Published in Transactions of the American Mathematical Society, Volume 247, January 1, 1979, pages 157-176. Copyright © 1979 American Mathematical Society. The definitive version is available at http://dx.doi.org/10.1090/S0002-9947-1979-0517690-9.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
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 ε-optimal Markov families (in which the strategy depends only on the current state and time) and also ε-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.