The Shapley value equals a playerâ€™s contribution to the potential of a game. The potential is a most natural one-number summary of a game, which can be computed as the expected accumulated worth of a random partition of the players. This computation integrates the coalition formation of all players and readily extends to games with externalities. We investigate those potential functions for games with externalities that can be computed this way. It turns out that the potential that corresponds to the MPW solution introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339â€“356) is unique in the following sense. It is obtained as the expected accumulated worth of a random partition, it generalizes the potential for games without externalities, and it induces a solution that satisfies the null player property even in the presence of externalities.

Consumer Choice Under Limited Attention When Alternatives Have Different Information Costs

Consumers often do not have complete information about the choices they face and, therefore, have to spend time and effort acquiring information. Because information acquisition is costly, consumers trade off the value of better information against its cost and make their final product choices based on imperfect information. We model this decision using the rational inattention approach and describe the rationally inattentive consumerâ€™s choice behavior when the consumer faces alternatives with different information costs. To this end, we introduce an information cost function that distinguishes between direct and implied information. We then analytically characterize the optimal choice probabilities. We find that nonuniform information costs can have a strong impact on product choice, which gets particularly conspicuous when the product alternatives are otherwise very similar. There are significant implications on how a seller should provide information about its products and how changes to the product set impacts consumer choice. For example, nonuniform information costs can lead to situations in which it is disadvantageous for the seller to provide easier access to information for a particular product and to situations in which the addition of an inferior (never chosen) product increases the market share of another existing product (i.e., failure of regularity). We also provide an algorithm to compute the optimal choice probabilities and discuss how our framework can be empirically estimated from suitable choice data.

The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some playerâ€™s marginal contribution to all coalitions increase or stay the same, then this playerâ€™s payoff should not decrease. We investigate the class of values that satisfy efficiency, symmetry, and weak monotonicity. It turns out that this class coincides with the class of egalitarian Shapley values. Thus, weak monotonicity reflects the nature of the egalitarian Shapley values in the same vein as strong monotonicity reflects the nature of the Shapley value. An egalitarian Shapley value redistributes the Shapley payoffs as follows: First, the Shapley payoffs are taxed proportionally at a fixed rate. Second, the total tax revenue is distributed equally among all players.

Axiomatic arguments for decomposing goodness of fit according to Shapley and Owen values

We advocate the decomposition of goodness of fit into contributions of (groups of) regressor variables according to the Shapley value orâ€”if regressors are exogenously groupedâ€”the Owen value because of the attractive axioms associated with these values. A wage regression model with German data illustrates the method.