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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.

We study the asymptotic stability in replicator dynamics derived from TU games using the dual Lovász-Shapley value and the Shapley2 value for non-negatively weighted games. In particular, we provide a complete description of asymptotically stable population profiles in both dynamics. In the dual Lovász-Shapley replicator dynamic, for example, asymptotically stable populations for simple monotonic games correspond to their minimal blocking coalitions.

We suggest two alternatives to the Lovász-Shapley value for non-negatively weighted TU games, the dual Lovász-Shapley value and the Shapley2 value. Whereas the former is based on the Lovász extension operator for TU games, the latter two are based on extension operators that share certain economically plausible properties with the Lovász extension operator, the dual Lovász extension operator and the Shapley extension operator, respectively.

We suggest two economically plausible alternatives to the Lovász-Shapley value for non-negatively weighted TU games (Casajus and Wiese, 2017. Int. J. Game Theory 46 , 295-310), the dual Lovász-Shapley value and the Shapley² value. Whereas the former is based on the Lovász extension operator for TU games (Lovász, 1983. Mathematical Programming: The State of the Art, Springer, 235.256; Algaba et al., 2004. Theory Decis. 56, 229.238.), the latter two are based on the dual Lovász extension operator and the Shapley extension operator (Casajus and Kramm, 2021. Discrete Appl. Math. 294, 224.232), respectively.

Asymptotic stability in the dual Lovász-Shapley and the Shapley² replicator dynamics for TU games
(2022)

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.

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. It satisfies three properties: linearity in the game, homogeneity of extensions, and the extension property. The latter requires the indicator vector of any coalition to be assigned the worth generated by this coalition in the underlying TU game. Algaba et al. (2004) advocate the Lovász extension (Lovász, 1983) as a natural extension operator. We show that it is the unique extension operator that satisfies two desirable properties. Resources of players outside a carrier of the game do not affect the worth generated. For monotonic games, extensions are monotonic. Further, we discuss generalizations of the Lovász extension using CES production functions.

We introduce the concepts of the components' second-order productivities in cooperative games with transferable utility (TU games) with a coalition structure (CS games) and of the components' second-order payoffs for one-point solutions for CS games as generalizations of the players' second-order productivities in TU games and of the players' second-order payoffs for one-point solutions for TU games (Casajus, 2021, Discrete Appl. Math. 304, 212-219). The players' second-order productivities are conceptualized as second-order marginal contributions, that is, how one player affects another player's marginal contributions to coalitions containing neither of them by entering these coalitions. The players' second-order payoffs are conceptualized as the effect of one player leaving the game on the payoff of another player. Analogously, the components' second-order productivities are conceptualized as their second-order productivities in the game between components; the components' second-order payoffs are conceptualized as their second-order payoffs in the game between components. We show that the Owen value is the unique efficient one-point solution for CS games that reflects the players' and the components' second-order productivities in terms of their second-order payoffs.

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.

We introduce the concepts of the players’ second-order productivities in cooperative games with transferable utility (TU games) and of the players’ second-order payoffs for one-point solutions for TU games. Second-order productivities are conceptualized as second-order marginal contributions, that is, how one player affects another player’s marginal contributions to coalitions containing neither of them by entering these coalitions. Second-order payoffs are conceptualized as the effect of one player leaving the game on the payoff of another player. We show that the Shapley value is the unique efficient one-point solution for TU games that reflects the players’ second-order productivities in terms of their second-order payoffs.

We provide a concise characterization of the class of positively weighted Shapley values by three properties, two standard properties, efficiency and marginality, and a relaxation of the balanced contributions property called the weak balanced contributions property. Balanced contributions: the amount one player gains or loses when another player leaves the game equals the amount the latter player gains or loses when the former player leaves the game. Weakly balanced contributions: the direction (sign) of the change of one player’s payoff when another player leaves the game equals the direction (sign) of the change of the latter player’s payoff when the former player leaves the game. Given this characterization, the symmetric Shapley value can be “extracted”from the class of positively weighted Shapley values by either replacing the weak balanced contributions property with the standard symmetry property or by strengthening the former into the balanced contributions property.

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.

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. It satisfies three properties: linearity in the game, homogeneity of extensions, and the extension property. The latter requires the indicator vector of any coalition to be assigned the worth generated by this coalition in the underlying TU game. Algaba et al. (2004, Theor Decis 56, 229-238) advocate the Lovász extension (Lovász, 1983, Mathematical Programming: The State of the Art, Springer, 235-256) as a natural extension operator. We show that it is the unique extension operator that satisfies two desirable properties.
Resources of players outside a carrier of the TU game do not affect the worth generated. For monotonic TU games, extensions are monotonic. Further, we discuss generalizations of the Lovász extension using CES production functions.

We show that the Shapley value is the unique efficient one-point solution for cooperative games with transferable utility that reflects the players‘ second-order productivities in terms of their second-order payoffs. Second-order productivities are conceptualized as second-order marginal contributions, that is, how one player affects another player‘s marginal contributions to coalitions containing neither of them by entering these coalitions. Second-order payoffs are conceptualized
as the effect of one player leaving the game on the payoff of another player.

We derive population dynamics from finite cooperative games with transferable utility, where the players are interpreted as types of individuals. We show that 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. The average productivity of such a stable coalition must be greater than the average productivity of any proper sub- or supercoalition. In simple monotonic games, this means that exactly the minimal winning coalitions are stable. Possible applications are the analysis of the organizational structure of businesses or the population constitution of eusocial species.

We suggest a new component efficient solution for monotonic TU games with a coalition structure, the conditional Shapley value. Other than other such solutions, it satisfies the null player property. Nevertheless, it accounts for the players‘ outside options in productive components of coalition structures. For all monotonic games, there exist coalition structures that are stable under the conditional Shapley value. For voting games, the stability of coalition structures under
the conditional Shapley value supports Gamson‘s theory of coalition formation (Gamson, Am Sociol Rev 26, 1961, 373-382).

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.

We revisit Kalai and Samet's (Int J Game Theory 16, 1987, 205-222) first characterization of the class of weighted Shapley values. While keeping efficiency, additivity, and the null player property from the original characterization of the symmetric Shapley value, they replace symmetry with positivity and partnership consistency. The latter two properties, however, are neither implied by nor related to symmetry. We suggest relaxations of symmetry that together with efficiency, additivity, and the null player property characterize classes of weighted Shapley values. For example, weak sign symmetry requires the payoffs of mutually dependent players to have the same sign. Mutually dependent players are symmetric players whose marginal contributions to coalitions containing neither of them are zero.

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.

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.

We provide a concise characterization of the class of positively weighted Shapley values by three properties, two standard properties, efficiency and marginality, and a relaxation of the balanced contributions property called the weak balanced contributions property. Balanced contributions: the amount one player gains or loses when another player leaves the game equals the amount the latter player gains or loses when the former player leaves the game. Weakly balanced contributions: the direction (sign) of the change of one player's payoff when another player leaves the game equals the direction (sign) of the change of the latter player's payoff when the former player leaves the game. Since the (symmetric) Shapley value is characterized by efficiency and the balanced contributions property and satisfies marginality, we pinpoint position of the Shapley value within the class of positively weighted Shapley values to obeying the balanced contributions property versus just obeying the weak balanced contributions property.