Infinite Games and Analytic SetsAuthor(s): David BlackwellSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 58, No. 5 (Nov. 15, 1967), pp. 1836-1837Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/58114 .

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INFINITE GAMES AND ANALYTIC SETS* BY DAVID BLACKWELL

UNIVERSITY OF CALIFORNIA, BERKELEY

Communicated September 21, 1967

We show in this note that Kuratowski's coreduction principle for analytic sets" 3 is a consequence of a theorem of Gale and Stewart2 on infinite games of perfect in- formation.

A subset A of a complete separable metric space Y is analytic if there is a continu- ous function f from Q to Y with A = fQ where Q is the space of infinite sequences of positive integers.

Kuratowski's coreduction principle asserts that (1) If A,B are analytic subsets of Y, there are analytic sets A,,B1 such that

(a) AlDA,B,DB, (b) A1 U B,= Y, (c) A, nB1 =AnB.

To obtain (1) from the result of Gale and Stewart, let A = fU, B = go, where f,g are continuous, and for any finite sequence x = (n,, . . ., nQ) of positive integers, denote by Q(x) the set of all co E& Q with x as initial segment, and by R(x), S(x) the closures of fQ(x), gg(x), respectively.

We associate with each y E Y a game G(y) with two players, a and A3, played as follows. The players alternately choose positive integers, a choosing first, each choice made with complete information about previous choices. A play

T = (ml,nl,m2,n2, ... )

will be called a win for a (in G(y)) if there is a positive integer k for which y C R (ml, ... X mO) y (Z S(n,, . ., nk); a win for /3 if there is a positive integer k for which y C S(n,, ..., nk-1), y X R(ml, . . ., M,O), and a draw if for every positive integer k,

y C R(ml, ..., nMk) and y C S(n,, ..., nk).

Informally, a is trying to produce an X = (mi, M2, . . . ) with f(w) = y, and A is trying to produce an co = (n,, n2, ... ) with g(c) = y. The first player to fail drastically, in the sense of producing a finite sequence x for which y is not even in the closure of the image of the set of w consistent with x, loses. If neither ever fails drastically, so that, as is easily checked, both succeed, the game is a draw.

Denote by A, the set of y such that a can force a draw or better in G(y), and by B, the set of y such that A can force a draw or better in G(y). Then A,,Bl have the properties asserted in (1). We sketch the proof.

First, A, is analytic. For the set D of all pairs (y,o) such that 4 is a strategy for a that forces at least a draw in G(y) is a Borel set, and Al is its projection on Y. Similarly, B, is analytic.

Second, A, v A, for if y C A, there is an w = (Mi,M2, ...) with f(w) = y. So a, by playing Mi,12, ..., never drastically fails, so secures at least a draw. Similarly, B, D B.

Third, A, U B, = Y, for the set Q of plays r that win for a in G(y) is open, so

1836

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VoL. 58, 1967 MATHEMATICS: D. BLACKWELL 1837

that, according to the Gale-Stewart theorem, either a can force ir E Q, so that y C A1, or 0 can force ir - Q?, so that y C B1.

Finally, A1 f B1 = A n B, for if y C A1 f B1, either player can force a draw or better in G(y). If both do this, the resulting play

7 = (mi,ni, m2,n2, . .)

is a draw. Then

y = f(ml,m2, ... ) = g(n1,n2, ... ), so that y C AfnB. * Research sponsored by the U.S. Air Force Office of Scientific Research, Office of Aerospace

Research, under AFOSR grant 1312-67. 1 Addison, J. W., "Separation principles in the hierarchies of classical and effective descriptive

set theory," Fund. Math., 46, 123-135 (1958). 2 Gale, D., and F. M. Stewart, "Infinite games with perfect information," Ann. Math. Studies,

28, 245-266 (1953). 3 Kuratowski, K., "Sur les th6or4mes de s6paration dans la theorie des ensembles," Fund. Math.,

26, 183-191 (1936).

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