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$p$-Dominance and Belief Potential

Stephen Morris
Rafael Rob
Hyun Song Shin

Abstract

This paper elucidates the logic behind recent papers which show that a unique equilibrium is selected in the presence of higher order uncertainty, i.e., when players lack common knowledge. We introduce two new concepts: belief potential of the information system and $p$-dominance of Nash-equilibria of the game, and show that a Nash-equilibrium is uniquely selected whenever its $p$-dominance is below the belief potential. This criterion applies to many-action games, not merely $2 \times 2$ games. It also applies to games without dominant strategies, where the set of equilibria is shown to be smaller and simpler than might be initially conjectured. Finally, the new concepts help understand the circumstances under which the set of equilibria varies with the amount of common knowledge among players.