Walrasian Equilibrium with Gross Substitutes

Gül, Faruk; Stacchetti, Ennio

doi:10.1006/jeth.1999.2531

Cited by 527 publications

(525 citation statements)

References 9 publications

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“…We say that a valuation profile is Walrasian if it admits a Walrasian equilibrium and non-Walrasian otherwise. Walrasian equilibria always exist when valuations meet the gross substitutes property, but not generally otherwise (see [11,14]). The pure Nash equilibria of a CA with item bidding and the first-price payment rule correspond to the Walrasian equilibria (if any) in a natural way, and are fully efficient when they exist [12].…”

Section: Overviewmentioning

confidence: 99%

Simultaneous Single-Item Auctions

Bhawalkar

Roughgarden

2012

Lecture Notes in Computer Science

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Abstract. In a combinatorial auction (CA) with item bidding, several goods are sold simultaneously via single-item auctions. We study how the equilibrium performance of such an auction depends on the choice of the underlying single-item auction. We provide a thorough understanding of the price of anarchy, as a function of the single-item auction payment rule. When the payment rule depends on the winner's bid, as in a first-price auction, we characterize the worst-case price of anarchy in the corresponding CAs with item bidding in terms of a sensitivity measure of the payment rule. As a corollary, we show that equilibrium existence guarantees broader than that of the first-price rule can only be achieved by sacrificing its property of having only fully efficient (pure) Nash equilibria. For payment rules that are independent of the winner's bid, we prove a strong optimality result for the canonical second-price auction. First, its set of pure Nash equilibria is always a superset of that of every other payment rule. Despite this, its worst-case POA is no worse than that of any other payment rule that is independent of the winner's bid.

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Section: Overviewmentioning

confidence: 99%

Simultaneous Single-Item Auctions

Bhawalkar

Roughgarden

2012

Lecture Notes in Computer Science

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“…In many circumstances, for example, when the domain of interest is a discrete set or the presence of indivisibility in an economic model is significant, the continuity (upper semicontinuity) or convexity requirement of Brouwer's (Kakutani's) theorem is no longer fulfilled; see e.g., Kelso and Crawford [27], Fudenberg and Tirole [28], Scarf [29], Gul and Stacchetti [30], Sun and Yang [31]. The purpose of this section is to investigate (i) what conditions guarantee the existence of a solution to a discrete system of nonlinear equations…”

Section: Solving Discrete Systems Of Nonlinear Equationsmentioning

confidence: 99%

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Laan

Talman

Yang

2009

J Optim Theory Appl

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Tucker's well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, . . . , ±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1, ±2, . . . , ±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0, 1} n + q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.

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“…CABOB then returns from the call without searching further under that node. Sufficient conditions under which the LP provides an integer-valued solution are described in Gul and Stacchetti (1999), Nisan (2000), and Bikhchandani and Ostroy (2002).…”

Section: The Integer Special Casementioning

confidence: 99%

CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions

Sandholm

Suri

Gilpin³

et al. 2005

Management Science

229

180

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C ombinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations, but determining the winners is -complete and inapproximable. We present CABOB, a sophisticated optimal search algorithm for the problem. It uses decomposition techniques, upper and lower bounding (also across components), elaborate and dynamically chosen bid-ordering heuristics, and a host of structural observations. CABOB attempts to capture structure in any instance without making assumptions about the instance distribution. Experiments against the fastest prior algorithm, CPLEX 8.0, show that CABOB is often faster, seldom drastically slower, and in many cases drastically faster-especially in cases with structure. CABOB's search runs in linear space and has significantly better anytime performance than CPLEX.We also uncover interesting aspects of the problem itself. First, problems with short bids, which were hard for the first generation of specialized algorithms, are easy. Second, almost all of the CATS distributions are easy, and the run time is virtually unaffected by the number of goods. Third, we test several random restart strategies, showing that they do not help on this problem-the run-time distribution does not have a heavy tail.

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Walrasian Equilibrium with Gross Substitutes

Cited by 527 publications

References 9 publications

Simultaneous Single-Item Auctions

Simultaneous Single-Item Auctions

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions

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