Imitation games and computation

Abstract

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.

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

“…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. Our main result is that these problems are N P-hard for symmetric win-lose bimatrix games (Theorems 7.1 and 7.7).…”

“…encompassing those from [20], for symmetric bimatrix games; their reduction yields games with Nash equilibria mirroring satisfiability parsimoniously. McLennan and Tourky [25,Theorem 1] refined some of these N P-hardness results for imitation bimatrix games, where the utility of the imitator is 1 if and only if she chooses the same strategy as the mover [26]. The problems (vii) and (viii) are considered here for the first time; (viii) is trivial for symmetric 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. Our main result is that these problems are N P-hard for symmetric win-lose bimatrix games (Theorems 7.1 and 7.7).…”

“…Among the most fundamental computational problems in Algorithmic Game Theory are those concerning the Nash equilibria [29,30] of a game, where no player could unilaterally deviate to increase her expected utility. Such problems have been studied extensively [1,4,7,9,10,11,12,13,14,20,25,26] for 2-player games with rational utilities given by a bimatrix. There are two prominent special cases of such general bimatrix games: win-lose and symmetric.…”

A repeated imitation model with dependence between stages: Decision strategies and rewards

“…Existing works on imitation games (McLennan and Tourky, 2006;2010a;2010b) usually focus on theoretical aspects of these games with complete information, such as complexity issues (imitation games are proved to be as complex as any general bi-personal game) and the equivalence of the computation of optimal solutions to the game, i.e., Nash equilibria, with other problems such as proving Kakutani's fixed-point theorem (McLennan and Tourky, 2006). However, none of these works take into account incomplete information based merely on observations, like we will do in our model.…”