A dynamic epistemic characterization of backward induction without counterfactuals

The analysis of rational play in dynamic games is usually done within a static framework that specifies a player’s initial beliefs as well as his disposition to revise those beliefs conditional on hypothetical states of information. We suggest a simpler approach, where the rationality of a player’s choice is judged on the basis of the actual beliefs that the player has at the time he has to make that choice. We propose a dynamic framework where the set of “possible worlds” is given by state-instant pairs (ω, t). Each state ω specifies the entire play of the game and, for every instant t, (ω, t) specifies the history that is reached at that instant (in state ω). A player is said to be active at (ω, t) if the history reached in state ω at date t is a decision history of his. At every state-instant pair (ω, t) the beliefs of the active player provide an answer to the question “what will happen if I take action a?”, for every available action a. A player is said to be rational at (ω, t) if either he is not active there or the action he ends up taking at state ω is optimal given his beliefs at (ω, t). We provide a characterization of backward induction in terms of the following event: the first mover (i) is rational and has correct beliefs, (ii) believes that the active player at date 1 is rational and has correct beliefs, (iii) believes that the active player at date 1 believes that the active player at date 2 is rational and has correct beliefs, etc. Thus our epistemic characterization does not rely on dispositional belief revision or on (objective or subjective) counterfactuals.