Optimal Strategies of Constrained Repeated Games

Abstract

We examine a specific framework of a repeated game that asks for the player to repeatedly choose between a constant and a two-point distribution, where the two-point distribution has higher variance and expected value. The revenue is persistently subject to a linear minimum constraint. Under these conditions, we seek to maximize the expected value of \[\left(\sum\limits_{i=1}^T X_i\right)\cdot I\left(\left\{\sum\limits_{i=1}^t X_i> f(t)\right\}_ {1\leq t\leq T}\right)\]

We are able to give specific asymptotic results on the best adaptive and non-adaptive strategies for this game and solve it completely for linear \(f(\cdot)\). In doing so we draw upon methods from random walks and calculus and even find a novel variant of the zero-one law.