We propose a new and simple adaptive procedure for playing a game: ''regret-matching.'' In this procedure, players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past. It is shown that our adaptive procedure guarantees that, with probability one, the empirical distributions of play converge to the set of correlated equilibria of the game.

Let P be a real-valued function defined on the space of cooperative games with transferable utility, satisfying the following condition: In every game, the marginal contributions of all players (according to P) are efficient (i.e., add up to the worth of the grand coalition). It is proved that there exists just one such function P-called thepotentiul-and moreover that the resulting payoff vector coincides with the Shapley value. The potential approach is also shown to yield other characterizations for the Shapley value, in particular, in terms of a new internal consistency property. Further results deal with weighted Shapley values (which emerge from the above consistency) and with the nontransferable utility case (where the egalitarian solutions and the Harsanyi value are obtained).

Andreu Mas-Colell has been doing pioneering work using differential topology in the analysis of general equilibrium. This work is regarded as outstanding and one of the major contributions to the development of rigorous economic theory in the last twenty years. The articles he has written have been difficult and technically demanding. In this book he brings this work together to show its scope and its power. He presents the analysis in a way which makes it accessible to the broader range of economic theorists and advanced students. This book has been long awaited as a seminal contribution to the subject.

We propose a new and simple adaptive procedure for playing a game: ''regret-matching.'' In this procedure, players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past. It is shown that our adaptive procedure guarantees that, with probability one, the empirical distributions of play converge to the set of correlated equilibria of the game.

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