The Complexity of Decision Problems about Nash Equilibria in Win-Lose Games

Abstract

We revisit the complexity of deciding, given a bimatrix game, whether it has a Nash equilibrium with certain natural properties; such decision problems were early known to be N P-hard [20]. We show that N P-hardness still holds under two significant restrictions in simultaneity: the game is win-lose (that is, all utilities are 0 or 1) and symmetric. To address the former restriction, we design win-lose gadgets and a win-lose reduction; to accomodate the latter restriction, we employ and analyze the classical GHR-symmetrization [21] in the win-lose setting. Thus, symmetric win-lose bimatrix games are as complex as general bimatrix games with respect to such decision problems.As a byproduct of our techniques, we derive hardness results for search, counting and parity problems about Nash equilibria in symmetric win-lose bimatrix games.An additional decision problem, which is trivial for bimatrix games, becomes N P-hard already for 3-player games [4, Theorem 2]:(xi) A Nash equilibrium where all probabilities are rational? [4] It is natural to ask whether these problems remain N P-hard when restricted to win-lose games.To the best of our knowledge, this important question has been addressed only in [7,12]. It was shown in [12, Theorem 1] that the decision problem (i) with k = 1 is N P-hard for win-lose bimatrix games; there so is a variant of (ii) for imitation win-lose bimatrix games. The decision problem (vi) was shown N P-hard for imitation win-lose bimatrix games in [7, Theorem 1].The counting problem and the parity problem for Nash equilibria ask for the number and for the parity of the number of Nash equilibria, respectively, for a given game. To each decision problem there corresponds a counting problem and a parity problem, asking for the number and the parity of the number of Nash equilibria with the corresponding property, respectively. By the parsimonious property of the reduction in [13], these counting (resp., parity) problems are #P-hard (resp., ⊕P-hard) for general bimatrix games; the #P-hardness (resp., ⊕P-hardness)is inherited from the #P-hardness [35] (resp., ⊕P-hardness [33]) of computing the number (resp., the parity of the number) of satisfying assignments for a CNF SAT formula. State-of-the-Art and Statement of ResultsThe polynomial time transformation of a general bimatrix game into a win-lose bimatrix game from [1] gave no guarantee on the preservation of properties of Nash equilibria; so, it had no implication on the complexity of deciding the properties for win-lose bimatrix games. Thus, the composition of a polynomial time reduction from an N P-hard problem to a decision problem about Nash equilibria for general bimatrix games (cf. [13, 20]) with the polynomial time transformation from [1] does not yield a polynomial time reduction from the N P-hard problem to the decision problems for win-lose bimatrix games, and their complexity remained open.In this work, we settle the complexity of the decision problems about Nash equilibria [4,7,12,13,20,25,26,28] for symmetric win-lose bimatrix games. ...

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“…In G ′ , the expected payoff for the row player for i is at least 5 8 + ǫ * irrespective of the column player's strategy. Similarly, the expected payoff for j is at least 5 8 + ǫ * irrespective of the row player's strategy. This means that: • If G s possesses an ǫ * -NE with social welfare 10 8 , then G ′ possesses at least one ǫ * -NE where the players do not play the pure strategies i and j.…”

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