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- W3135339944 abstract "Abstract We study subgame $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> . First, we provide necessary and sufficient conditions for a strategy to be a subgame $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function $$phi ^*$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> with the following property: if a player has a subgame $$phi ^*$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> -maxmin strategy for every positive tolerance function $$phi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>ϕ</mml:mi> </mml:math> is equivalent to the existence of a subgame maxmin strategy." @default.
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- W3135339944 date "2021-03-02" @default.
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- W3135339944 title "Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels" @default.
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- W3135339944 doi "https://doi.org/10.1007/s13235-021-00378-z" @default.
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