We consider an analytic formulation of the class of efficient, linear, and symmetric values for TU games that, in contrast to previous approaches, which rely on the standard basis, rests on the linear representation of TU games by unanimity games. Unlike most of the other formulae for this class, our formula allows for an economic interpretation in terms of taxing the Shapley payoffs of unanimity games. We identify those parameters for which the values behave economically sound, i.e., for which the values satisfy desirability and positivity. Put differently, we indicate requirements on fair taxation in TU games by which solidarity among players is expressed.

The states of the qubit, the basic unit of quantum information, are 2×2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

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.

An extension operator assigns to any TU game its extension, a mapping that assigns a worth to any non-negative resource vector for the players. Algaba et al. (2004) advocate the Lovász extension (Lovász, 1983) as a natural extension operator. This operator is determined by the minimum operator representing one particular CES (constant elasticity of substitution) technology. We explore alternative extension operators, the dual Lovász extension and the Shapley extension, that are based on the only two alternative CES technologies that induce an economically sound behavior of extensions in some sense, the maximum operator and the average operator.

We relax the assumption that the grand coalition must form by imposing the axiom of Cohesive efficiency: the total payoffs that the players can share is equal to the maximal total worth generated by a coalition structure. We determine how the three main axiomatic characterizations of the Shapley value are affected when the classical axiom of Efficiency is replaced by Cohesive efficiency. We introduce and characterize two variants of the Shapley value that are compatible with Cohesive efficiency. We show that our approach can also be applied to the variants of more egalitarian values.

Casajus, Kramm, and Wiese (2020, J. Econ. Theory 186, 104993) study the asymptotic stability in population dynamics derived from finite cooperative games with transferable utility using the Lovász-Shapley value (Casajus and Wiese, 2017, Int. J. Game Theory 45, 1-16) for non-negatively weighted games, where the players are interpreted as types of individuals. We extend their analysis to the population dynamics derived using the dual Lovász-Shapley value and the Shapley² value for non-negatively weighted games (Casajus and Kramm, HHL Working Paper 196, HHL Leipzig Graduate School of Management, Leipzig, Germany). As the former, we provide a complete description of asymptotically stable population profiles in both dynamics. In the dual Lovász-Shapley replicator dynamic, for example, any asymptotically stable population profile is characterized by a coalition: while the types in the coalition have the same positive share, the other types vanish. In the dual of the game, the per-capita productivity of such a stable coalition must be greater than the per-capita productivity of any proper sub- or supercoalition. In simple monotonic games, this means that exactly the minimal blocking coalitions are stable.

Also published in: Ostashevksy, Eugene; La Mura, Pierfrancesco (traduttore): Vita e opinioni di dj Spinoza: the life and opinions of Dj Spinoza. Ed. bilingue. Campanotto, 2019

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The principle of differential monotonicity for cooperative games states that the differential of two players' payoffs weakly increases whenever the differential of these players' marginal contributions to coalitions containing neither of them weakly increases. Together with the standard efficiency property and a relaxation of the null player property, differential monotonicity characterizes the egalitarian Shapley values, i.e., the coex mixtures of the Shapley value and the equal division value for games with more than two players. For games that contain more than three players, we show that, cum grano salis, this characterization can be improved by using a substantially weaker property than differential monotonicity. Weak differential monotonicity refers to two players in situations where one player's change of marginal contributions to coalitions containing neither of them is weakly greater than the other player's change of these marginal contributions. If, in such situations, the latter player's payoff weakly/strictly increases, then the former player's payoff also weakly/strictly increases.