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

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

Weighted Shapley levels values

This paper presents a collection of four different classes of weighted Shapley levels values. All classes contain generalisations of the weighted Shapley values to cooperative games with a level structure. The first class is an upgrade of the weighted Shapley levels value in Gómez-Rúa and Vidal-Puga (2011), who use the size of components as weights. The following classes contain payoff vectors from the Harsanyi set. Hence they satisfy the dummy axiom, in contrary to the values in the first class in general. The second class contains extensions of the McLean weighted coalition structure values (Dragan, 1992; Levy and McLean, 1989; McLean, 1991). The first two classes satisfy the level game property (the payoff to all players of a component sum up to the payoff to the component in a game where components are the players) and the last two classes meet a null player out property. As a special case, the first three classes include the Shapley levels value and the last class contains a new extension of the Shapley value

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

Axiomatizations of the proportional Shapley value

We provide new axiomatic characterizations of the proportional Shapley value, a weighted TU-value with the worths of the singletons as weights. The presented characterizations are proportional counterparts to the famous characterizations of the Shapley value by Shapley (1953b) and Young (1985a). We introduce two new axioms, called proportionality and player splitting respectively. Each of them makes a main difference between the proportional Shapley value and the Shapley value. If the stand-alone worths are plausible weights, the proportional Shapley value is a convincing alternative to the Shapley value, for example in cost allocation. Especially the player splitting property, which states that the players’ payoffs do not change if another player splits into two new players who have the same impact to the game as the original player, justifies the use of the proportional Shapley value in many economic situations

Player splitting, players merging, the Shapley set value and the Harsanyi set value

We discuss a value, proposed in the context of cost allocation by Shapley (1981) and Dehez (2011) and in general by Radzik (2012). This value, we call it Shapley set value, covers the weighted Shapley values all at once. It is defined on weighted TU-games in the form of two constituent parts, a weight system and a classical TU-game, where the weights and the coalition function may vary at the same time. In addition, similar to the Shapley set value, we introduce the Harsanyi set value. It captures all TU-values from the Harsanyi set, called Harsanyi payoffs. A player splitting and a players merging property enable new axiomatizations. Examples recommend both solution concepts for profit distribution and cost allocation

Parallel axiomatizations of weighted and multiweighted Shapley values, random order values, and the Harsanyi set

We present new axiomatic characterizations of five classes of TU-values, the classes of the weighted, positively weighted, and multiweighted Shapley values, random order values, and the Harsanyi set. The axiomatizations are given in parallel, i.e., they differ only in one axiom. In conjunction with marginality, a new property, called coalitional differential dependence, is the key that allows us to dispense with additivity. In addition, we propose new axiomatizations of the above five classes, in which, in part new, different versions of monotonicity, associated with the strong monotonicity in Young (1985), are decisive