Weighted Shapley hierarchy levels values

In this paper we present a new class of values for cooperative games with level structure. We use a multi-step proceeding, suggested first in Owen (1977), applied to the weighted Shapley values. Our first axiomatization is an generalisation of the axiomatization given in Gómez-Rúa and Vidal-Puga (2011), itselves an extension of a special case of an axiomatization given in Myerson (1980) and Hart and Mas-Colell (1989) respectively by efficiency and weighted balanced contributions. The second axiomatization is completely new and extends the axiomatization of the weighted Shapley values introduced in Hart and Mas-Colell (1989) by weighted standardness for two player games and consistency. As a corollary we obtain a new axiomatization of the Shapley levels value

Weighted Shapley hierarchy levels values

In this paper we present a new class of values for cooperative games with level structure. We use a multi-step proceeding, suggested first in Owen (1977), applied to the weighted Shapley values. Our first axiomatization is an generalisation of the axiomatization given in Gómez-Rúa and Vidal-Puga (2011), itselves an extension of a special case of an axiomatization given in Myerson (1980) and Hart and Mas-Colell (1989) respectively by efficiency and weighted balanced contributions. The second axiomatization is completely new and extends the axiomatization of the weighted Shapley values introduced in Hart and Mas-Colell (1989) by weighted standardness for two player games and consistency. As a corollary we obtain a new axiomatization of the Shapley levels value

Weighted Shapley hierarchy levels values

In this paper we present a new class of values for cooperative games with level structure. We use a multi-step proceeding, suggested first in Owen (1977), applied to the weighted Shapley values. Our first axiomatization is an generalisation of the axiomatization given in Gómez-Rúa and Vidal-Puga (2011), itselves an extension of a special case of an axiomatization given in Myerson (1980) and Hart and Mas-Colell (1989) respectively by efficiency and weighted balanced contributions. The second axiomatization is completely new and extends the axiomatization of the weighted Shapley values introduced in Hart and Mas-Colell (1989) by weighted standardness for two player games and consistency. As a corollary we obtain a new axiomatization of the Shapley levels value

Two classes of weighted values for coalition structures with extensions to level structures

In this paper we introduce two new classes of weighted values for coalition structures with related extensions to level structures. The values of both classes coincide on given player sets with Harsanyi payoffs and match therefore adapted standard axioms for TU-values which are satisfied by these values. Characterizing elements of the values from the new classes are a new weighted proportionality within components property and a null player out property, but on different reduced games for each class. The values from the first class, we call them weighted Shapley alliance coalition structure values (weighted Shapley alliance levels values), satisfy the null player out property on usual reduced games. By contrast, the values from the second class, named as weighted Shapley collaboration coalition structure values (weighted Shapley collaboration levels values) have this property on new reduced games where a component decomposes in the components of the next lower level if one player of this component is removed from the game. The first class contains as a special case the Owen value (Shapley levels value) and the second class includes a new extension of the Shapley value to coalition structures (level structures) as a special case

Values for level structures with polynomial-time algorithms, relevant coalition functions, and general considerations

Exponential runtimes of algorithms for TU-values like the Shapley value are one of the biggest obstacles in the practical application of otherwise axiomatically convincing solution concepts of cooperative game theory. We discuss how the hierarchical structure of a level structure improves the runtimes compared to an unstructured set of players. As examples, we examine the Shapley levels value, the nested Shapley levels value, and, as a new LS-value, the nested Owen levels value. Polynomial-time algorithms for these values (under ordinary conditions) are provided. Furthermore, we introduce relevant coalition functions where all coalitions which are not relevant for the payoff calculation have a Harsanyi dividend of zero. By these coalition functions, our results shed new light on the computation of values of the Harsanyi set and many values from extensions of this set

Values for level structures with polynomial-time algorithms, relevant coalition functions, and general considerations

Exponential runtimes of algorithms for TU-values like the Shapley value are one of the biggest obstacles in the practical application of otherwise axiomatically convincing solution concepts of cooperative game theory. We discuss how the hierarchical structure of a level structure improves the runtimes compared to an unstructured set of players. As examples, we examine the Shapley levels value, the nested Shapley levels value, and, as a new LS-value, the nested Owen levels value. Polynomial-time algorithms for these values (under ordinary conditions) are provided. Furthermore, we introduce relevant coalition functions where all coalitions which are not relevant for the payoff calculation have a Harsanyi dividend of zero. By these coalition functions, our results shed new light on the computation of values of the Harsanyi set and many values from extensions of this set

Two classes of weighted values for coalition structures with extensions to level structures

In this paper we introduce two new classes of weighted values for coalition structures with related extensions to level structures. The values of both classes coincide on given player sets with Harsanyi payoffs and match therefore adapted standard axioms for TU-values which are satisfied by these values. Characterizing elements of the values from the new classes are a new weighted proportionality within components property and a null player out property, but on different reduced games for each class. The values from the first class, we call them weighted Shapley alliance coalition structure values (weighted Shapley alliance levels values), satisfy the null player out property on usual reduced games. By contrast, the values from the second class, named as weighted Shapley collaboration coalition structure values (weighted Shapley collaboration levels values) have this property on new reduced games where a component decomposes in the components of the next lower level if one player of this component is removed from the game. The first class contains as a special case the Owen value (Shapley levels value) and the second class includes a new extension of the Shapley value to coalition structures (level structures) as a special case

Harsanyi support levels payoffs and weighted Shapley support levels values

This paper introduces a new class of values for level structures. The new values, called Harsanyi support levels payoffs, extend the Harsanyi payoffs from the Harsanyi set to level structures and contain the Shapley levels value (Winter, 1989) as a special case. We also look at extensions of the weighted Shapley values to level structures. These values, we call them weighted Shapley support levels values, constitute a subset of the class of Harsanyi support levels payoffs and coincide on a level structure with only two levels with a class of weighted coalition structure values, already mentioned in Levy and McLean (1989) and discussed in McLean (1991). Axiomatizations of the studied classes are provided for both exogenously and endogenously given weights

Harsanyi support levels solutions

We introduce a new class of values with transferable utility for level structures. In these hierarchical structures, each level corresponds to a partition of the player set, which becomes increasingly coarse from the trivial partition containing only singletons to the partition containing only the grand coalition. The new values, called Harsanyi support levels solutions, extend the Harsanyi solutions to level structures. As an important subset of these values, we present the class of weighted Shapley support levels values as a further result. The values from this class extend the weighted Shapley values to level structures and contain the Shapley levels value as a special case. Axiomatizations of the studied classes are provided