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The determinacy of Blackwell games

  • Donald A. Martin (a1)
Abstract

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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[4] Harvey M. Friedman , Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2 (1971), pp. 325357.

[5] Kurt F. Gödel , The consistency of the axiom of choice and the generalized continuum-hypothesis, Proceedings of the National Academy of Science USA, vol. 24 (1938), pp. 556557.

[9] A. Maitra and W. Sudderth , An operator solution of stochastic games, Israel Journal of Mathematics, vol. 78 (1992), pp. 3349.

[10] A. Maitra and W. Sudderth , Borel stochastic games with lim sup payoff, Annals of Probability, vol. 21 (1993), pp. 861885.

[12] Donald A. Martin , Borel determinacy, Annals of Mathematics, vol. 102 (1975), pp. 363371.

[13] Donald A. Martin , A purely inductive proof of Borel determinacy, Recursion theory (Providence) ( Anil Nerode and Richard A. Shore , editors), Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, 1985, pp. 303308.

[14] Donald A. Martin , An extension of Borel determinacy, Annals of Pure and Applied Logic, vol. 49 (1990), pp. 279293.

[15] Donald A. Martin and John R. Steel , A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71125.

[17] M. Orkin , Infinite games with imperfect information, Transactions of the American Mathematical Society, vol. 171 (1972), pp. 501507.

[19] John von Neumann , Zur Theorie der Gesellschaftsspiele, Mathemathische Annalen, vol. 100 (1928), pp. 295320.

[20] W. Hugh Woodin , Supercompact cardinals, sets of reals and weakly homogeneous trees, Proceedings of the National Academy of Sciences USA, vol. 85 (1988), pp. 65876591.

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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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