Structural Indifference in Normal Form Games

Refinements of the Nash equilibrium concept differ in which indifferences between strategies they select for evaluation. In this paper, we suggest that "structural" indifferences, or indifferences that arise out of the structure of the payoffs of the game independently of opponents' strategies, are worthy of special attention. We define an order over a player's strategies, called the structured order, by ranking strategies according to expected payoff under a belief about opponents' play and requiring that (only) structural indifferences be evaluated by appealing to higher-order beliefs about opponents' play. This order is robust to trembles in payoffs and beliefs and ranks strategy r [sub]i ahead of s[sub]i if and only if r[sub]i receives a higher payoff along every sequence of trembles that converges (in a certain sense) to the beliefs. We use the structural order to define an equilibrium concept called the structural indifference respecting equilibrium (SIRE). A proper equilibrium is SIRE but not conversely. We show that thge lexicographic probability system used to describe beliefs about opponents' play when defining SIRE can always be taken to have disjoing supports. Finally, we argue that SIRE can be viewed as a normal form extention of the sequential equilibrium concept. Keywords: refinements, proper equilibrium, sequential equilibrium, trembles lexicographic probability systems, indifferences. JEL Classification Numbers: C70, C72.