Inapproximability of pure nash equilibria

Skopalik, Alexander; Vöcking, Berthold

doi:10.1145/1374376.1374428

Cited by 95 publications

(107 citation statements)

References 18 publications

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“…Our work differs from all the above as none of them capture lookahead dynamics. In another line of work, convergence of best-response dynamics to (approximate) equilibria and the complexity of game dynamics and sink equilibria have been studied [23,1,15,73,22,50], but our paper does not focus on these types of dynamics or convergence to equilibria.…”

Section: Background and Related Workmentioning

confidence: 99%

A Theoretical Examination of Practical Game Playing: Lookahead Search

Mirrokni

Thain

Vetta

2012

Algorithmic Game Theory

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Abstract. Lookahead search is perhaps the most natural and widely used game playing strategy. Given the practical importance of the method, the aim of this paper is to provide a theoretical performance examination of lookahead search in a wide variety of applications.To determine a strategy play using lookahead search, each agent predicts multiple levels of possible re-actions to her move (via the use of a search tree), and then chooses the play that optimizes her future payoff accounting for these re-actions. There are several choices of optimization function the agents can choose, where the most appropriate choice of function will depend on the specifics of the actual game -we illustrate this in our examples. Furthermore, the type of search tree chosen by computationally-constrained agent can vary. We focus on the case where agents can evaluate only a bounded number, k, of moves into the future. That is, we use depth k search trees and call this approach k-lookahead search.We apply our method in five well-known settings: industrial organization (Cournot's model); AdWord auctions; congestion games; valid-utility games and basic-utility games; cost-sharing network design games. We consider two questions. First, what is the expected social quality of outcome when agents apply lookahead search? Second, what interactive behaviours can be exhibited when players use lookahead search?Myopic game playing (whose corresponding equilibria are Nash equilibria), where each player can only foresee the immediate effect of her own actions, is the special case of 1-lookahead search. Thus, for the first question, it is natural to ask whether social outcomes improve when players use more foresight than in myopic behaviour. The answer depends on the game played: (i) For the Cournot game, applying 2-lookahead leads to a 12.5% increase in output and a 5.5% increase in social surplus compared with myopic competition. Similar bounds arise as the length k of foresight increases.(ii) In AdWord auctions (or generalized second-price auctions), we show that 2-lookahead game playing results in outcomes that are always optimal to within a constant factor; in contrast, myopic game play can produce arbitrarily poor equilibrium outcomes.(ii) For congestion games, as with myopic game playing, lookahead search leads to constant factor qualitative guarantees. (iv) For basic-utility games, on the other hand, whilst myopic game playing always leads to constant factor approximations, additional foresight can lead to arbitrarily bad solutions! (v) In a simple Shapley network design game, qualitative guarantees improve with the length of foresight. Regarding the second question, a variety of interesting game playing characteristics also arise with lookahead search. Stackelberg leader-follower behaviours can be induced when the players have asymmetric computational power. For example, Stackelberg equilibria can be produced in the Cournot game. Lookahead search can also generate "uncoordinated" cooperative behaviour!

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Section: Background and Related Workmentioning

confidence: 99%

A Theoretical Examination of Practical Game Playing: Lookahead Search

Mirrokni

Thain

Vetta

2012

Algorithmic Game Theory

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“…This result makes a compelling argument for the approximate pure Nash equilibrium as a credible solution concept in symmetric congestion games. 25 On the negative side, Skopalik and Vöcking [124] prove that computing even an ǫ-pure Nash equilibrium is P LS-hard in asymmetric congestion games, and that ǫ-better-response dynamics can require an exponential number of steps to converge in such games. This P LS-hardness result implies that any dynamics for reaching an approximate equilibrium that requires only polynomial-time computation per iteration does not always converge in a polynomial number of iterations, unless P LS ⊆ P .…”

Section: Discussionmentioning

confidence: 99%

Computing equilibria: a computational complexity perspective

Roughgarden

2009

Econ Theory

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“…These results have been strengthened to hold even when the cost functions are non-decreasing and linear [1]. Also finding α-approximate PNEs in congestion games is PLS-complete for any α > 1 [22].…”

Section: Literature Reviewmentioning

confidence: 91%

A logarithmic approximation for polymatroid congestion games

Harks

Oosterwijk

Vredeveld

2016

Operations Research Letters

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We study the problem of computing a social optimum (minimum cost solution) in polymatroid congestion games, where the strategy space of every player consists of the set of vectors in a playerspecific integral polymatroid base polyhedron defined on the ground set of resources. For general non-decreasing cost functions we devise an H rk -approximation algorithm, where rk is the sum of the ranks of the player-specific polymatroids and H rk denotes the rk-th harmonic number. The main idea of our algorithm is to iteratively increase resource utilization in a greedy fashion and to invoke a polynomial covering oracle that checks feasibility of every computed resource utilization. The approximation guarantee is best possible up to a constant factor. As a special case, our result 2) where the complexity of computing a socially optimal solution for matroid congestion games with non-decreasing cost functions is considered. Here, the approximation guarantee is best possible up to a constant factor if the number of resources is polynomially bounded in the number of players.

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Inapproximability of pure nash equilibria

Cited by 95 publications

References 18 publications

A Theoretical Examination of Practical Game Playing: Lookahead Search

A Theoretical Examination of Practical Game Playing: Lookahead Search

Computing equilibria: a computational complexity perspective

A logarithmic approximation for polymatroid congestion games

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