Adaptive Game Playing Using Multiplicative Weights

Yoav Freund, Robert E. Schapire

Research output: Contribution to journalArticlepeer-review

371 Scopus citations


We present a simple algorithm for playing a repeated game. We show that a player using this algorithm suffers average loss that is guaranteed to come close to the minimum loss achievable by any fixed strategy. Our bounds are nonasymptotic and hold for any opponent. The algorithm, which uses the multiplicative-weight methods of Littlestone and Warmuth, is analyzed using the Kullback-Liebler divergence. This analysis yields a new, simple proof of the min-max theorem, as well as a provable method of approximately solving a game. A variant of our game-playing algorithm is proved to be optimal in a very strong sense. Journal of Economic Literature Classification Numbers: C44, C70, D83.

Original languageEnglish (US)
Pages (from-to)79-103
Number of pages25
JournalGames and Economic Behavior
Issue number1
StatePublished - Oct 1999
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Finance
  • Economics and Econometrics


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