Numerical Computation for Identifying Nash Equilibria Using Shapley's Labeling Method

This paper proposes a method to find equilibria approximately using Shapley's labeling method for 2-person bimatrix games. It is well known fact that n-person finite games in strategic forms always have equilibria (Nash equilibria) in mixed strategies. A method for identifying equilibria is well-known for 2-person games with 2 × 2 bimatrices. Furthermore, the Shapley's label method is famous for games with more strategies. He presented a graphical method so called "Shapley's labeling method" which draws a diagram called "label diagram". However, drawing a label diagram for n × m bimatrix games is a complicated task, and it's not easy to find equilibria in the procedural way. The implemented program can calculate all equilibria for any 3 × 3 bimatrix games in less than 1 second, and for any 4 × 4 games in about less than 20 minutes on ordinary available PCs.