New algorithms for approximate Nash equilibria in bimatrix games

Abstract

We consider the problem of computing additively approximate Nash equilibria in non-cooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. Our work improves the previously best known (0.38197 +)-approximation algorithm of Daskalakis, Mehta and Papadimitriou [6]. First, we provide a simpler algorithm, which also achieves 0.38197. This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point.

Computing equilibria: a computational complexity perspective

“…No equally fast algorithm is known for computing an exact Nash equilibrium. There are also polynomial-time algorithms for computing an ǫ-Nash equilibrium with relatively large ǫ (around 1/3) [15,129]. On the negative side, Theorem 4.2 can be strengthened to rule out polynomial-time algorithms that compute an ǫ-Nash equilibrium with ǫ going to zero inverse polynomially (or faster) with the size of the game's description (assuming P P AD ⊆ P ) [22].…”

On the Complexity of Approximating a Nash Equilibrium

“…It was shown in [8] that there is a constant c such that the Approximate Circuit Evaluation problem is PPAD-complete. But, given any circuit, it is easy to set up, using the gad- Lemmas 3.3,3.4 and 3.5, a bipartite graphical polymatrix game GG with the same functionality as the circuit: every node of the circuit corresponds to a player, the players participate in arithmetic, comparison and logical gadgets depending on the types of gates with which the corresponding nodes of the circuit are connected, and given any relative -Nash equilibrium of the graphical game we can obtain an approximate circuit evaluation by interpreting the probabilities with which every player plays strategy 1 as the value of the corresponding node of the circuit. To make sure that every node in our graphical game has positive minimax value we can use in our construction the sophisticated versions G…”

“…A positive outcome of this investigation would be useful for applications since it would provide algorithmic tools for computing approximate equilibria; but, most importantly, it would alleviate the negative implications of the aforementioned hardness results to the predictive power of the Nash equilibrium concept. Unfortunately, since the appearance of the original hardness results, and despite considerable effort in providing upper [20,9,10,21,15,3,27,28] and lower [12,18] bounds for the approximation problem, the approximation complexity of the Nash equilibrium has remained unknown. This paper obtains the first constant inapproximability results for the problem.…”

“…This provides a quasi-polynomial-time algorithm for normalized bimatrix games (games with payoffs scaled in a unit-length interval), by establishing that, for any fixed , there exists an additiveapproximate Nash equilibrium of support-size logarithmic in the total number of strategies. 3 The LMM algorithm performs an exhaustive search over pairs of strategies with logarithmic support and can therefore also optimize some objective over the output equilibrium. This property of the algorithm has been exploited in recent lower bounds for the problem [18,12], albeit these fall short from a quasi-polynomial-time lower bound for additive approximations.…”

On Nash-Equilibria of Approximation-Stable Games

“…There has also been a series of results [8,12,5] on polynomial-time algorithms for computing approximate equilibria for larger values of ǫ. The best polynomial-time approximation guarantee known is 0.3393 [12].…”