Fixed Points for Strong and Weak Dominance

In this note, we provide fixed-point characterizations for two solution concepts in finite games: bestresponse sets (BRS’s) and self-admissible sets (SAS’s). The BRS concept is due to Pearce [7, 1984]. The SAS concept is a weak-dominance analog to a BRS, and is due to BrandenburgerFriedenberg-Keisler [4, 2006]. BRS’s are important because they characterize the epistemic condition of rationality and common belief of rationality (RCBR) in a game. Similarly, SAS’s characterize the condition of rationality and common assumption of rationality (RCAR). (See [4, 2006] for the meanings of “rationality” in a weak-dominance setting and of “assumption.”) We imagine that the material here on BRS’s is well known to researchers in the area. In particular, Apt [1, 2006] is a lattice-theoretic treatment of strong dominance in general (infinite) games. The focus of most work is on a particular BRS—the iteratively undominated (IU) set, i.e., the set of strategies that survive iterated deletion of strongly dominated strategies. We cover all BRS’s in this note, to make clear the comparison with SAS’s. Here is a summary on BRS’s: BRS’s are the fixed points of a certain map defined on the complete lattice of rectangular subsets of the product of the players’ strategy sets. There is a greatest fixed point, which is the IU set. The map Φ is monotone increasing: If x ≤ y, then Φ(x) ≤ Φ(y). So, the existence of a greatest fixed point also follows from the Knaster-Tarski Theorem [5, 1928], [8, 1955]. (This map already appears in Apt [1, 2006].) SAS’s, too, are the fixed points of a map Ψ. Unlike Φ, the map Ψ need not be monotone increasing and so need not have a greatest fixed point. But Ψ does satisfy: Ψ( ) ≥ Ψ(Ψ( )) ≥ . . ., where is the top element of the lattice. So, induction gives a fixed point. The fixed point obtained this way is the iteratively admissible (IA) set, i.e., the set of strategies that survive