Rational Dynamics and Epistemic Logic in Games

I propose a bare-bones look at epistemic models for games, with a focus on update procedures for reaching equilibrium 'zones'. Connections are given with standard update and fixedpoint logics. This is just a 'methods' paper, as readers will want to play with models, uncertainty relations, and announcements different from those used here for the purposes of illustration. 1 Reaching equilibrium Nash equilibrium describes optimal strategy profiles for players of a game. But it has some well-known problems, where predicted equilibria do not match most plausible behaviour. Examples arise both in strategic games such as Prisoner's Dilemma, and extensive ones such as the Centipede (cf. Binmore 1992, Osborne & Rubinstein 1994). One important line of analysis tries to analyze this situation by identifying suitable epistemic assertions about players' behaviour which imply the usual equilibrium solutions. For this purpose, game theorists have introduced epistemic models for a given game, representing what players know and do not know about possible strategy profiles – plus perhaps other features of the game, its players, and its environment. One such type of behaviour is rationality: choosing one's best available response to the action profile of the other players. And common knowledge of rationality is taken to imply the Nash equilibrium solution concept for strategic games, and suitably formulated, also the backward induction solution for extensive games. This Note takes a closer look at the logic of the implication from rationality to equilibrium. In this first part, we focus on strategic games, where players choose one action each, simultaneously – as in the familiar 2-player game matrices. There may be a second part later. The main thrust of what follows is a familiar point of 'logical dynamics'. In addition to the standard static notion of equilibrium, game-theoretic intuitions seem to be just as much about dynamic procedures that produce equilibria. E.g., in Iterated Removal of Dominated Strategies for strategic games, or Backward Induction for extensive games, equilibrium arises as a fixed-point of some process