Bayesian Subjunctive Conditionals for Games and Decisions

Abstract

The theory of rational decision has always been implicitly involved with subjunctive and counterfactual conditionals. “If I were to do A, this would happen; if I were to do B that would happen. ” When I have done A, I use the counterfactual: “If I had done B, the outcome would have been worse. ” Counterfactuals are handled so smoothly in decision theory and game theory that they are hardly ever explicitly discussed except in cases where they cause problems. For many philosophers, Newcomb’s paradox was such a critical case (See Gibbard and Harper (1978)). In game theory, questions about beliefs off the equilibrium path in extensive form games and related issues of backward and forward induction provoked explicit discussion of counterfactuals (See Aumann (1995), Bicchieri (1988), Binmore (1987), Harsanyi and Selten (1988), Reny (1993), Stalnaker (1994), (forthcoming)). Where subjunctive conditionals cause problems, we need an explicit theory. What theory should we use? There is the selection function semantics of Stalnaker (1968) based on the idea of “most similar possible world in which the antecedent is true”, and related theories of Woodruff (1969), Lewis (1973), Pollock (1976), Nute (1980), which relax some of Stalnaker’s assumptions.