Since the seminal PPAD-completeness result for computing a Nash equilibrium
even in two-player games, an important line of research has focused on
relaxations achievable in polynomial time. In this paper, we consider the
notion of ε-well-supported Nash equilibrium, where ε∈[0,1] corresponds to the approximation guarantee. Put simply, in an
ε-well-supported equilibrium, every player chooses with positive
probability actions that are within ε of the maximum achievable
payoff, against the other player's strategy. Ever since the initial
approximation guarantee of 2/3 for well-supported equilibria, which was
established more than a decade ago, the progress on this problem has been
extremely slow and incremental. Notably, the small improvements to 0.6608, and
finally to 0.6528, were achieved by algorithms of growing complexity. Our main
result is a simple and intuitive algorithm, that improves the approximation
guarantee to 1/2. Our algorithm is based on linear programming and in
particular on exploiting suitably defined zero-sum games that arise from the
payoff matrices of the two players. As a byproduct, we show how to achieve the
same approximation guarantee in a query-efficient way