Gambling in a rigged casino: the adversarial multi-armed bandit problem

Peter Auer, Nicolo Cesa-Bianchi, Yoav Freund, Robert E. Schapire

Research output: Contribution to journalConference articlepeer-review

530 Scopus citations


In the multi-armed bandit problem, a gambler must decide which arm of K non-identical slot machines to play in a sequence of trials so as to maximize his reward. This classical problem has received much attention because of the simple model it provides of the trade-off between exploration (trying out each arm to find the best one) and exploitation (playing the arm believed to give the best payoff). Past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines. In this work, we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines. We give a solution to the bandit problem in which an adversary, rather than a well-behaved stochastic process, has complete control over the payoffs. In a sequence of T plays, we prove that the expected per-round payoff of our algorithm approaches that of the best arm at the rate O(T-1/3), and we give an improved rate of convergence when the best arm has fairly low payoff. We also consider a setting in which the player has a team of 'experts' advising him on which arm to play; here, we give a strategy that will guarantee expected payoff close to that of the best expert. Finally, we apply our result to the problem of learning to play an unknown repeated matrix game against an all-powerful adversary.

Original languageEnglish (US)
Pages (from-to)322-331
Number of pages10
JournalAnnual Symposium on Foundations of Computer Science - Proceedings
StatePublished - 1995
Externally publishedYes
EventProceedings of the 1995 IEEE 36th Annual Symposium on Foundations of Computer Science - Milwaukee, WI, USA
Duration: Oct 23 1995Oct 25 1995

All Science Journal Classification (ASJC) codes

  • Hardware and Architecture


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