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Institute
The Shapley value equals a player'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, 339-356) 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.
In the absence of externalities, marginality is equivalent to an independence property that rests on Harsanyi's dividends. These dividends identify the surplus inherent to each coalition. Independence states that a player's payoff stays the same if only dividends of coalitions to which this player does not belong to change. We introduce notions of marginality and independence for games with externalities. We measure a player's contribution in an embedded coalition by the change in the worth of this coalition that results when the player is removed from the game. We provide a characterization result using efficiency, anonymity, and marginality or independence, which generalizes Young's characterization of the Shapley value. An application of our result yields a new characterization of the solution put forth by Macho-Stadler et al. (J Econ Theor, 135, 2007, 339-356) without linearity, as well as for almost all generalizations put forth in the literature. The introduced method also allows us to iestigate egalitarian solutions and to reveal how accounting for externalities may result in a deviation from the Shapley value. This is exemplified with a new solution that is designed in a way to not reward external effects, while at the same time it cannot be assumed that any partition is the default partition.
The Coleman–Shapley index
(2019)
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 index—the Coleman–Shapley index (CSI)—indicating each voter’s 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 voters’ power equals the CPCA. Similar to the Shapley–Shubik index (SSI) and the Penrose–Banzhaf 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 voter’s 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 voter’s power as such and the voter’s 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.
We study values for transferable utility games enriched by a communication graph. The most well-known such values are component-efficient and characterized by some link-deletion property. We study efficient extensions of such values: for a given component-efficient value, we look for a value that (i) satisfies efficiency, (ii) satisfies the link-deletion property underlying the original component-efficient value, and (iii) coincides with the original component-efficient value whenever the underlying graph is connected. Béal et al. (2015) prove that the Myerson value (Myerson, 1977) admits a unique efficient extension, which has been introduced by van den Brink et al. (2012). We pursue this line of research by showing that the average tree solution (Herings et al., 2008) and the compensation solution (Béal et al., 2012b) admit similar unique efficient extensions, and that there exists no efficient extension of the position value (Meessen, 1988; Borm et al., 1992). As byproducts, we obtain new characterizations of the average tree solution and the compensation solution, and of their efficient extensions. Keywords: Efficient extension, average tree solution, compensation solution, position value, component fairness, relative fairness, balanced link contributions, Myerson value, component-wise egalitarian solution
The resolution of a solution for cooperative games is a recently developed tool to decompose a solution into a player's direct contribution in a game and her (higher-order) indirect contribution, i.e., her contribution to other players' 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.
We suggest foundations for the Shapley value and for the naïve 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 player's contribution in a game as such, the indirect part indicates how she affects the other players' direct contributions by leaving the game. The Shapley value turns out to be unique decomposable decomposer of the naïve solution.
The Coleman-Shapley-index
(2018)
The Coleman Power of the Collectivity to Act (CPCA) is a popular statistic that reflects the ability of a committee to pass a proposal. Applying the Shapley value to this measure, we derive a new power index that indicates each voter’s contribution to the CPCA. This index is characterized by four axioms: anonymity, the null voter property, transfer property, and a property that stipulates that sum of the voters’ power equals the CPCA. Similar to the Shapley-Shubik index (SSI) and the Penrose-Banzhaf index (PBI), our new index emerges as 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, the voters are not ordered according to their agreement with a potential bill but according to their vested interest in it. Among the most interested voters, the power is then measured in a similar way as with the PBI. Althoug we advocate the CSI against the PBI to capture a voter’s influence on whether a proposal passes, the CSI gives new meaning to the PBI. The CSI is the decomposer of the PBI, splitting it into a voter’s power as such and a her impact on the power of the other voters by threatening to block any proposal. We apply the index to the EU Council and the UN Security Council. Keywords: Decomposition, Shapley value, Shapley-Shubik Index, power index, Coleman Power of the Collectivity to Act, Penrose-Banzhaf Index, EU Council, UN Security Council
We suggest a foundation of the Shapley value via 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 player‘s contribution in a game as such, the indirect part indicates how she affects the other players‘ direct contributions by leaving the game. The Shapley value turns out to be unique decomposable decomposer of the naïve solution, where the naïve solution assigns to any player the difference between the worth of the grand coalition and its worth after this player left the game.