The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game

Etessami, Kousha

doi:10.1016/j.geb.2019.03.006

Cited by 8 publications

(6 citation statements)

References 39 publications

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“…It was proved recently by Hansen and Lund [2018] that the task of approximating a proper equilibrium is complete for the class FIXP a of [Etessami and Yannakakis, 2010]. This work follows a line of similar results [Etessami et al, 2014;Etessami, 2021] for approximating other notions of equilibrium refinements, e.g. Selten's trembling hand perfect equilibrium [Selten, 1975], that are, like proper equilibria, defined as limit points of certain ε-equilibria.…”

Section: Computing An ε-Proper Equilibrium Via Systems Of Conditional...supporting

confidence: 55%

“…Fixed point computation. Besides the applications above, the class FIXP also captures the complexity of other problems, such as branching process and context-free grammars [Etessami and Yannakakis, 2010], equilibrium refinements [Etessami et al, 2014;Etessami, 2021], and more recently the complexity of computing a Bayes-Nash equilibrium in the first-price auction with subjective priors [Filos-Ratsikas et al, 2021]. Besides FIXP, there are some other computational classes that capture the complexity of different fixed point problems, namely the classes BU [Deligkas et al, 2021] and BBU [Batziou et al, 2021] which correspond to the Borsuk-Ulam theorem [Borsuk, 1933], and the class HB [Goldberg and Hollender, 2021], which corresponds to the Hairy Ball theorem [Poincaré, 1885].…”

Section: Related Workmentioning

confidence: 99%

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FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

et al. 2023

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We introduce a new technique for proving membership of problems in FIXP -the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets.In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with very general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

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Section: Computing An ε-Proper Equilibrium Via Systems Of Conditional...supporting

confidence: 55%

Section: Related Workmentioning

confidence: 99%

FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

et al. 2023

View full text Add to dashboard Cite

show abstract

“…It was proved recently by Hansen and Lund [2018] that the task of approximating a proper equilibrium is complete for the class FIXP a of [Etessami and Yannakakis, 2010]. This work follows a line of similar results [Etessami et al, 2014;Etessami, 2021] for approximating other notions of equilibrium refinements, e.g. Selten's trembling hand perfect equilibrium [Selten, 1988], that are, like proper equilibria, defined as limit points of certain ε-equilibria.…”

Section: Computing An ε-Proper Equilibrium Via Systems Of Conditional...mentioning

confidence: 59%

Section: Related Workmentioning

confidence: 99%

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

Filos-Ratsikas¹,

Hansen²,

Høgh³

et al. 2021

Preprint

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We introduce a new technique for proving membership of problems in FIXP -the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets.In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

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“…In such games the basic notion of equilibrium is a subgame perfect equilibrium. Unfortunately, these equilibria are, in general, hard to compute; see, for example, [7,23,8]. Intriguing structural properties can be derived for the equilibria of sequential auctions, but the recursive nature of this structure makes reasoning about equilibria complex.…”

Section: Introductionmentioning

confidence: 99%

Risk-Free Bidding in Complement-Free Combinatorial Auctions

Narayan

Rayaprolu

Vetta

2019

Algorithmic Game Theory

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We study risk-free bidding strategies in combinatorial auctions with incomplete information. Specifically, what is the maximum profit that a complement-free (subadditive) bidder can guarantee in a multi-item combinatorial auction? Suppose there are n bidders and Bi is the value that bidder i has for the entire set of items. We study the above problem from the perspective of the first bidder, Bidder 1. In this setting, the worst case profit guarantees arise in a duopsony, that is when n = 2, so this problem then corresponds to playing an auction against a budgeted adversary with budget B2. We present worst-case guarantees for two simple and widely-studied combinatorial auctions; namely, the sequential and simultaneous auctions, for both the first-price and second-price case. In the general case of distinct items, our main results are for the class of fractionally subadditive (XOS) bidders, where we show that for both first-price and second-price sequential auctions Bidder 1 has a strategy that guarantees a profit of at least ( √ B1 − √ B2) 2 when B2 ≤ B1, and this bound is tight. More profitable guarantees can be obtained for simultaneous auctions, where in the first-price case, Bidder 1 has a strategy that guarantees a profit of at least, and in the second-price case, a bound of B1 − B2 is achievable. We also consider the special case of sequential auctions with identical items, for which we provide tight guarantees for bidders with subadditive valuations.

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The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game

Cited by 8 publications

References 39 publications

FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

Risk-Free Bidding in Complement-Free Combinatorial Auctions

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