Dating Strategies Are Not Obvious Itai Ashlagi∗
Yannai A. Gonczarowski† October 29, 2015
arXiv:1511.00452v1 [cs.GT] 2 Nov 2015
Abstract The Gale-Shapley (men-proposing) stable matching algorithm (1962) has been shown by Dubins and Freedman (1981) to be strategy proof for each man. In recent years, this algorithm has been used extensively across many real-life matching markets, such as matching students to schools in many cities across the world. Interestingly, despite applicants being advised that it is in their best interest to state their true preferences, there is evidence to suggest that a significant amount of applicants are nonetheless attempting to strategically misreport their preferences (Hassidim et al., 2015). In this note, we show that while strategy proof for each man, no algorithm implementing the men-optimal stable matching mechanism is “obviously strategy proof” for the men — a term recently defined by Li (2015) — showing that indeed to be convinced that no strategic opportunities exist for any man, regardless of whether this mechanism is described using the Gale-Shapley algorithm or in any other way, requires significant cognitive effort, and contingent reasoning.
1
Model and Background
Let M and W be disjoint finite sets, s.t. |M | = |W |. We think of M as a set of agents and of W as a set of resources.
1.1
Matching with Single-Sided Preferences
Definition 1 (Preferences). 1. A preference list over W is a total ordering of W . We denote the set of all preference lists for over W by P(W ). 2. A preference profile for M over W is a specification of a preference list over W for each agent m ∈ M . (So the set of all preference profiles for M over W is P(W )M .) 3. Given a preference profile for M over W , an agent m ∈ M is said to prefer w ∈ W over w0 ∈ W , denoted by w m w0 , if w precedes w0 on the preference list of m. We write w m w0 if either w m w0 or w = w0 . Definition 2 (Matching). A matching between M and W is a one-to-one mapping between M and W . We denote the set of all matchings between M and W by M(M, W ). Definition 3 (Matching Mechanism). A (single-sided) matching mechanism is a function C : P(W )M → M(M, W ), from preference profiles for M over W to matchings between M and W . ∗
Department of Management Science and Engineering, Stanford University, E-mail :
[email protected]. Einstein Institute of Mathematics, Rachel & Selim Benin School of Computer Science & Engineering and Federmann Center for the Study of Rationality, The Hebrew University of Jerusalem, Israel; and Microsoft Research, E-mail :
[email protected]. †
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Definition 4 (Strategy Proofness). A matching mechanism C is said to be strategy proof if for every preference profile p¯ = (pm )m∈M ∈ P(W )M , for every agent m ∈ M and for every alternative preference list p0m ∈ P(W ), it is the case that Cm (p) m Cm (p0m , p¯−m ) according to pm , where (p0m , p¯−m ) denotes that preference profile obtained from p¯ by setting the preferences of m to be p0m . In other words, m would not be better off misrepresenting his preference list to be p0m .
1.2
Obvious Strategy Proofness
In this section, we briefly describe machinery recently developed in great generality by Li (2015), which we heavily rephrase for the special case of deterministic matching mechanisms with finite preference and outcomes sets. For ease of presentation, we furthermore focus on mechanism implementations with complete information. We emphasize that the results of this note, and their proofs, still hold (mutatis mutandis) for the general definitions of Li (2015). Definition 5 (Extensive-Form Mechanism Implementation). An extensive-form mechanism implementation for M over W consists of: 1. A rooted tree T . 2. A map X : L(T ) → M(M, W ) from the leaves of T to matchings between M and W . 3. A map Q : V (T ) \ L(T ) → M , from internal nodes of T to M . 4. A map A : E(T ) → 2P(W ) , from edges of T to predicates over P(W ), s.t. both of the following hold: • The predicates corresponding to edges outgoing from the same node are disjoint. • The disjunction (i.e., set union) of all predicates corresponding to edges outgoing from a node n equals the predicate corresponding to the last edge outgoing from a node labeled Q(n) along the path from the root to n, or to the predicate matching all elements of P(W ) if no such edge exists. Definition 6 (Pass Through). We say that a preference profile p¯ ∈ P(W )M passes through a node n ∈ V (T ) if for each edge e along the path from the root to n, it is the case that pQ(n0 ) ∈ A(e), where n0 is the source node of e. Definition 7 (Implemented Mechanism). Given an extensive-form mechanism implementation I, we denote by C I , called the mechanism implemented by I, the matching mechanism mapping a preference profile p¯ ∈ P(W )M to the matching X(n), where n is the unique leaf through which p¯ passes. Equivalently, n is the node in T obtained by traversing T from its root, and from each node n0 , following the edge outgoing from n0 whose predicate matches the preference list of Q(n0 ). Definition 8 (Divergence). We say that p, p0 ∈ P(W ) diverge at a node n ∈ V (T ) if there exist two distinct edges e, e0 outgoing from n s.t p ∈ A(e) and p0 ∈ A(e0 ). Definition 9 (Obvious Strategy Proofness). An extensive-form mechanism implementation I is said to be obviously strategy proof (OSP) if for every p¯ = (pm )m∈M ∈ P(W )M and p¯0 = (p0m )m∈M ∈ P(W )M , for every m ∈ M and for every node n with Q(n) = m through which I (¯ I (¯ both p¯ and p¯0 pass and in which pm and p0m diverge, it is the case that Cm p) m Cm p0 ) according to pm .
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Li (2015) shows that obviously-strategy-proof mechanism implementations are, in a precise sense, mechanism implementations that can be shown to implement strategy-proof mechanisms under a cognitively limited model that does not allow for contingent reasoning. Remark 1. To observe that strategy proofness of C I indeed is a weaker condition than obvious strategy proofness of I, note that C I is strategy proof iff for every p¯ = (pm )m∈M ∈ P(W )M , for every m ∈ M , for every p¯0m ∈ P(W ), and for every node n with Q(n) = m through which p¯ I (¯ I (¯ passes and in which pm and p0m diverge,1 it is the case that Cm p) m Cm p) according to pm . Definition 10 (OSP Implementability). A matching mechanism C : P(W )M → M(M, W ) is said to be OSP implementable if C = C I for some obviously strategy proof implementation I. In this case, we say that I OSP-implements C.
1.3
The Stable Matching Problem
Gale and Shapley (1962) consider a matching problem with two-sided preferences; they think of M as a set of men, and of W as a set of women. We consider a simplified version of their model. Definition 11 (Preferences for W over M ). We define preference lists over M and preference profiles for W over M analogously to Definition 1. Definition 12 (Stability). Given a preference profile p¯ = (pm )m∈M ∈ P(W )M for M over W and a preference profile q¯ = (qw )m∈M ∈ P(M )W for W over M , a matching µ ∈ M(M, W ) is said to be unstable w.r.t. p¯ and q¯ if there exist m ∈ M and w ∈ W , each preferring the other over the partner matched to them by µ. If µ is not unstable, then it is said to be stable. Theorem 1 (Gale and Shapley (1962)). A stable matching between M and W always exists for every pair of preference profiles p¯ ∈ P(W )M and q¯ ∈ P(M )W . Moreover, there exists an M -optimal stable matching, i.e., a stable matching where each man weakly prefers his match in this stable matching over his match in any other stable matching. Definition 13 (C q¯). For a preference profile q¯ ∈ P(M )W for W over M , we denote by C q¯ : P(W )M → M(M, W ) the (single-sided) matching mechanism mapping each preference profile p¯ ∈ P(W )M for M over W to the M -optimal stable matching w.r.t. p¯ and q¯. Dubins and Freedman (1981) have shown that Gale and Shapley’s algorithm for finding the M -optimal stable matching is strategy proof for each man m ∈ M (and in fact weakly group strategy proof for the men). We now rephrase their result in the notation of this note. Theorem 2 (Dubins and Freedman (1981)). C q¯ is strategy proof for every preference profile q¯ ∈ P(M )W for W over M . In this note, we focus on the question of whether C q¯ is not only strategy proof, as Dubins and Freedman have shown, but also OSP implementable.
2
OSP-Implementable Special Cases
Before we phrase our main impossibility result, we first review a few special cases in which C q¯, the men-optimal stable matching mechanism for fixed women’s preferences q¯, is in fact OSP implementable. This is, for instance, the case when women’s preferences are perfectly aligned. 1
These conditions imply that (p0m , p¯−m ) also passes through n.
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Example 1 (C q¯ is OSP Implementable when Women’s Preferences are Perfectly Aligned). Let q ∈ P(M ) and let q¯ = (q)w∈W be the preference profile in which all women share the same preference list q. C q¯ is OSP implementable by the following “serial-dictatorship” mechanism: ask the man most preferred according to q¯ which woman he prefers most, and assign that woman to this man (in all leaves of the subtree corresponding to this response), ask the man secondmost preferred according to q¯ which woman he prefers most out of those not yet assigned to any man, and assign that woman to this man (in all leaves of the subtree corresponding to this response), etc. The proof that this implementation is OSP is similar to the proof of Li (2015) that random serial dictatorship (where the order of the dictatorship is determined uniformly at random rather than using an externally given order q¯) is OSP. Another noteworthy example is that of arbitrary preferences in a very small matching market. Example 2 (C q¯ is OSP Implementable when |M | = |W | = 2). When |W | = |M | = 2, C q¯ is OSP implementable for every q¯ ∈ P(M )W . Indeed, let W = {x, y} and M = {a, b}. If qx = qy , then C q¯ is OSP implementable as explained in Example 1. Otherwise, w.l.o.g. a x b and b y a; for this case, Fig. 1 describes an OSP implementation of C q¯.
a
y a x
b
x a y
a⇐x b⇐y
x b y
a⇐y b⇐x
y b x
a⇐x b⇐y
Figure 1: An OSP implementation of C q¯ for |W | = |M | = 2 and for q¯ where a x b and b y a. Examples 1 and 2 can be generalized together for acyclical preferences, a preference-profile structure first defined by Ergin (2002). Definition 14 (Acyclicality). A preference profile q¯ ∈ P(M )W for W over M is said to be cyclical if there exist a, b, c ∈ M and x, y ∈ W s.t. a x b x c y a. If q¯ is not cyclical, then it is said to be acyclical. Ergin (2002) shows that acyclicality of q¯ is necessary and sufficient for C q¯ to be group strategy proof (and not only weakly group strategy proof) and Pareto efficient. When a preference profile q¯ satisfies this condition (as do the preference profiles in Examples 1 and 2), we can show that C q¯ is OSP implementable. Theorem 3. C q¯ is OSP implementable for every acyclical preference profile q¯ ∈ P(M )W for W over M . Proof sketch. We prove the result by induction over |M | = |W |. By acyclicality, at most two men are ranked by some woman as her top choice. If only one such man m ∈ M exists, then 4
he is ranked by all women as their top choice — in this case, we ask this man for his top choice w ∈ W , assign her to him, and then continue by induction (finding in an OSP manner the men-optimal stable matching between M \ {m} and W \ {w}). Otherwise, there are precisely two men a ∈ M and b ∈ M who are ranked by some woman as her top choice. By acyclicality, each woman either has a as her top choice and b as her second-best choice, or vice versa. For each woman w ∈ W that prefers a most, we ask a whether he prefers w most; if so, we assign w to a and continue by induction. Otherwise, for each woman w ∈ W that prefers b most, as ask b whether he prefers b most; if so, we assign w to b and continue by induction. Otherwise, we ask each of a and b for his top choice, assign each of them his top choice, and continue by induction. We do note, however, that acyclicality of q¯ is not a necessary condition for OSP implementability of C q¯, as demonstrated by the following example. Example 3 (OSP-Implementable C q¯ with Cyclical q¯). Let W = {x, y, z} and M = {a, b, c}. We claim that C q¯, for the following cyclical preference profile q¯, is OSP implementable: a x b x c a y c y b b z a z c. We begin by noting that q¯ is indeed cyclical, as a y c y b z a. We now note that the following mechanism OSP-implements C q¯: 1. Ask a whether he prefers x the most; if so, assign x to a and continue as in Example 2 (finding in an OSP manner the men-optimal stable matching between {y, z} and {b, c}). 2. Ask a whether he prefers y the most; if so, assign y to a and continue as in Example 2. (Otherwise, we deduce that 1) a prefers z the most and therefore 2) c will not end up being matched to z.) 3. Ask b whether he prefers z the most; if so, assign z to b and continue as in Example 2. 4. Ask b whether he prefers x the most; if so, assign x to b, z to a, and y to c. (Otherwise, we deduce that b prefers y the most.) 5. Ask c whether he prefers x over y. If so, assign x to c, y to b, and z to a. (Otherwise, we deduce that b will not end up being matched to y.) 6. Ask b whether he prefers z over x. Assign b to his most-preferred choice between z and x and continue as in Example 2. Nonetheless, as we show in the next section, when |W | > 2 and women’s preferences are sufficiently unaligned, C q¯ is not OSP implementable.
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Impossibility Result for General Preferences
We now phrase our main impossibility result. Theorem 4. a. If |W | > 2, there exists a preference profile q¯ ∈ P(M )W , s.t. C q¯ is not OSP implementable. b. As |W | grows, with high probability C q¯ is not OSP implementable, where q¯ ∼ U P(M )W . 5
Before proving Theorem 4, we first prove a special case of Theorem 4(a), which cleanly demonstrates the construction underlying our proof of both parts of Theorem 4. Lemma 1. For |W | = 3, there exists a preference profile q¯ ∈ P(M )W s.t. C q¯ is not OSP implementable. Proof. Let W = {x, y, z} and M = {a, c, b}. Let q¯ be the following preference profile: a x b x c b y c y a c z a z b.
(1)
Assume for contradiction that an OSP implementation I for C q¯ exists. We define: p1a , z y x p2a , y x z
p1b , x z y p2b , z y x
p1c , y x z p2c , x z y,
and set Pa , {p1a , p2a }, Pb , {p1b , p2b }, and Pc , {p1c , p2c }. Following a proof technique of Li (2015), we note that if we “prune” the tree of I by replacing, for each edge e, the predicate A(e) with the conjunction (i.e., set intersection) of A(e) with the predicate matching all elements of PQ(n) , where n is the source node of e, and by consequently deleting all edges e for which A(e) = ⊥, we obtain, in a precise sense, an OSP implementation of C q¯ where the preference list of each agent m ∈ M is a priori restricted to be in Pm . See Appendix A for the precise definition of implementation and OSP when the domain of preferences is restricted. Let n be the earliest (i.e., closest to the root) node in the pruned tree that has more than one outgoing edge (such a node clearly exists, since C q¯ is not constant over Pa × Pb × Pc ). By symmetry of q¯, Pa , Pb , Pc , w.l.o.g. Q(n) = a. By definition of pruning, it must be the case that n has two outgoing edges, one labeled p1a , and the other — p2a . We claim that the mechanism of the pruned tree is in fact not OSP. Indeed, for pa = p2a (the “true preferences”), pb = p2b , and pc = p1c , we have that CaI (¯ p) = Caq¯(¯ p) = x, yet for p0a = p1a (a “possible manipulation”), p0 ) = Caq¯(¯ p0 ) = y, even though CaI (¯ p0 ) = y a x = CaI (¯ p) p0b = p1b , and p0c = p2c , we have that CaI (¯ 0 according to pa (by definition of n, both p¯ and p¯ pass through n, and pa and pb diverge at n), and so the mechanism of the pruned tree indeed is not OSP — a contradiction. Proof sketch of Theorem 4. Part a follows from the proof of Lemma 1. Indeed, for |M | = |W | > 2, we can choose three distinct men a, b, c and three distinct women x, y, z whose preferences will be set as in Lemma 1 (with all men in M \ {a, b, c} less preferred by x, y, and z than a, b, and c), and pair up all other men and women into pairs (m, w) where we set the preference list of w to be m followed by all other men arbitrarily (and in the pruning in the proof, set only one possible preference list for m, which favors w the most). Part b also follows from the proof of Lemma 1. Indeed, if there exist a, b, c ∈ M and x, y, z ∈ W s.t. all parts of Eq. (1) hold, then by a similar argument (setting a, b, c to prefer each of x, y, z over all women in W \{x, y, z}, and pairing up all other men and women into pairs (m, w) where we set the preference list of m to prefer w the most), q¯ is not OSP implementable.
References L. E. Dubins and D. A. Freedman. Machiavelli and the Gale-Shapley algorithm. American Mathematical Monthly, 88(7):485–494, 1981.
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H. I. Ergin. Efficient resource allocation on the basis of priorities. Econometrica, 70(6):2489– 2497, 2002. D. Gale and L. S. Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15, 1962. A. Hassidim, D. Marciano-Romm, A. Romm, and R. I. Shorrer. “Strategic” behavior in a strategy-proof environment. mimeo, 2015. S. Li. Obviously strategy-proof mechanisms. mimeo, 2015.
A
Mechanism Implementations with Restricted Domains
In this appendix, we explicitly provide the adaptation of the definitions of Section 1.2 for a restricted domain of preferences, as used in the proof of Lemma 1. The differences from the definitions of Section 1.2 are marked with an underscore. For every m ∈ M , fix a subset Pm ⊆ P(W ). Furthermore, define P , ×m∈M Pm . Definition 15 (Extensive-Form Mechanism Implementation). An extensive-form mechanism implementation for M over W w.r.t. P consists of: 1. A rooted tree T . 2. A map X : L(T ) → M(M, W ) from the leaves of T to matchings between M and W . 3. A map Q : V (T ) \ L(T ) → M , from internal nodes of T to M . 4. A map A : E(T ) → 2P(W ) , from edges of T to predicates over P(W ), s.t. both of the following hold: • The predicates corresponding to edges outgoing from the same node are disjoint. • The disjunction (i.e., set union) of all predicates corresponding to edges outgoing from a node n equals the predicate corresponding to the last edge outgoing from a node labeled Q(n) along the path from the root to n, or to the predicate matching all elements of PQ(n) if no such edge exists. Definition 16 (Pass Through). We say that a preference profile p¯ ∈ P passes through a node n ∈ V (T ) if for each edge e along the path from the root to n, it is the case that pQ(n0 ) ∈ A(e), where n0 is the source node of e. Definition 17 (Implemented Mechanism). Given an extensive-form mechanism implementation I w.r.t. P, we denote by C I , called the mechanism implemented by I, the mechanism mapping a preference profile p¯ ∈ P to the matching X(n), where n is the unique leaf through which p¯ passes. Equivalently, n is the node in T obtained by traversing T from its root, and from each node n0 , following the edge outgoing from n0 whose predicate matches the preference list of Q(n0 ). Definition 18 (Divergence). Let n ∈ V (T ). We say that p, p0 ∈ PQ(n) diverge at n if there exist two distinct edges e, e0 outgoing from n s.t p ∈ A(e) and p0 ∈ A(e0 ). Definition 19 (Obvious Strategy Proofness). An extensive-form mechanism implementation I w.r.t. P is said to be obviously strategy proof (OSP) if for every p¯ = (pm )m∈M ∈ P and p¯0 = (p0m )m∈M ∈ P, for every m ∈ M and for every node n with Q(n) = m through which both I (¯ I (¯ p¯ and p¯0 pass and in which pm and p0m diverge, it is the case that Cm p) m Cm p0 ) according to pm . 7