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The determinacy of Blackwell games

  • Donald A. Martin (a1)

Games of infinite length and perfect information have been studied for many years. There are numerous determinacy results for these games, and there is a wide body of work on consequences of their determinacy.

Except for games with very special payoff functions, games of infinite length and imperfect information have been little studied. In 1969, David Blackwell [1] introduced a class of such games and proved a determinacy theorem for a subclass. During the intervening time, there has not been much progress in proving the determinacy of Blackwell's games. Orkin [17] extended Blackwell's result to a slightly wider class. Blackwell [2] found a new proof of his own result. Maitra and Sudderth [9, 10] improved Blackwell's result in a different direction from that of Orkin and also generalized to the case of stochastic games. Recently Vervoort [18] has obtained a substantial improvement. Nevertheless, almost all the basic questions have remained open.

In this paper we associate with each Blackwell game a family of perfect information games, and we show that the (mixed strategy) determinacy of the former follows from the (pure strategy) determinacy of the latter. The complexity of the payoff function for the Blackwell game is approximately the same as the complexity of the payoff sets for the perfect information games. In particular, this means that the determinacy of Blackwell games with Borel measurable payoff functions follows from the known determinacy of perfect information games with Borel payoff sets.

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[2] Blackwell David, Operator solution of infinite Gδ games of imperfect information, Probability, statistics, and mathematics: Papers in honor of S. Karlin (Anderson T. W., Athreya K. B., and Iglehart D. L., editors), Academic Press, New York, 1989, pp. 85101.
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[16] Neeman Itay, Optimal proofs of determinacy, Bulletin of Symbolic Logic, vol. 1 (1995), pp. 327339.
[17] Orkin M., Infinite games with imperfect information, Transactions of the American Mathematical Society, vol. 171 (1972), pp. 501507.
[18] Vervoort Marco R., Blackwell games, Statistics, probability, and game theory: Papers in honor of David Blackwell (Ferguson Thomas, Shapley Lloyd, and MacQueen James, editors), IMS Lecture Notes-Monograph Series, vol. 30, Institute of Mathematical Statistics, 1996, pp. 369390.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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