥
ai n X X i=1 `=1
ai n X (2) X Ki> (`) − δ ≥ vi (`) − δ i=1 `=1
ai bi bi n n n (3) X X (4) X X (5) X X ⊥ ≥ vi (`) − mδ ≥ vi (`) − mδ ≥ Ki (`) − mδ (6)
≥
Above
i=1 `=1 bi n X X i=1 `=1
i=1 `=1
i=1 `=1
(7) (θi (`) − δ) − mδ ≥ MSW(θ) − 2mδ.
• Inequality (1) holds because by hypothesis θ ∈ K and accordingly θi (`) ≥ Ki⊥ (`) ≥ Ki> (`) − δ;
• Inequality (2) holds by Lemma 5.2;
• Inequality (3) holds because we have only m copies of the good, that is, m;
Pn
i=1
ai ≤
• Inequality (4) holds because the Vickrey mechanism maximizes social welfare
with respect to v, and thus relative to v, (a0 , . . . , an ) is no worse than any other allocation, and in particular (b0 , . . . , bn );
• Inequality (5) holds again by Lemma 5.2;
• Inequality (6) holds because by hypothesis θ ∈ K and accordingly θi (`) ≤ Ki> (`) ≤ Ki⊥ (`) + δ; and finally
• Inequality (7) holds by the definition of MSW(θ) and
5.4
The Bound of Theorem 2 is Tight
Pn
i=1 bi
≤ m.
Having established Theorem 2, let us now illustrate that its performance bound is actually tight by means of the following simple example. Example 5.3. Consider the two-player δ-approximate auction in which, for some x > 0, the candidate sets are K1 = θ1 ∈ Θ1 ∀j ∈ [m] θ1 (j) ∈ [x, x + δ] K2 = θ2 ∈ Θ2 ∀j ∈ [m] θ2 (j) ∈ [x + δ, x + 2δ] .
In this case, the Vickrey mechanism may miss the maximum social welfare by 2δm
17
as follows. Player 1 is ‘optimistic’ and bids the valuation v1 = (x + δ, . . . , x + δ); player 2 is ‘pessimistic’ and bids v2 = (x + δ, . . . , x + δ); the Vickrey mechanism (with the lexicographic tie-breaking rule) allocates all copies of the good to player 1; the true valuation θ1∗ of player 1 is (x, . . . , x); and the true valuation θ2∗ of player 2 is (x + 2δ, . . . , x + 2δ). Accordingly, the realized social welfare is xm, while the maximum one is xm+2mδ.
6
Third Theorem
Although the social welfare performance of the Vickrey mechanism, as guaranteed by Theorem 2, can be considered good, it is legitimate to wonder whether a different undominated-strategy mechanism might enjoy an even better performance. Consider again Example 5.3. In principle, there may be another undominated strategy mechanism missing the maximum social welfare by at most δm. Indeed, even a mechanism M allowing each player to report a single valuation might be able to do so, if it ensures that the undominated strategies for player 1 (respectively, player 2) consist of bidding a value close to x + δ/2 (respectively, x + 3δ/2) for each copy. More generally, there may exist an undominated-strategy mechanism that, allowing each player to report a set of valuations, guarantees social welfare ≥ MSW − δm, not only in the above example but in all δ-approximate multi-unit Knightian auctions.
Theorem 3, however, rules out the existence of such mechanisms, so long as they give each player a finite set of strategies. The optimality of the Vickrey mechanism relative to all finite mechanisms is significant, because in practice all mechanisms are finite, and because finite mechanisms may perform very well in Knightian auctions. The latter mechanisms include not only the discretized Vickrey mechanism,22 which elicits a single valuation from each player, but also more general mechanisms, which, 22
The Vickrey mechanism for an m-unit auction can be discretized by restricting each player i def 20 to report a valuation in Fi = vi ∈ Θi : vi (j) ∈ {0, 0.01, 0.02, . . . , 10 } ∀j ∈ [m] . (Indeed, one may prefer that all payments be made in cash, and thus ultimately in cents, rather than —say— implemented via probabilistic lotteries.) Then, it is easy to check that Theorem 2 continues to hold with only an additive 0.01m loss in the social welfare guarantee, provided that no player valuation exceeds 1020 . Since the discretized Vickrey mechanism performs well in undominated strategies, in principle finite mechanisms that are free to choose their outcome functions and strategy sets may perform even better. Thus, Theorem 3 highlights how clever the Vickrey mechanism really is.
18
in particular, may allow a player to report a subset of a finite set of valuations.23 Remark.
We note that this is the only place in our paper where the mechanism
is assumed to be finite. In Section 6.1, when providing intuition for Theorem 3, we highlight how the finiteness of the pure strategies spaces is relied upon in our proof. Theorem 3. Let M be a finite mechanism in a δ-approximate multi-unit Knightian auction with n > 1 players and δ > 0.24 Then, for every ε > 0, there exist products K of δ-approximate candidate sets, valuation profiles θ ∈ K, and undominated strategy profiles s ∈ UD(K), such that E SW(θ, M (s)) ≤ MSW(θ) − 2δm(1 − 1/n) + ε .
Above, the expectation is over the possible random choices of the mechanism M .
6.1
Intuition for Proving Theorem 3
The key idea behind our proof is best illustrated when there is only one copy of the good for sale. In this special case, as in the general one, there are two main steps in the proof. Let M be a finite mechanism in a δ-approximate single-good Knightian auction. In the first step (corresponding to our structural lemma 6.1), we show that if (a) e i , of the same player i, overlap non-trivially, then (b) two candidate sets Ki and K e i ) essentially have a common strategy. UDi (Ki ) and UDi (K 23
For instance, a finite mechanism may also allow each player to report an arbitrary subset of Fi , as defined in Footnote 22. Such a strategy space clearly enables each player to report, if he so wishes, a reasonable approximation to his own candidate set Ki . For instance, in a single-good auction where i’s true candidate set is Ki = [100, 110], i would be able to report {100.00, 100.01, 100.02, . . . , 110.00}. It should also be appreciated that a finite mechanism may allow each player to report an ‘infinite object’. For instance, again in single-good auctions, although candidate sets may be arbitrary subsets of the reals, a finite mechanism may allow a player to report a set of intervals, overlapping or not, of the form [a, b], where a and b are multiples of 0.01 and less than 1020 . In fact, there are finitely many such intervals. (Although each such [a, b] consists of infinitely many reals, it can be reported by means of its two extreme points, that is, by means of two numbers of at most 22 decimal digits.) Of course, each player i should pay close attention to his Ki and to the outcome function of the mechanism, in order to choose which of these finitely many intervals to report. 24 Note that M may be probabilistic, and may even know δ in advance. The latter point is important because in a single-good auction one expects the distance between every two valuations of the same Ki to be upper-bounded by a small value δ, which would therefore be de facto known to an auction mechanism.
19
e i has at In the case of a single good auction, property (a) means that Ki ∩ K
least two values in common. The strongest way to express property (b) is that e i ) 6= ∅. A weaker way to express property (b), which is nonetheless UDi (Ki ) ∩ UDi (K
sufficient for our second step, is the following: there exist mixed strategies σi ∈ e i )) that are sufficiently close, meaning that for any ∆(UDi (Ki )) and σ ei ∈ ∆(UDi (K strategy subprofile s−i ∈ S−i , in the outcomes M (σi , s−i ) and M (e σi , s−i ), i’s winning
probability and price are close enough. This is the only place in our proof that relies on the finiteness of the pure strategy space Si .25
In the second step, we construct a candidate set profile K = (K1 , . . . , Kn ), a valuation profile θ ∈ K, and a strategy profile s ∈ UD(K) as follows.
For each player i, we first consider two possible δ-approximate candidate sets: e i = [v, v − δ] and K e i0 = [v − 2δ + ε, v − δ + ε], where ε > 0 is some sufficiently K e i and K e 0 overlap nontrivially, the first small positive value, and v > 2δ. Since K i
step applies. For simplicity, suppose that it implies the existence of a pure strategy e i ) ∩ UDi (K e 0 ). Again for simplicity, let M be deterministic. Let the si ∈ UDi (K i
outcome ω = M (s1 , . . . , sn ), and, without loss of generality, assume that player 1 e 0 ) and θ = (v, v − e1, K e0, . . . , K does not win the good in ω. We now choose K = (K 2
n
2δ + ε, . . . , v − 2δ + ε) ∈ K. Under the strategy profile s = (s1 , . . . , sn ), the social welfare is SW(θ, ω) = v − 2δ + ε, while the maximum social welfare is MSW(θ) = v.
This shows SW(θ, M (s)) ≤ MSW(θ) − 2δ + ε. When M is probabilistic, this bound
becmes E[SW(θ, M (s))] ≤ MSW(θ) − 2δ(1 − 1/n) + ε, which coincides with the bound of Theorem 3 for m = 1.
6.2
A Structural Lemma
The following lemma applies to all finite mechanisms, including those that allow players to report sets of valuations, or anything else.26 e i ) = ∅, then we We prove the first step by way of contradiction. Suppose that UDi (Ki ) ∩ UDi (K (1) (2) (k) exhibit an infinite chain of mixed strategies σi ≺ σi ≺ · · · , where loosely speaking, each σi is (k+1) dominated by σi . This leads to a contradiction if the set of pure strategies Si is finite. 26 As we are not dealing with dominant-strategy or ex-post Nash mechanisms, the revelation principle no longer holds, and thus we have no control on what the strategy spaces of the players might consist of. 25
20
In a mechanism M , for a (possibly mixed) strategy profile σ = (σ1 , . . . , σn ), denote A by Mi,j (σ) the probability of player i winning j copies of the good, and by MiP (σ)
the expected price that player i pays. Then, Lemma 6.1. Let M be a finite mechanism, i ∈ [n] a player, x = (x1 , . . . , xm ) and
y = (y1 , . . . , ym ) two valuations in Θi such that xj > yj for all j ∈ [m], and Ki and e i two candidate sets for i such that, K ∀ t ∈ {0, 1, . . . , m}
e i .27 (x1 , . . . , xt , yt+1 , . . . , ym ) ∈ Ki ∩ K
Then, for every ε > 0, there are mixed strategies σi ∈ ∆(UDi (Ki )) and σ ei ∈ e i )) such that, for all s−i ∈ S−i and all j ∈ [m], ∆(UDi (K A A Mi,j (σi , s−i ) − Mi,j (e σi , s−i ) < ε and MiP (σi , s−i ) − MiP (e σi , s−i ) < ε . The proof of Lemma 6.1 can be found in Appendix C.
6.3
Proof of Theorem 3
Arbitrarily choose two numbers v and ε such that v > 2δ and ε ∈ (0, δ), and let def
def
V = [v − δ, v] and W = [v − 2δ + ε00 , v − δ + ε00 ].
For each player i, consider the following two δ-approximate candidate sets e 0 def e i def = {θi ∈ Θi | ∀j, θi (j) ∈ V } and K K i = {θi ∈ Θi | ∀j, θi (j) ∈ W } ,
and the following two valuations in Θi
x = (v − δ + ε00 , . . . , v − δ + ε00 ) and y = (v − δ, . . . , v − δ) . e i and K e i0 satisfy the hypothesis of Lemma 6.1. Thus, It is simple to verify that x, y, K for any ε0 > 0, we have
e i )) and σi0 ∈ ∆(UDi (K e i0 )) such that : ∀i ∈ [n] there exist σi ∈ ∆(UDi (K A A ∀s−i ∈ S−i , ∀j ∈ [m], Mi,j (σi , s−i ) − Mi,j (σi0 , s−i ) < ε0 .
(6.1)
Consider the allocation of mechanism M under the strategy profile σ 0 = (σ10 , σ20 , . . . , σn0 ). Because there are m copies of the good, there ought to be one player who, in expec27
Recall that all valuations in Θi , and thus in every candidate set, are non-increasing. Note that each vector (x1 , . . . , xt , yt+1 , . . . , ym ) is indeed non-increasing, because we have xj > yj and both x and y are non-increasing.
21
tation, receives no more than m copies. Without loss of generality, let him be player n Pm A . Thus, by (6.1) and multiple applications of 1: that is, j=1 j · M1,j (σ10 , . . . , σn0 ) ≤ m n
the triangle inequality,
Pm
A 0 · M1,j (σ1 , σ−1 )≤
Pm
A · M1,j (s1 , s−1 ) ≤
m + ε0 m2 . n By averaging, there exists a pure strategy profile s = (s1 , s−1 ) in the support of j=1 j
0 ) satisfying (σ1 , σ−1
j=1 j
Now let def
def
K = (K1 , . . . , Kn ) where Ki =
def
def
θ = (θ1 , . . . , θn ) where θi =
K e1
m + ε0 m 2 . n
(6.2)
if i = 1
e0 K i
if i = 2, . . . , n v, . . . , v
v − 2δ + ε00 , . . . , v − 2δ + ε00
if i = 1 if i = 2, . . . , n.
0 ) ∈ ∆(UD1 (K1 )) × · · · × ∆(UDn (Kn )) from (6.1), we Because we know (σ1 , σ−1
deduce that s ∈ UD(K). It is also obvious that θ ∈ K, and MSW(θ) = mv.
We now show that s, K, and θ satisfy the desired inequality of Theorem 3. Indeed, h i (∗) m m E SW θ, M (s) ≤ + ε0 m2 · v + m − − ε0 m2 · (v − 2δ + ε00 ) n n m m 0 2 00 0 2 = mv − 2δ m − −εm +ε m− −εm n n m = MSW(θ) − 2δm 1 − 1/n + 2δε0 m2 + ε00 m − − ε0 m2 . n Above, inequality (∗) holds because, when θ is the true-valuation profile, all players except player 1 value each copy of the good as v − 2δ + ε00 , while only player 1 values it as v. However, in the outcome M (s), player 1 can receive, in expectation, at most m n
+ ε0 m2 copies of the good (cf. (6.2)).
Lastly, notice that ε0 > 0 and ε00 > 0 can both be arbitrarily small, so we can choose them to satisfy 2δε0 m2 + ε00 m − m − ε0 m2 < ε. n This concludes the proof of Theorem 3.
22
7
Beyond Multi-Unit Auctions
As a social planner may never be certain about how precisely the players know their own preferences, the Knightian robustness of the Vickrey mechanism may be a welcome piece of news. A social planner’ desiderata, however, go beyond social welfare in multi-unit auctions. It is therefore legitimate to ask whether or not Knightian robustness is enjoyed by other mechanisms in different settings. So far, we have found the answer for provision of a public good (more generally, for single-parameter domains), and for unrestricted combinatorial auctions.
7.1
Knightian Single-Parameter Domains
In a classical single-parameter domain, there is a set A, the set of all possible allocations; for each player i there exists a publicly known subset Si ⊆ A; and the set of possible valuations for player i, Θi , consists of all functions mapping A to the reals,
subject to the following constraints: for each θi ∈ Θi , (1) θi (x) = 0 ∀x 6∈ Si and (2) θi (x) = θi (y) ∀x, y ∈ Si .
We continue to denote the true valuation of player i by θi∗ . An outcome consists of an allocation and a price profile. (The term “single-parameter” derives from the fact that each θi ∈ Θi coincides
with a single number: i’s value for, say, the lexicographically first element of Si . The term “classical” emphasizes that each player knows exactly his own true valuation.)
Single-parameter domains are general enough to include several settings of interest: in particular, provision of a public good28 [Cla71], bilateral trades [MS83], and buying a path in a network [NR01]. 28
Indeed, in the provision of a public good, A has just two elements, a (i.e., the good is provided), which different players may value differently, and b (i.e., the good is not provided), which all players value 0.
23
In a classical single-parameter domain, we say that a mechanism M is weakly dominant-strategy-truthful if for each player i: (0) Si = Θi (1) ∀vi ∈ Θi ∀vi0 ∈ Θi ∀v−i ∈ Θ−i
(2) ∀vi ∈ Θi ∀vi0 ∈ Θi \ {vi } ∃v−i ∈ Θ−i
Ui vi , M (vi , v−i ) ≥ Ui vi , M (vi0 , v−i ) Ui vi , M (vi , v−i ) > Ui vi , M (vi0 , v−i ) .
Several weakly dominant-strategy-truthful mechanisms have been proposed for classical single-parameter domains. (For a characterization of dominant-strategy truthfulness in such domains, see [AT01].) In this setting we again define Knightian self uncertainty by starting, for each player i, with a set of distributions for θi∗ , and then considering their expected values. Definition 7.1. In a single-parameter domain with Knightian valuation uncertainty, the only information that each player i has about the true valuation profile θ∗ consists of a set of reals, Ki , containing θi∗ . Such a domain is δ-approximate if for each player i, sup Ki − inf Ki ≤ δ. Here, we prove the Knightian robustness of many mechanisms at once as follows. Theorem 6.1. Let M be a weakly dominant-strategy-truthful mechanism for classical single-parameter domains. Then, in a domain with Knightian valuation uncertainty, for every player i, UD(Ki ) ⊆ inf Ki , sup Ki . A proof of this theorem can be found in [CMZ14c].
Theorem 6.1 is, for single-parameter domains, the counterpart of Lemma 4.1. That lemma implied that the social-welfare performance of the Vickrey mechanism in a δapproximate multi-unit auction gracefully degrades with δ. Theorem 6.1 implies the same for the VCG mechanism in single-parameter domains. In particular, it implies that, when applied to the provision of a public good in the presence of n Knightian players, the VCG mechanism guarantees, in undominated strategies, a social welfare ≥ MSW − 2nδ. As another example, when applied to buying paths in a network, the VCG mechanism guarantees a social welfare ≥ MSW − 2mδ, where m is the number
of edges in the network. Finally, we note that the proof of Theorem 6.1 easily extends to imply an analogous result for the VCG mechanism for single-minded combinatorial 24
auctions, which are not quite single-parameter domains.29 More generally, Theorem 6.1 implies that, for all weakly dominant-strategy mechanisms M (which include those of [Cla71], [MS83], and [NR01]) ‘the outcome M (θ) is sufficiently good whenever maxi |θi − θi∗ | is sufficiently small’.
7.2
Unrestricted Combinatorial Auctions
In an unrestricted combinatorial auction, there are n players and m distinct goods. The set of possible allocations A consists of all possible partitions A of [m] into 1 + n
subsets, A = (A0 , A1 , . . . , An ), where A0 is the (possibly empty) set of unassigned goods and Ai is the (possibly empty) set of goods assigned to player i. For each player i, a valuation is a function mapping each possible subset of the goods to a non-negative real, and the set of all possible valuations is Θi = {θi : 2[m] → R≥0 | θi (∅) = 0}.
In such auctions, the VCG mechanism guarantees efficiency in dominant strategies.
Definition 7.2. In an unrestricted combinatorial auction with Knightian valuation uncertainty, a player i’s candidate set is a subset Ki ⊂ Θi . def
For each subset of the goods T ⊂ [m], we let Ki (T ) = {θi (T ) | θi ∈ Ki }.
We say that Ki is δ-approximate if sup Ki (T ) − inf Ki (T ) ≤ δ for all T ⊆ [m]. The auction is δ-approximate if each candidate set Ki is δ-approximate.
It is immediately clear that, with Knightian valuation uncertainty, the VCG mechanism is no longer dominant strategy, and Theorem 1 can be extended to prove that no dominant strategy mechanism can guarantee social welfare greater that that obtainable by assigning the goods at random.The novelty, however, is that this time the social-welfare performance of the VCG mechanism is actually very poor. Namely, Theorem 6.2. In a δ-approximate combinatorial Knightian auction with n ≥ 2 play-
ers and m goods, the VCG mechanism cannot, in undominated strategies, guarantee social welfare greater than MSW − (2m − 3)δ. 29
In such an auction, there are m distinct goods, and each player i values, positively and for the same amount θi∗ , only the supersets of a given subset Si of the goods. This auction is not singleparameter because Si is private, that is, known solely to i. Accordingly, i’s true valuation can be fully described only by the number θi∗ and the subset Si . The VCG mechanism for single-minded auctions ensures, in undominated strategies, a social welfare that is at least MSW − 2 min{n, m}δ.
25
A proof of Theorem 6.2 can be found in [CMZ14a]. The reason for the above poor social-welfare performance is that Theorem 6.1 does not extend to unrestricted Knightian combinatorial auctions, and the undominated strategies of the VCG mechanism can be very counterintuitive. For example, for a player i knowing that his true value for some subset of the goods T lies in some interval [x, x + δ], reporting a valuation θi such that θi (T ) is very far from [x, x + δ] can be an undominated strategy. The Knightian fragility of the VCG mechanism in unrestricted combinatorial auctions may be disappointing, but then the latter auctions are difficult in many ways. That said, in [CMZ14b] we extend the notion of regret to the Knightian valuation model, and prove that the social welfare performance of the VCG mechanism continues to be excellent if each player (1) chooses a regret-minimizing strategy, or (2) resorts to regret minimization only to further refine his set of undominated strategies, if needed.
8
Conclusions
The literature on auction design so far concentrated on players having ambiguous beliefs about the valuations of their opponents. We believe it natural to assume that such ambiguity may extend to a player’s own valuation. We thus find it natural to investigate the performance of the Vickrey mechanism in such an environment, where it is no longer a dominant-strategy mechanism. Fragility is inherent to sophisticated endeavors, which certainly include mechanism design. It is thus reasonable to expect that many mechanisms fail when their fundamental underlying assumptions do not exactly hold. In multi-unit auctions, it is a fundamental assumption that the players know their own valuations. We have proven, however, that the classical Vickrey mechanism is robust and continues to perform very well even when the players have Knightian uncertainty about their own valuations. In fact, we have proven that the socialwelfare performance of the Vickrey mechanism is essentially optimal in Knightian undominated strategies. 26
In sum, as most things classical, the Vickrey mechanism outlives the confines in which it was conceived, and continues to be relevant in new and unforeseen settings.
Appendix A
Knightian Revelation Principle
Let us explicitly show that a version of the revelation principle [Gib73, DHM79, Mye79] holds also in the Knightian setting. We do so directly for auctions, so as not to introduce any additional notation. Recall that Ki is the set of possible candidate sets for player i. Definition A.1. Let M be a mechanism; for each player i, let Si be the set of actions of i in M ; and let si : Ki → ∆(Si ) be a function such that ∀K ∈ K, ∀i, ∀ai ∈ Si , ∀θi ∈ Ki ,
Ui (θi , M (si (Ki ), s−i (K−i ))) ≥ Ui (θi , M (ai , s−i (K−i ))) .
Then the function profile s = (s1 , . . . , sn ) is an ex-post Nash equilibrium. Lemma A.2 (Revelation Principle). Let M , Si for each player i, and s be as in Definition A.1. Then, there exists a Knightian DST mechanism M 0 such that ∀K ∈ K M 0 (K1 , . . . , Kn ) = M (s1 (K1 ), . . . , sn (K1 )) . Proof. Let M 0 be the Knightian direct mechanism so defined: def
∀K ∈ K M 0 (K1 , . . . , Kn ) = M (s1 (K1 ), . . . , sn (K1 )). (The above equality is between distributions if M is probabilistic.) All that is left to prove is that mechanism M 0 is dominant-strategy-truthful. To this end, let Ki be the true candidate set of a player i. Then, for all Ki0 ∈ Ki , all θi ∈ Ki , and all K−i ∈ K−i ,
U θi , M 0 (Ki , K−i ) = U θi , M (si (Ki ), s−i (K−i )) ≥ U θi , M (si (Ki0 ), s−i (K−i )) = U θi , M 0 (Ki0 , K−i ) ,
where the inequality follows from the definition of ex-post Nash by letting ai = si (Ki0 ).
This completes the proof. 27
Because every dominant-strategy mechanism is also ex-post Nash, the theorem holds also when M is dominant-strategy.
B
Proof of Theorem 1
Let M be an arbitrarily chosen Knightian DST mechanism, and Si the set of pure strategies for player i. Recall that Si = Ki is the set of all possible δ-approximate candidate sets. We start by proving a separate claim. Namely, for every player i, fixing the candidate sets reported by his opponents, i’s probability of winning the good does not change with the candidate sets he might report, as long as they share at least two valuations. More precisely, denoting by MiA (K) the probability that i wins the good under a profile of strategies (i.e., candidate sets) K, and by MiP (K) the expected price of i under K, ei, K e 0 ∈ Ki such that Claim B.1. For every player i, every two candidate sets K i 0 e −i of candidate sets for i’s opponents, ei ∩ K e | ≥ 2, and every subprofile K |K i
ei, K e −i ) = M A (K e 0, K e −i ) .30 MiA (K i i
ei, Proof of Claim B.1. Because the true candidate set of player i may coincide with K e i should dominate reporting K e i0 , the and because when this is the case reporting K following inequality holds: ei, ∀ θi0 ∈ K
ei, K e −i ) · θ0 − M P (K ei, K e −i ) ≥ M A (K e 0, K e −i ) · θ0 − M P (K e 0, K e −i ) . MiA (K i i i i i i i
(B.1)
e 0 , and Similarly, because the true candidate set of player i may coincide with K i 0 e e because when this is the case reporting Ki should dominate reporting Ki , also the
following inequality holds:
e 0 , M A (K e 0, K e −i )·θ00 −M P (K e 0, K e −i ) ≥ M A (K ei, K e −i )·θ00 −M P (K ei, K e −i ). (B.2) ∀ θi00 ∈ K i i i i i i i i i ei ∩ K e 0 such that x > y. Then, setting Now let us pick two valuations x, y ∈ K i
θi0 = x in (B.1) and θi00 = y in (B.2), and summing up (the corresponding terms 30
ei, K e −i ) = M P (K e 0, K e −i ). Although we do not care about prices, it is also the case that MiP (K i i
28
of) the resulting inequalities, the MiP price terms cancel out, yielding the following inequality: ei, K e −i ) · x + M A (K e 0, K e −i ) · y ≥ M A (K e 0, K e −i ) · x + M A (K ei, K e −i ) · y , MiA (K i i i i i
that is,
e i0 , K e −i ) · (x − y). ei, K e −i ) · (x − y) ≥ MiA (K MiA (K
Because x > y, the last inequality implies that
e 0, K e −i ) ≥ M A (K ei, K e −i ) . MiA (K i i
(B.3)
Similarly, setting θi0 = y in (B.1) and θi00 = x in (B.2), summing up the resulting inequalities, regrouping, and using again the fact x > y, we deduce that: ei, K e −i ) ≥ MiA (K e i0 , K e −i ) MiA (K
(B.4)
Together, inequalities (B.3) and (B.4) imply our claim.
Let us now prove a second claim. Namely, given any DST mechanism M , and any profile K = (K1 , . . . , Kn ) of candidate sets, there exists a player j ∈ [n] such that M is insensitive to j’s report. That is, when j’s opponents collectively report the
subprofile K−j , j can report any set Vj ∈ Kj he wants, but cannot increase beyond 1/n his probability of getting the good.
Claim B.2. If K is a profile of δ-approximate candidate sets of cardinality greater than 1, then there exists j ∈ [n] so that, for every δ-approximate set Vj ∈ Kj of cardinality greater than 1,
MjA (Vj , K−j ) ≤
1 . n
Proof of Claim B.2. When the players report a profile of strategies K as in the stateP ment of the claim, because i∈[n] MiA (K) ≤ 1, there must exist a player j such that MjA (K) ≤ 1/n.
(0)
(1)
(k)
Next, let us consider a sequence of δ-approximate candidate sets Kj , Kj , . . . , Kj
such that: (0)
Kj
= Kj ,
(k)
Kj
= Vj ,
(`−1) (`) and Kj ∩ Kj ≥ 2 for all `.
For instance, we can construct the above sequence as follows. 29
(1) def (k−1) def • Let Kj = inf Kj , inf Kj + δ be an interval containing Kj , and Kj = inf Vj , inf Vj + δ be an interval containing Vj . (2)
• Let Kj
(3)
def
= [inf Kj + δ/2, inf Kj + 3δ/2], Kj
def
= [inf Kj + δ, inf Kj + 2δ], and so
(k−1)
on, until an interval intersecting Kj
(on more than 1 value) is generated. e −j = K−j , K e j = K (`) , and Then, invoking Claim B.1 for each ` ∈ [k], with i = j, K j (`−1) 0 e =K K , we obtain that j
j
(k)
(1)
MjA (Vj , K−j ) = MjA (Kj , K−j ) = · · · = MjA (Kj , K−j ) (0)
= MjA (Kj , K−j ) = MjA (Kj , K−j ) ≤
1 . n
Let us now see that Theorem 1 holds, that is, let us see that MSW(θ) ∀θ ∈ K1 × · · · × Kn E SW(θ, M (K)) ≤ +δ . n Apply Claim B.2 by choosing Ki = [0, δ] for each i ∈ [n], and Vj = [B − δ, B] for any
B > 2δ. Consider the scenario where the profile of true candidate sets is (Vj , K−j ), and the profile θ of true valuations is as follows: θj = B, and θi = 0 for all i 6= j . Then, the expected social-welfare E[SW(θ, M (Vj , K−j ))] is at most Bn , because Claim B.2 shows that MjA (Vj , K−j ) ≤
1 . n
On the other hand, the maximum social welfare
MSW(θ) = B. This completes the proof of Theorem 1.
C
Proof of Lemma 6.1
e i ) are both nonIt is a simple exercise to verify that the sets UDi (Ki ) and UDi (K
empty (but anyways proved in Appendix E). If there exists a common (pure) strategy e i ), then setting σi = σ si ∈ UDi (Ki ) ∩ UDi (K ei = si completes the proof. Therefore, let e i ) are totally disjoint. us assume that UDi (Ki ) and UDi (K
e i ). Accordingly, let si be a pure strategy in UDi (Ki ). Then, necessarily, si 6∈ UDi (K e i ) implies the existence of a (possibly mixed) strategy By definition, si 6∈ UDi (K e i )) that (weakly) dominates si for player i under candidate set K ei, σ ei ∈ ∆(UDi (K
which is denoted by σ ei (i,Ke i ) si (see Definition 5.1). 30
Next, we argue that ∃ τi ∈ ∆(UDi (Ki )) such that τi (i,Ki ) σ ei .31
(C.1)
P (t) (t) (t) Let σ ei = ei , where X is a finite index set, each sei is a pure strategy t∈X α s (t) e i ), and P of UDi (K = 1. Invoking again the disjointedness of UDi (Ki ) and t∈X α e i ), we deduce that, for each t ∈ X, se(t) 6∈ UDi (Ki ). This implies the existence UDi (K i
(t) τi
(t)
of a strategy ∈ ∆(UDi (Ki )) such that τi P (t) (t) t∈X α τi , (C.1) holds.
(t)
def
(i,Ki ) sei . Thus, by defining τi =
e i )) such that Similarly, we could argue that there exists some τei ∈ ∆(UDi (K
τei (i,Ke i ) τi . Continuing in this fashion, going back and forth between ∆(UDi (Ki )) e i )), we obtain an infinite chain of strategies, and ∆(UDi (K (1)
σi
(1)
≺(i,Ke i ) σ ei
(2)
≺(i,Ki ) σi
(2)
≺(i,Ke i ) σ ei
≺(i,Ki ) · · ·
This (weak) dominance chain implies the following utility inequalities: for all s−i ∈ S−i and all k ∈ N:
P j A (σ (k) , s ) ei (l) − M P (σ (k) , s−i ) M θ −i i,j i i j∈[m] i Pl=1 P (k) (k) (k) j A P e e ≤ Ui θi , M (e σi , s−i ) σi , s−i ) = σi , s−i ) j∈[m] Mi,j (e l=1 θi (l) − Mi (e P Pj (k) A σ (k) , s ) P σ (k) , s ) ∀θi ∈ Ki , Ui θi , M (e σi , s−i ) = −i −i j∈[m] Mi,j (e i i l=1 θi (l) − Mi (e P P (k+1) (k+1) j A P (k+1) , s ) ≤ Ui θi , M (σi , s−i ) = , s−i ) −i j∈[m] Mi,j (σi l=1 θi (l) − Mi (σi (C.2) ei, ∀θei ∈ K
(k) Ui θei , M (σi , s−i )
=
P
Now, pick some t ∈ {0, 1, . . . , m}, and let def ei . θi = θei = (zt,1 , zt,2 , . . . , zt,m ) = (x1 , x2 , . . . , xt , yt+1 , . . . , ym ) ∈ Ki ∩ K
For this choice of θi and θei , we see from (C.2) that for all s−i ∈ S−i and all k ∈ N (k) (k) (k) (k+1) Ui θei , M (σi , s−i ) ≤ Ui θei , M (e σi , s−i ) = Ui θi , M (e σi , s−i ) ≤ Ui θi , M (σi , s−i ) 31
Note that, while we have only defined what it means for a pure strategy to be dominated by a possibly mixed one, the definition trivially extends to the case of dominated strategies that are mixed, as is the case in “τi (i,Ki ) σ ei ” in (C.1).
31
which then implies that for all s−i ∈ S−i and all k ∈ N P P (k) (k) j A − MiP (σi , s−i ) l=1 zt,l j∈[m] Mi,j (σi , s−i ) P P (k) (k) j A σi , s−i ) (e σ , s ) z − MiP (e ≤ M −i i,j i l=1 t,l j∈[m] P Pj (k+1) (k+1) A ≤ , s−i ) − MiP (σi , s−i ) . j∈[m] Mi,j (σi l=1 zt,l
Considering the above inequalities for k = 1, 2, . . . , we get the following infinite and non-decreasing sequence of real numbers (for each s−i ∈ S−i ): P P (1) (1) j A (σ , s ) z − MiP (σi , s−i ) M −i i,j i l=1 t,l j∈[m] P P (1) (1) j A (e σ , s ) z − MiP (e σi , s−i ) ≤ M −i t,l i,j i l=1 j∈[m] P P (2) (2) j A ≤ M (σ , s ) z − MiP (σi , s−i ) ≤ · · · −i i,j i j∈[m] l=1 t,l P A This sequence is upperbounded by x1 + · · · + xm . (Indeed, zt,l ≤ xl , j∈[m] Mi,j (·) ≤
1, and each price is non-negative.) Thus, by the Bolzano-Weierstrass theorem, for def
every s−i ∈ S−i and t ∈ {0, 1, . . . , m}, denoting by X = max{1, x1 + · · · + xm } and (s−i ,t)
def
D = min{1, minl∈[m] {xl − yl }}, there must exist some Hε k>
(s ,t) Hε −i :
∈ N such that for all
P Pj (k) (k) P A , s ) z (σ , s ) − M (σ M −i −i i i i l=1 t,l j∈[m] i,j P Pj (k) (k) P A σi , s−i ) ≤ − σi , s−i ) l=1 zt,l − Mi (e j∈[m] Mi,j (e
εD . (C.3) 12m · X Notice that, because the mechanism is finite and thus so is the set S−i , we can let def Hε = max Hε(s−i ,t) : s−i ∈ S−i , t ∈ {0, 1, . . . , m} ∈ N . (k)
Next, we pick arbitrarily k > Hε , and prove that σi
(k)
and σ ei
are the two strategies
that we are looking for. To this end, pick arbitrarily s−i ∈ S−i , consider the two
inequalities obtained by invoking (C.3) twice (once for t and once for t − 1), and
32
combine them with the triangle inequality to obtain:32 P Pj εD (k) (k) A P M (σ , s ) z − M (σ , s ) ≥ −i −i i i i j∈[m] i,j l=1 t,l 6m · X P Pj (k) (k) P A σi , s−i ) σi , s−i ) − l=1 zt,l − Mi (e j∈[m] Mi,j (e P Pj (k) (k) P A (σ , s ) (σ , s ) z − M − M −i −i i i i l=1 t−1,l j∈[m] i,j P Pj (k) (k) P A σi , s−i ) σi , s−i ) + l=1 zt−1,l − Mi (e j∈[m] Mi,j (e Pm P (k) (k) m A A σi , s−i ) xt − yt − = j=t Mi,j (e j=t Mi,j (σi , s−i ) xt − yt P Pm (k) (k) m A A σi , s−i ) , = (xt − yt ) j=t Mi,j (σi , s−i ) − j=t Mi,j (e
which further implies that (using xt − yt ≥ D): P Pm ε (k) (k) m A A σi , s−i ) ≤ . j=t Mi,j (e j=t Mi,j (σi , s−i ) − 6m · X
(C.4)
Now we use triangle inequality again on (C.4) for t and t − 1, and deduce that: ε A (k) (k) A σi , s−i ) ≤ ≤ε . (C.5) Mi,t (σi , s−i ) − Mi,t (e 3m · X That is, the first inequality of Lemma 6.1 has been proven. Let us now consider the
price terms. We begin with a simple application of (C.5). P Using the fact that jl=1 zt,l ≤ X, we obtain, Pj Pj ε A (k) (k) A σi , s−i ) . Mi,j (σi , s−i ) l=1 zt,l ≤ l=1 zt,l − Mi,j (e 3m Then, using the triangle inequality again, we get P ε P Pj Pj (k) (k) A A z − M (e σ , s ) z M (σ , s ) . j∈[m] i,j i −i −i i l=1 t,l l=1 t,l ≤ j∈[m] i,j 3 Expressing the above inequality as |a − c| ≤ 3ε , (C.3) can be expressed as |(a − b) − (c − d)| ≤
εD . 12m·X
Thus, we use the triangle inequality again and obtain ε εD P (k) (k) P + 0, there exist profiles K of δ-approximate candidate sets such that MSW(θ) E SW(θ, M (K)) ≤ n Theorem 2’. In a m-unit Knightian auction, for all multiplicative δ, all products K ∀θ ∈ K1 × · · · × Kn
of δ-approximate candidate sets, all profiles v ∈ UD(K), and all θ ∈ K 2 1−δ SW θ, Vickrey(v) ≥ · MSW(θ) . 1+δ Theorem 3’. Let M be a finite mechanism in a δ-approximate multi-unit Knightian auction with n > 1 players and multiplicative δ > 0. Then, for every ε > 0, there exist products K of δ-approximate candidate sets, valuation profiles θ ∈ K, and undominated strategy profiles s ∈ UD(K), such that ! (1 − δ)2 + 4δ n E SW(θ, M (s)) ≤ · MSW(θ) + ε . (1 + δ)2
E
The Set of Undominated Strategies is Non-Empty
It is trivial to see that, no matter what candidate set Ki a player i may have, UDi (Ki ) is non-empty in the Vickrey mechanism. Indeed, for each θi ∈ Ki , θi ∈ UDi (Ki ).
We wish to point out that UDi (Ki ) is non-empty for all finite mechanisms, a
simple fact relied upon in the proof of Theorem 3. This fact is trivial if a Knightian player is restricted to consider only pure strategies, and not hard to prove when mixed strategies are allowed. As we shall see, the proof follows the analogous one for the classical setting, that is, when Ki is a singleton. Fact E.1. Let M be a finite mechanism, i a player, and Ki a candidate set of i. Then, UDi (Ki ) 6= ∅. 34
Proof. Let Si = {s1 , . . . , st } be the finite pure strategy set of player i. We proceed by contradiction. Suppose that every strategy in Si is (weakly) dominated, with respect
to Ki , by some strategy in ∆(Si ). Then, s1 is dominated, and thus there exists a P mixed strategy tk=1 αk sk ∈ ∆(Si ) such that s1 ≺(i,Ki )
def
where α ∈ ∆ = {x ∈ [0, 1]t :
Pt
t X
αk sk ,
(E.1)
k=1
xk = 1}. Notice that, by condition (2) in
k=1
Definition 5.1, we cannot have s1 ≺(i,Ki ) s1 . Therefore, we must have α1 < 1. Now, we simplify (E.1) by canceling α1 s1 on both sides, and obtain s1 ≺(i,Ki ) Next, since s2 is dominated, let
t X k=2
s2 ≺(i,Ki )
αk sk . 1 − α1
t X
βk sk
(E.2)
(E.3)
k=1
for some β ∈ ∆. By substituting (E.2) into (E.3), we further simplify (E.3) with s2 ≺(i,Ki )
t X
βk0 sk
(E.4)
k=2
for some β 0 ∈ ∆ such that β10 = 0. Again, by subtracting β20 s2 on both sides and rescaling, we obtain
s2 ≺(i,Ki )
t X
βk00 sk ,
(E.5)
k=3
for some β 00 ∈ ∆ such that β100 = β200 = 0. We substitute (E.5) into (E.2), and obtain s1 ≺(i,Ki ) 0
for some α ∈ ∆ such that
α10
=
α20
= 0.
t X
αk0 sk ,
k=3
This process, quite similar to Gaussian elimination for linear systems, can be
continued until we obtain sk ≺(i,Ki ) st for every k = 1, . . . , t − 1. This implies that
st must be an undominated strategy for player i, contradicting the hypothesis that
UDi (Ki ) = ∅.
35
References [AM06] Lawrence M. Ausubel and Paul Milgrom. The lovely but lonely vickrey auction. In Peter Cramton, Yoav Shoham, and Richard Steinberg, editors, Combinatorial Auctions, page Ch. 1. MIT Press, 2006. ´ [AT01] Aaron Archer and Eva Tardos. Truthful mechanisms for one-parameter agents. In Proceedings of the 2001 IEEE 42nd Annual Symposium on Foundations of Computer Science, FOCS ’01, pages 482–491. IEEE Computer Society, 2001. [Aum62] Robert J. Aumann. Utility theory without the completeness axiom. Econometrica, 30(3):445–462, July 1962. [Bew02] Truman F. Bewley. Knightian decision theory. Part I. Decisions in Economics and Finance, 25(2):79–110, 2002. Earlier version appeared as a discussion paper no. 807 of the Cowles Foundation at Yale University, November 1986. [BLP06] Moshe Babaioff, Ron Lavi, and Elan Pavlov. Single-value combinatorial auctions and implementation in undominated strategies. In SODA ’06: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1054–1063, New York, NY, USA, 2006. ACM. [Bod12] Aaron L. Bodoh-Creed. Ambiguous beliefs and mechanism design. Games and Economic Behavior, 75(2):518–537, 2012. [BOP06] Subir Bose, Emre Ozdenoren, and Andreas Pape. Optimal auctions with ambiguity. Theoretical Economics, 1(4):411–438, December 2006. [Cla71] Edward H. Clarke. Multipart pricing of public goods. Public Choice, 11:17– 33, 1971. [CMZ14a] Alessandro Chiesa, Silvio Micali, and Zeyuan Allen Zhu. Knightian analysis of the VCG mechanism in unrestricted combinatorial auctions. ArXiv e-prints, abs/1403.6410, March 2014. http://arxiv.org/abs/1403.6410. [CMZ14b] Alessandro Chiesa, Silvio Micali, and Zeyuan Allen Zhu. Knightian robustness from regret minimization. ArXiv e-prints, abs/1403.6409, March 2014. http://arxiv.org/abs/1403.6409.
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[CMZ14c] Alessandro Chiesa, Silvio Micali, and Zeyuan Allen Zhu. Knightian robustness of single-parameter domains. ArXiv e-prints, abs/1403.6411, March 2014. http://arxiv.org/abs/1403.6411. [Dan10] Eric Danan. Randomization vs. selection: How to choose in the absence of preference? Management Science, 56:503–518, March 2010. [DHM79] Partha S. Dasgupta, Peter J. Hammond, and Eric S. Maskin. The implementation of social choice rules: Some general results on incentive compatibility. Review of Economic Studies, 46(2):185–216, April 1979. [DMO04] Juan Dubra, Fabio Maccheroni, and Efe A. Ok. Expected utility theory without the completeness axiom. Journal of Economic Theory, 115(1):118– 133, March 2004. [FO79] Vincy Fon and Yoshihiko Otani. Classical welfare theorems with nontransitive and non-complete preferences. Journal of Economic Theory, 20(3):409–418, June 1979. [FT91] Drew Fudenberg and Jean Tirole. Game theory. 1991. MIT Press, 1991. [Gib73] Allan Gibbard. Manipulation of voting schemes: a general result. Econometrica: journal of the Econometric Society, pages 587–601, 1973. [GM75] D. Gale and A. Mas-Colell. An equilibrium existence theorem for a general model without ordered preferences. Journal of Mathematical Economics, 2(1):9–15, March 1975. [GS89] Itzhak Gilboa and David Schmeidler. Maxmin expected utility with nonunique prior. Journal of Mathematical Economics, 18(2):141–153, April 1989. [Jac92] Matthew O Jackson. Implementation in undominated strategies: A look at bounded mechanisms. Review of Economic Studies, 59(4):757–75, October 1992. [JPS94] Matthew O. Jackson, Thomas Palfrey, and Sanjay Srivastava. Undominated Nash implementation in bounded mechanisms. Games and Economic Behavior, 6(3):474–501, 1994. [Kni21] Frank H. Knight. Risk, Uncertainty and Profit. Houghton Mifflin, 1921.
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[LRS09] Giuseppe Lopomo, Luca Rigotti, and Chris Shannon. Uncertainty in mechanism design, 2009. http://www.pitt.edu/~luca/Papers/ mechanismdesign.pdf. [LRS11] Giuseppe Lopomo, Luca Rigotti, and Chris Shannon. Knightian uncertainty and moral hazard. Journal of Economic Theory, 146(3):1148 – 1172, 2011. Incompleteness and Uncertainty in Economics. [LS08]
Kevin Leyton-Brown and Yoav Shoham. Essentials of game theory: A concise multidisciplinary introduction. Synthesis Lectures on Artificial Intelligence and Machine Learning, 2(1):1–88, 2008.
[Mas74] Andrew Mas-Colell. An equilibrium existence theorem without complete or transitive preferences. Journal of Mathematical Economics, 1(3):237–246, December 1974. [MS83] Roger B Myerson and Mark A Satterthwaite. Efficient mechanisms for bilateral trading. Journal of economic theory, 29(2):265–281, 1983. [Mye79] Roger B Myerson. Incentive compatibility and the bargaining problem. Econometrica: journal of the Econometric Society, pages 61–73, 1979. [Nas11] Leandro Nascimento. Remarks on the consumer problem under incomplete preferences. Theory and Decision, 70(1):95–110, January 2011. [NR01] Noam Nisan and Amir Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35:166–196, 2001. [Ok02] Efe A. Ok. Utility representation of an incomplete preference relation. Journal of Economic Theory, 104:429–449, 2002. [RS05] Luca Rigotti and Chris Shannon. Uncertainty and risk in financial markets. Econometrica, 73(1):203–243, 01 2005. [Sch89] David Schmeidler. Subjective probability and expected utility without additivity. Econometrica, 57(3):571–87, May 1989. [SS75]
Wayne Shafer and Hugo Sonnenschein. Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics, 2(3):345– 348, December 1975.
[Vic61] William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16(1):8–37, 1961.
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arXiv:1403.6413v2 [cs.GT] 1 Apr 2014
Alessandro Chiesa
Silvio Micali
Zeyuan Allen Zhu
MIT April 1, 2014 Abstract We investigate the resilience of some classical mechanisms to alternative specifications of preferences and information structures. Specifically, we analyze the Vickrey mechanism for auctions of multiple identical goods when the only information a player i has about the profile of true valuations, θ∗ , consists of a set of distributions, from one of which θi∗ has been drawn. In this setting, the players no longer have complete preferences, and the Vickrey mechanism is no longer dominant-strategy. However, we prove that its efficiency performance is excellent, and essentially optimal, in undominated strategies.
Keywords: Knightian uncertainty, implementation in undominated strategies, incomplete preferences
1
Introduction
We prove that some classical mechanisms are more robust than previously thought: namely, that they guarantee desirable outcomes even when the players have no beliefs about their opponents, and very limited knowledge about themselves. In particular, we prove that the Vickrey mechanism enjoys such robustness. The Vickrey mechanism efficiently allocates multiple identical goods by ensuring that it is a dominant strategy for each player i to report his true valuation θi∗ . Being dominant-strategy truthful, this mechanism of course (1) works even when the players have no beliefs about their opponents, but also (2) assumes that each player knows his own true valuation precisely. In real life, however, a player i may be uncertain about his own θi∗ , as it may depend on variables that are not directly observable by him. For example, in an auction for the exclusive right to a newly discovered oil well, a player’s valuation may depend on the exact quality of the crude, on the likelihood that other oil wells will be discovered, on the political stability of the country where the well resides, etc. Knightian Valuation Uncertainty. A simple way to capture a player i’s uncertainty about his own valuation is the ‘single-distribution’ model, where i does not know θi∗ , but only the true distribution Di∗ , over the set Θi of all his possible valuations, from which θi∗ has been drawn. We instead investigate a more general form of self uncertainty. Namely, i only knows a set of distributions, from one of which θi∗ has been drawn. We refer to this model as Knightian valuation uncertainty, as it is a special case of the uncertainty model envisaged by Frank H. Knight almost a century ago [Kni21], and later formalized by Truman F. Bewley [Bew02]. Various reasons sometimes make this model more realistic than the single-distribution one. First of all, before the auction starts, a player i may not have the time necessary to compute the true probability distribution Di∗ for θi∗ , but may have sufficient time to distill a set of possible distributions for θi∗ , Ki , which is guaranteed to include Di∗ .1 For instance, assume that, in the time available, i has been able to determine that Di∗ is a Gaussian distribution, but has not been able to learn its mean µ and its standard deviation σ beyond the fact that the first is between 9 and 10, and the second between 1 and 2. Then, Ki = {N (µ, σ) : µ ∈ [9, 11], σ ∈ [1, 3]}. 1
1
Secondly, sometimes no single distribution for θi∗ may be very meaningful. Consider an auction of m BitCoins. Bitcoin is a recent, hard to exchange, and wildly fluctuating, cryptocurrency, whose security depends on an unproven mathematical assumption, M A.2 This is an auction of identical goods, and in it a player i’s valuation depends not only on hard-to-observe variables, such as the future legality of BitCoins and the likelihood of the rise of new crypto-currencies, but also on whether M A is true or false. Since the correctness of a mathematical statement is not a probabilistic event, i may find it hard to form Bayesian beliefs about M A. Accordingly, even if all other variables of BitCoins can be precisely captured by a single distribution, i’s valuation is best modeled as being drawn from one of two distinct distributions, Dtrue if M A is true, and Dfalse if M A is false. Thirdly, Knightian valuation uncertainty may also arise from conflicting expert opinions. Consider an auction of a novel technology, where a firm i’s valuation depends on how much the technology might increase i’s productivity. Due to the novelty of the technology, firm i does not know how useful it will prove, and hires k (properly incentivized) independent experts to figure it out, trusting that at least one of them will be right. If each of them reports a different distribution for θi∗ , either because time was limited or because some of the experts made an error, then i is faced with a set of distributions, and believes that θi∗ has been drawn from one of them. Findings. The Vickrey mechanism can certainly be used in multi-unit auctions with Knightian valuation uncertainty, but it would no longer be dominant-strategy. We prove, however, that this is no ‘big loss’. Although dominant-strategy mechanisms for multi-unit auctions do exist,3 we prove that all of them (including those that allow a player to report a set of valuation distributions rather than a single valuation) perform poorly in terms of social-welfare. For the special case of single-good auctions, Theorem 1 points out that no such mechanism, as well as no ex-post Nash mecha2
BitCoins are ‘minted” via a given easy-to-compute function h, and are unforgeable only under the assumption that it is computationally infeasible to find two different strings x and y such that h(x) = h(y). 3 At least the ‘degenerate’ mechanism, which assigns each copy of the good at random, is one of them. In a sense, we prove that all dominant-strategy mechanisms must be de facto degenerate in the Knightian valuation model, even for single-good auctions.
2
nism, whether deterministic or randomized, can guarantee higher efficiency than that obtainable by allocating the good at random. Dominant-strategy mechanisms provide designers with the best guarantee that the players will choose the desired strategies. However, one may also be quite confident that a player will not choose a strategy outside his undominated set, and we prove that, no matter which undominated strategies the players might choose, the Vickrey mechanism is guaranteed to return an allocation of very high social welfare. More precisely, Theorem 2 guarantees that, in the Knightian valuation model, the social-welfare performance of the Vickrey mechanism gracefully degrades with the uncertainty level of the players (naturally measured) and the number of copies of the good. Let us emphasize that this performance guarantee is proved without ‘inertia’.4 Of course, the fact that the Vickrey mechanism performs well in undominated strategies does not exclude that different mechanisms may perform even better. Theorem 3, however, shows that the performance of the Vickrey mechanism is essentially optimal among all finite undominated-strategy mechanisms, probabilistic or not. Here, a mechanism is ‘finite’ if it assigns a finite set of strategies to each player. Together, Theorem 1, Theorem 2 and Theorem 3 show that the Vickrey mechanism is very robust to alternative specifications of preferences and information structures. We believe that such robustness is an important property of a mechanism. In our last section, we point out that this robustness is also enjoyed by additional, but not all, classical mechanisms. 4 As originally put forward by Bewley, the inertia assumption states that a player with incomparable options will chose a ‘reference point’ unless a strictly better alternative exists. (Lopomo, Rigotti, and Shannon rely on the inertia assumption when putting forward maximal incentive compatibility.) As we shall argue in our technical sections, in the Vickrey mechanism, a player i with k possible distributions for θi∗ , D1 , . . . , Dk , has k natural reference points to report: namely, E[D1 ], . . . , E[Dk ], where each expectation E[Dj ] is an m-dimensional vector if there are m copies of the good. Indeed, reporting each of these valuations is undominated, and when all the players report these reference points, the Vickrey mechanism performs well. However, these are not the only undominated strategies, and Theorem 2 guarantees that the Vickrey mechanism continues to perform well no matter which undominated strategy each player may report, without any behavioral assumptions.
3
2
Related Work
Knightian players have received much attention in decision theory. Aumann [Aum62], Dubra, Maccheroni and Ok [DMO04], Ok [Ok02], and Nascimento [Nas11] investigate decision with incomplete orders of preferences. Various criteria for selecting a single distribution out of a set of distributions have been studied by Danan [Dan10], Schmeidler [Sch89], and Gilboa and Schmeidler [GS89]. (In fact, Bose, Ozdenoren and Pape [BOP06] and Bodoh-Creed [Bod12] use the model from [GS89] to study auctions.) General equilibrium models with incompletely ordered preferences have been considered by Mas-Colell [Mas74], Gale and Mas-Colell [GM75], Shafer and Sonnenschein [SS75], and Fon and Otani [FO79]. More recently, Rigotti and Shannon [RS05] have characterized the set of equilibria in a financial market problem.5 Knightian mechanisms were first considered by Lopomo, Rigotti, and Shannon [LRS09], for the rental extraction problem, in a model different from ours. In their model, there is a single player, whose Knightian uncertainty arises from a variable that he cannot observe, but is exactly known to the mechanism. More precisely, the player’s utility for a given outcome depends on (a) his own type, t ∈ T , which the player knows exactly, and (b) the state of the world, s ∈ S, which only the mechanism knows exactly, while the
player only knows that it is drawn from a distribution in some set Π(t) ⊆ ∆(S).
They proposed two notions of implementation in their Knightian model: optimal incentive compatibility and maximal incentive compatibility. Their first notion corresponds to our Knightian dominant-strategy truthfulness, but does not coincide with it, due to the difference between theirs and our model.6 Their second notion 5
A strategy profile is an equilibrium if no player can deviate and strictly benefit no matter which distribution is picked from his set. Notice that such an equilibrium is not a notion of dominance. 6 Optimal incentive compatibility applies to mechanisms φ : T × S → O mapping the reported type of the player t ∈ T and the true state of the world s ∈ S to an outcome φ(t, s) ∈ O. Formally (see [LRS09, Definition 3]), such a mechanism φ is optimal incentive for all types t and compatible if all mixed strategies σ ∈ ∆(T ): Es∼π U (φ(t, s), t, s) ≥ Es∼π Eθ∼σ U (φ(θ, s), t, s) ∀π ∈ Π(t). By contrast, we consider auctions with multiple players, so that to express Knightian dominance the strategy subprofiles of a player’s opponents must be universally quantified. Moreover, and more importantly, our mechanisms have no information about the players’ true valuations.
4
corresponds to a weaker version of our implementation of Knightian undominated strategies, and does not apply to the Vickrey mechanism in our model.7 Lopomo, Rigotti, and Shannon also studied variants of their notions in a principal-agent model with Knightian uncertainty [LRS11]. Implementation in (traditional) undominated strategies was originally proposed by Jackson [Jac92, JPS94]. An example of of such an implementation in the exactvaluation model is given by the mechanism of Babaioff et al. [BLP06] for efficiency in single-value multi-minded auctions.
3
Notation and Model
3.1
Notation for Multi-Unit Auctions
In a multi-unit auction there are identical copies of the same good. We denote by n the number of players, and by m the number of copies of the good. The set of all P def possible allocations is A = {A ∈ Zn≥0 : ni=0 ai = m}. In an allocation A, A0 is the
number of unallocated copies and Ai the number of copies allocated to player i.
The set of all possible valuation profiles is Θ = Θ1 × · · · × Θn . Throughout this def
paper, for each player i, Θi = {θi : [m] → R≥0 | θi (1) ≥ · · · ≥ θi (m) ≥ 0}. That is, we
never assume any restrictions on the players’ possible valuations, except for marginal
valuations being non-increasing, the standard assumption envisaged by Vickrey.8 The profile of the players’ true valuations is θ∗ = (θ1∗ , . . . , θn∗ ) ∈ Θ. def
The set of possible outcomes is Ω = A × Rn≥0 . If (A, P ) ∈ Ω, we refer Pi as the
7 In maximal incentive compatibility, truthful reporting is not strictly dominated, there may be additional undominated strategies, for which the mechanism may fail to produce a desired outcome. By contrast, in implementation in Knightian undominated strategies, a mechanism must produce a desired outcome no matter which undominated strategy each player may choose. Moreover, their notion applies only to ‘Knightian direct’ mechanisms, but not to the Vickrey mechanism in our model. Indeed, the Vickrey mechanism allows a player to report only a single valuation, while in our Knightian model a player has a set of possible valuations. Thus, he could not ‘truthfully’ report his candidate set even if he wanted. 8 Multi-unit auctions are sometimes referred to as Vickrey auctions (in particular, see Ausubel and Milgrom [AM06]). Indeed, Vickrey was the first one to provide an efficient mechanism for such auctions [Vic61], and his mechanism was later generalized to become the VCG mechanism. If a player i has a valuation θi ∈ Θi , then θi (j) is his marginal value of receiving the j-th copy of the good. Non-decreasing marginal valuations in multi-unit auctions were indeed envisaged by Vickrey.
5
price charged to player i. The utility of a player i, with valuation θi , for an outcome ω = (A, P ) is def
Ui (θi , ω) =
PA i
j=0 θi (j)
− Pi .
If ω is a distribution over outcomes, then Ui (θi , ω) is the expected utility of player i. The social welfare of an outcome ω = (A, P ), and that of an allocation A, where A = (a0 , a1 , . . . , an ), relative to a valuation profile θ, is SW(θ, ω) = SW(θ, A) = P Pai i j=1 θi (j). The maximum social welfare relative to θ, MSW(θ), is maxA∈A SW(θ, A).
The maximum social welfare is MSW = MSW(θ∗ ).
A mechanism M specifies, for each player i, a set Si . We interchangeably refer to each member of Si as a pure strategy/action/report of i, and similarly, a member of ∆(Si ) as a mixed strategy/action/report of i.9 After each player i, simultaneously with his opponents, reports a strategy si in Si , M maps the reported strategy profile s to an outcome M (s) ∈ Ω. If M is probabilistic, then M (s) ∈ ∆(Ω).
3.2
Knightian Valuation Uncertainty
In our model, a player i’s sole information about θ∗ consists of Ki , a set of distributions
over Θi , from one of which θi∗ has been drawn. (The true valuations are uncorrelated.) That is, Ki is i’s sole (and private) information about his own true valuation θi∗ .
Furthermore, for every opponent j, i has no information (or beliefs) about θj∗ or Kj .
For auctions, because each Θi is convex, this model has an equivalent, but non-
distributional formulation. Definition 3.1 (Knightian valuation model). For each player i, i’s sole information about θ∗ is a set Ki , the candidate (valuation) set of i, such that θi∗ ∈ Ki ⊂ Θi . We refer to an element of Ki as a candidate valuation.
def
The set of all possible candidate sets of a player i is Ki = {Ki ⊆ Θi : Ki 6= ∅}. 9
Often, in pre-Bayesian settings, the notion of a strategy and that of an action are distinct. Indeed, a strategy si of a player i maps the set of all possible types of i to the set of i’ possible actions/reports. But since strategies are universally quantified in all relevant definitions of this paper, we have no need to separate (and for simplicity refrain from separating) the notions of strategies and actions.
6
The reason behind the above equivalence is that, in auctions, given that all a player i cares about is his expected (quasi-linear) utility, he may ‘collapse’ each distribution Di ∈ Ki to its expectation Eθi ∼Di [θi ].10
In the Knightian valuation model, a mechanism’s performance will of course de-
pend on the inaccuracy of the players’ candidate sets, which we measure as follows. Definition 3.2. For all players i, candidate set Ki , and copies j ∈ [m], we let def
Ki (j) = {θi (j) | θi ∈ Ki },
def
Ki⊥ (j) = inf Ki (j),
and
def
Ki> (j) = sup Ki (j).
A candidate set Ki is (at most) δ-approximate if Ki> (j) − Ki⊥ (j) ≤ δ for all j ∈ [m].
An auction is (at most) δ-approximate if the set of all possible candidate sets of a def
player i is Kδi = {Ki ∈ Ki : Ki is δ-approximate}. Remarks 1. Non-Convexity. Note that a candidate set Ki may not be convex. This fact may be puzzling, because in a typical single-good auction we expect that a player having a and b as two possible valuations must also have
a+b 2
as another possible
valuation. Let us stress that ‘the possibility of holes’ in Ki is not a restriction, nor a speciousness of our model. To the contrary, it is the natural sub-product of the generality of the Knightian setting. To make our results stronger, when proving that a mechanism performs well, we consider all possible candidate sets, including non convex ones. When proving that a mechanism performs poorly, we consider candidate sets that are convex. 10
For example, in a 2-unit auction, a player’s valuation consists of a pair (v(1), v(2)) where v(1) ≥ v(2) ≥ 0. Suppose a player i knows that θi∗ is drawn from a distribution Di∗ over such pairs. In def addition, let (e1 , e2 ) = Eθi ∼Di θi (1), θi (2) ∈ Θi , and let ω be an outcome in which i wins exactly one copy of the good. Then, i’s expected utility for ω is Eθi ∼Di [θi (1)] − p = e1 − p, as if his θi∗ were (e1 , e2 ). Since this is true for every (even probabilistically chosen) outcome, i may very well act as if his true valuation were exactly θi∗ = (e1 , e2 ). Let us now be more abstract. The desired equivalence could be formalized in three steps. Consider a player i only knowing a set of distributions Ki from one of which θi∗ has been drawn. In a first step, i can be identified with a player only knowing a multiset of valuations Ki0 , containing his true valuation, if there exists a bijection φ : Ki ↔ Ki0 such that, for all outcome ω and Di ∈ Ki , Eθi ∼Di [Ui (θi , ω)] = Ui (φ(Di ), ω). In a second step, the latter player can be identified with a player only knowing a set of valuations Ki , containing his true valuation and coinciding with Ki0 after removing redundant valuations. In a third step, we choose φ to be the function mapping a distribution D to its expectation. In the above proof sketch we rely on the fact that φ(Di ) = E[Di ] exists in Θi for all Di ∈ ∆(Θi ), which is indeed the case because Θi is convex. The above equivalence thus holds not only for multi-unit auctions, but also for any auction where each Θi is a convex set.
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2. Beliefs. Assume that Ki = {a, b}. Then, despite our prior remark, it may still
be hard to accept that player i may know his true valuation to be either a or b, without forming some (even partial) beliefs. For instance, without exactly knowing the probabilities of a and b, i may believe that his true valuation is more probable to be a than b. Note, however, that this belief corresponds to the player having a set of distributions Ki0 = {Dp : p ∈ [0.5, 1]} where each Dp is a distribution taking value a with probability p, and value b with probability 1 − p. Again, i may as well collapse each distribution Dp to its expected value, so as to end up, de facto, with a set of candidate valuations: namely, Ki0 = {pa + (1 − p)b : p ∈ [0.5, 1]} ⊆ Θi . In other words, candidate sets may be very expressive. When we say that the candidate set is Ki , we assume that all (partial) beliefs that player i may have about his own valuation θi∗ have already been taken into account.
˙ 3. Additive vs. Multiplicative Inaccuracy. We could have measured Knightian uncertainty via a multiplicative, rather than additive, parameter δ. Although all results in this paper can be restated for a multiplicative δ (see Appendix D), the additive version provides the simplest proofs and is more meaningful.11 4. Richness of Candidate Sets. In a δ-approximate auction, the mechanism could face all possible δ-approximate candidate sets Ki : for instance, {2, 2 + δ},
[100, 100 + δ/2], or [200, 200 + δ] \ {200 + δ/i : i = 2, 3, . . .}.
This richness of candidate sets is important, because the Vickrey mechanism performs well in the classical model for all possible valuations, that is, for Θi as defined in section 3.1. 11 For simplicity, consider a single-copy (i.e., single-good) auction. Here, the multiplicative version of the inaccuracy can be defined as follows. Let δ be in [0, 1]. A candidate set Ki is δ-approximate if inf Ki ≥ (1 − δ) · sup Ki . The auction is δ-approximate if each Ki is δ-approximate. Suppose now that K1 = [1000, 1010] and K2 = [10, 20]. Then, the maximum social welfare of this auction is at least 1000, no matter what the true valuations of the two players might be. Using our additive definition, this auction is only 10-approximate, where 10 is small relative to the maximum social welfare. By contrast, using the multiplicative definition, the auction is 50%-approximate. That is, it has the same inaccuracy as when K1 = [1000, 2000] and K2 = [10, 20].
8
The richness of candidate sets makes the robustness of the Vickrey mechanism —that is Theorem 2— stronger. Moreover, as we shall see, it is crucial for our negative results —that is, for Theorems 1 and 3. 5. Social welfare. Note that the true social welfare of an outcome ω = (A, P ), that of an allocation A, and the true maximum social welfare continue to be SW(θ∗ , ω), SW(θ∗ , A), and MSW(θ∗ ), whether or not every player i knows θi∗ exactly.
4
First Theorem
It is easy to see (but anyway proved by Lopomo, Rigotti and Shannon [LRS09] for their model, and in Appendix A for our model) that, in line with the revelation principle [Gib73, DHM79, Mye79], if a social choice correspondence f is implementable in Knightian dominant strategies or ex-post Nash equilibrium, then f is also implementable by a Knightian dominant-strategy-truthful mechanism. We thus state our first theorem in terms of Knightian DST mechanisms, which we define below. Definition 4.1. A mechanism is Knightian direct if Si = Ki for each player i. Such a mechanism M is Knightian dominant-strategy-truthful (Knightian DST) if ∀Ki , Ki0 ∈ Ki ∀K−i ∈ K−i ∀θi ∈ Ki Ui θi , M (Ki , K−i ) ≥ Ui θi , M (Ki0 , K−i ) .
Theorem 1. For all (possibly probabilistic) Knightian DST mechanisms M for single-good auctions, and for all δ > 0, there exist profiles K of δ-approximate candidate sets such that MSW(θ) +δ . E SW(θ, M (K)) ≤ n Above, the expectation is over the possible random choices of the mechanism M . ∀θ ∈ K1 × · · · × Kn
The proof of Theorem 1 can be found in Appendix B.12 12
Note that our proof of Theorem 1 continues to hold even when M is allowed to know δ in advance. In other words, the proof holds when M knows that every Ki comes from Kδi . (The Knightian DST requirement can be easily weakened by a δ-DST requirement, where the mechanism only takes as input δ-approximate candidate sets.)
9
Here, we just wish to make three remarks. 1. Visualization of Bad Cases. To appreciate how intrinsically inadequate DST (and ex-post Nash) mechanisms are in single-good Knightian auctions, consider the following example. Let n = 3, δ = 10, let M be any dominant-strategy (or ex-post Nash) mechanism, and let the players have the following δ-approximate candidate sets: Case 1: K1 = [9, 10]
K2 = [9, 10]
Case 2: K1 = [9, 10]
K2 = [1000, 1010] K3 = [9, 10]
Case 3: K1 = [1000, 1010] K2 = [9, 10]
K3 = [1000, 1010] K3 = [9, 10].
Then, when the players choose a profile of dominant (or ex-post Nash equilibrium) strategies, Claim B.2 shows that (a) if M is deterministic, then it must assign the good to a ‘low-valuation player’ for at least one of the three cases above; and (b) if M is probabilistic, then it must assign the good to the ‘high-valuation player’ with probability ≥ 1/3 for at least one of the three cases above. 2. Richness of Candidate Sets. In our model, where candidate sets are unconstrained, it is natural to require that a good mechanism must perform well for all possible candidate sets of the players.13 Only for analyzing a mechanism’s performance may we restrict our attention to δ-approximate candidate sets. This richness of candidate sets is crucially relied upon by our proof of Theorem 1, but is not a limitation of our proof techniques. Indeed, if the players’ possible candidate sets were sufficiently restricted, then dominant-strategy mechanisms could actually guarantee maximum social welfare.14 13
In the model of [LRS09] this is unnecessary. Indeed, the optimal incentive compatible mechanism of their Theorem 4 achieves full rent extraction if, in our language, each possible candidate set of the single player is disjoint from the convex hull of the union of all other candidate sets he may have. 14 For instance, assume that, in an n-player auction of a single good, for some δ < 1/3, the only possible candidate sets for player i are of the form [kn + i, kn + i + δ] where k is an arbitrary positive integer. Let M be the mechanism such that (1) Si = {kn + i : k ∈ Z+ } for each player i, (2) for all s ∈ S1 × · · · × Sn , M (s) = 2P(s), where 2P is the second-price mechanism. Then it is clear that (a) for player i whose candidate set is [kn + i, kn + i + δ], reporting kn + i is a dominant strategy, and (b) when dominant strategies are played, M produces an outcome with maximum social welfare.
10
3. Ubiquity of Bad Cases. The proof of Theorem 1 not only shows the existence of bad cases for every possible dominant-strategy truthful mechanism M for single-good auctions, but also that such bad cases are ‘extremely abundant’. In particular, if M is deterministic, Claim B.1 implies the following. For all players i and strategy subprofiles K−i of i’s opponents, if there exists a candidate set Ki of cardinality > 1 such that i does not get the good in M (Ki , K−i ), then, for all candidate sets Ki0 of cardinality > 1, i does not get the good in M (Ki0 , K−i ).
4.1
Intuition for Proving Theorem 1
Denote by MiA (Ki , K−i ) the probability that player i wins the good in mechanism M when players report the profile (Ki , K−i ). The key point in our proof is to show that, for every K−i , MiA (Ki , K−i ) = MiA (Ki0 , K−i ) whenever |Ki ∩ Ki0 | ≥ 2. Fixing the choice of K−i , let us view all the candidate sets of player i as nodes in a graph, and connect two nodes Ki and Ki0 whenever they share two candidate valuations. If the player’s candidate sets are rich enough so that this graph is connected, then the probability MiA ( · , K−i ) must be constant regardless of player i’s report. From this, it immediately follows that M cannot guarantee non-trivial social welfare.
5
Second Theorem
We wish to prove that the social-welfare performance of the Vickrey mechanism gracefully degrades with δ, no matter which undominated strategies the players may choose. Recall that the Vickrey mechanism, denoted by Vickrey, is a direct mechanism (i.e., satisfies Si = Θi ) and maps a profile of valuations θ ∈ Θ1 × · · · × Θn , to an outcome P Pk (A, P ); where A ∈ arg maxA∈A SW(θ, A), Pi = MSW(θ−i ) − k6=i aj=1 θk (j), and
possible ties are broken lexicographically.15 15
More precisely, on a reported valuation profile θ, the Vickrey mechanism sorts all the values {θi (j) : i ∈ [n], j ∈ [m]} in a non-increasing order, and then chooses the m largest entries to assign the m copies of the good. If ties occur in this ordering, that is, θi (j) = θi0 (j 0 ), then θi (j) precedes θi0 (j 0 ) if and only if either (1) i < i0 or (2) i = i0 and j < j 0 . This technical assumption can be replaced with any other order that only resorts to the players’ names to break ties.
11
For the Knightian valuation model, in this section, we prove in Theorem 2 that Vickrey delivers high social welfare in (Knightian) undominated strategies, which we define below. Definition 5.1. In a mechanism M , a pure strategy si ∈ Si of a player i is (weakly) dominated by another possibly mixed strategy σi ∈ ∆(Si ) of i with respect to his Ki , in symbols si ≺(i,Ki ) σi , if
(1) ∀θi ∈ Ki ∀s−i ∈ S−i (2) ∃θi ∈ Ki ∃s−i ∈ S−i
Ui θi , M (σi , s−i ) ≥ Ui θi , M (si , s−i ) , and Ui θi , M (σi , s−i ) > Ui θi , M (si , s−i ) .16
A strategy si ∈ Si is (weakly Knightian) undominated, if there exists no σi ∈ ∆(Si )
such that si ≺(i,Ki ) σi . We denote the set of such undominated strategies by UDi (Ki ). If K is a product or a profile of candidate sets, that is, if K = (K1 , . . . , Kn ) or def
K = K1 × · · · × Kn , then UD(K) = UD1 (K1 ) × · · · × UDn (Kn ). Our notion of an undominated strategy intends to capture the ‘weakest condition’ for which si should be discarded in favor of σi . Note that it is a natural extension of its classical counterpart. In particular, we allow for the possibility of a pure strategy si to be dominated by a mixed one σi .17 Of course, other extensions are also possible. However, they either (1) fail to capture our intended ‘weakest condition’, (2) fail to coincide with the classical notion when players know their valuations exactly, or (3) are equivalent to ours under mild 16
This notion is thus different from strong dominance, where inequality (1) is always strict. For strong dominance in the the exact-valuation case, see, for instance, [FT91, LS08]. 17 We allow for this possibility because, also in the Knightian setting, a pure strategy si may not be dominated by any other pure strategy ti , while being dominated by some mixed strategy σi . Let us construct a simple example, outside auctions, by modifying a classical example for strategic-form games. Consider two players, where the row player has two actions, Up and Down, the column player has three actions, Left, Middle and Right, and the column player’s utilities 1 3 0 are as follows: . Then, for the column player, Left is not dominated by Middle or 1 0 3
Right, but is dominated by half Middle plus half Right. This phenomenon generalizes to that when the column player has Knightian uncertainties, so that his utilities are now intervals as follows: [0.8, 1.2] [2.7, 3.3] [0, 0.3] . In this game, Left is still dominated by ‘1/2 Middle + 1/2 Right’, [0.8, 1.2] [0, 0.3] [2.7, 3.3] but not by Middle or Right alone.
12
conditions.18 Finally, the difficulty of defining undominated strategies pointed out by Jackson [Jac92] does not apply to our paper.19 We are now ready to state our second theorem. Theorem 2. In a m-unit Knightian auction, for all δ, all products K of δ-approximate candidate sets, all profiles v ∈ UD(K), and all θ ∈ K SW θ, Vickrey(v) ≥ MSW(θ) − 2mδ .
5.1
Intuition for Proving Theorem 2
The good social-welfare performance of the Vickrey mechanism follows from a structural lemma, Lemma 5.2, guaranteeing that, for all players i, all candidate sets Ki , and all copies j: If a strategy vi ∈ UD(Ki ), then vi (j) ∈ Ki⊥ (j), Ki> (j) .
It is immediate to see that this condition holds in the simpler case of a single-copy (i.e., single-good) auction, where the Vickrey mechanism coincides with the secondprice one. Indeed, assume that for some player i, Ki⊥ = a and Ki> = b. Then, it is easy to check that reporting any value greater than b (respectively, smaller than a) is (weakly) dominated by reporting precisely b (respectively, a). 18 Our extension desires to capture the ‘weakest condition’ for which si should be discarded in favor of σi . To express condition (2) in Definition 5.1, we must quantify the true valuation θi ∈ Ki and the pure strategy subprofile of i’s opponents s−i ∈ S−i . There are three alternative possibilities to consider. Namely, (a) ∀θi ∀s−i , (b) ∃θi ∀s−i , and (c) ∀θi ∃s−i . Alternatives (a) and (b) do not yield the classical notion of (weak) dominance when Ki is a singleton. Alternative (c) fails to capture our desideratum. (Indeed, since σi is already no worse than si , for player i to discard strategy si in favor of σi , it should suffice that si be strictly worse than σi for a single possible valuation θi ∈ Ki .) Alternatively, one may also consider defining an undominated strategy in the spirit of an equilibrium notion in a pre-Bayesian game. Here, a strategy of player i is a function fi mapping i’s candidate set Ki to an action. Accordingly, a profile of strategies (f1 , . . . , fn ) is ‘undominated’ if, for each player i, all K, and all θ ∈ K , there is no σ such that U θ , M (σ , f (K ) ≥ i i i i i i −i −i Ui θi , M (fi (Ki ), f−i (K−i ) , where at least one inequality is strict. Note, however, that this notion coincides with ours if each strategy fi is onto. That is, if for every possible action si , there exists some Ki such that si = fi (Ki ). Also note that, in the case of the Vickrey mechanism, fi should be onto. Indeed, when Ki is a singleton, Ki = {si }, player i should only want to play si , and thus fi ({si }) should coincide with si . 19 As pointed out by Jackson [Jac92] in the exact-valuation case, the general notion of an undominated strategy is more complex. However, for bounded mechanisms, the simpler notion above coincides with the general notion, even in the Knightian setting. Since this class of mechanisms includes the Vickrey mechanism and all finite ones, we adopt this simpler notion for this paper.
13
Consider now the Vickrey mechanism in a multi-unit auction, and let aj = Ki⊥ (j) and bj = Ki> (j). Here, it is again intuitively clear that reporting a valuation vi0 such that vi0 (j) < aj (resp. vi0 (j) > bj ) for all j ∈ [m] is a dominated strategy for player
i. However, i may consider some strategy vi00 that bids above bj for some copy j and below ak for some other copy k. Such a strategy vi00 is, in general, not dominated by “bidding bj for all j ∈ [m]” or “bidding aj for all j ∈ [m]”. However, one can still carefully construct a strategy vi∗ that dominates vi00 .
More specifically, we construct vi∗ from vi00 by changing only one, but carefully chosen, coordinate j ∗ ∈ [m]. This allows us to argue that, for all v−i ∈ Θ−i , the
outcomes Vickrey(vi∗ , v−i ) and Vickrey(vi00 , v−i ) differ minimally. That is, if player i receives a different number of copies in these two outcomes, then it must be that i has received precisely j ∗ copies in the first outcome, and precisely j ∗ − 1 (or j ∗ + 1) in
the second. By taking into account the Vickrey prices, it becomes easy to conclude that vi∗ dominates vi00 .
5.2
A Structural Lemma
Let us now formally state and prove our structural lemma. Lemma 5.2. In Vickrey, for each player i, candidate set Ki , and copy j vi ∈ UDi (Ki ) ⇒ vi (j) ∈ [Ki⊥ (j), Ki> (j)]. Proof of Lemma 5.2. Recall that the Vickrey mechanism is direct, that is, Si = Θi for all players i, and that multi-unit auctions have non-increasing marginal valuations, that is, θi (1) ≥ θi (2) ≥ · · · ≥ θi (m) for each θi ∈ Θi . ⊥
⊥
>
This implies
>
that Ki (1), . . . , Ki (m) and Ki (1), . . . , Ki (m) are non-decreasing sequences. Thus, K > , K ⊥ ∈ Θ. That is, both Ki> and Ki⊥ are valid strategies in the Vickrey mechanism. We start by proving, by contradiction, that
vi ∈ UDi (Ki ) =⇒ vi (j) ≥ Ki⊥ (j) for all j ∈ [m].
(5.1)
Assume that implication (5.1) is false; let j ∗ ∈ [m] be the first coordinate j such that
14
vi (j) < Ki⊥ (j); and define the function vi∗ : [m] → R≥0 as follows: v (j), if j 6= j ∗ ; i ∗ vi (j) = K ⊥ (j), if j = j ∗ . i
Since vi and K ⊥ are monotonically non-increasing, so is vi∗ . Indeed, • if j ∗ > 1, then vi∗ (j ∗ − 1) = vi (j ∗ − 1) ≥ Ki⊥ (j ∗ − 1) ≥ Ki⊥ (j ∗ ) = vi∗ (j ∗ ) • if j ∗ < m, then vi∗ (j ∗ ) = Ki⊥ (j ∗ ) > vi (j ∗ ) ≥ vi (j ∗ + 1) = v ∗ (j ∗ + 1) . Thus also vi∗ is a valid valuation in Θi . We now reach a contradiction by showing that vi∗ weakly dominates vi , that is, ∀ θi ∈ Ki , ∀ v−i , 0 ∃ θi0 ∈ Ki , ∃ v−i ,
Ui θi , Vickrey(vi∗ , v−i ) ≥ Ui θi , Vickrey(vi , v−i ) , 0 0 Ui θi0 , Vickrey(vi∗ , v−i ) > Ui θi0 , Vickrey(vi , v−i ) .
(5.2) (5.3)
To show (5.2), choose arbitrarily v−i ∈ Θ−i , and consider the following two cases: (1) In Vickrey(vi∗ , v−i ) and Vickrey(vi , v−i ), i receives the same number of copies. In this case, inequality (5.2) holds because its two sides are equal for all θ. (2) In Vickrey(vi∗ , v−i ) and Vickrey(vi , v−i ), i receives a different number of copies. In this case, one can carefully verify that player i wins j ∗ copies of the good in Vickrey(vi∗ , v−i ) and only j ∗ − 1 copies in Vickrey(vi , v−i ).20 Thus, (5.2) holds
because of the following two reasons.
(a) i’s price for his extra j ∗ -th copy of the good is ≤ K ⊥ (j ∗ ).
Indeed, being dominant-strategy in the classical setting, Vickrey guar-
antees that i pays for his j ∗ -th copy at most the value he reports for it. 20
Recall that, when each player reports a valuation vi , the Vickrey mechanism orders the nm values {vi (j) : i ∈ [n], j ∈ [m]} (and breaking ties lexicographically), and allocates the m copies of the good by looking at the first m values in this order. Since the only difference between vi∗ and vi is that vi∗ (j ∗ ) > vi (j ∗ ), the ordering of the reported nm values is minimally affected. That is, if player i receives different numbers of copies in outcome (vi , v−i ) and outcome (vi∗ , v−i ), then it must be that vi (j ∗ ) is outside the largest m sorted numbers under (vi , v−i ), but vi∗ (j ∗ ) is within the largest m entries under (vi∗ , v−i ). This implies that i wins j ∗ − 1 copies in Vickrey(vi , v−i ) but j ∗ copies in Vickrey(vi∗ , v−i ).
15
(b) i’s value for this j ∗ -th copy is ≥ K ⊥ (j ∗ ).
Indeed, for any possible choice of θi in Ki , θi (j ∗ ) ≥ K ⊥ (j ∗ ).
Therefore, inequality (5.2) holds. Let us now show that also inequality (5.3) holds. To do so, we need to construct a ‘witness’ candidate valuation θi0 ∈ Ki and a ‘witness’
0 0 strategy sub-profile v−i . In fact, we will construct some v−i so that (5.3) holds for all 0 θi0 . Let v−i be such that for any player k 6= i,
vi (j ∗ ) + Ki⊥ (j ∗ ) < Ki⊥ (j ∗ ) ∀j ∈ [m] =x= 2 ∗ 0 Then, player i wins exactly j copies in Vickrey(vi∗ , v−i ) and exactly j ∗ − 1 copies vk0 (j)
def
def
0 ). Indeed, there are exactly j ∗ − 1 numbers greater than x in vi , in Vickrey(vi , v−i
0 while exactly j ∗ of such in v ∗ . Moreover, i’s price in Vickrey(vi∗ , v−i ) is < Ki⊥ (j ∗ ). In
fact, his price for the j ∗ -th copy of the good coincides with the maximum reported
valuation of another player for such an extra copy. Thus, (5.3) holds for all possible 0 θi0 ∈ Ki and the so constructed v−i .
Since both (5.2) and (5.3) hold, valuation vi∗ (weakly) dominates vi , contradicting
the hypothesis that vi ∈ UDi (Ki ). The contradiction proves (5.1). An absolutely symmetrical argument shows that21
vi ∈ UDi (Ki ) =⇒ vi (j) ≤ Ki> (j) for all j ∈ [m]. Together, statements (5.1) and (5.4) imply that Lemma 5.2 holds.
5.3
(5.4)
Proof of Theorem 2
Let v ∈ UD(K) be any profile of undominated strategies, and a0 , a1 , . . . , an represent the allocation in the outcome Vickrey(v), where each player i receives ai
copies of the goods, and a0 is the number of unallocated copies. For any θ ∈ K,
let b0 , b1 , . . . , bn represent the allocation that maximizes social welfare under θ, i.e., In this symmetrical case, one needs to define j ∗ ∈ [m] to be the last coordinate such that vi (j ) > Ki> (j). 21
∗
16
MSW(θ) =
Pn Pbi i=1
`=1 θi (`). ai n X X
SW(θ, Vickrey(v)) =
i=1 `=1
Then, (1)
θi (`) ≥
ai n X X i=1 `=1
ai n X (2) X Ki> (`) − δ ≥ vi (`) − δ i=1 `=1
ai bi bi n n n (3) X X (4) X X (5) X X ⊥ ≥ vi (`) − mδ ≥ vi (`) − mδ ≥ Ki (`) − mδ (6)
≥
Above
i=1 `=1 bi n X X i=1 `=1
i=1 `=1
i=1 `=1
(7) (θi (`) − δ) − mδ ≥ MSW(θ) − 2mδ.
• Inequality (1) holds because by hypothesis θ ∈ K and accordingly θi (`) ≥ Ki⊥ (`) ≥ Ki> (`) − δ;
• Inequality (2) holds by Lemma 5.2;
• Inequality (3) holds because we have only m copies of the good, that is, m;
Pn
i=1
ai ≤
• Inequality (4) holds because the Vickrey mechanism maximizes social welfare
with respect to v, and thus relative to v, (a0 , . . . , an ) is no worse than any other allocation, and in particular (b0 , . . . , bn );
• Inequality (5) holds again by Lemma 5.2;
• Inequality (6) holds because by hypothesis θ ∈ K and accordingly θi (`) ≤ Ki> (`) ≤ Ki⊥ (`) + δ; and finally
• Inequality (7) holds by the definition of MSW(θ) and
5.4
The Bound of Theorem 2 is Tight
Pn
i=1 bi
≤ m.
Having established Theorem 2, let us now illustrate that its performance bound is actually tight by means of the following simple example. Example 5.3. Consider the two-player δ-approximate auction in which, for some x > 0, the candidate sets are K1 = θ1 ∈ Θ1 ∀j ∈ [m] θ1 (j) ∈ [x, x + δ] K2 = θ2 ∈ Θ2 ∀j ∈ [m] θ2 (j) ∈ [x + δ, x + 2δ] .
In this case, the Vickrey mechanism may miss the maximum social welfare by 2δm
17
as follows. Player 1 is ‘optimistic’ and bids the valuation v1 = (x + δ, . . . , x + δ); player 2 is ‘pessimistic’ and bids v2 = (x + δ, . . . , x + δ); the Vickrey mechanism (with the lexicographic tie-breaking rule) allocates all copies of the good to player 1; the true valuation θ1∗ of player 1 is (x, . . . , x); and the true valuation θ2∗ of player 2 is (x + 2δ, . . . , x + 2δ). Accordingly, the realized social welfare is xm, while the maximum one is xm+2mδ.
6
Third Theorem
Although the social welfare performance of the Vickrey mechanism, as guaranteed by Theorem 2, can be considered good, it is legitimate to wonder whether a different undominated-strategy mechanism might enjoy an even better performance. Consider again Example 5.3. In principle, there may be another undominated strategy mechanism missing the maximum social welfare by at most δm. Indeed, even a mechanism M allowing each player to report a single valuation might be able to do so, if it ensures that the undominated strategies for player 1 (respectively, player 2) consist of bidding a value close to x + δ/2 (respectively, x + 3δ/2) for each copy. More generally, there may exist an undominated-strategy mechanism that, allowing each player to report a set of valuations, guarantees social welfare ≥ MSW − δm, not only in the above example but in all δ-approximate multi-unit Knightian auctions.
Theorem 3, however, rules out the existence of such mechanisms, so long as they give each player a finite set of strategies. The optimality of the Vickrey mechanism relative to all finite mechanisms is significant, because in practice all mechanisms are finite, and because finite mechanisms may perform very well in Knightian auctions. The latter mechanisms include not only the discretized Vickrey mechanism,22 which elicits a single valuation from each player, but also more general mechanisms, which, 22
The Vickrey mechanism for an m-unit auction can be discretized by restricting each player i def 20 to report a valuation in Fi = vi ∈ Θi : vi (j) ∈ {0, 0.01, 0.02, . . . , 10 } ∀j ∈ [m] . (Indeed, one may prefer that all payments be made in cash, and thus ultimately in cents, rather than —say— implemented via probabilistic lotteries.) Then, it is easy to check that Theorem 2 continues to hold with only an additive 0.01m loss in the social welfare guarantee, provided that no player valuation exceeds 1020 . Since the discretized Vickrey mechanism performs well in undominated strategies, in principle finite mechanisms that are free to choose their outcome functions and strategy sets may perform even better. Thus, Theorem 3 highlights how clever the Vickrey mechanism really is.
18
in particular, may allow a player to report a subset of a finite set of valuations.23 Remark.
We note that this is the only place in our paper where the mechanism
is assumed to be finite. In Section 6.1, when providing intuition for Theorem 3, we highlight how the finiteness of the pure strategies spaces is relied upon in our proof. Theorem 3. Let M be a finite mechanism in a δ-approximate multi-unit Knightian auction with n > 1 players and δ > 0.24 Then, for every ε > 0, there exist products K of δ-approximate candidate sets, valuation profiles θ ∈ K, and undominated strategy profiles s ∈ UD(K), such that E SW(θ, M (s)) ≤ MSW(θ) − 2δm(1 − 1/n) + ε .
Above, the expectation is over the possible random choices of the mechanism M .
6.1
Intuition for Proving Theorem 3
The key idea behind our proof is best illustrated when there is only one copy of the good for sale. In this special case, as in the general one, there are two main steps in the proof. Let M be a finite mechanism in a δ-approximate single-good Knightian auction. In the first step (corresponding to our structural lemma 6.1), we show that if (a) e i , of the same player i, overlap non-trivially, then (b) two candidate sets Ki and K e i ) essentially have a common strategy. UDi (Ki ) and UDi (K 23
For instance, a finite mechanism may also allow each player to report an arbitrary subset of Fi , as defined in Footnote 22. Such a strategy space clearly enables each player to report, if he so wishes, a reasonable approximation to his own candidate set Ki . For instance, in a single-good auction where i’s true candidate set is Ki = [100, 110], i would be able to report {100.00, 100.01, 100.02, . . . , 110.00}. It should also be appreciated that a finite mechanism may allow each player to report an ‘infinite object’. For instance, again in single-good auctions, although candidate sets may be arbitrary subsets of the reals, a finite mechanism may allow a player to report a set of intervals, overlapping or not, of the form [a, b], where a and b are multiples of 0.01 and less than 1020 . In fact, there are finitely many such intervals. (Although each such [a, b] consists of infinitely many reals, it can be reported by means of its two extreme points, that is, by means of two numbers of at most 22 decimal digits.) Of course, each player i should pay close attention to his Ki and to the outcome function of the mechanism, in order to choose which of these finitely many intervals to report. 24 Note that M may be probabilistic, and may even know δ in advance. The latter point is important because in a single-good auction one expects the distance between every two valuations of the same Ki to be upper-bounded by a small value δ, which would therefore be de facto known to an auction mechanism.
19
e i has at In the case of a single good auction, property (a) means that Ki ∩ K
least two values in common. The strongest way to express property (b) is that e i ) 6= ∅. A weaker way to express property (b), which is nonetheless UDi (Ki ) ∩ UDi (K
sufficient for our second step, is the following: there exist mixed strategies σi ∈ e i )) that are sufficiently close, meaning that for any ∆(UDi (Ki )) and σ ei ∈ ∆(UDi (K strategy subprofile s−i ∈ S−i , in the outcomes M (σi , s−i ) and M (e σi , s−i ), i’s winning
probability and price are close enough. This is the only place in our proof that relies on the finiteness of the pure strategy space Si .25
In the second step, we construct a candidate set profile K = (K1 , . . . , Kn ), a valuation profile θ ∈ K, and a strategy profile s ∈ UD(K) as follows.
For each player i, we first consider two possible δ-approximate candidate sets: e i = [v, v − δ] and K e i0 = [v − 2δ + ε, v − δ + ε], where ε > 0 is some sufficiently K e i and K e 0 overlap nontrivially, the first small positive value, and v > 2δ. Since K i
step applies. For simplicity, suppose that it implies the existence of a pure strategy e i ) ∩ UDi (K e 0 ). Again for simplicity, let M be deterministic. Let the si ∈ UDi (K i
outcome ω = M (s1 , . . . , sn ), and, without loss of generality, assume that player 1 e 0 ) and θ = (v, v − e1, K e0, . . . , K does not win the good in ω. We now choose K = (K 2
n
2δ + ε, . . . , v − 2δ + ε) ∈ K. Under the strategy profile s = (s1 , . . . , sn ), the social welfare is SW(θ, ω) = v − 2δ + ε, while the maximum social welfare is MSW(θ) = v.
This shows SW(θ, M (s)) ≤ MSW(θ) − 2δ + ε. When M is probabilistic, this bound
becmes E[SW(θ, M (s))] ≤ MSW(θ) − 2δ(1 − 1/n) + ε, which coincides with the bound of Theorem 3 for m = 1.
6.2
A Structural Lemma
The following lemma applies to all finite mechanisms, including those that allow players to report sets of valuations, or anything else.26 e i ) = ∅, then we We prove the first step by way of contradiction. Suppose that UDi (Ki ) ∩ UDi (K (1) (2) (k) exhibit an infinite chain of mixed strategies σi ≺ σi ≺ · · · , where loosely speaking, each σi is (k+1) dominated by σi . This leads to a contradiction if the set of pure strategies Si is finite. 26 As we are not dealing with dominant-strategy or ex-post Nash mechanisms, the revelation principle no longer holds, and thus we have no control on what the strategy spaces of the players might consist of. 25
20
In a mechanism M , for a (possibly mixed) strategy profile σ = (σ1 , . . . , σn ), denote A by Mi,j (σ) the probability of player i winning j copies of the good, and by MiP (σ)
the expected price that player i pays. Then, Lemma 6.1. Let M be a finite mechanism, i ∈ [n] a player, x = (x1 , . . . , xm ) and
y = (y1 , . . . , ym ) two valuations in Θi such that xj > yj for all j ∈ [m], and Ki and e i two candidate sets for i such that, K ∀ t ∈ {0, 1, . . . , m}
e i .27 (x1 , . . . , xt , yt+1 , . . . , ym ) ∈ Ki ∩ K
Then, for every ε > 0, there are mixed strategies σi ∈ ∆(UDi (Ki )) and σ ei ∈ e i )) such that, for all s−i ∈ S−i and all j ∈ [m], ∆(UDi (K A A Mi,j (σi , s−i ) − Mi,j (e σi , s−i ) < ε and MiP (σi , s−i ) − MiP (e σi , s−i ) < ε . The proof of Lemma 6.1 can be found in Appendix C.
6.3
Proof of Theorem 3
Arbitrarily choose two numbers v and ε such that v > 2δ and ε ∈ (0, δ), and let def
def
V = [v − δ, v] and W = [v − 2δ + ε00 , v − δ + ε00 ].
For each player i, consider the following two δ-approximate candidate sets e 0 def e i def = {θi ∈ Θi | ∀j, θi (j) ∈ V } and K K i = {θi ∈ Θi | ∀j, θi (j) ∈ W } ,
and the following two valuations in Θi
x = (v − δ + ε00 , . . . , v − δ + ε00 ) and y = (v − δ, . . . , v − δ) . e i and K e i0 satisfy the hypothesis of Lemma 6.1. Thus, It is simple to verify that x, y, K for any ε0 > 0, we have
e i )) and σi0 ∈ ∆(UDi (K e i0 )) such that : ∀i ∈ [n] there exist σi ∈ ∆(UDi (K A A ∀s−i ∈ S−i , ∀j ∈ [m], Mi,j (σi , s−i ) − Mi,j (σi0 , s−i ) < ε0 .
(6.1)
Consider the allocation of mechanism M under the strategy profile σ 0 = (σ10 , σ20 , . . . , σn0 ). Because there are m copies of the good, there ought to be one player who, in expec27
Recall that all valuations in Θi , and thus in every candidate set, are non-increasing. Note that each vector (x1 , . . . , xt , yt+1 , . . . , ym ) is indeed non-increasing, because we have xj > yj and both x and y are non-increasing.
21
tation, receives no more than m copies. Without loss of generality, let him be player n Pm A . Thus, by (6.1) and multiple applications of 1: that is, j=1 j · M1,j (σ10 , . . . , σn0 ) ≤ m n
the triangle inequality,
Pm
A 0 · M1,j (σ1 , σ−1 )≤
Pm
A · M1,j (s1 , s−1 ) ≤
m + ε0 m2 . n By averaging, there exists a pure strategy profile s = (s1 , s−1 ) in the support of j=1 j
0 ) satisfying (σ1 , σ−1
j=1 j
Now let def
def
K = (K1 , . . . , Kn ) where Ki =
def
def
θ = (θ1 , . . . , θn ) where θi =
K e1
m + ε0 m 2 . n
(6.2)
if i = 1
e0 K i
if i = 2, . . . , n v, . . . , v
v − 2δ + ε00 , . . . , v − 2δ + ε00
if i = 1 if i = 2, . . . , n.
0 ) ∈ ∆(UD1 (K1 )) × · · · × ∆(UDn (Kn )) from (6.1), we Because we know (σ1 , σ−1
deduce that s ∈ UD(K). It is also obvious that θ ∈ K, and MSW(θ) = mv.
We now show that s, K, and θ satisfy the desired inequality of Theorem 3. Indeed, h i (∗) m m E SW θ, M (s) ≤ + ε0 m2 · v + m − − ε0 m2 · (v − 2δ + ε00 ) n n m m 0 2 00 0 2 = mv − 2δ m − −εm +ε m− −εm n n m = MSW(θ) − 2δm 1 − 1/n + 2δε0 m2 + ε00 m − − ε0 m2 . n Above, inequality (∗) holds because, when θ is the true-valuation profile, all players except player 1 value each copy of the good as v − 2δ + ε00 , while only player 1 values it as v. However, in the outcome M (s), player 1 can receive, in expectation, at most m n
+ ε0 m2 copies of the good (cf. (6.2)).
Lastly, notice that ε0 > 0 and ε00 > 0 can both be arbitrarily small, so we can choose them to satisfy 2δε0 m2 + ε00 m − m − ε0 m2 < ε. n This concludes the proof of Theorem 3.
22
7
Beyond Multi-Unit Auctions
As a social planner may never be certain about how precisely the players know their own preferences, the Knightian robustness of the Vickrey mechanism may be a welcome piece of news. A social planner’ desiderata, however, go beyond social welfare in multi-unit auctions. It is therefore legitimate to ask whether or not Knightian robustness is enjoyed by other mechanisms in different settings. So far, we have found the answer for provision of a public good (more generally, for single-parameter domains), and for unrestricted combinatorial auctions.
7.1
Knightian Single-Parameter Domains
In a classical single-parameter domain, there is a set A, the set of all possible allocations; for each player i there exists a publicly known subset Si ⊆ A; and the set of possible valuations for player i, Θi , consists of all functions mapping A to the reals,
subject to the following constraints: for each θi ∈ Θi , (1) θi (x) = 0 ∀x 6∈ Si and (2) θi (x) = θi (y) ∀x, y ∈ Si .
We continue to denote the true valuation of player i by θi∗ . An outcome consists of an allocation and a price profile. (The term “single-parameter” derives from the fact that each θi ∈ Θi coincides
with a single number: i’s value for, say, the lexicographically first element of Si . The term “classical” emphasizes that each player knows exactly his own true valuation.)
Single-parameter domains are general enough to include several settings of interest: in particular, provision of a public good28 [Cla71], bilateral trades [MS83], and buying a path in a network [NR01]. 28
Indeed, in the provision of a public good, A has just two elements, a (i.e., the good is provided), which different players may value differently, and b (i.e., the good is not provided), which all players value 0.
23
In a classical single-parameter domain, we say that a mechanism M is weakly dominant-strategy-truthful if for each player i: (0) Si = Θi (1) ∀vi ∈ Θi ∀vi0 ∈ Θi ∀v−i ∈ Θ−i
(2) ∀vi ∈ Θi ∀vi0 ∈ Θi \ {vi } ∃v−i ∈ Θ−i
Ui vi , M (vi , v−i ) ≥ Ui vi , M (vi0 , v−i ) Ui vi , M (vi , v−i ) > Ui vi , M (vi0 , v−i ) .
Several weakly dominant-strategy-truthful mechanisms have been proposed for classical single-parameter domains. (For a characterization of dominant-strategy truthfulness in such domains, see [AT01].) In this setting we again define Knightian self uncertainty by starting, for each player i, with a set of distributions for θi∗ , and then considering their expected values. Definition 7.1. In a single-parameter domain with Knightian valuation uncertainty, the only information that each player i has about the true valuation profile θ∗ consists of a set of reals, Ki , containing θi∗ . Such a domain is δ-approximate if for each player i, sup Ki − inf Ki ≤ δ. Here, we prove the Knightian robustness of many mechanisms at once as follows. Theorem 6.1. Let M be a weakly dominant-strategy-truthful mechanism for classical single-parameter domains. Then, in a domain with Knightian valuation uncertainty, for every player i, UD(Ki ) ⊆ inf Ki , sup Ki . A proof of this theorem can be found in [CMZ14c].
Theorem 6.1 is, for single-parameter domains, the counterpart of Lemma 4.1. That lemma implied that the social-welfare performance of the Vickrey mechanism in a δapproximate multi-unit auction gracefully degrades with δ. Theorem 6.1 implies the same for the VCG mechanism in single-parameter domains. In particular, it implies that, when applied to the provision of a public good in the presence of n Knightian players, the VCG mechanism guarantees, in undominated strategies, a social welfare ≥ MSW − 2nδ. As another example, when applied to buying paths in a network, the VCG mechanism guarantees a social welfare ≥ MSW − 2mδ, where m is the number
of edges in the network. Finally, we note that the proof of Theorem 6.1 easily extends to imply an analogous result for the VCG mechanism for single-minded combinatorial 24
auctions, which are not quite single-parameter domains.29 More generally, Theorem 6.1 implies that, for all weakly dominant-strategy mechanisms M (which include those of [Cla71], [MS83], and [NR01]) ‘the outcome M (θ) is sufficiently good whenever maxi |θi − θi∗ | is sufficiently small’.
7.2
Unrestricted Combinatorial Auctions
In an unrestricted combinatorial auction, there are n players and m distinct goods. The set of possible allocations A consists of all possible partitions A of [m] into 1 + n
subsets, A = (A0 , A1 , . . . , An ), where A0 is the (possibly empty) set of unassigned goods and Ai is the (possibly empty) set of goods assigned to player i. For each player i, a valuation is a function mapping each possible subset of the goods to a non-negative real, and the set of all possible valuations is Θi = {θi : 2[m] → R≥0 | θi (∅) = 0}.
In such auctions, the VCG mechanism guarantees efficiency in dominant strategies.
Definition 7.2. In an unrestricted combinatorial auction with Knightian valuation uncertainty, a player i’s candidate set is a subset Ki ⊂ Θi . def
For each subset of the goods T ⊂ [m], we let Ki (T ) = {θi (T ) | θi ∈ Ki }.
We say that Ki is δ-approximate if sup Ki (T ) − inf Ki (T ) ≤ δ for all T ⊆ [m]. The auction is δ-approximate if each candidate set Ki is δ-approximate.
It is immediately clear that, with Knightian valuation uncertainty, the VCG mechanism is no longer dominant strategy, and Theorem 1 can be extended to prove that no dominant strategy mechanism can guarantee social welfare greater that that obtainable by assigning the goods at random.The novelty, however, is that this time the social-welfare performance of the VCG mechanism is actually very poor. Namely, Theorem 6.2. In a δ-approximate combinatorial Knightian auction with n ≥ 2 play-
ers and m goods, the VCG mechanism cannot, in undominated strategies, guarantee social welfare greater than MSW − (2m − 3)δ. 29
In such an auction, there are m distinct goods, and each player i values, positively and for the same amount θi∗ , only the supersets of a given subset Si of the goods. This auction is not singleparameter because Si is private, that is, known solely to i. Accordingly, i’s true valuation can be fully described only by the number θi∗ and the subset Si . The VCG mechanism for single-minded auctions ensures, in undominated strategies, a social welfare that is at least MSW − 2 min{n, m}δ.
25
A proof of Theorem 6.2 can be found in [CMZ14a]. The reason for the above poor social-welfare performance is that Theorem 6.1 does not extend to unrestricted Knightian combinatorial auctions, and the undominated strategies of the VCG mechanism can be very counterintuitive. For example, for a player i knowing that his true value for some subset of the goods T lies in some interval [x, x + δ], reporting a valuation θi such that θi (T ) is very far from [x, x + δ] can be an undominated strategy. The Knightian fragility of the VCG mechanism in unrestricted combinatorial auctions may be disappointing, but then the latter auctions are difficult in many ways. That said, in [CMZ14b] we extend the notion of regret to the Knightian valuation model, and prove that the social welfare performance of the VCG mechanism continues to be excellent if each player (1) chooses a regret-minimizing strategy, or (2) resorts to regret minimization only to further refine his set of undominated strategies, if needed.
8
Conclusions
The literature on auction design so far concentrated on players having ambiguous beliefs about the valuations of their opponents. We believe it natural to assume that such ambiguity may extend to a player’s own valuation. We thus find it natural to investigate the performance of the Vickrey mechanism in such an environment, where it is no longer a dominant-strategy mechanism. Fragility is inherent to sophisticated endeavors, which certainly include mechanism design. It is thus reasonable to expect that many mechanisms fail when their fundamental underlying assumptions do not exactly hold. In multi-unit auctions, it is a fundamental assumption that the players know their own valuations. We have proven, however, that the classical Vickrey mechanism is robust and continues to perform very well even when the players have Knightian uncertainty about their own valuations. In fact, we have proven that the socialwelfare performance of the Vickrey mechanism is essentially optimal in Knightian undominated strategies. 26
In sum, as most things classical, the Vickrey mechanism outlives the confines in which it was conceived, and continues to be relevant in new and unforeseen settings.
Appendix A
Knightian Revelation Principle
Let us explicitly show that a version of the revelation principle [Gib73, DHM79, Mye79] holds also in the Knightian setting. We do so directly for auctions, so as not to introduce any additional notation. Recall that Ki is the set of possible candidate sets for player i. Definition A.1. Let M be a mechanism; for each player i, let Si be the set of actions of i in M ; and let si : Ki → ∆(Si ) be a function such that ∀K ∈ K, ∀i, ∀ai ∈ Si , ∀θi ∈ Ki ,
Ui (θi , M (si (Ki ), s−i (K−i ))) ≥ Ui (θi , M (ai , s−i (K−i ))) .
Then the function profile s = (s1 , . . . , sn ) is an ex-post Nash equilibrium. Lemma A.2 (Revelation Principle). Let M , Si for each player i, and s be as in Definition A.1. Then, there exists a Knightian DST mechanism M 0 such that ∀K ∈ K M 0 (K1 , . . . , Kn ) = M (s1 (K1 ), . . . , sn (K1 )) . Proof. Let M 0 be the Knightian direct mechanism so defined: def
∀K ∈ K M 0 (K1 , . . . , Kn ) = M (s1 (K1 ), . . . , sn (K1 )). (The above equality is between distributions if M is probabilistic.) All that is left to prove is that mechanism M 0 is dominant-strategy-truthful. To this end, let Ki be the true candidate set of a player i. Then, for all Ki0 ∈ Ki , all θi ∈ Ki , and all K−i ∈ K−i ,
U θi , M 0 (Ki , K−i ) = U θi , M (si (Ki ), s−i (K−i )) ≥ U θi , M (si (Ki0 ), s−i (K−i )) = U θi , M 0 (Ki0 , K−i ) ,
where the inequality follows from the definition of ex-post Nash by letting ai = si (Ki0 ).
This completes the proof. 27
Because every dominant-strategy mechanism is also ex-post Nash, the theorem holds also when M is dominant-strategy.
B
Proof of Theorem 1
Let M be an arbitrarily chosen Knightian DST mechanism, and Si the set of pure strategies for player i. Recall that Si = Ki is the set of all possible δ-approximate candidate sets. We start by proving a separate claim. Namely, for every player i, fixing the candidate sets reported by his opponents, i’s probability of winning the good does not change with the candidate sets he might report, as long as they share at least two valuations. More precisely, denoting by MiA (K) the probability that i wins the good under a profile of strategies (i.e., candidate sets) K, and by MiP (K) the expected price of i under K, ei, K e 0 ∈ Ki such that Claim B.1. For every player i, every two candidate sets K i 0 e −i of candidate sets for i’s opponents, ei ∩ K e | ≥ 2, and every subprofile K |K i
ei, K e −i ) = M A (K e 0, K e −i ) .30 MiA (K i i
ei, Proof of Claim B.1. Because the true candidate set of player i may coincide with K e i should dominate reporting K e i0 , the and because when this is the case reporting K following inequality holds: ei, ∀ θi0 ∈ K
ei, K e −i ) · θ0 − M P (K ei, K e −i ) ≥ M A (K e 0, K e −i ) · θ0 − M P (K e 0, K e −i ) . MiA (K i i i i i i i
(B.1)
e 0 , and Similarly, because the true candidate set of player i may coincide with K i 0 e e because when this is the case reporting Ki should dominate reporting Ki , also the
following inequality holds:
e 0 , M A (K e 0, K e −i )·θ00 −M P (K e 0, K e −i ) ≥ M A (K ei, K e −i )·θ00 −M P (K ei, K e −i ). (B.2) ∀ θi00 ∈ K i i i i i i i i i ei ∩ K e 0 such that x > y. Then, setting Now let us pick two valuations x, y ∈ K i
θi0 = x in (B.1) and θi00 = y in (B.2), and summing up (the corresponding terms 30
ei, K e −i ) = M P (K e 0, K e −i ). Although we do not care about prices, it is also the case that MiP (K i i
28
of) the resulting inequalities, the MiP price terms cancel out, yielding the following inequality: ei, K e −i ) · x + M A (K e 0, K e −i ) · y ≥ M A (K e 0, K e −i ) · x + M A (K ei, K e −i ) · y , MiA (K i i i i i
that is,
e i0 , K e −i ) · (x − y). ei, K e −i ) · (x − y) ≥ MiA (K MiA (K
Because x > y, the last inequality implies that
e 0, K e −i ) ≥ M A (K ei, K e −i ) . MiA (K i i
(B.3)
Similarly, setting θi0 = y in (B.1) and θi00 = x in (B.2), summing up the resulting inequalities, regrouping, and using again the fact x > y, we deduce that: ei, K e −i ) ≥ MiA (K e i0 , K e −i ) MiA (K
(B.4)
Together, inequalities (B.3) and (B.4) imply our claim.
Let us now prove a second claim. Namely, given any DST mechanism M , and any profile K = (K1 , . . . , Kn ) of candidate sets, there exists a player j ∈ [n] such that M is insensitive to j’s report. That is, when j’s opponents collectively report the
subprofile K−j , j can report any set Vj ∈ Kj he wants, but cannot increase beyond 1/n his probability of getting the good.
Claim B.2. If K is a profile of δ-approximate candidate sets of cardinality greater than 1, then there exists j ∈ [n] so that, for every δ-approximate set Vj ∈ Kj of cardinality greater than 1,
MjA (Vj , K−j ) ≤
1 . n
Proof of Claim B.2. When the players report a profile of strategies K as in the stateP ment of the claim, because i∈[n] MiA (K) ≤ 1, there must exist a player j such that MjA (K) ≤ 1/n.
(0)
(1)
(k)
Next, let us consider a sequence of δ-approximate candidate sets Kj , Kj , . . . , Kj
such that: (0)
Kj
= Kj ,
(k)
Kj
= Vj ,
(`−1) (`) and Kj ∩ Kj ≥ 2 for all `.
For instance, we can construct the above sequence as follows. 29
(1) def (k−1) def • Let Kj = inf Kj , inf Kj + δ be an interval containing Kj , and Kj = inf Vj , inf Vj + δ be an interval containing Vj . (2)
• Let Kj
(3)
def
= [inf Kj + δ/2, inf Kj + 3δ/2], Kj
def
= [inf Kj + δ, inf Kj + 2δ], and so
(k−1)
on, until an interval intersecting Kj
(on more than 1 value) is generated. e −j = K−j , K e j = K (`) , and Then, invoking Claim B.1 for each ` ∈ [k], with i = j, K j (`−1) 0 e =K K , we obtain that j
j
(k)
(1)
MjA (Vj , K−j ) = MjA (Kj , K−j ) = · · · = MjA (Kj , K−j ) (0)
= MjA (Kj , K−j ) = MjA (Kj , K−j ) ≤
1 . n
Let us now see that Theorem 1 holds, that is, let us see that MSW(θ) ∀θ ∈ K1 × · · · × Kn E SW(θ, M (K)) ≤ +δ . n Apply Claim B.2 by choosing Ki = [0, δ] for each i ∈ [n], and Vj = [B − δ, B] for any
B > 2δ. Consider the scenario where the profile of true candidate sets is (Vj , K−j ), and the profile θ of true valuations is as follows: θj = B, and θi = 0 for all i 6= j . Then, the expected social-welfare E[SW(θ, M (Vj , K−j ))] is at most Bn , because Claim B.2 shows that MjA (Vj , K−j ) ≤
1 . n
On the other hand, the maximum social welfare
MSW(θ) = B. This completes the proof of Theorem 1.
C
Proof of Lemma 6.1
e i ) are both nonIt is a simple exercise to verify that the sets UDi (Ki ) and UDi (K
empty (but anyways proved in Appendix E). If there exists a common (pure) strategy e i ), then setting σi = σ si ∈ UDi (Ki ) ∩ UDi (K ei = si completes the proof. Therefore, let e i ) are totally disjoint. us assume that UDi (Ki ) and UDi (K
e i ). Accordingly, let si be a pure strategy in UDi (Ki ). Then, necessarily, si 6∈ UDi (K e i ) implies the existence of a (possibly mixed) strategy By definition, si 6∈ UDi (K e i )) that (weakly) dominates si for player i under candidate set K ei, σ ei ∈ ∆(UDi (K
which is denoted by σ ei (i,Ke i ) si (see Definition 5.1). 30
Next, we argue that ∃ τi ∈ ∆(UDi (Ki )) such that τi (i,Ki ) σ ei .31
(C.1)
P (t) (t) (t) Let σ ei = ei , where X is a finite index set, each sei is a pure strategy t∈X α s (t) e i ), and P of UDi (K = 1. Invoking again the disjointedness of UDi (Ki ) and t∈X α e i ), we deduce that, for each t ∈ X, se(t) 6∈ UDi (Ki ). This implies the existence UDi (K i
(t) τi
(t)
of a strategy ∈ ∆(UDi (Ki )) such that τi P (t) (t) t∈X α τi , (C.1) holds.
(t)
def
(i,Ki ) sei . Thus, by defining τi =
e i )) such that Similarly, we could argue that there exists some τei ∈ ∆(UDi (K
τei (i,Ke i ) τi . Continuing in this fashion, going back and forth between ∆(UDi (Ki )) e i )), we obtain an infinite chain of strategies, and ∆(UDi (K (1)
σi
(1)
≺(i,Ke i ) σ ei
(2)
≺(i,Ki ) σi
(2)
≺(i,Ke i ) σ ei
≺(i,Ki ) · · ·
This (weak) dominance chain implies the following utility inequalities: for all s−i ∈ S−i and all k ∈ N:
P j A (σ (k) , s ) ei (l) − M P (σ (k) , s−i ) M θ −i i,j i i j∈[m] i Pl=1 P (k) (k) (k) j A P e e ≤ Ui θi , M (e σi , s−i ) σi , s−i ) = σi , s−i ) j∈[m] Mi,j (e l=1 θi (l) − Mi (e P Pj (k) A σ (k) , s ) P σ (k) , s ) ∀θi ∈ Ki , Ui θi , M (e σi , s−i ) = −i −i j∈[m] Mi,j (e i i l=1 θi (l) − Mi (e P P (k+1) (k+1) j A P (k+1) , s ) ≤ Ui θi , M (σi , s−i ) = , s−i ) −i j∈[m] Mi,j (σi l=1 θi (l) − Mi (σi (C.2) ei, ∀θei ∈ K
(k) Ui θei , M (σi , s−i )
=
P
Now, pick some t ∈ {0, 1, . . . , m}, and let def ei . θi = θei = (zt,1 , zt,2 , . . . , zt,m ) = (x1 , x2 , . . . , xt , yt+1 , . . . , ym ) ∈ Ki ∩ K
For this choice of θi and θei , we see from (C.2) that for all s−i ∈ S−i and all k ∈ N (k) (k) (k) (k+1) Ui θei , M (σi , s−i ) ≤ Ui θei , M (e σi , s−i ) = Ui θi , M (e σi , s−i ) ≤ Ui θi , M (σi , s−i ) 31
Note that, while we have only defined what it means for a pure strategy to be dominated by a possibly mixed one, the definition trivially extends to the case of dominated strategies that are mixed, as is the case in “τi (i,Ki ) σ ei ” in (C.1).
31
which then implies that for all s−i ∈ S−i and all k ∈ N P P (k) (k) j A − MiP (σi , s−i ) l=1 zt,l j∈[m] Mi,j (σi , s−i ) P P (k) (k) j A σi , s−i ) (e σ , s ) z − MiP (e ≤ M −i i,j i l=1 t,l j∈[m] P Pj (k+1) (k+1) A ≤ , s−i ) − MiP (σi , s−i ) . j∈[m] Mi,j (σi l=1 zt,l
Considering the above inequalities for k = 1, 2, . . . , we get the following infinite and non-decreasing sequence of real numbers (for each s−i ∈ S−i ): P P (1) (1) j A (σ , s ) z − MiP (σi , s−i ) M −i i,j i l=1 t,l j∈[m] P P (1) (1) j A (e σ , s ) z − MiP (e σi , s−i ) ≤ M −i t,l i,j i l=1 j∈[m] P P (2) (2) j A ≤ M (σ , s ) z − MiP (σi , s−i ) ≤ · · · −i i,j i j∈[m] l=1 t,l P A This sequence is upperbounded by x1 + · · · + xm . (Indeed, zt,l ≤ xl , j∈[m] Mi,j (·) ≤
1, and each price is non-negative.) Thus, by the Bolzano-Weierstrass theorem, for def
every s−i ∈ S−i and t ∈ {0, 1, . . . , m}, denoting by X = max{1, x1 + · · · + xm } and (s−i ,t)
def
D = min{1, minl∈[m] {xl − yl }}, there must exist some Hε k>
(s ,t) Hε −i :
∈ N such that for all
P Pj (k) (k) P A , s ) z (σ , s ) − M (σ M −i −i i i i l=1 t,l j∈[m] i,j P Pj (k) (k) P A σi , s−i ) ≤ − σi , s−i ) l=1 zt,l − Mi (e j∈[m] Mi,j (e
εD . (C.3) 12m · X Notice that, because the mechanism is finite and thus so is the set S−i , we can let def Hε = max Hε(s−i ,t) : s−i ∈ S−i , t ∈ {0, 1, . . . , m} ∈ N . (k)
Next, we pick arbitrarily k > Hε , and prove that σi
(k)
and σ ei
are the two strategies
that we are looking for. To this end, pick arbitrarily s−i ∈ S−i , consider the two
inequalities obtained by invoking (C.3) twice (once for t and once for t − 1), and
32
combine them with the triangle inequality to obtain:32 P Pj εD (k) (k) A P M (σ , s ) z − M (σ , s ) ≥ −i −i i i i j∈[m] i,j l=1 t,l 6m · X P Pj (k) (k) P A σi , s−i ) σi , s−i ) − l=1 zt,l − Mi (e j∈[m] Mi,j (e P Pj (k) (k) P A (σ , s ) (σ , s ) z − M − M −i −i i i i l=1 t−1,l j∈[m] i,j P Pj (k) (k) P A σi , s−i ) σi , s−i ) + l=1 zt−1,l − Mi (e j∈[m] Mi,j (e Pm P (k) (k) m A A σi , s−i ) xt − yt − = j=t Mi,j (e j=t Mi,j (σi , s−i ) xt − yt P Pm (k) (k) m A A σi , s−i ) , = (xt − yt ) j=t Mi,j (σi , s−i ) − j=t Mi,j (e
which further implies that (using xt − yt ≥ D): P Pm ε (k) (k) m A A σi , s−i ) ≤ . j=t Mi,j (e j=t Mi,j (σi , s−i ) − 6m · X
(C.4)
Now we use triangle inequality again on (C.4) for t and t − 1, and deduce that: ε A (k) (k) A σi , s−i ) ≤ ≤ε . (C.5) Mi,t (σi , s−i ) − Mi,t (e 3m · X That is, the first inequality of Lemma 6.1 has been proven. Let us now consider the
price terms. We begin with a simple application of (C.5). P Using the fact that jl=1 zt,l ≤ X, we obtain, Pj Pj ε A (k) (k) A σi , s−i ) . Mi,j (σi , s−i ) l=1 zt,l ≤ l=1 zt,l − Mi,j (e 3m Then, using the triangle inequality again, we get P ε P Pj Pj (k) (k) A A z − M (e σ , s ) z M (σ , s ) . j∈[m] i,j i −i −i i l=1 t,l l=1 t,l ≤ j∈[m] i,j 3 Expressing the above inequality as |a − c| ≤ 3ε , (C.3) can be expressed as |(a − b) − (c − d)| ≤
εD . 12m·X
Thus, we use the triangle inequality again and obtain ε εD P (k) (k) P + 0, there exist profiles K of δ-approximate candidate sets such that MSW(θ) E SW(θ, M (K)) ≤ n Theorem 2’. In a m-unit Knightian auction, for all multiplicative δ, all products K ∀θ ∈ K1 × · · · × Kn
of δ-approximate candidate sets, all profiles v ∈ UD(K), and all θ ∈ K 2 1−δ SW θ, Vickrey(v) ≥ · MSW(θ) . 1+δ Theorem 3’. Let M be a finite mechanism in a δ-approximate multi-unit Knightian auction with n > 1 players and multiplicative δ > 0. Then, for every ε > 0, there exist products K of δ-approximate candidate sets, valuation profiles θ ∈ K, and undominated strategy profiles s ∈ UD(K), such that ! (1 − δ)2 + 4δ n E SW(θ, M (s)) ≤ · MSW(θ) + ε . (1 + δ)2
E
The Set of Undominated Strategies is Non-Empty
It is trivial to see that, no matter what candidate set Ki a player i may have, UDi (Ki ) is non-empty in the Vickrey mechanism. Indeed, for each θi ∈ Ki , θi ∈ UDi (Ki ).
We wish to point out that UDi (Ki ) is non-empty for all finite mechanisms, a
simple fact relied upon in the proof of Theorem 3. This fact is trivial if a Knightian player is restricted to consider only pure strategies, and not hard to prove when mixed strategies are allowed. As we shall see, the proof follows the analogous one for the classical setting, that is, when Ki is a singleton. Fact E.1. Let M be a finite mechanism, i a player, and Ki a candidate set of i. Then, UDi (Ki ) 6= ∅. 34
Proof. Let Si = {s1 , . . . , st } be the finite pure strategy set of player i. We proceed by contradiction. Suppose that every strategy in Si is (weakly) dominated, with respect
to Ki , by some strategy in ∆(Si ). Then, s1 is dominated, and thus there exists a P mixed strategy tk=1 αk sk ∈ ∆(Si ) such that s1 ≺(i,Ki )
def
where α ∈ ∆ = {x ∈ [0, 1]t :
Pt
t X
αk sk ,
(E.1)
k=1
xk = 1}. Notice that, by condition (2) in
k=1
Definition 5.1, we cannot have s1 ≺(i,Ki ) s1 . Therefore, we must have α1 < 1. Now, we simplify (E.1) by canceling α1 s1 on both sides, and obtain s1 ≺(i,Ki ) Next, since s2 is dominated, let
t X k=2
s2 ≺(i,Ki )
αk sk . 1 − α1
t X
βk sk
(E.2)
(E.3)
k=1
for some β ∈ ∆. By substituting (E.2) into (E.3), we further simplify (E.3) with s2 ≺(i,Ki )
t X
βk0 sk
(E.4)
k=2
for some β 0 ∈ ∆ such that β10 = 0. Again, by subtracting β20 s2 on both sides and rescaling, we obtain
s2 ≺(i,Ki )
t X
βk00 sk ,
(E.5)
k=3
for some β 00 ∈ ∆ such that β100 = β200 = 0. We substitute (E.5) into (E.2), and obtain s1 ≺(i,Ki ) 0
for some α ∈ ∆ such that
α10
=
α20
= 0.
t X
αk0 sk ,
k=3
This process, quite similar to Gaussian elimination for linear systems, can be
continued until we obtain sk ≺(i,Ki ) st for every k = 1, . . . , t − 1. This implies that
st must be an undominated strategy for player i, contradicting the hypothesis that
UDi (Ki ) = ∅.
35
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