Constant Rank Two-player Games Are Ppad-hard
نویسنده
چکیده
Finding Nash equilibrium in a two-player normal form game (2-Nash) is one of the most extensively studied problem within mathematical economics as well as theoretical computer science. Such a game can be represented by two payoff matrices A and B, one for each player. 2Nash is PPAD-complete in general, while in case of zero-sum games (B = −A) the problem reduces to LP and hence is in P. Extending the notion of zero-sum, in 2005, Kannan and Theobald defined rank of game (A,B) as rank(A + B), e.g., rank-0 are zero-sum games. They gave an FPTAS for constant rank games, and asked if there exists a polynomial time algorithm to compute an exact Nash equilibrium (NE). Adsul et al. (2011) answered this question affirmatively for rank-1 games, leaving rank-2 and beyond unresolved. In this paper we show that NE computation in games with rank ≥ 3 is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets, and provides a simpler proof of PPAD-hardness for NE computation in bimatrix games. In addition, we get: • An equivalence between 2D-Linear-FIXP and PPAD, improving on a result of Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD. • NE computation in a bimatrix game with convex set of Nash equilibria is as hard as solving a simple stochastic game [16]. • Computing a symmetric NE of a symmetric bimatrix game with rank ≥ 6 is PPAD-hard. • Computing a 1 poly(n) -approximate fixed-point of a (Linear-FIXP) piecewise-linear function
منابع مشابه
A Polynomial Time Algorithm for Rank-1 Bimatrix Games
Two player normal form game is the most basic form of game, studied extensively in game theory. Such a game can be represented by two payoff matrices (A,B), one for each player, hence they are also known as bimatrix games. The rank of a bimatrix game (A,B) is defined as the rank of matrix (A+B). For zero-sum games, i.e., rank-0, von Neumann (1928) [13] showed that Nash equilibrium are min-max s...
متن کاملOn minmax theorems for multiplayer games Citation
We prove a generalization of von Neumann’s minmax theorem to the class of separable multiplayer zerosum games, introduced in [Bregman and Fokin 1998]. These games are polymatrix—that is, graphical games in which every edge is a two-player game between its endpoints—in which every outcome has zero total sum of players’ payoffs. Our generalization of the minmax theorem implies convexity of equili...
متن کاملSparse Games Are Hard
A two-player game is sparse if most of its payoff entries are zeros. We show that the problem of computing a Nash equilibrium remains PPAD-hard to approximate in fully polynomial time for sparse games. On the algorithmic side, we give a simple and polynomial-time algorithm for finding exact Nash equilibria in a class of sparse win-lose games.
متن کاملThree-Player Games Are Hard
We prove that computing a Nash equilibrium in a 3-player game is PPAD-complete, solving a problem left open in (2).
متن کاملFinding Equilibria in Games of No Chance
We consider finding maximin strategies and equilibria of explicitly given extensive form games with imperfect information but with no moves of chance. We show: 1. A maximin pure strategy for a two-player extensive form game with perfect recall and no moves of chance can be found in time linear in the size of the game tree. In contrast, it is known that this problem is NP-hard for games with cha...
متن کامل