Two-Armed Dirichlet Bandits with Discounting

Abstract

Sequential selections are to be made from two independent stochastic processes, or "arms." At each stage we choose which arm to observe based on past selections and observations. The observations on arm $i$ are conditionally i.i.d. given their marginal distribution $P_i$ which has a Dirichlet process prior with parameter $\alpha_i, i = 1, 2$. Future observations are discounted: at stage $m$, the payoff is $a_m$ times the observation $Z_m$ at that stage. The discount sequence $A_n = (a_1, a_2,\cdots, a_n, 0,0,\cdots)$ is a nonincreasing sequence of nonnegative numbers, where the "horizon" $n$ is finite. The objective is to maximize the total expected payoff $E(\sum^n_1a_iZ_i)$. It is shown that optimal strategies continue with an arm when it yields a sufficiently large observation, one larger than a "break-even observation." This generalizes results of Clayton and Berry, who considered two arms with one arm known and assumed $a_m = 1 \forall m \leq n$.