A note on the two-armed bandit problem with finite memory

Abstract

Robbins has proposed a finite memory constraint on the two-armed bandit problem in which the coin to be tossed at each stage may depend on the history of the previous tosses only through the outcomes of the last r tosses. Letting the choice of coin depend on the time at which the toss is made, we exhibit a deterministic rule with memory r = 2, the description of which is independent of the coin biases p1 and p2 , which achieves, with probability one, a limiting proportion of heads equal to max \{{ p1 , p2 \}}. Thus this rule is asymptotically uniformly best among the class of time-varying finite memory rules.