Summary
For games with uncountably many pure strategies there exists in general not only a uniquely determined mixed extension, but a system of different “randomizations”. A randomization for which the interval between lower and upper value is as small as possible is called “minimal indefinite”. A minimal indefinite randomization is either definite (strictly determined) or the game is “essentially indefinite”. The construction of increasing sequences of sets of minimal indefinitely randomized games by means of iterated “composition” leads to criteria for a randomization to be minimal indefinite. As a by-product we get an arsenal of necessary conditions for the existence of a definite randomization. The notion of randomization is understood here in a rather wide sense: Instead of using a condition adapted to convexity only a much weaker composibility-condition is required; conditions on measurability are reduced to a minimum.
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Bierlein, D. Minimal indefinite Randomisierungen von Spielen. Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 193–207 (1969). https://doi.org/10.1007/BF00536380
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DOI: https://doi.org/10.1007/BF00536380