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Register a new userThe Multilinear Minimax Relaxation of Bimatrix Games and Comparison with Nash Equilibria via Lemke-Howson
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Bahman Kalantari
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Extensive study on the complexity of computing Nash Equilibrium has resulted in the definition of the complexity class PPAD by Papadimitriou cite{Papa2}, Subsequently shown to be PPAD-complete, first by Daskalakis, Goldberg, and Papadimitriou cite{Papa} for $3$ or more and even for the bimatrix case by Chen and Deng cite{Chen}. On the other hand, it is well known that Nash equilibria of games with smooth payoff functions are generally Pareto-inefficient cite{Dubey} In the spirit of Von Neumanns Minimax Theorem and its polynomial-time solvability via Linear Programming, Kalantari cite{Kalantari} has described a multilinear minimax relaxation (MMR) that provides an approximation to a convex combination of expected payoffs in any Nash Equilibrium via LP. In this paper, we study this relaxation for the bimatrix game, solving its corresponding LP formulation and comparing its solution to the solution computed by the Lemke-Howson algorithm. We also give a game theoretic interpretation of MMR for the bimatrix game involving a meta-player. Our relaxation has the following theoretical advantages: (1) It can be computed in polynomial time; (2) For at least one player, the computed MMR payoff is at least as good any Nash Equilibrium payoff; (3) There exists a convex scaling of the payoff matrices giving equal payoffs. Such a solution is a satisfactory compromise. Computationally, we have compared our approach with the state-of-the-art implementation of the Lemke-Howson algorithm cite{Lemke}. We have observed the following advantages: (i) MMR outperformed Lemke-Howson in time complexity; (ii) In about $80%$ of the cases the MMR payoffs for both players are better than any Nash Equilibria; (iii) in the remaining $20%$, while one players payoff is better than any Nash Equilibrium payoff, the other players payoff is only within a relative error of $17%$.
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We prove that computing a Nash equilibrium of a two-player ($n times n$) game with payoffs in $[-1,1]$ is PPAD-hard (under randomized reductions) even in the smoothed analysis setting, smoothing with noise of constant magnitude. This gives a strong negative answer to conjectures of Spielman and Teng [ST06] and Cheng, Deng, and Teng [CDT09]. In contrast to prior work proving PPAD-hardness after smoothing by noise of magnitude $1/operatorname{poly}(n)$ [CDT09], our smoothed complexity result is not proved via hardness of approximation for Nash equilibria. This is by necessity, since Nash equilibria can be approximated to constant error in quasi-polynomial time [LMM03]. Our results therefore separate smoothed complexity and hardness of approximation for Nash equilibria in two-player games. The key ingredient in our reduction is the use of a random zero-sum game as a gadget to produce two-player games which remain hard even after smoothing. Our analysis crucially shows that all Nash equilibria of random zero-sum games are far from pure (with high probability), and that this remains true even after smoothing.
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