A Simple Ascending Generalized Vickrey Auction - CiteSeerX

A Simple Ascending Generalized Vickrey Auction∗ David C. Parkes†

Debasis Mishra‡

Lyle H. Ungar§

December 31, 2004

Abstract We design a simple ascending Vickrey auction for the combinatorial allocation problem. We make use of a concept called universal competitive equilibrium (UCE) price to design our auction. Our auction searches for a UCE price, which (along with an efficient allocation) provides enough information to compute the Vickrey payments of buyers. Our auction maintains non-linear and non-anonymous prices. At every iteration, we identify the set of “unsatisfied” buyers in either the main economy or in one of the “marginal” economies and increase prices of the bundles demanded by these buyers. The auction terminates at a UCE price when there are no unsatisfied buyers in either the main economy or in any of the marginal economies. When buyers are substitutes, considering unsatisfied buyers from the main economy alone terminates the auction at a UCE price. Truthful bidding is an ex post Nash equilibrium for buyers in our auction. In a variant of this auction, we introduce dynamic price discrimination, thus maintaining anonymous prices in the auction as long as possible. This becomes a true generalization of the English auction to the combinatorial allocation problem.

Keywords: combinatorial auctions; multi-item auctions; universal competitive equilibrium; Vickrey auctions

∗

The first author is supported in part by NSF grant IIS-0238147. Thanks to participants at the Stanford Institute for Theoretical Economics Summer Workshop on The Economics of the Internet, 2002, INFORMS 2004 Annual Meeting, and CORE Mathematical Programming Seminar. † Division of Engineering and Applied Sciences, Harvard University, [email protected] ‡ Center for Operations Research and Econometrics (CORE), Universit´e Catholique de Louvain, Belgium, [email protected] § Computer and Information Science Department, University of Pennsylvania, [email protected]

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1

Introduction

A seller is selling a set of heterogeneous indivisible items to a set of buyers. Buyers may have value on bundles of items. The problem of finding an efficient allocation (allocation that maximizes the total payoff of buyers and the seller) in this setting is also called the combinatorial allocation problem (CAP) [34]. The famous incentive compatible mechanism which solves the CAP is the Vickrey-Clarke-Groves (VCG) [35, 10, 17] mechanism. In its natural form, the VCG mechanism is a sealed-bid mechanism which takes valuations from buyers, calculates the efficient allocation and payments of buyers such that every buyer gets his marginal product as payoff. Truthful bidding is a dominant strategy equilibrium in the VCG mechanism. Iterative auctions are often preferred over sealed-bid auctions in practical settings [11]. Iterative auctions have better privacy properties, are more transparent for bidders, and can have higher revenue in correlated value settings. For the CAP, iterative auctions are invaluable because they allow bidders to follow simple equilibrium strategies without reporting, or even knowing, their exact value for all bundles [30]. Iterative auctions provide price discovery, which is absolutely essential if auctions are to extend to problems with many items. For these reasons, there has been great interest in implementing the VCG outcome using an ascending auction [15, 2, 8, 14, 25]. We refer to such auctions as ascending Vickrey auctions. This paper designs a simple ascending Vickrey auction, iBundle Extend and Adjust (iBEA) which implements the VCG outcome for the CAP. We view the auction as a direct generalization of the English auction. The underlying auction dynamics are simple: bids are collected at current prices, a “winner determination problem” is solved to provide a provisional allocation, and prices are increased based on losing bids. A recent negative result due to de Vries et al. [14] serves to motivate the design. These authors demonstrated the impossibility of an ascending Vickrey auction, for a class of ascending auctions. We relax their definition to allow an ascending price trajectory, but coupled with a final downwards price adjustment once the auction terminates. In this aspect, our auction is related to the only other ascending Vickrey auction [25], in that final payments are computed as an adjustment from final ask prices. The main contribution in iBEA is that this auction couples a simple price update rule with a dynamic method to determine when it is necessary to introduce non-anonymous prices. All previous ascending Vickrey auctions, even for the special case of submodular valuations [14], maintain non-anonymous prices throughout the auction and thus cannot be considered a generalization of the English auction. The significance of the anonymous price update rules in iBEA goes beyond the simple appeal of generalizing a well-known single item auction. Anonymous prices bring the benefit of more effective elicitation. Price discovery is more tightly coupled across bidders, with the prices to one bidder increasing as a direct result of unsuccessful bids from another bidder.

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1.1

Earlier Results on Implementing the VCG Outcome

With one exception, all ascending auctions in the literature are only able to implement the VCG outcome for restricted valuation profiles. The auctions in Demange et al. [15] implement the VCG outcome when every buyer demands at most one item. The auction in Ausubel [2] implements the VCG outcome for homogeneous units when marginal values of buyers are non-increasing. The auctions in de Vries et al. [14], Ausubel and Milgrom [4], Parkes [29], and Parkes and Ungar [31] implement the VCG outcome for the CAP when a submodularity condition on valuations holds. Indeed, de Vries et al. [14] define a class of ascending auctions for which buyer submodularity is necessary for implementing the VCG outcome for the CAP. To implement the VCG outcome on all possible valuation profiles, Mishra and Parkes [25] provide a slight relaxation to the definition of ascending auctions in [14]. The definitions agree on the following features of an ascending auction: 1) Prices on all bundles (weakly) increase across rounds while buyers submit bids. 2) Auction dynamics are history-free, with the final allocation and the final payments determined based on the bids in the final round. 3) The final allocation should be a solution to the winner determination problem in the final round. However, de Vries et al. [14] require that the final payments be the final ask prices, which Mishra and Parkes [25] relax as follows: 4) The final payments can be computed based on the prices in the final round and bids submitted in the final round. Mishra and Parkes [25] introduced the concept of a universal competitive equilibrium (UCE). A UCE price is a competitive equilibrium (CE) price of the main economy as well as of every marginal economy. They show that UCE price (along with an efficient allocation) gives enough information to implement the VCG outcome and any CE price (along with an efficient allocation) that gives enough information to implement the VCG outcome has to be a UCE price. Given UCE prices, VCG payments are determined as a discounted price to each buyer. Subsequently, Lahaie and Parkes [23] prove that UCE price information must necessarily be elicited in any mechanism implementing VCG outcome, whether or not the mechanism is explicitly price-based. Mishra and Parkes [25] present an auction that maintains a single non-linear and nonanonymous price path, and has a price adjustment step that is both difficult to describe and computationally difficult. The complexity of the price update is shared with the primal-dual based methods in de Vries et al. [14]. The auction in this paper, in comparison, has a simple price update step and maintains non-linear but anonymous prices with non-anonymous prices introduced dynamically to a subset of buyers during the auction.

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1.2

Our Contributions

In this work, we generalize the iBundle auction in Parkes [29], and Parkes and Ungar [31] to implement the VCG outcome for generic valuation profiles. Submitting true demand sets in every round of the auction is an ex post Nash equilibrium for buyers. We summarize our contributions in Table 1. Settings

Primal-Dual Ascending Price Vickrey Auction

Subgradient Ascending Price Vickrey Auction

Discounts Used From the Final Price

Optimization Algorithm of

Single Item

English (Japanese) Auction

English Auction

0

Main economy

Heterogeneous Items Unit Demand

Demange et al.[15] (Exact Version)

0

Main economy

Homogeneous Units Non-Increasing Marginal Values Heterogeneous Items Buyers are Submodular Heterogeneous Items Buyers are Substitutes

Ausubel[1] Bikhchandani and Ostroy [8] de Vries et al.[14]

Demange et al.[15] (Approximate Version) Bertsekas[7] Crawford and Knoer[12] -

0

Main economy

Parkes [29], Parkes and Ungar[31] Ausubel and Milgrom[4] Parkes [29], Parkes and Ungar[31] modified in this paper

0

Main economy

Marginal contribution to revenue Marginal contribution to revenue

Main economy

Multi-Item Auctions

Heterogeneous Items Generic Valuations

de Vries et al.[14] modified in Mishra and Parkes[25] Mishra and Parkes[25]

This paper

Main economy and marginal economies

Table 1: Ascending Price Vickrey Auctions In its simplest form, the new iBEA auction maintains non-anonymous prices throughout the auction. This variation is very simple to describe to bidders and completely transparent. Simplicity in the price adjustment process can be very important to generate transparency in the auction. It can also encourage bidder participation in the auction, thus improving revenue and efficiency [11]. In each round, we identify a pivot economy for which the efficient allocation problem is not yet solved. We solve the winner determination problem for the pivot economy at the current price. Based on this, we find out the set of unsatisfied buyers in a round. We increase the price of the unsatisfied buyers on all bundles demanded by them. The auction converges to a UCE price and implements the VCG outcome by giving discounts from the final UCE price. When buyers are substitutes, the only pivot economy considered in our auction is the main economy and our auction converges to a UCE price immediately after solving the efficient allocation problem of the main economy. In its more interesting variation, iBEA is initialized with anonymous prices and introduces non-anonymous (discriminatory) prices as necessary. We identify sufficient conditions under which anonymous prices can be maintained in our auction. Unlike conditions on valuation functions of buyers, these are conditions on the demand sets of buyers and are easy to detect by the seller. Moreover, the transition from anonymous to non-anonymous prices is continual. Individual buyers can be removed from the set of buyers that still face anonymous prices, as necessary to ensure convergence to UCE prices. For a single-item allocation problem, the auction maintains a single (anonymous) price throughout and is exactly the English auction. A technical condition on buyers (“buyers are 4

substitutes”) is necessary for an anonymous price trajectory [25]. We provide a sufficient characterization (in terms of a covering idea) to explain when iBEA can maintain anonymous prices throughout the auction. The underlying optimization algorithm in i Bundle is a subgradient algorithm [14]. iBEA is running iBundle sequentially on different economies, and therefore the underlying optimization algorithm in iBEA is also a subgradient algorithm. This stands in contrast to the underlying optimization algorithm implemented in previous ascending Vickrey auctions [14, 25], which are based on a primal-dual algorithm and have more complicated priceadjustment rules. We adopt a general proof framework to establish the economic properties of iBEA, which emphasizes the specific role of quasi-CE prices in establishing important revenue monotonicity properties. A quasi-CE price is a price at which there exists a feasible allocation of bundles to buyers such that every buyer gets a bundle either from his payoff-maximizing set of bundles or the empty bundle and the seller maximizes his revenue from such an allocation. Auctions in deVries et al. [14], Parkes [29], and Ausubel [1] maintain quasi-CE prices across rounds. Before proceeding we should make a couple of comments about the economic properties of the VCG mechanism for general combinatorial allocation problems. Ausubel and Milgrom [4, 5] note that the VCG mechanism is vulnerable to manipulation by coalitions of losing buyers and also observe that the VCG outcome can be outside of the core of the coalitional game induced by the CAP. While acknowledging these weaknesses, the VCG mechanism nevertheless remains unique amongst efficient mechanisms. The VCG mechanism maximizes seller revenue across all individual rational and efficient mechanisms [22], and thus the vulnerabilities of the VCG mechanism are not specific to the VCG mechanism per se but apply to the class of efficient mechanisms. Efficiency and not revenue is of primary concern for many combinatorial auction problems. For instance, consider the use of combinatorial auctions to allocate resources in computational systems such as computational grids [16]. Typically deployed by national and international research agencies, the primary concern is that of efficient allocation. For instance, consider the use of combinatorial auctions to allocate take-off and landing slots at a congested airport [33, 6]. Again, mandates from congress may be expected to place the primary emphasis on efficient usage. The chief advantage of the VCG mechanism over other combinatorial auctions (such as ascending-proxy [4]) in these settings, is that it implements the efficient outcome in an equilibrium that is invariant to the distribution of private information across buyers. In comparison, the ascending-proxy and iBundle [29, 31] auctions are efficient only in a complete-information Nash equilibrium. Thus, when efficiency is the primary concern, and when implemented as an ascending auction to promote price-discovery and efficient preference elicitation, VCG mechanisms remain compelling in many settings. Opportunities for collusive manipulation should be mitigated through providing only the minimal feedback that is necessary to guide a bidder 5

in his own bidding, for instance through price information that gives an aggregate signal about the bids from other buyers.

1.3

Outline

The rest of the paper is organized as follows. Section 2 introduces our model, and the concepts of competitive equilibrium and universal competitive equilibrium. In Section 3 we design an ascending auction that implements the VCG outcome for any valuation profile of buyers. In Section 4, we design an ascending auction that implements the VCG outcome with dynamic price discrimination, striving to maintain an anonymous price trajectory. Both Sections 3 and 4 first describe an ascending auction that terminates with CE prices, and then extend the design to terminate with UCE prices from which VCG payments can be determined. Section 5 contains some useful discussions on special cases when our auctions can maintain anonymous prices throughout and how to speed up our auctions. We conclude with a summary in Section 6.

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Competitive Equilibrium and Extensions

In this section, we introduce a formal model for the combinatorial allocation problem and define CE and UCE prices. Connections are established between UCE prices and the VCG outcome, from which a class of ascending auctions for which straightforward bidding is an ex post Nash equilibrium is defined.

2.1

The Model

A seller has n heterogeneous indivisible items to sell to m buyers denoted by the set B = {1, . . . , i, . . . , m}. We will use the terms buyer(s) and bidder(s) interchangeably. The set of items is denoted by A and Ω = {S : S ⊆ A}. Elements of Ω are called bundles and ∅ is a bundle by definition. Buyers have valuation on bundles. The valuation of buyer i on bundle S is denoted as vi (S) (assumed to be a non-negative integer). The valuation on the ∅ bundle is zero for all buyers. We adopt a private-values model. Each buyer exactly knows his own valuation function and it does not depend on the valuation of other buyers. The utility (or payoff) of a buyer i on a bundle S is given by vi (S) − p, where p is the price paid. We will also assume that valuations are monotonic, i.e. for every buyer i ∈ B, vi (S) ≤ vi (T ) if S ⊆ T . The seller is assumed to have zero value on items and thus his utility or payoff or revenue is the total payment he receives. An economy E(M ) is simply a collection of buyers M ⊆ B, their valuation functions and items in A. A feasible allocation in economy E(M ) is a vector of bundles on buyers in M and is denoted as X = (X1 , . . . , X|M | ) such that Xi ∩ Xk = ∅ for any i 6= k. Realize that Xi = ∅ is allowed and ∪i∈M Xi ⊆ A. Allocation X allocates bundle Xi to buyer i. The set 6

of all feasible allocations in economy E(M ) is denoted by X(M ). An allocation X ∈ X(M ) is efficient in economy E(M ) if there does not exist another allocation Y ∈ X(M ) such P P that i∈M vi (Yi ) > i∈M vi (Xi ). If X ∗ is an efficient allocation of economy M , we will let P V (M ) = i∈M vi (Xi∗ ). We will denote B \ {i} as B−i and B = {B, B−1 , . . . , B−m }. For any i ∈ B, economy E(B−i ) is called a marginal economy.

2.2

Competitive Equilibrium

Let p ∈ RΩ×B be a non-linear and non-anonymous price vector. Throughout the paper, price + p has components for every buyer i ∈ B. But, when we talk about economy E(M ) for some M ⊆ B, the components of p associated with buyers in M only are considered. We define the demand set of a buyer i at price p as Di = {S ∈ Ω : vi (S) − pi (S) ≥ vi (T ) − pi (T ), ∀ T ∈ Ω}. P Similarly, define the supply set of the seller as L(M ) = {X ∈ X(M ) : i∈M pi (Xi ) ≥ P i∈M pi (Yi )}. Definition 1 (Competitive Equilibrium) A price and an allocation tuple (p, X) is a competitive equilibrium (CE) of economy E(M ) for M ⊆ B if X ∈ L(M ) and for every buyer i ∈ M , Xi ∈ Di . The price p is called a CE price. The allocation corresponding to a CE is an efficient allocation [9]. The CE concept is very useful in designing efficient iterative auctions. In general, efficient auctions search for CE price of economy E(B). We will show connections between CE concept and Vickrey payments next.

2.3

Universal Competitive Equilibrium and Vickrey Payments

The VCG mechanism is an efficient, individually rational and strategy-proof direct revelation mechanism. A direct revelation mechanism asks each buyer to reveal the complete valuation function. If vˆ is the reported valuation profile, then the VCG mechanism solves the efficient allocation problem for every M ∈ B with respect to this reported valuation profile. The final allocation X is an efficient h allocation ofieconomy E(B) and the payment for buyer i is vcg calculated as pi = vˆi (Xi )− V (B)−V (B−i ) , where V (B) and V (B−i ) are determined with respect to reported values. As the VCG mechanism is strategy-proof, truth-revelation is a dominant strategy equilibrium. The payment of a buyer is often referred to as the Vickrey payment and the payoff as the Vickrey payoff. Now, consider the following “demand game”. The seller announces a (non-linear and . The buyers report their demand sets at price p (assume non-anonymous) price p ∈ RΩ×B + buyers report demand sets truthfully). Given the demand set information, if p is a CE price, then we can implement an efficient allocation. What price gives us enough information to implement VCG outcome? To answer this question, Mishra and Parkes [25] introduced the notion of universal competitive equilibrium price. 7

Definition 2 (Universal Competitive Equilibrium Price) A price p is a universal competitive equilibrium (UCE) price if p is a CE price of economy E(M ) for every M ∈ B. A UCE price always exists in our model. Consider the price p with pi (S) = vi (S) ∀ i ∈ B, ∀ S ∈ Ω. Clearly, p is a UCE price since the efficient allocation in economy E(M ) for every M ∈ B trivially satisfies the CE conditions for that economy. P For every M ⊆ B, define π s (M ) = i∈M pi (Xi ), where X ∈ L(M ). The following result shows that knowledge of a UCE price is sufficient and necessary to implement VCG outcome. Theorem 1 (Mishra and Parkes [25]) Let (p, X) be a CE of E(B). (i) If p is a UCE price, then for every buyer h i i ∈ B, the Vickrey payment of i can be vcg s s calculated as pi = pi (Xi ) − π (B) − π (B−i ) . (ii) If there is enough information to compute the VCG payment for every buyer from (p, X), then p is a UCE price. h i The term π s (B)−π s (B−i ) is the marginal contribution of buyer i to the payoff (revenue) of the seller and is the discount given to buyer i. The price after the discount is called the discounted price. The form of this adjustment is closely analogous to the standard term V (B) − V (B−i ) in the VCG payments. However, with a UCE price the VCG payments are computed as adjustments from prices and without direct revelation. Recently, Lahaie and Parkes [23] extended this result to prove that a UCE price is generally necessary for any VCG mechanism, whether or not the mechanism is explicitly pricebased. This parallels the result due to Nisan and Segal [28], that demonstrates that a CE price is both necessary and sufficient to implement the efficient outcome in any mechanism.

2.4

Incentives

Although the VCG mechanism is strategy-proof, it is well known that an iterative auction implementing VCG outcome may not support straightforward bidding (where buyers bid truthfully in each iteration) in a dominant strategy equilibrium [19, 32, 8, 14, 25]. Rather, straightforward bidding forms an ex post equilibrium for buyers, i.e. if all buyers except buyer i bid truthfully, then buyer i has no incentive to follow any strategy other than the truthful bidding strategy. We formalize this idea for a general ascending auction. For this, we adopt the definition of ascending auctions from Mishra and Parkes [25]. Definition 3 (Ascending Auction) An ascending auction assigns to each profile of B×Ω , an ascending price path P v (t) in each round t, and a final buyer valuations v ∈ R+ allocation X v , such that: C1 At every iteration t, buyers make claims about their respective demand sets, Dit , at price P v (t). 8

C2 Every buyer gets a bundle (possibly ∅) from his demand set at the end of the auction, T . Xiv ∈ DiT for all i ∈ B. C3 The final payment of buyers in the auction is determined from the final allocation X v and the final prices P v (T ). C4 For every iteration t in the auction, the prices in t + 1 are determined by P v (t) and the demand set information Dit . This definition is a slight relaxation of that in de Vries et al. [14], retaining the temporallocality of updates but allowing for the final payments to be adjusted from the final prices. Ascending auctions that implement the VCG outcome must be “UCE-based”, by Theorem 1. By this, we mean that the auctions must terminate with UCE prices and then adjust to VCG payments using the revenue-based discounts. So, we will now state the equilibrium property for UCE-based ascending auctions. The proof that straightforward bidding is an ex post Nash equilibrium of an ascending UCE-based auction requires consistency, meaning that each buyer follows a straightforward bidding strategy for some (perhaps untruthful) valuation function vˆi . This is readily achieved, for instance via revealed-preference activity rules [3] or via proxy agents that follow straightforward bidding strategies on the basis of partial value information [32]. Revealed-preference activity rules simply interpret bids in each round as placing constraints on a buyer’s valuation (given a model of straightforward bidding) and ensure that bids in each round are consistent with some valuation function. Buyers must conform to the activity rule to remain active in the auction. Theorem 2 Straightforward bidding is an ex post Nash equilibrium for buyers in a UCEbased auction with consistency requirements. Proof : Fix the valuations vˆ−i that are implied by the strategies of all buyers except i. Recall that such valuations exist by the consistency requirements. By bidding truthfully, buyer i receives payoff π vcg (vi , vˆ−i ) from the equivalence between adjusted UCE prices and the VCG payment. Consider some non-truthful strategy from buyer i, that is consistent with valuation vˆi . The iterative auction will implement the outcome of the VCG auction for valuations (ˆ vi , vˆ−i ), but buyer i will always prefer the VCG outcome for valuations (vi , vˆ−i ) because VCG mechanism is strategy-proof. ¥ In this paper, we design UCE-based ascending auctions that together with proper consistency requirements support straightforward bidding in an ex post Nash equilibrium. Therefore, without loss of generality, we will assume in the sequel that buyers will report their true demand sets in each round of our auctions. Three comments are in order. First, truthful bidding is an ex post equilibrium, meaning that this equilibrium is invariant to the private valuations of buyers. Truthful bidding is a 9

best-response for buyer i, whatever the values of other buyers. Buyers need not have models of the values of other buyers. Second, this is not a dominant strategy equilibrium because buyers are able to condition their strategy on the strategy of other buyers. This is where the equilibrium analysis of an iterative VCG auction (with consistency requirements) differs from that of a sealed-bid VCG auction. Third, truthful bidding is also a subgame perfect (ex post) equilibrium because every subgame is equivalent to a subgame that would be reached in equilibrium for some valuation profile. This follows from the consistency requirements.

3

A Simple Ascending Vickrey Auction

In this section, we design a simple ascending auction which implements the VCG outcome for any valuation profile. We call this auction i Bundle, Extend and Adjust (i BEA). iBEA maintains non-linear and non-anonymous prices. iBEA is a simple extension to the iBundle auction [29, 31]. For instance, in its first stage iBEA can run iBundle on the main economy. As subsequent stages, iBEA then runs iBundle from the current price on each marginal economy in turn. The auction always maintains a single, and ascending, price trajectory. The auction terminates with a price that is simultaneously a CE price in the main economy and in each marginal economy, and therefore a UCE price. From these prices the VCG payments are determined as revenue-based adjustments. The more interesting variation of iBEA, in which price discrimination is introduced dynamically during the auction, will be presented in the next section. As a preliminary, we review the iBundle auction, which has identical price dynamics to the ascending-proxy auction of Ausubel and Milgrom [4]. Once iBundle is introduced, iBEA is defined as a simple extension. We prove that iBEA converges to a UCE price, and present an example of the auction on a small problem.

3.1

The iBundle Auction

iBundle [29, 31] is a simple ascending combinatorial auction. The auction maintains quasiCE prices, defined below, and terminates with CE prices and the efficient allocation when buyers follow a straightforward bidding strategy. We define the auction for a general economy E(M ) with buyers in M ⊆ B only. Given prices p, let M + = {i ∈ M : ∅ ∈ / Di } denote the buyers with positive payoff at the current prices. As before, L(M ) denotes the supply set of the seller at the current prices. Let L∗ (M ) = {X ∈ L(M ) : Xi ∈ Di ∪ {∅}, ∀ i ∈ M }, denote the set of seller-maximal allocations that are aligned with the demand sets of buyers, in that every buyer receives the empty bundle or a bundle in his demand set. In general, prices in iBundle are not CE but quasiCE, defined as follows: Definition 4 (Quasi-CE Price) A price p is a quasi-CE price of economy E(M ) for any M ⊆ B, if L∗ (M ) is non-empty. 10

Quasi-CE prices are useful in designing ascending auctions, and maintained in every round of iBundle, as well as in the auctions of de Vries et al. [14] and Mishra and Parkes [25]. Given quasi-CE prices, the price-adjustment in each round of iBundle is determined by solving a winner determination problem.1 In every round, the seller chooses a revenue maximizing allocation X. If Xi ∈ / Di , then i is called an unsatisfied buyer. If Xi ∈ Di , then i is called a winner. Definition 5 (Winner Determination) The winner determination problem for economy E(M ) at a price selects a revenue maximizing allocation X ∈ L∗ (M ) such that the total number of unsatisfied buyers is minimized. At an arbitrary price, there may not exist a revenue maximizing allocation X such that X ∈ L∗ (M ). But, if quasi-CE prices are maintained, we can always find a solution to the winner determination problem. Such a solution is often referred to as a provisional allocation in a round of the auction. We will denote the set of unsatisfied buyers of economy E(M ) as U (M ). These are buyers which have positive payoff but do not receive any bundle from their demand set in the current provisional allocation. When there are no unsatisfied buyers, we reach a CE price of economy E(M ). Otherwise, the solution to winner determination problem defines the price adjustment, which depends on the bids of unsatisfied buyers. iBundle maintains non-anonymous and non-linear prices, and increases the prices in each round to unsatisfied buyers on the basis of their losing bids. Definition 6 (iBundle) i Bundle for economy E(M ) is an iterative auction with the following steps: S0 Start the auction from zero price. S1 Collect the demand set of buyers M ⊆ B at the current price. S2 Solve the winner determination problem given current demand, and identify the unsatisfied buyers. S3 If there are any unsatisfied buyers, increase the price of each such buyer by 1 on all bundles in his current demand set. Go back to Step S1. S4 The auction ends with the final solution to winner determination implemented as the allocation, and with winners making payments equal to the final ask price on the bundle allocated to them. The following properties are useful for i Bundle: 1

Determining effective ways to solve the winner determination problem has been an active area of research. Any existing computationally feasible method can be applied to find the set of unsatisfied buyers in our case. For a study on computational aspects of the winner determination problem, the readers are referred to [24].

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Lemma 1 Let U (M ) be the set of unsatisfied buyers in a round of i Bundle for economy E(M ) for some M ∈ B. Assume straightforward bidding. (i) If i ∈ U (M ), the change in maximum payoff of i is −1 and zero for every other buyer due to price adjustment. (ii) The demand set of each buyer weakly increases in each round. (iii) The price is a quasi-CE price of economy E(M 0 ) for every M 0 ∈ B. Proof : (i) Starting from zero price, the prices in the auction remain integral. For every buyer i ∈ U (M ), the price is increased by unity on all the bundles he demands. This decreases the payoff by 1. Prices of buyers in B \ U (M ) is unchanged and thus the change in payoff is zero. (ii) Since prices are integral and price increase is only 1 on the bundles demanded, the claim follows from (i). (iii) Consider any X ∈ L(M 0 ) for some M 0 ∈ B. Let Q = {i ∈ M 0 : Xi ∈ Di } denote the buyers that receive a bundle in their demand set. All bundles have zero price at the start of the auction. From (ii), if a bundle is not demanded in the current round it was never demanded before this round and its price is still zero. Thus, at the current price p, we have pi (Xi ) = 0 for all i ∈ M 0 \ Q and we can define partition Y as Yi = Xi for all i ∈ Q and Yi = ∅ otherwise. Clearly, Y ∈ L∗ (M 0 ) because it has the same revenue as X. This means, L∗ (M 0 ) is nonempty and p is a quasi-CE price of economy E(M 0 ). ¥ Theorem 3 [31] i Bundle for economy E(M ) maintains a quasi-CE price of economy E(M ) throughout the auction and converges to a CE price of that economy, with straightforward bidding. Proof : Lemma 1 already establishes that iBundle maintains a quasi-CE price in each round. For convergence to a CE price, note that the prices of bundles are bounded above by the valuations of buyers, since buyers are straightforward and by the price-adjustment rule. So, the auction will converge to a quasi-CE price where there are no unsatisfied buyers in economy E(M ), which is a CE price. ¥

3.2

iBundle, Extend and Adjust

In this section, we extend iBundle to search for a UCE price, and define the simplest variant of iBundle Extend and Adjust. This allows us to implement the VCG outcome for any valuation profile, and brings straightforward bidding into an ex post Nash equilibrium. Our auction extends the idea of unsatisfied buyers to marginal economies and thus defines a price adjustment that can search for a UCE price. Definition 7 (iBundle, Extend and Adjust) i Bundle, Extend and Adjust (i BEA) is an iterative auction with the following steps: 12

S0 Start the auction from zero prices. S1 Choose each economy in B in some order, taking the economy as the pivot economy, and perform the following steps: S1.1

Collect the demand set from all buyers at the current price.

S1.2

Solve the winner determination problem in the pivot economy given current demand, and identify the set of unsatisfied buyers in the pivot economy.

S1.3

If there are one or more unsatisfied buyers, increase the price on all bundles in his current demand set. Go back to Step S1.1.

S1.4

If all the economies have been pivoted, then stop and go to Step S2. Else, repeat by choosing the next economy in B as the pivot.

S2 The auction ends with the solution to winner determination problem in the main economy implemented as the final allocation, and the payment for every winner i defined as the final ask price on his bundle discounted by π s (B)−π s (B−i ), which is the marginal revenue to the seller from his presence at the final prices. The auction is an immediate extension of iBundle. For instance, suppose that the economies in B are ordered with the main economy B, followed by B−1 , B−2 , . . . , B−m . Then, iBEA first runs iBundle on the main economy. On termination of this stage, it then runs iBundle on economy E(B−1 ), with prices initialized to those at the termination of iBundle on the main economy. On termination of this stage, it then runs iBundle on economy E(B−2 ), with prices initialized to those at the termination of iBundle on economy E(B−1 ). This continues, until iBundle terminates for economy E(B−m ). The final price is a UCE price. The following result shows that iBEA is an ascending Vickrey auction for general valuations. Theorem 4 Straightforward bidding is an ex post Nash equilibrium of i BEA (with consistency requirements) and the auction implements the VCG outcome. Proof : From Lemma 1, the price in every round of iBEA is a quasi-CE price of economy E(M ) for every M ∈ B. On termination, there are no unsatisfied buyers in economy E(M ) for every M ∈ B and the final price is a UCE price. From Theorem 1 and the payment rules of iBEA, it implements the VCG outcome. Finally, the auction is UCE-based and has straightforward bidding in an ex post Nash equilibrium by Theorem 2. ¥ The consistency requirements are as described in Section 2.4, and can be satisfied through revealed-preference activity rules. We can also identify some special terminationhconditions. Consider the case of buyers i P are substitutes, when V (B) − V (K) ≥ i∈B\K V (B) − V (B−i ) for all K ⊆ B. There is 13

a very close correspondence between buyers are substitutes and substitutes valuations [4].2 The following result shows that iBEA will terminate without pivoting, but at UCE prices that may require a non-zero adjustment. Theorem 5 If buyers are substitutes, then i BEA implements the VCG outcome by pivoting only in the main economy, and has straightforward bidding in an ex post Nash equilibrium (with consistency requirements). Proof : If buyers are substitutes, any price that is a quasi-CE price in all marginal economies and a CE price in the main economy is a UCE price [25]. Since iBEA maintains quasi-CE price of marginal economies (Lemma 1), we can terminate it as soon as a CE price of main economy is discovered and converge to a UCE price. From the payment rule of iBEA, we implement the VCG outcome. ¥ Consider the case of buyers are submodular, when V (K) − V (K−i ) ≥ V (L) − V (L−i ) for every K ⊆ L ⊆ B and every hi ∈ B \ L. It isi easy to verify that buyer submodularity P implies V (L) − V (K) ≥ i∈L\K V (L) − V (L−i ) for every K ⊆ L ⊆ B, and thus buyers are substitutes as well. We have a stronger termination result: iBEA will converge to prices that are buyer-optimal CE prices and equal to VCG payments on bundles in the efficient allocation without adjustment.3 Theorem 6 If buyers are submodular, then i BEA implements the VCG outcome in an ex post Nash equilibrium by only pivoting in the main economy and without any discounts. Proof : Without loss of generality, consider a buyer i. Since the final price is a UCE price, we have πivcg ≥ πi , where πi is the final payoff of buyer i in the auction and πivcg is the VCG payoff of buyer i. Assume for contradiction that πivcg > πi . Now, πivcg and πi are integers because valuations are integer and prices are always integer. Since price increases monotonically, there is some round t in which πit = πivcg and buyer i is unsatisfied, where πit is the payoff of buyer i in round t. Let K be the set of winning buyers in round t, and X denote the revenue-maximizing allocation chosen by the seller (solution to the winner determination problem) so that X ∈ L∗ (B). Notice that X ∈ L∗ (K), X ∈ L∗ (K ∪ {i}), and the revenues of the seller in economy E(K) and E(K ∪ {i}) are related as π s (K) = π s (K ∪ {i}). Since prices are always quasiCE, then they are a CE price for E(K) in this round. By strong duality we can write, P s t V (K) = j∈K πj + π (K). Since buyer i is unsatisfied, we cannot satisfy the buyers in K ∪ {i} simultaneously. So, the current solution is not a CE for economy E(K ∪ {i}). This 2

Moreover, this condition is necessary and sufficient for VCG payments to be supported in the buyeroptimal competitive equilibrium [9]. 3 This result is closely related to that of Ausubel and Milgrom [4], who demonstrate convergence to the VCG outcome in their ascending-proxy auction (analogous to iBundle) in the limit as the bid increment approaches zero.

14

P means, by weak duality we can write, V (K ∪{i}) < j∈K∪{i} πjt +π s (K ∪{i}) = V (K)+πit = V (K) + V (B) − V (B \ {i}). Since buyers are submodular, this gives us a contradiction. ¥ Thus, if buyers are substitutes, then by only considering the main economy as the pivot economy, we will converge to a UCE price. For buyer submodularity, these prices support the VCG payments without discounts. One potential concern in iBEA is that buyers may lose interest in the auction once the main economy E(B) is adopted as the pivot. Beyond this phase, continued bidding from a buyer only serves to adjust the final payments made by other buyers, and has no effect on a buyer’s own final payment. Thus, if bidding is costly then a buyer would prefer to just stop bidding once the efficient allocation is determined. This can be partially mitigated by suppressing the feedback that allows a buyer to know the current pivot economy. For instance, a buyer need not be informed about the other buyers that are actively participating in any particular phase of iBEA. This problem can be fully mitigated by selecting the main economy as the final pivot economy. This is an elegant method to keep buyers interested till the end of the auction.

3.3

An Example

The example in Table 2 has three buyers and two items. The values of buyers are shown in the third row. Subsequent rows illustrate the progress of the auction in each round. Each row provides the prices on each bundle to each buyer, and the seller revenue in the main economy and in each marginal economy (i.e. for M ∈ {B, B−1 , B−2 , B−3 }). The bid of each buyer is indicated with parentheses. Comments in each round indicate which allocation is selected to solve the winner determination (WD) problem. The main economy E(B) is adopted as the initial pivot economy. iBundle for E(B) terminates in round 7, at which point the price is also a CE price of economies E(B−2 ) and E(B−3 ). Pivot economy E(B−1 ) is adopted for the final two rounds, at which point iBEA terminates with a UCE price.

4

Anonymous Price Adjustments

In this section we present a variant of the simple ascending auction that introduces nonanonymous prices only as necessary, but still converges to a UCE price. We refer to this auction as iBEA with Dynamic Price Discrimination, or simply iBEA(d). Significantly, this is the first ascending auction for CAP that is able to dynamically introduce non-anonymous prices, and this allows the auction to reduce to the English auction for the special case of a single item allocation problem. The main benefit that arises from iBEA(d) is a more concise price space that facilitates more efficient price discovery. For instance, with anonymous prices the unsuccessful bids from a buyer will directly increase the prices of other buyers. This is in contrast to i BEA, in which unsuccessful bid from a buyer 15

Values 1 2 3 4 5 6 7

8 9

Buyer 1 {1} {2} {1, 2} 3 0 3

Buyer 2 {1} {2} {1, 2} 0 6 6

Buyer 3 {1} {2} {1, 2} 0 2 4

Seller revenue in main and marginal economies

(0) 0 (0) 0 (0) (0) 0 0 (0) {0,0,0,0} Pivot: E(B). WD selects {{1}, {2}, ∅}. Buyer {3} is unsatisfied. (0) 0 (0) 0 (0) (0) 0 0 (1) {1,1,1,0} Pivot: E(B). WD selects {∅, ∅, {1, 2}}. Buyers {1, 2} are unsatisfied. (1) 0 (1) 0 (1) (1) 0 0 (1) {2,1,1,2} Pivot: E(B). WD selects {{1}, {2}, ∅}. Buyer {3} is unsatisfied. (1) 0 (1) 0 (1) (1) 0 (0) (2) {2,2,2,2} Pivot: E(B). WD selects {{1}, {2}, ∅}. Buyer {3} is unsatisfied. (1) 0 (1) 0 (1) (1) 0 (1) (3) {3,3,3,2} Pivot: E(B). WD selects {∅, ∅, {1, 2}}. Buyers {1, 2} are unsatisfied. (2) 0 (2) 0 (2) (2) (0) (1) (3) {4,3,3,4} Pivot: E(B). WD selects {{1}, {2}, ∅}. Buyer {3} is unsatisfied. (2) 0 (2) 0 (2) (2) (0) (2) (4) {4,4,4,4} CEs of economies E(B), E(B−2 ), E(B−3 ) are reached. {{1}, {2}, ∅} is an efficient allocation of E(B). Pivot: E(B−1 ). WD selects {∅, ∅, {1, 2}}. Buyer {2} is unsatisfied. (2) 0 (2) 0 (3) (3) (0) (2) (4) {5,4,4,5} Pivot: E(B−1 ). WD selects {∅, ∅, {1, 2}}. Buyer {2} is unsatisfied. (2) 0 (2) 0 (4) (4) (0) (2) (4) {6,4,4,6} An UCE price is reached. Final allocation: {{1}, {2}, ∅}. Final payment: p1 ({1}) = 2 − [6 − 4] = 0, p2 ({2}) = 4 − [6 − 4] = 2, p3 (∅) = 0. Table 2: Progress of i BEA for an example

only increases his own ask price. The cost of this concise price space and more tightly-coupled price dynamics is that the price adjustment step in iBEA(d) is a little more complicated to explain than in iBEA and requires some additional computation. However, this new complexity is at least isolated to one particular part of the price update: that of determining the buyers that will continue to face anonymous prices in the next round. It is the unsuccessful bids from these buyers that drive increases to anonymous prices. The additional complexity in the price updates brings a huge benefit: simple, anonymous prices in many interesting settings. In each round of the auction a test is performed to determine whether an unsatisfied buyer can continue to face anonymous prices. Once this is determined, then the actual price increase rule is as in iBundle: anonymous prices are increased on all bundles in bids from unsatisfied buyers facing anonymous prices, while prices to buyers facing non-anonymous 16

prices are increased when they are unsatisfied and on bundles in their own bids. The auction is initialized with every buyer facing anonymous prices. The presentation of the variation of iBEA with dynamic price discrimination will mirror that of Section 3. We will first define an ascending-price auction that terminates with CE prices for straightforward bidding and facilitates anonymous price adjustments. Then, we describe an extension to this auction that pivots on marginal economies and terminates with UCE prices from which VCG payments are determined. We characterize some sufficient conditions under which we can maintain anonymous prices throughout the auction in Section 5.1.

4.1

i Bundle with Dynamic Price Discrimination

Following the structure of Section 3 we first define a variation of iBundle with dynamic price discrimination. Parkes and Ungar [31] previously described a variation of iBundle with dynamic price discrimination that relied on the safety property. This section provides a slight generalization of this idea, defining a covering property that is satisfied by safety but also holds for additional cases. Following Parkes and Ungar [31], we still refer to this auction as iBundle(d). The basic idea is to start with every buyer facing anonymous prices, and then incrementally identify buyers who face non-anonymous prices. Let panon denote the anonymous prices and let pi denote the prices of buyer i which faces non-anonymous prices. Whenever it is not confusing, we will talk about one non-linear and non-anonymous price vector, p. If a buyer i is facing anonymous prices, then pi is simply panon . Let Anont denote the buyers facing anonymous prices in round t, and call these the anonymous buyers. Let Dit denote the demand set of buyer i in round t. We will denote X = (X1 , X2 , . . .) as a partition of items such that Xi ∩ Xj = ∅ for all Xi , Xj ∈ X and ∪i Xi ⊆ A. Since we will deal with anonymous prices, we do not associate buyers with partitions any more. We define iBundle(d) for an economy E(M ), for some M ∈ B. This will allow for an easy generalization to iBEA(d). Since every anonymous price is also a non-anonymous price, all previous notations (such as L(M ), L∗ (M ), π s (M ), U (M ) for every economy M ∈ B) can be extended to this setting. For instance, at a price p, L∗ (M ) consists of all feasible partitions that maximize the revenue of the seller in economy E(M ), where a feasible partition is defined as follows. Definition 8 (Feasible Partition) A partition X in economy E(M ) is a feasible partition at a price if every bundle S in X which has positive price is demanded by a unique buyer in M . Let U t (M ) denote the unsatisfied buyers in round t, given current prices and the current solution to winner determination.

17

         

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6

6

8QVDWLVILHG %X\HU

6

6

6 

:LQQHU

:LQQHU

Figure 1: Simple Example to Illustrate Buyer Covering The main task is to determine Anont+1 from Anont given bids (demand sets). Initially all buyers see anonymous prices, and Anon1 = M . Given Anont , we define Anont+1 as the winners in Anont together with a subset of unsatisfied and currently anonymous buyers, J t ⊆ U t (M ) ∩ Anont , so that we have Anont+1 := (Anont \ U t (M )) ∪ J t . It is the set of buyers J t that define the increases to anonymous prices in the next round, while buyers in Anont \ Anont+1 see non-anonymous prices for the first time. The set J t is defined in terms of a covering property, which is defined with respect to a set of covering buyers. We say a partition is i-consistent if it contains a bundle from the demand set of buyer i. A set of bundles S is K-feasible if each bundle in S is demanded by a unique buyer from K. Definition 9 (Covered Property) An anonymous and unsatisfied buyer i is covered by buyers in K in a round of the auction (denoted cover(i, K)), if every i-consistent partition on bundles for which the anonymous price increases is K-feasible. Buyers in K are called covering buyers. As an example, consider Figure 1. The figure shows a partition with bundles S1 , . . . , S5 . There are four anonymous buyers, two of which are unsatisfied and the other two are winners. The black arrows in the figure points to the bundles demanded by respective buyers. Now, all the bundles except bundle S3 will see a price increase in this partition according to the auction rules. Assume that these are the only unsatisfied anonymous buyers, so that bundles S1 , S2 , S4 and S5 are the only bundles to increase in price. Now, if all buyers are defined as covering buyers the both unsatisfied buyers are covered because this partition is covered (for instance, S1 is demanded by unsatisfied buyer 1, S2 by winner 1, S4 by unsatisfied buyer 2, and S5 by winner 2.) Observe that if we exclude winner 1 from the set of covering buyers, then unsatisfied buyer 1 does not satisfy the covered property because it is only able to cover one of bundles S1 and S2 itself. We construct the maximal set of buyers J t , such that every buyer i ∈ J t satisfies cover(i, J t ∪ W t ). The covering buyers are J t and the set of covering winners W t ⊆

18

(Anont \ U t (M )), defined as: W t := {i : Dit ⊆

[

Djt , i ∈ (Anont \ U t (M ))}

j∈J t

The covering winners, W t , includes all winners that face anonymous prices and see price increase on all bundles in their demand set given the final set of anonymous losers J t . Set J t is a maximal fixed point of this system of equations, and computed through a simple iterative procedure outlined below. Let us revisit the example in Figure 1. As before, assume that buyers 1 and 2 are the only unsatisfied anonymous buyers and that winners 1 and 2 are the only winners, and currently face anonymous prices. Since bundle S3 does not see a price change, winner 1 is not a covering winner. The only covering winner is winner 2. Thus, unsatisfied buyer 1 is not covered by himself, unsatisfied buyer 2 and covering winner 2 because buyer 1 is unable to cover both bundle S1 and S2 . So, this buyer begins to face non-anonymous prices in the next round. On the other hand, unsatisfied buyer 2 is covered by winner 2 and himself, and this buyer can continue to face non-anonymous prices. We provide a simple iterative procedure to determine the set J t , and thus Anont+1 . The procedure starts with J t including all anonymous but unsatisfied buyers and removes buyers from the set as necessary: Definition 10 (Finding Anonymous Buyers in i Bundle(d)) The maximal set of anonymous but unsatisfied buyers that are covered is found with the following iterative procedure: S0. J t := U t (M ) ∩ Anont . S1. Determine the set of covering winners W t given J t . S2. If every buyer in J t is covered then stop. J t gives the maximal set of anonymous but unsatisfied buyers that are covered. S3. Else, remove all buyers not covered from J t . Repeat from Step S1. The procedure identifies a maximal set of buyers that continue to face anonymous prices.4 Given this, we can now modify iBundle to allow for dynamic price discrimination: Definition 11 (i Bundle(d)) i Bundle(d) is an iterative auction for economy E(M ) with the following steps: 4

Consider J ∗ , which is the set of unsatisfied buyers satisfying the covering property at the end of the procedure in Definition 10. By contradiction, suppose some S ⊃ J ∗ and S ⊆ U t (M ) ∩ Anont for which covering holds. But, there must be an iteration in the procedure of Definition 10 at which S ⊆ J t and buyers in J t \ S were eliminated from the set J t . This means that buyers in J t \ S cannot satisfy covering property in this iteration. Since the set of covering buyers weakly decreases in each iteration, buyers in J t \ S cannot satisfy covering property at the end of the procedure. Thus, the set J ∗ is a maximal (although perhaps not unique) set.

19

S0. Start the auction from zero price. Initially all buyers are anonymous buyers. S1. Collect the demand set of buyers at the current price. S2. Solve the winner determination problem given current demand, and identify the unsatisfied buyers. If there are no unsatisfied buyers go to Step S5. S3. Identify the set of anonymous buyers Anont+1 using the covering test. For each buyer i ∈ (Anont \ Anont+1 ), initialize his prices to ptanon . S4. For unsatisfied buyers in set Anont+1 , increase the anonymous price on all bundles in the demand sets of these buyers by 1. For unsatisfied buyers facing non-anonymous prices, increase the price of each such buyer by 1 on all bundles in his current demand set. Go back to Step S1. S5. The auction ends with the final solution to winner determination problem implemented as the allocation, and with winners making payments equal to the final ask price on their bundles. Given the covering property, then iBundle(d) maintains quasi-CE price in each round. The crucial property provided by the covering check is that each bundle in a partition which sees an anonymous price increase is in the demand set of a separate buyer. This ensures that all partitions remain feasible throughout the auction. Theorem 7 iBundle(d) for economy E(M ) maintains quasi-CE price of economy E(M ) throughout the auction and converges to a CE price of that economy, for straightforward bidding. Proof : Every bundle in the demand set of a covering buyer sees an anonymous price increase. This means the demand set of a covering buyer grows weakly due to anonymous price adjustment and the payoff of covering buyers and the unsatisfied buyers facing nonanonymous prices decreases by one in every round of the auction. This also implies that in every round, every bundle with positive anonymous price is demanded by some buyer. Assume that every partition is feasible in round t. We will show that it remains feasible in round t+1. Let K t be the set of covering buyers in round t. Consider a partition X. If the price of none of the bundles in X are increased in round t, then it remains feasible in round t + 1. If the price of some bundles in X are increased non-anonymously, then those bundles continue to be demanded in round t + 1 by the buyers who demanded them in round t. The bundles in X which see anonymous price increase are each demanded by a separate buyer from K t by the covering property. These bundles continue to be demanded by the covering buyers at the new prices since the demand set of covering buyers grow from round to round. Thus, X remains a feasible partition in round t + 1. In the first round, every partition is feasible as the prices of all bundles are zero. By induction, every partition is feasible in every 20

round. Now, consider a partition X ∈ L(M ) in a round. Since X is feasible, X ∈ L∗ (M ) and we have quasi-CE prices. The prices of the bundles are bounded above by the valuations of buyers, by straightforward bidding and the price-adjustment rules. So, the auction will converge to a quasi-CE price where there are no unsatisfied buyers in economy E(M ), which is a CE price. ¥ Earlier, Parkes and Ungar [31] proposed the condition of bid safety, to determine when an unsatisfied buyer can continue to face anonymous prices. A buyer satisfies bid safety (or simply safety) if no pairs of bundles in his demand set are disjoint. Any buyer for which safety holds will be covered. To see this, let J t := Anont ∩ U t (M ) and consider a partition X constructed from the bundles that see price increases. We know that every such bundle is in the demand set of some buyer j ∈ J t , and that no buyer j ∈ J t can have more than one bundle in X in his demand set because this would violate safety.5 However, the covering property is more general than bid safety. In the example, the safety condition is clearly violated by both the unsatisfied buyers. But covering can still hold for unsatisfied buyer 2 because of the presence of winner 2. 4.1.1

An Example of i Bundle(d)

Consider an economy with four buyers and suppose all buyers currently face anonymous prices. Suppose there are two items. The demand sets of buyers at this round are the following: D1 = {{1, 2}}, D2 = {{1}, {1, 2}}, D3 = {{2}, {1, 2}}, and D4 = {{∅}, {1}}. Suppose buyers 1 and 4 are the winning buyers. So, prices on bundles {1}, {2}, and {1, 2} will increase. The payoff of all buyers except buyer 4 will decrease by 1 due to this price increase. So, if the price for all the buyers is increased anonymously, buyers 1, 2, and 3 will continue to demand the bundles they demanded before. Observe that every feasible partition (except the partition in which only ∅ is selected) has a bundle which sees price increase. To test covering, we need that the bundles that see price increases in every such partition are demanded by a separate buyer from buyers 1, 2, and 3. Covering holds, and after price adjustment every such partition will be feasible given the demand sets of buyers. This ensures that quasi-CE prices are maintained in the auction. We make two quick observations. First, when buyers are single-minded the safety condition is trivially satisfied. A buyer is single-minded if his valuation function is of the following form: there exists Si ∈ Ω such that vi (S) = vi (Si ) if Si ⊆ S and vi (S) = 0 otherwise. Second, iBundle(d) generalizes the English auction. In a single item setting, all buyers are single-minded, and safety is trivially satisfied in each round. The auction maintains a single price trajectory and exactly mimics the English auction. On the other hand, as observed in the simple example, cover(i, J t ∪W t ) can fail for J t = Anont ∩U t (M ) when an unsatisfied buyer i has two or more disjoint bundles in his demand set. 5

21

4.2

i BEA with Dynamic Price Discrimination

Now, we will introduce dynamic price discrimination in iBEA. Just as in the simple variation of iBEA, we will be selecting a pivot economy E(M ) for M ∈ B, and adjusting prices until we have a UCE price. The additional feature in iBEA with dynamic price discrimination, iBEA(d), is that we will be introducing non-anonymous prices only as necessary and incrementally throughout the auction. The test for covering is a little more difficult than in i Bundle(d). This is because when checking for covering, we have to worry not only about the current pivot economy but also about the other economies in B. Loosely, we would like any anonymous price increases that are made in the current pivot economy to maintain quasi-CE prices in every economy, and will require covering by the appropriate set of buyers in each economy. Let E(P ) for P ∈ B denote the pivot economy in round t of iBEA(d). Let U t (P ) denote the unsatisfied buyers in this pivot economy E(P ). In determining which set of anonymous and unsatisfied buyers can remain in Anont+1 we must check that every buyer is covered by the appropriate set of covering buyers in every marginal economy. This will ensure that price changes in pivot E(P ) are valid in other economies. The new (inter-economy) checks are required because price changes are to anonymous prices, and thus affect other economies in which an unsatisfied buyer is not present. By case analysis on which economy we use as a pivot: Pivoting in Main Economy. Consider round t. We need to find the maximal set of covered but unsatisfied buyers, J t , as defined in iBundle(d) and the covering check is that all buyers i ∈ J t continue to be covered in every marginal economy. For instance, in marginal economy E(B−k ), if k ∈ J t then the set of covering buyers is W t ∪ (J t \ {k}), where W t is the set of covering winners as before. Similarly, if k ∈ W t then the set of covering buyers is (W t \ {k}) ∪ J t . In fact, a buyer i that is covered in every marginal economy is also covered in the main economy and this latter check becomes redundant. Pivoting in a Marginal Economy. Consider round t, and pivot economy E(B−k ). Again, we must construct a set of unsatisfied buyers J t that are covered in this pivot economy, in the main economy, and in other marginal economies. First, we note that buyer k can himself be included in the covering set in pivot E(B−k ), within the set of coverS ing winners W t as long as k is facing anonymous prices and Dk ⊆ i∈J t Dit . Given this, a buyer j ∈ J t that is covered in marginal economy E(B−i ) (for some i 6= k) is also covered in pivot economy E(B−k ) because the set of covering buyers in E(B−i ) is (W t ∪ J t ) \ {i}. Similarly, the covering set in pivot economy E(B−k ) is exactly that in the main economy and this check becomes redundant. Combining, we see that we need only check cover(j, K−i ) for all i 6= k, and all j ∈ J t , where K = W t ∪ J t . Putting this together, we provide a simple iterative procedure to determine the set J t , and thus Anont+1 in iBEA(d) when pivoting in economy E(P ) for P ∈ B. The procedure applies 22

irrespective of whether the pivot is currently the main economy or a marginal economy. The key difference from the procedure used to find anonymous buyers in iBundle(d) is step S2, in which covering must be checked for every marginal economy. Definition 12 (Finding Anonymous Buyers in i BEA(d)) The maximal set of anonymous but unsatisfied buyers that are covered, and continue to face anonymous prices, when pivoting in economy E(P ) in iBEA(d) is found using the following iterative procedure: S0. J t := U t (P ) ∩ Anont S1. Set W t to the set of covering winners given J t . (Note, when E(P ) = E(B−k ) then this S will include k if k ∈ Anont and Dkt ⊆ i∈J t Dit ). S2. If every buyer j ∈ J t satisfies cover(j, (W t ∪ J t ) \ {i}), for all i ∈ P (including i = j), then stop. J t is the maximal set of anonymous but unsatisfied buyers that are covered. S3. Else, remove all buyers from J t who are not covered and repeat from step S1. Finally, Anont+1 := (Anont \ U t (P )) ∪ J t . For a pivot P = B−k , this new set of anonymous prices will continue to include buyer k if k ∈ Anont because k ∈ / U t (B−k ) by definition. This procedure identifies anonymous buyers by doing the covering test for the main and marginal economies. Observe that the safety condition is no longer sufficient to ensure all unsatisfied anonymous buyers are covered, and thus allow anonymous prices throughout the auction. This is because step S2 requires cover(j, (W t ∪ J t ) \ {j}) for j ∈ J t , and an unsatisfied buyer j can no longer be used to cover the effect of its own price increase. The new requirement arises because price changes must still maintain quasi-CE prices in the marginal economy without buyer j. Given this new procedure to determine which buyers can continue to face anonymous prices, we can define iBEA with dynamic price discrimination as follows: Definition 13 (i BEA(d)) i BEA(d) is an iterative auction with the following steps: S0 Start the auction from zero prices. Initially all buyers are anonymous. S1 Choose each economy in B in some order, taking the economy as the pivot economy, and perform the following steps: S1.1 S1.2

Collect the demand set from all buyers at the current price. Solve the winner determination problem in the pivot economy given current demand, and identify the set of unsatisfied buyers in the pivot economy. If there are no unsatisfied buyers go to step S1.5.

23

S1.3

Identify the set of anonymous buyers Anont+1 using the covering test for main and marginal economies. For each buyer i ∈ Anont \ Anont+1 , initialize his prices to ptanon .

S1.4

For unsatisfied buyers in set Anont+1 , increase the anonymous price on bundles in the demand sets of these buyers by 1. For unsatisfied buyers facing nonanonymous prices, increase the price of each buyer by 1 on all bundles in his demand set. Go back to Step S1.1.

S1.5

If all the economies have been pivoted, then stop and go to Step S2. Else, repeat from Step S1.1 by choosing the next economy in B as the pivot.

S2 The auction ends with the solution to winner determination in the main economy implemented as the final allocation, and the payment for every winner i defined as the final ask price on his bundle discounted by π s (B) − π s (B−i ), which is the marginal revenue to the seller from his presence at the final price. Our main result is that this auction is an ascending Vickrey auction and has straightforward bidding in an ex post Nash equilibrium. Theorem 8 Straightforward bidding is an ex post Nash equilibrium of iBEA(d) (with consistency requirements) and the auction implements the VCG outcome. Proof : First, we show that i BEA(d) maintains quasi-CE prices for economy E(M ) for every M ∈ B in every round. Consider economy E(M ), for some M ∈ B. Assume that every partition of economy E(M ) is feasible in round t. We will show that it remains feasible in round t + 1. Consider a partition X. If the price of none of the bundles in X are increased in round t, then it remains feasible in round t + 1. If the price of some bundles in X are increased non-anonymously, then those bundles continue to be demanded in round t + 1 by the buyers who demanded them in round t (since prices of bundles in their demand set is increased by unity). The bundles in X which see anonymous price increases are each demanded by a separate covering buyer, because of the covering property. Then, every such bundle is demanded by a covering buyer since the demand set of covering buyers grow from round to round (due to unit price increase on all bundles in their demand set). Thus, X remains a feasible partition in round t + 1. In the first round, every partition is feasible as prices of all bundles are zero. By induction, every partition is feasible in all rounds of iBEA(d). In particular, X ∈ L∗ (M ), and iBEA(d) maintains quasi-CE prices for economy E(M ) for every M ∈ B in every round. The auction ends when there are no unsatisfied buyers in economy E(M ) for every M ∈ B. Such a price is a UCE price, given that prices are always quasi-CE. From Theorem 1 and the payment rule of the auction, it implements VCG outcome. Finally, the auction is UCE-based and has straightforward bidding in an ex post Nash equilibrium by Theorem 2. ¥

24

4.2.1

Examples of i BEA(d)

We illustrate the idea of dynamic price discrimination in i BEA(d) using two examples. In the first example anonymous prices can be maintained throughout the auction. Consider a unit-demand setting, in which each buyer demands at most one item. There are three buyers and two items with the valuations as follows: v1 ({1}) = 3, v1 ({2}) = v1 ({1, 2}) = 4; v2 ({1}) = v2 ({1, 2}) = 4, v2 ({2}) = 3; v3 ({1}) = v3 ({2}) = v3 ({1, 2}) = 3. iBEA(d) starts with all buyers being anonymous and with anonymous price set at zero. At that price D1 = {{2}, {1, 2}}, D2 = {{1}, {1, 2}}, and D3 = {{1}, {2}, {1, 2}}. Considering price adjustment for main economy, we can assign item 1 to buyer 1 and item 2 to buyer 2. This makes buyer 3 unsatisfied. So, U (B) = {3}, J = {3}, and G = {1, 2}. Since buyers 1 and 2 are not in U (B), they continue to face anonymous prices. To check covered property of buyer 3 in all marginal economies, observe that the possible partitions of items are ({1}), ({2}), ({1}, {2}), ({1, 2}). Since buyer 3 demands all bundles, every bundle sees a price increase. This means, in every partition all the bundles see a price increase. Buyer 3 is covered in economy E(B−1 ) because for any partition, the bundles in that partition are demanded by buyers 2 and 3 separately. For example, in partition ({1}, {2}) bundle {1} is demanded by buyer 2 and bundle {2} is demanded by buyer 3. A similar argument shows that buyer 3 is covered in economies E(B−2 ) and E(B−3 ). Thus, all buyers see anonymous prices in the next round. The demand sets of buyers do not change until the price reaches (3, 3, 3), and can remain anonymous in all rounds. At price (3, 3, 3), we reach a UCE price and the auction terminates. In the second example, which revisits the example in Table 2, prices are initially anonymous but by round 3 all buyers face non-anonymous prices. In the first round, all the buyers face anonymous prices. Buyer 3 is an unsatisfied buyer in the first round. But buyer 3 cannot be covered in economy E(B−3 ) (only buyer 3 is the covering buyer in the main economy). So, only buyers 1 and 2 face anonymous prices in round 2 and buyer 3 faces non-anonymous prices. In round 2, buyers 1 and 2 are unsatisfied buyers. Buyers 1 and 2 are the only covering buyers in main economy. It is easy to verify that buyer 1 does not satisfy covering in economy E(B−1 ), and buyer 2 does not satisfy covering in economy E(B−2 ). Thus, all the buyers face non-anonymous prices from round 3 onwards, and the price dynamics in iBEA(d) is then the same as in Table 2 from round 3 onwards.

5

Special Cases and Discussion

In this section, we make some observations about some special cases of iBEA. We also discuss methods to speed up the convergence of the auction.

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5.1

Special Cases

We can characterize some special cases in which iBEA(d) can maintain anonymous prices throughout the auction. This said, it is already understood that non-anonymous prices are often required in UCEbased auctions, for instance even in the simple case of substitutes valuations [25]. This is especially revealing because linear CE prices are known to exist for substitutes valuation functions. The negative result arises when one also cares about implementing the VCG outcome. First, it is useful to define a class of valuations that identify the role of non-disjoint bundles in allowing for anonymous prices. Definition 14 (k-valuation) A valuation vi is a k-valuation function if it admits a maximum of k disjoint bundles in a demand set at any price, where 1 ≤ k ≤ n. For instance, the valuations of single-minded buyers are 1-valuations while the valuations of double-minded buyers with disjoint interesting bundles are 2-valuations. On the other hand, any buyer with a “primary+optional” valuation function that demands at least a primary set of items, and then some additional optional items has a 1-valuation. A general valuation is a n-valuation given n items, because it permits a maximum of n disjoint bundles in a demand set. Definition 15 (Replica Economy) An economy is a replica economy if for every buyer, i, if i has a k-valuation function (1 ≤ k ≤ n), then there exists at least k other buyers with the valuation function vi .6 In a replica economy, it is easy to satisfy covering. In particular, iBEA(d) can maintain anonymous prices throughout. Proposition 1 If the main economy is a replica economy, then all buyers will face anonymous prices throughout i BEA(d). Proof : Consider any round of i BEA(d). Consider a buyer i who has a k-valuation function for 1 ≤ k ≤ n. This means there are at least k other buyers who demand the same bundles that i demands. Since vi is a k-valuation function, i will not have more than k disjoint bundles in his demand set. This means, in any partition i will not have more than k bundles and these bundles are demanded by at least k other buyers. Since these buyers have the same valuation function as i, they must have the same demand set as i. This means they can be covering buyers (either winners or unsatisfied buyers) of i if i is an unsatisfied buyer. So, 6

In Gul and Stacchetti [18], a k-replica economy is defined as an economy in which there are k identical copies of each item and k identical copies of each buyer. Our notion of replica economy is different than this since we do not require items to be replicated.

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if i is an unsatisfied buyer, then these buyers can cover i in main and marginal economies. For every such unsatisfied buyer, we have separate set of covering buyers which have same valuation function. Thus, every buyer satisfies covered property and iBEA(d) maintains anonymous prices for all the buyers. ¥ As a direct corollary to this proposition, we observe that in an economy with singleminded buyers, for every buyer we require another buyer with the same valuation function to maintain anonymous prices throughout i BEA(d). A similar proposition can also be stated for iBundle(d). Proposition 2 For every buyer i in the main economy, if vi is a k-valuation function (1 ≤ k ≤ n) and there are at least k − 1 other buyers having the valuation function vi , then iBundle(d) maintains anonymous prices throughout the auction. Proof : To determine anonymous buyers in iBundle(d), we only to check the covering property in main economy. In any partition, a buyer having a k-valuation function can demand k separate bundles. One of them can be covered by the buyer himself, while the others can be covered by k−1 other buyers having same valuation function. Thus, iBundle(d) can maintain anonymous prices throughout the auction in this setting. ¥ If the economy consists of buyers with only 1-valuation function (for example, a valuation function satisfying bid safety [31]), then iBundle(d) maintains anonymous prices throughout the auction. But an economy with only 1-valuation function does not ensure anonymous prices throughout iBEA(d). As an example, consider the case when an economy consists of only single-minded buyers. If there is only one unsatisfied buyer demanding a bundle that no other buyer is demanding, then in the marginal economy without this buyer, no buyer covers this bundle. Hence, non-anonymous prices have to be introduced in iBEA(d). Also, neither iBundle(d) nor iBEA(d) can maintain anonymous prices throughout the auction for the assignment problem setting (where every buyer is interested in at most one item). As an example, consider an economy with four buyers and three items. Every buyer wants at most one item and so we give valuations on individual items only: v1 ({1}) = v1 ({2}) = v2 ({2}) = v3 ({2}) = v3 ({3}) = v4 ({1}) = v4 ({3}) = 2; v1 ({3}) = v2 ({1}) = v2 ({3}) = v3 ({1}) = v4 ({2}) = 0. Since buyers demand at most one item, we can just maintain prices on items and the price of a bundle can be obtained by adding the prices of items in that bundle. i Bundle(d) starts with anonymous price of zero on all items. One possible provisional allocation at this price is buyer 1 gets item 1, buyer 2 gets item 2, and buyer 3 gets item 3, which leaves buyer 4 unsatisfied. Buyer 4 demands item 1 and item 3 at this price and no other buyer demands these items. Thus, there are no covering winners. In a partition including items 1 and 3, buyer 4 does not satisfy covering property since items 1 and 3 are not demanded by separate covering buyers. Thus, according to iBundle(d), buyer 4 will face non-anonymous prices in the next round. 27

5.2

Speeding up Ascending Price Auctions

A practical issue with all the auctions described in this paper is the use of “small” bid increments. We note that we only require in our auctions that the payoff of unsatisfied buyers decrease such that their demand sets grow weakly from iteration to iteration. This property can be preserved if we increase the price of unsatisfied buyers by an integral amount less than or equal to the difference between their highest payoff and second highest payoff. Clearly, the bid increment of 1, suggested in this paper, is such an amount. Such ideas first appeared in Bertsekas [7] for auction algorithms in assignment problem setting (where every buyer demands at most one item). Day and Raghavan [13] and Hoffman et al. [20] have proposed similar ideas to accelerate the implementation of ascending CAs. We also observe that it is possible to define i BEA with a finite bid increment ² > 1. In that case, iBEA will not converge a UCE price exactly, but the for every buyer the difference between his payoff in the auction and the VCG payoff can be bounded as a linear function of ². The straightforward bidding strategy becomes an ²-ex post Nash equilibrium strategy.7 We omit details on the new analysis that is required to formalize the effect of such an ²-bid increment, but refer an interested reader to the ²-analysis of iBundle in Parkes [31] and the ²-equilibrium analysis of a VCG-based mechanism with an approximate allocation rule in Kothari et al. [21]. Finally, Ausubel et al. [3] discuss the idea of intra-round bidding in ascending combinatorial auctions. Intra-round bidding allows a buyer to respond with a demand curve, describing how his demand changes as prices are increased linearly from current prices to the new ask price. Intra-round bidding can be useful because it allows a larger bid increment without compromising allocative efficiency. To fix ideas, a simple version of intra-round bidding announces new prices pt+1 but asks a buyer to bid at pt +α(pt+1 −pt ) for α ∈ {0.25, 0.5, 0.75, 1.0}. The idea is that the seller can consider the bids at each price point, and back-off the full price increase if demand drops below supply. In our context, one can define an initial bid increment ²1 , and reduce this bid increment each time the auction must back-off its full price increase. Let ²t denote the bid increment in round t. The auction rule can be modified as follows. First, determine the provisional allocation for every α. The actual price increase that is adopted can be defined as the maximal α at which prices are still ²t -quasi-CE. The bid increment, ²t+1 can be set equal to ²t unless prices for some α < 1 are adopted, in which case ²t+1 := γ · ²t for some γ < 1. The rate of decrease in bid increment, γ, can be defined heuristically to allow for convergence to a suitably accurate final outcome. 7

The ²-Nash equilibrium is defined similar to Nash equilibrium except that best reponse strategies include all strategies which give payoff within ² of the maximum payoff.

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6

Conclusion

This paper proposes a simple ascending Vickrey auction. Unlike most ascending combinatorial auctions, straightforward bidding is an ex post equilibrium for general valuations and the auction is efficient. It is interesting to compare the auction with an earlier ascending Vickrey auction for general valuations [25]. This auction also has an ascending price trajectory followed by a final adjustment to implement the VCG payments. Our new auction, iBEA(d), enjoys two main advantages. First, the price update step is computationally simple. Second, the auction maintains an anonymous price trajectory whenever possible and only introduces non-anonymous prices as necessary to ensure termination in the VCG outcome. This allows for more concise prices and more efficient price discovery, and provides what we believe to be a good generalization of the single item English auction to the combinatorial setting. The auction presented in this paper extends an earlier ascending combinatorial auction, iBundle [31]. The iBundle auction also has simple price dynamics, but is only able to converge to CE prices and can implement the VCG outcome only for a restricted class of valuations. The iBEA auction extends this earlier auction, and considers the marginal economies together with the main economies when adjusting prices while still maintaining a single price trajectory. The auction is able to converge to Universal CE prices when buyers follow a straightforward bidding strategy, from which the VCG payments are determined as a final adjustment. The auction shares a defining feature with previously defined ascending auctions, in that it maintains quasi-CE prices in each round. These are prices that ensure demand exceeds supply, so that the seller can always maximize his revenue with an allocation that is consistent with buyer demand sets. The price dynamics in iBEA are defined to maintain quasi-CE prices in the main economy and the marginal economies across all rounds, so that convergence to Universal CE prices is guaranteed. The most interesting line of future research is to design ascending auctions for CAP that facilitate new representation languages, both for the reporting of demand sets to the auctioneer and for the quoting of prices to participants. There has been plenty of research into languages that are both concise and expressive [27, 26], but no work at this point on closely integrating these ideas into the design of ascending auctions for CAP. A related question is to ask whether one can design ascending Vickrey auctions that introduce nonlinear prices incrementally, just as we introduce non-anonymous prices incrementally in this auction. Finally, a compelling extension to this current work is to define an asynchronous variation in which prices can be updated without requiring bids from every bidder. This would then provide a generalization of the familiar “open cry” style of ascending auction.

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