Complexity and effective prediction

Let G = 〈I, J, g〉 be a two-person zero-sum game. We examine the two-person zero-sum repeated game G(k, m) in which player 1 and 2 place down finite state automata with k, m states respectively and the payoff is the average per stage payoff when the two automata face off. We are interested in the cases in which player 1 is “smart” in the sense that k is large but player 2 is “much smarter” in the sense that m k. Let S(g) be the value of G were the second player is clairvoyant, i.e., would know the player 1’s move in advance. The threshold for clairvoyance is shown to occur for m near min(|I|, |J |)k. For m of roughly that size, in the exponential scale, the value is close to S(g). For m significantly smaller (for some stage payoffs g) the value does not approach S(g). ∗Institute of Mathematics and Center for the Study of Rationality, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (e-mail: [email protected]). †Department of Computer Science, and Department of Mathematics, New York University, USA (email: [email protected]). This research was supported in part by Israel Science Foundation grant 263/03. The authors enjoyed the hospitality, generosity and stimulating atmosphere of Microsoft Research (Redmond) in the initial phases of this research.