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Maximin equilibrium

Ismail, Mehmet (2014): Maximin equilibrium.

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Abstract

We introduce a new solution concept called maximin equilibrium which extends von Neumann's maximin strategy idea to n-player non-cooperative games by incorporating common knowledge of 'rationality' of the players. Our rationality assumption is, however, stronger than the one of maximin strategy and weaker than the one of Nash equilibrium. Maximin equilibrium, just like maximin strategies, is a method for evaluating the uncertainty that players are facing by playing the game. We show that maximin equilibrium is invariant under strictly increasing transformations of the payoff functions. Notably, every finite game possesses a maximin equilibrium in pure strategies. Considering the games in von Neumann and Morgenstern mixed extension, we show that maximin equilibrium is a generalization of Nash equilibrium. In addition, we demonstrate that maximin equilibria and Nash equilibria coincide in two-player zero-sum games. We propose a refinement of maximin equilibrium called strong maximin equilibrium. Accordingly, we show that for every Nash equilibrium that is not a strong maximin equilibrium there exists a strong maximin equilibrium that Pareto dominates it. In addition, no strong maximin equilibrium is ever Pareto dominated by a Nash equilibrium. Finally, we discuss maximin equilibrium predictions in several games including the traveler's dilemma. (Submitted to MPRA for stable archival purposes. This is the first Maastricht University version.)

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