Abstract.
We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v ∞ (p) for all 0≤p≤1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v ∞ (p) (and will converge from above to v ∞ (p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs-to the value v ∞ (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG ∞ (p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v n (p)− v ∞ (p) is of the order of magnitude of . In that case the value of SG ∞ (p) is of the order of magnitude of . We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛∞={X k }∞ k=1, one defines for each n : V n (𝔛∞) ≔E∑n k=1 |X k+1 − X k|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛∞, V n (𝔛∞) can be of the order of magnitude of n 1/2−ε, for arbitrarily small ε>0.
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Received January 1999/Final version April 2002
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Nowik, I., Zamir, S. The game for the speed of convergence in repeated games of incomplete information. Game Theory 31, 203–222 (2003). https://doi.org/10.1007/s001820200101
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DOI: https://doi.org/10.1007/s001820200101