Centre for Discrete and Applicable Mathematics |
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CDAM Research Report, LSE-CDAM-2005-18December 2005 |
Enumeration of All Extreme Equilibria of Bimatrix Games with Integer Pivoting and Improved Degeneracy Check
Gabriel D. Rosenberg
The "Enumeration of All Extreme Equilibria of Bimatrix
Games" algorithm of Audet et al. (2001) uses the best
response condition of Nash Equilibria to create a search
tree in which pure strategies are forced to be either a best
response or played with zero probability. Finding sets of
constraints with no feasible solution allows the algorithm
to avoid searching all game supports and thereby speeds the
enumeration process. This paper presents two new
improvements to the EEE algorithm. First, the algorithm is
implemented in Java using only integer arithmetic, as
opposed to previous implementation using floating-point
arithmetic. This exact solution of linear programs for the
algorithm avoids potential rounding errors. Preliminary
running time results of this implementation, determining the
relative efficacy of objective functions for the lin- ear
program search, and a comparison to another enumeration
algorithm are reported. Second, the degeneracy check is
improved, drastically cutting running time for certain
classes of games and making the algorithm theoretically
clearer. The new approach introduces constraints until the
feasible set consists of only one strategy or is empty. The
combination of these two improvements increases EEE's
usefulness as a tool for bimatrix game solution.
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Last modified: 20th December 2005