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We show that if the row player knows an upper bound on the value of the game it can use a variant of MW to generate a sequence of mixed strategies that approach a strategy which

# Adaptive Game Playing Using Multiplicative Weights

GAMES AND ECONOMIC BEHAVIOR, no. 1-2 (1999): 79-103

EI

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...更多

• The authors present a simple algorithm for playing a repeated game. The authors 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.
• The authors use ¤ to denote the probability ¤ £§¦ © £ that ¤ associates with the row , and the authors write ¦¤ §¥ © ̈¤ T¥ to denote the expected loss when the two mixed strategies are used.
• 76 8@9#A) The learning algorithm MW starts with some initial mixed strategy ¤ 1 which it uses for the first round of the game.

• We present a simple algorithm for playing a repeated game
• In the analysis presented so far we have shown that the average of the strategies used by MW converges to an optimal strategy
• We show that if the row player knows an upper bound on the value of the game it can use a variant of MW to generate a sequence of mixed strategies that approach a strategy which
• In Section 7, we show that this dependence on , and ¢ cannot be improved by any constant factor
• For the purposes of the proof, we imagine choosing the matrix at random according to an appropriate distribution, and we show that the stated properties hold with strictly positive probability, implying that there must exist at least one matrix for which they hold
• ¥ In other words, property 2 holds with probability at least ¡ . ¥ We show that property 1 fails to hold with probability strictly smaller than ¡ so that both properties must hold simultaneously with positive probability

• From Theorem 1 and Corollary 4 the authors know that the expected per-iteration loss of MW approaches the optimal achievable value for any fixed strategy as #
• Lemma 6 Let the players of a matrix game use any pair of methods for choosing their mixed strategies
• The goal of the row player is the same as before—to minimize its expected average loss over a sequence of repeated games.
• In Section 6.2 the authors show that if an upper bound on the value of the game is known ahead of time one can use a variant of MW that generates a sequence of row distributions such that the expected loss of the th distribution approaches .
• The authors show that if the row player knows an upper bound on the value of the game it can use a variant of MW to generate a sequence of mixed strategies that approach a strategy which
• £§¦ © £ 0 achieves loss .1 To do that the authors have the algorithm select a different value of for each round of the game.
• A single application of the exponential weights algorithm yields approximate solutions for both the column and row players.
• The solution for game matrix is related to the on-line prediction ' £ problem described in Section 4, while the “dual” solution for T corresponds to a method of learning called “boosting.”
• They may be most appropriate for the setting the authors have described of approximately solving a game when an oracle is available for choosing columns of the matrix on every round.

• The authors show that this dependence of the rate of convergence on , and ¢ is optimal in the sense that no adaptive game-playing algorithm can beat this bound even by a constant factor.
• For any adaptive game-playing algorithm , there exists a game matrix M of rows and a sequence of column strategies such that: 1.
• £§¦ © % is chosen at random, the authors need the row player has sole control a lower bound on over the choice of the ¤ , probability the authors need a that lower

• Shows that a player using this algorithm suffers average loss that is guaranteed to come close to the minimum loss achievable by any fixed strategy
• Presents a simple algorithm for solving this problem, and give a simple analysis of the algorithm

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