Keywords and Synonyms
Approximate Nash equilibrium
Problem Definition
In this entry, the following two problems are considered: 1) the problem of finding an approximate Nash equilibrium in a positively normalized bimatrix (or two-player) game; and 2) the smoothed complexity of finding an exact Nash equilibrium in a bimatrix game. It turns out that these two problems are strongly correlated [3].
Let \( { \cal G=({\mathbf{A}},{\mathbf{B}}) } \) be a bimatrix game, where \( { {\mathbf{A}}=(a_{i,j}) } \) and \( { {\mathbf{B}}=(b_{i,j}) } \) are both \( { n\times n } \) matrices. Game \( { \cal G } \) is said to be positively normalized, if \( { 0\le a_{i,j},b_{i,j}\le 1 } \) for all \( { 1\le i,j\le n } \).
Let \( { \mathbb{P}^n } \) denote the set of all probability vectors in \( { \mathbb{R}^n } \), i. e., non-negative vectors whose entries sum to 1. A Nash equilibrium [8] of \( { \cal G=({\mathbf{A}},{\mathbf{B}}) } \) is a pair of mixed strategies \( { ({\mathbf{x}}^*\in...