The Coleman power of a collectivity to act (CPCA) is a popular statistic that reflects the ability of a committee to pass a proposal. Applying the Shapley value to that measure, we derive a new power indexâthe ColemanâShapley index (CSI)âindicating each voterâs contribution to the CPCA. The CSI is characterized by four axioms: anonymity, the null voter property, the transfer property, and a property stipulating that the sum of the votersâ power equals the CPCA. Similar to the ShapleyâShubik index (SSI) and the PenroseâBanzhaf index (PBI), our new index reflects the expectation of being a pivotal voter. Here, the coalitional formation model underlying the CPCA and the PBI is combined with the ordering approach underlying the SSI. In contrast to the SSI, voters are ordered not according to their agreement with a potential bill, but according to their vested interest in it. Among the most interested voters, power is then measured in a way similar to the PBI. Although we advocate the CSI over the PBI so as to capture a voterâs influence on whether a proposal passes, our index gives new meaning to the PBI. The CSI is a decomposer of the PBI, splitting the PBI into a voterâs power as such and the voterâs impact on the power of the other voters by threatening to block any proposal. We apply our index to the EU Council and the UN Security Council.

OR

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 consumerâs 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.

We suggest foundations for the Shapley value and for the naĂŻve solution, which assigns to any player the difference between the worth of the grand coalition and its worth after this player left the game. To this end, we introduce the decomposition of solutions for cooperative games with transferable utility. A decomposer of a solution is another solution that splits the former into a direct part and an indirect part. While the direct part (the decomposer) measures a playerâs contribution in a game as such, the indirect part indicates how she affects the other playersâ direct contributions by leaving the game. The Shapley value turns out to be unique decomposable decomposer of the naĂŻve solution.

The resolution of a solution for cooperative games is a recently developed tool to decompose a solution into a playerâs direct contribution in a game and her (higher-order) indirect contribution, i.e., her contribution to other playersâ direct contributions. We provide new formulae for resolutions and their potentials, which facilitate the calculation of them in large (voting) games. These formulae make use of the multi-linear extension of cooperative games with transferable utility.

2016

On the existence of efficient and fair extensions of communication values for connected graphs

We study values for TU games with a communication graph (CO-values). In particular, we show that CO-values for connected graphs that are fair and efficient allow for a unique efficient and fair extension to the full domain.

Characterizations of weighted and equal division values

New and recent axioms for cooperative games with transferable utilities are introduced. The non-negative player axiom requires to assign a non-negative payoff to a player that belongs to coalitions with non-negative worth only. The axiom of addition invariance on bi-partitions requires that the payoff vector recommended by a value should not be affected by an identical change in worth of both a coalition and the complementary coalition. The nullified solidarity axiom requires that if a player who becomes null weakly loses (gains) from such a change, then every other player should weakly lose (gain) too. We study the consequence of imposing some of these axioms in addition to some classical axioms. It turns out that the resulting values or set of values have all in common to split efficiently the worth achieved by the grand coalition according to an exogenously given weight vector. As a result, we also obtain new characterizations of the equal division value.

The Need for Permission, the Power to Enforce, and Duality in Cooperative Games with a Hierarchy

A cooperative game with transferable utility (TU game) captures a situation in which players can achieve certain payoffs by cooperating. We assume that the players are part of a hierarchy. In the literature, this invokes the assumption that subordinates cannot cooperate without the permission of their superiors. Instead, we assume that superiors can force their subordinates to cooperate. We show how both notions correspond to each other by means of dual TU games. This way, we capture the idea that a superiorsâ ability to enforce cooperation can be seen as the ability to neutralize her subordinateâs threat to abstain from cooperation. Moreover, we introduce the coercion value for games with a hierarchy and provide characterizations thereof that reveal the similarity to the permission value.

We introduce a solution concept for cooperative games with transferable utility and a coalition structure that is proportional for two-player games. Our value is obtained from generalizing a proportional value for cooperative games with transferable utility (Ortmann 2000) in a way that parallels the extension of the Shapley value to the Owen value. We provide two characterizations of our solution concept, one that employs a property that can be seen as the proportional analog to Myersonâs balanced contribution property; and a second one that relies on a consistency property.

The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some playerâs marginal contribution to all coalitions increase or stay the same, then this playerâs 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.

We study the consequences of a solidarity property that specifies how a value for cooperative games should respond if some player forfeits his productivity, i.e., becomes a null player. Nullified solidarity states that in this case either all players weakly gain together or all players weakly lose together. Combined with efficiency, the null game property, and a weak fairness property, we obtain a new characterization of the equal division value.

We suggest a new one-parameter family of solidarity values for TU-games. The members of this class are distinguished by the type of player whose removal from a game does not affect the remaining playersâ payoffs. While the Shapley value and the equal division value are the boundary members of this family, the solidarity value is its center. With exception of the Shapley value, all members of this family are asymptotically equivalent to the equal division value in the sense of Radzik (2013).

Null, nullifying, or dummifying players: The difference between the Shapley value, the equal division value, and the equal surplus division value

We provide new characterisations of the equal surplus division value. This way, the difference between the Shapley value, the equal surplus division value, and the equal division value is pinpointed to one axiom.

2013

JME :)

Null players, solidarity, and the egalitarian Shapley values

The Shapley value probably is the most eminent single-point solution concept for TU-games. In its standard characterization, the null player property indicates the absence of solidarity among the players. First, we replace the null player property by a new axiom that guarantees null players non-negative payoffs whenever the grand coalitionâs worth is non-negative. Second, the equal treatment property is strengthened into desirability. This way, we obtain a new characterization of the class of egalitarian Shapley values, i.e.,Â of convex combinations of the Shapley value and the equal division solution. Within this characterization, additivity and desirability can be replaced by strong differential monotonicity, which translates higher productivity differentials into higher payoff differentials.

2012

EJS

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 orâif regressors are exogenously groupedâthe Owen value because of the attractive axioms associated with these values. A wage regression model with German data illustrates the method.