Different Measures of Win Rate for Optimal Proportional Betting

It is well known that all betting systems applied to independent, repeated, and identically distributed trials will result in the same expected gain per average unit wagered as that which applies to a single trial. This paper develops the asymptotic and paradoxical manner in which that constant win rate is maintained for optimal proportional betting according to the Kelly criterion. The appropriateness of using this traditional measure of win rate for proportional betting is contrasted with that of various alternatives.