On the existence of good Markov strategies

Hill, Theodore Preston

doi:10.1090/S0002-9947-1979-0517690-9

On the existence of good Markov strategies
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by Theodore Preston Hill

Trans. Amer. Math. Soc. 247 (1979), 157-176

DOI: https://doi.org/10.1090/S0002-9947-1979-0517690-9

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Abstract:

In contrast to the known fact that there are gambling problems based on a finite state space for which no stationary family of strategies is at all good, in every such problem there always exist $\varepsilon$-optimal Markov families (in which the strategy depends only on the current state and time) and also $\varepsilon$-optimal tracking families (in which the strategy depends only on the current state and the number of times that state has been previously visited). More generally, this result holds for all finite state gambling problems with a payoff which is shift and permutation invariant.

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