Exponential lower bounds for finding Brouwer fixed points

Hirsch, Morris W.; Vavasis, Stephen A.

doi:10.1109/sfcs.1987.24

Cited by 47 publications

(130 citation statements)

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“…Gilboa and Zemel (1989) show that many decision problems related to Nash equilibria of two person games (Is there more than one Nash equilibrium?, Is there a Nash equilibrium assigning positive probability to a certain pure strategy?, etc.) are NP- The most important reason for this is the fact that 2-Nash is a fixed point problem that is seemingly a small step beyond linear programming, which is in P. Hirsch et al (1989) studied a discrete version of Brouwer's fixed point theorem that is based on function evaluation. Specifically, one is given an "oracle" or "black box" that computes the value of a function f : [0, 1] n → [0, 1] n , and the goal is to find a point x satisfying f (x)−x ≤ 2 −p .…”

Section: Computationmentioning

confidence: 99%

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Imitation games and computation

McLennan¹,

Tourky²

2010

Games and Economic Behavior

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An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al.(1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) Imitation Games and ComputationAndrew McLennan * and Rabee Tourky † December 31, 2007Abstract: An imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.

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Section: Computationmentioning

confidence: 99%

“…Especially in view of the result of Hirsch et al (1989), it seems extremely unlikely that there is a polynomial time algorithm for PPAD, so this finding is regarded as compelling evidence that there is no polynomial time algorithm for 2-Nash, even if it does not quite amount to a complete proof.…”

Section: Computationmentioning

confidence: 99%

Imitation games and computation

McLennan¹,

Tourky²

2010

Games and Economic Behavior

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“…However, no sub-exponential upper bounds are known for approximating equilibria using this algorithm. In fact, Scarf's algorithm is known to take exponential time in the worst case for a general fixed point approximation ( [6]). Polymomial time algorithms for exact or approximate equilibria but only for special classes of games have also been obtained in [10,19,9].…”

Section: Introductionmentioning

confidence: 99%

Playing large games using simple strategies

Lipton

Markakis

Mehta

2003

Proceedings of the 4th ACM Conference on Electronic Commerce

293

361

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We prove the existence of -Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be -approximated by the payoffs to the players in some such logarithmic support -Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an -Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an -Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payoffs to all players in any m-tuple of mixed strategies can be -approximated by the payoffs in some m-tuple of constant support strategies.We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an -equilibrium with small support. It also implies that if the payoff matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.

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“…They show that this problem is complete for the complexity class PPAD. Intuitively, this means that a polynomial-time algorithm would imply a similar algorithm, for example, for computing Brouwer fixpoints; note that this is a problem for which quite strong lower bounds for large classes of algorithms are known [48]. (A precise definition of the complexity class PPAD is beyond the scope of this chapter.)…”

Section: Mixed Nash Equilibriamentioning

confidence: 99%

Algorithmic Game Theory and Applications

Nayak

2007

Handbook of Applied Algorithms

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Most of the existing and foreseen complex networks, such as the Internet, are operated and built by thousands of large and small entities (autonomous agents), which collaborate to process and deliver end-to-end flows originating from and terminating at any of them. The distributed nature of the Internet implies a lack of coordination among its users. Instead, each user attempts to obtain maximum performance according to his own parameters and objectives.Methods from game theory and mathematical economics have been proven to be a powerful modeling tool, which can be applied to understand, control, and efficiently design such dynamic, complex networks. Game theory provides a good starting point for computer scientists in their endeavor to understand selfish rational behavior in complex networks with many agents (players). Such scenarios are readily modeled using techniques from game theory, where players with potentially conflicting goals participate in a common setting with well-prescribed interactions.Nash equilibrium [73,74] distinguishes itself as the predominant concept of rationality in noncooperative settings. So, game theory and its various concepts of equilibria provide a rich framework for modeling the behavior of selfish agents in these kinds of distributed or networked environments; they offer mechanisms to achieve efficient and desirable global outcomes in spite of the selfish behavior.Mechanism design, a subfield of game theory, asks how one can design systems so that agents' selfish behavior results to desired systemwide goals. Algorithmic mechanism design additionally considers computational tractability to the set of concerns of mechanism design. Work on algorithmic mechanism design has focused on the complexity of centralized implementations of game-theoretic mechanisms for distributed optimization problems. Moreover, in such huge and heterogeneous networks, each agent does not have access to (and may not process) complete information. 286 ALGORITHMIC GAME THEORY AND APPLICATIONSThe notion of bounded rationality for agents and the design of corresponding incomplete-information distributed algorithms have been successfully utilized to capture the aspect of lack of global knowledge in information networks.In this chapter, we review some of the most thrilling algorithmic problems and solutions, and corresponding advances, achieved on the account of game theory. The areas addressed are the following.Congestion games A central problem arising in the management of large-scale communication networks is that of routing traffic through the network. However, due to the large size of these networks, it is often impossible to employ a centralized traffic management. A natural assumption to make in the absence of central regulation is that network users behave selfishly and aim at optimizing their own individual welfare. One way to address this problem is to model this scenario as a noncooperative multiplayer game and formalize it using congestion game. Congestion games (either unweighted or weighted) offer ...

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Exponential lower bounds for finding Brouwer fixed points

Cited by 47 publications

References 9 publications

Imitation games and computation

Imitation games and computation

Playing large games using simple strategies

Algorithmic Game Theory and Applications

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