Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games

TL;DR

This paper presents a polynomial-time approximation scheme for computing Nash equilibria in anonymous games with fixed strategies, extending previous results and introducing a probabilistic approximation of sum distributions of independent vectors.

Contribution

It introduces a novel probabilistic approximation method for sum distributions of independent vectors, enabling efficient Nash equilibrium computation in a broad class of games.

Findings

Polynomial-time approximation scheme for Nash equilibria in anonymous games.

Probabilistic approximation of sum distributions with controlled variational distance.

Construction of a sparse epsilon-cover for distributions of sums of independent vectors.

Abstract

We show that there is a polynomial-time approximation scheme for computing Nash equilibria in anonymous games with any fixed number of strategies (a very broad and important class of games), extending the two-strategy result of Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a probabilistic result of more general interest: The distribution of the sum of n independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is the unit vector along dimension i of the k-dimensional Euclidean space, can be approximated by the distribution of the sum of another set of independent unit vectors whose probabilities of obtaining each value are multiples of 1/z for some integer z, and so that the variational distance of the two distributions is at most eps, where eps is bounded by an inverse polynomial in z and a function of k, but with no dependence on n. Our probabilistic result specifies the construction of a surprisingly sparse eps-cover -- under the total variation distance -- of the set of distributions of sums of independent unit vectors, which is of interest on its own right.