Renewal Plans and Persistent Optimality in Countably Additive Gambling

In countably additive gambling models with general utility functions, plans for play are constructed which persist in being conditionally ϵ-optimal along every history and which are conditionally optimal whenever possible. Such plans are formed by piecing together plans which are known to be good for the gambler at single time periods. Verification of the optimality properties of these plans uses transient renewal theoretic arguments.