A mathematical analysis of the card game of Betweenies through Kelly's criterion

In the game of Betweenies, the player is dealt two cards out of a deck and bets on the probability that the third card to be dealt will have a numerical value in between the values of the first two cards. In this work, we present the exact rules of the two main versions of the game, and we study the optimal betting strategies. After discussing the shortcomings of the direct approach, we introduce an information-theoretic technique, Kelly'scriterion, which basically maximizes the expected log-return of the bet: we offer an overview, discuss feasibility issues, and analyze the strategies it suggests. We also provide some gameplay simulations.