Multi-armed bandits with discount factor near one: the Bernoulli case

F. P. Kelly

Annals of Statistics, 9 (1981) 987--1001.

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Each of n arms generates an infinite sequence of Bernoulli random variables. The parameters of the sequences are themselves random variables, and are independent with a common distribution satisfying a mild regularity condition. At each stage we must choose an arm to observe (or pull) based on past observations, and our aim is to maximize the expected discounted sum of the observations. In this paper it is shown that as the discount factor approaches one the optimal policy tends to the rule of least failures, defined as follows: pull the arm which has incurred the least number of failures, or if this does not define an arm uniquely select from amongst the set of arms which have incurred the least number of failures an arm with the largest number of successes.

Keywords: Bernoulli bandit process, Markov decision process, multi-armed bandit, limit rule, play-the-winner rule, least failures rule, discount optimality

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