Abstract
We propose and investigate a hierarchy of bimatrix games (A, B), whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an ε-approximation.
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Part of this work was done while T. Theobald was a Feodor Lynen fellow of the Alexander von Humboldt Foundation at Yale University. A conference version appeared at the Symposium on Discrete Algorithms 2007.
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Kannan, R., Theobald, T. Games of fixed rank: a hierarchy of bimatrix games. Econ Theory 42, 157–173 (2010). https://doi.org/10.1007/s00199-009-0436-2
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DOI: https://doi.org/10.1007/s00199-009-0436-2