Indices for Families of Competing Markov Decision Processes with Influence

Nash obtained an important extension to the classical theory of Gittins indexation when he demonstrated that index policies were optimal for a class of multiarmed bandit problems with a multiplicatively separable reward structure. We characterise the relevant indices (herein referred to as Nash indices) as equivalent retirement rewards/penalties for appropriately defined maximisation/minimisation problems. We also give a condition which is sufficient to guarantee the optimality of index policies for a Nash-type model in which each constituent bandit has its own decision structure.