60 DOXASTIC PARADOX AND REPUTATION EFFECTS IN ITERATED GAMES Robert C. Koons Philosophy Dept. / Center for Cognitive Science University of Texas at Austin koons@ sygmund.cgs.utexas.edu The "chain-store paradox" of Reinhard Sehen is one of a number of scenarios involving the finite repetition of a certain kind of sub-game about which a paradoxical conclusion can be derived. In each of these cases, a backward-induction argument is used to prove that it is futile to try to establish a reputation for cooperative or punitive behavior through appropriate action in the early stages of the game, despite the fact that nearly all agree that it is intuitively reasonable to do so. Selten believed that this is paradoxical in the weak sense: a surprising, unexpected result of game theory. In this paper, I argue that it is paradoxical in the strong sense: a logical antinomy of rational belief or subjective probability, analogous to the paradox of the Liar. I use formal theories of rational belief and of higher-order probability to demonstrate the existence in such games of a Liar-like antinomy. This involves extending Montague's 1963 generalization of Tarski's theorem [Kaplan and Montague 1960, Montague 1963]. It also involves adding some assumptions about the belief-revision dispositions of the players. Then, I apply recently developed solutions to the Liar paradox by Burge, Gaifman, and Barwise & Etchemendy to the situation presented by Selten's game. The result demonstrates and explains the existence of context-determined cognitive "blindspots" afflicting the players of the game. This necessitates the use of interval-valued probability functions in prescribing the mental states of such players. Game theorists have discovered several scenarios involving the finite repetition of a non- cooperative game which give rise to a certain kind of "paradox". These include: Selten's "chain- store paradox" [Selten 1978], the problem of the finite series of "Prisoner's Dilemma" games [Luce and Raiffa 1957, Hardin 1982] and the controversy over the game-theoretic justifiability of deterrent punishment by known act-utilitarians [Hodgson 1967, Reagan 1980]. In each of these cases, a "backward-induction" argument is used to prove that it is futile to try to establish a reputation for cooperative or punitive behavior through appropriate action in the early stages of the game, despite the fact that nearly all agree that it is intuitively "reasonable" to do so. 1. Selten's Paradox of Reputation Selten's "chain-store paradox" arose from the attempt by game-theoretical economists to analyze and evaluate the rationality of predatory behavior by monopolists. The name derives from a standard example, that of a firm which monopolizes retailing in a region through ownership of a chain of stores. In each sub-game, the corresponding potential competitor has to decide whether to enter into competition with the monopolist. If the potential competitor does enter the market, then the monopolist faces a choice between two alternatives: (1) engage in predatory pricing, driving the competitor out of business, at a great cost to both the competitor and to the monopolist, or (2) reach