3 research outputs found
A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium
We present a direct reduction from k-player games to 2-player games that
preserves approximate Nash equilibrium. Previously, the computational
equivalence of computing approximate Nash equilibrium in k-player and 2-player
games was established via an indirect reduction. This included a sequence of
works defining the complexity class PPAD, identifying complete problems for
this class, showing that computing approximate Nash equilibrium for k-player
games is in PPAD, and reducing a PPAD-complete problem to computing approximate
Nash equilibrium for 2-player games. Our direct reduction makes no use of the
concept of PPAD, thus eliminating some of the difficulties involved in
following the known indirect reduction.Comment: 21 page
Computing Approximate Nash Equilibria in Polymatrix Games
In an -Nash equilibrium, a player can gain at most by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in , the best-known achievable in polynomial time is
0.3393. In general, for -player games an -Nash equilibrium can be
computed in polynomial time for an that is an increasing function of
but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant such that
computing an -Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a -Nash equilibrium of an -player
polymatrix game can be computed in time polynomial in the input size and
. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
-Nash equilibrium in a two-player Bayesian game