TU Games Research Articles

Redistribution to the less productive: parallel characterizations of the egalitarian Shapley and consensus values

In cooperative game theory with transferable utilities (TU games), there are two well-established ways of redistributing Shapley value payoffs: using egalitarian Shapley values, and using consensus values. We present parallel characterizations of these classes of solutions. Together with the (weaker) axioms that characterize the original Shapley value, those that specify the redistribution methods characterize the two classes of values. For the class of egalitarian Shapley values, we focus on redistributions in one-person unanimity games from two perspectives: allowing the worth of coalitions to vary, while keeping the player set fixed; and allowing the player set to change, while keeping the worth of coalitions fixed. This class of values is characterized by efficiency, the balanced contributions property for equal contributors, weak covariance, a proportionately decreasing redistribution in one-person unanimity games, desirability, and null players in unanimity games. For the class of consensus values, we concentrate on redistributions in $$(n-1)$$ -person unanimity games from the same two perspectives. This class of values is characterized by efficiency, the balanced contributions property for equal contributors to social surplus, complement weak covariance, a proportionately decreasing redistribution in $$(n-1)$$ -person unanimity games, desirability, and null players in unanimity games.

Constrained welfare egalitarianism in surplus-sharing problems

The constrained equal welfare rule, fCE, distributes the surplus according to the uniform gains method and, hence, equalizes the welfare of the agents subsequent to the allocation process, subject to making nobody worse off. We show that fCE is the unique rule on the domain of surplus-sharing problems that satisfies efficiency, welfare monotonicity, path independence, and weak less first imposing an egalitarian bound for allowing positive payoffs to particular players. We provide an additional axiomatization employing consistency, a classical invariance property with respect to changes of the population. Finally, we show that the set of efficient solutions for cooperative TU games that support constrained welfare egalitarianism, i.e., distribute increments in the worth of the grand coalition according to fCE, is characterized by aggregate monotonicity and bounded pairwise fairness requiring that a player can only gain if his initial payoff does not exceed the initial payoff of any other player by the amount to be divided.

Extension operators for TU games and the Lovász extension

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.

The Two-Step Average Tree Value for Graph and Hypergraph Games

We introduce the two-step average tree value for transferable utility games with restricted cooperation represented by undirected communication graphs or hypergraphs. The solution can be considered as an alternative for both the average tree solution for graph games and the average tree value for hypergraph games. Instead of averaging players' marginal contributions corresponding to all admissible rooted spanning trees of the underlying (hyper)graph, which determines the average tree solution or value, we consider a two-step averaging procedure, in which in the first step for each player the average of players' marginal contributions corresponding to all admissible rooted spanning trees that have this player as the root is calculated, and in the second step the average over all players of all the payoff is obtained in the first step is computed. In general these two approaches lead to different solution concepts. When each component in the underlying communication structure is cycle-free, a linear cactus with cycles, or the complete graph, the two-step average tree value coincides with the average tree value. A comparative analysis of both solution concepts is done and an axiomatization of the the two-step average tree value on the subclass of TU games with semi-cycle-free hypergraph communication structure, which is more general than that given by a cycle-free hypergraph, is obtained.

On the characterizations of viable proposals

Sengupta and Sengupta (Int Econ Rev 35:347–359, 1994) consider a payoff configuration of a TU game as a viable proposal if it challenges each legitimate contender. Lauwers (Int Econ Rev 43:1369–1371, 2002) prove that the set of viable proposals is nonempty for every game. In the present paper, we prove that the set of viable proposals coincides with the coalition structure core if there exists an undominated proposal; otherwise, it coincides with the set of accessible proposals. This characterization result implies that a proposal is a viable proposal if and only if it is undominated or accessible. Moreover, we prove that the set of viable proposals includes the minimal dominant set, which is another nonempty extension of the coalition structure core introduced by Koczy and Lauwers (Games Econ Behav 61:277–298, 2007). In particular, we prove that the set of viable proposals of a cohesive game coincides with the minimal dominant set.

Consistency, anonymity, and the core on the domain of convex games

We show that neither Peleg’s nor Tadenuma’s well-known axiomatizations of the core by non-emptiness, individual rationality, super-additivity, and max consistency or complement consistency, respectively, hold when only convex rather than balanced TU games are considered, even if anonymity is required in addition. Moreover, we show that the core and its relative interior are the only two solutions that satisfy Peleg’s axioms together with anonymity and converse max consistency on the domain of convex games.

Similarities in axiomatizations: equal surplus division value and first-price auctions

We compare axiomatizations between a value for cooperative games with transferable utilities (TU games), and a rule for auctions. The equal surplus division value on the set of zero-monotonic TU games is characterized by the following: individual rationality, Pareto efficiency, and equal effect of players’ nullification on others. Meanwhile, first-price auctions, on the general preference domain, are characterized by individual rationality, envy-freeness, and weak equal effect of buyers’ nullification on others. Here, envy-freeness implies Pareto efficiency in the model of auctions. Given the agents’ general preferences in the auction model, the characteristic of a weak equal effect of buyers’ nullification on others weakens the requirement of equal effect of players’ nullification on others. Although the two models are different, the corresponding axioms in both models require conditions corresponding to each other. In particular, individual rationality requires voluntary participation of agents, Pareto efficiency (or its stronger axiom of envy-freeness in the model of auctions) requires outcomes with no waste, and (weak) equal effect of players’ (buyers’) nullification on others requires equal treatment of agents when an agent is nullified. Therefore, in terms of axiomatizations, we can similarly interpret the equal surplus division value and first-price auctions.

Shapley's Value and Its Axiomatization in Games with Prior Probabilities of Coalition Formation

A new model of TU game is introduced and studied. Here apart from the characteristic function, two additional functions are used: relations between the players and the probability of coalition formation. We suggest and study the sufficient properties of the probability function to transform it into a probability measure. The generalized Shapley's value is defined for a new class of games by analogy with the original Shapley approach. For the games under study, we give the axiomatization of value functions. It unambiguously characterizes the (generalized) Shapley's value.

Reactive and Semi-Reactive Bargaining Sets for Games with Restricted Cooperation

Generalizations of reactive and semi-reactive bargaining sets of TU games are defined for the case when objections and counter-objections are permitted not between singletons but between elements of a family of coalitions [Formula: see text] and can use coalitions from [Formula: see text]. Necessary and sufficient conditions on [Formula: see text], [Formula: see text] that ensure existence results for generalizations of the reactive bargaining set and of the semi-reactive barganing set at each TU game [Formula: see text] with nonnegative values are obtained. The existence conditions for the generalized reactive bargaining set do not coincide with existence conditions for the generalized kernel and coincide with conditions for the generalized semi-reactive bargaining set only if [Formula: see text] and [Formula: see text]. The conditions for the generalized semi-reactive bargaining set coincide with conditions for the generalized classical bargaining set that were described in the previous papers of the author. For monotonic [Formula: see text], the condition on [Formula: see text] for existence of the generalized semi-reactive bargaining sets on the class of games with nonnegative values is also necessary and sufficient on the class of simple games, but similar result for the generalized classical bargaining sets is proved only for [Formula: see text].

The Average Tree Value for Hypergraph Games

We consider transferable utility cooperative games (TU games) with limited cooperation introduced by hypergraph communication structure, the so-called hypergraph games. A hypergraph communication structure is given by a collection of coalitions, the hyperlinks of the hypergraph, for which it is assumed that only coalitions that are hyperlinks or connected unions of hyperlinks are able to cooperate and realize their worth. We introduce the average tree value for hypergraph games, which assigns to each player the average of the player's marginal contributions with respect to a particular collection of rooted spanning trees of the hypergraph. We also provide axiomatization of the average tree value for hypergraph games on the subclasses of cycle-free hypergraph games, hypertree games and cycle hypergraph games.

Individual Weighted Excess and Least Square Values

This work deals with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that lexicographically minimizing the individual weighted excesses of players gives the same minimal weighted excess for every player. Moreover, we show that the associated payoff vector is the corresponding least square value. Second, we show that minimizing the variance of the players' weighted excesses on the preimputation set, again yields the corresponding least square value. Third, we show that these results give rise to lower and upper bounds for the core payoff vectors and, using these bounds, we define the weighted super core as a polyhedron that contains the core. It turns out that the least square values can be seen as a center of this weighted super core, giving a third new characterization of the least square values. Finally, these lower and upper bounds for the core inspire us to introduce a new solution for cooperative TU games that has a strong similarity with the Shapley value.

Characterization of TU Games with Stable Cores by Nested Balancedness

A balanced transferable utility game (N, v) has a stable core if its core is externally stable, that is, if each imputation that is not in the core is dominated by some core element. Given two payoff allocations x and y, we say that x outvotes y via some coalition S of a feasible set if x dominates y via S and x allocates at least v(T) to any feasible T that is not contained in S. It turns out that outvoting is transitive and the set M of maximal elements with respect to outvoting coincides with the core if and only if the game has a stable core. By applying the duality theorem of linear programming twice, it is shown that M coincides with the core if and only if a certain nested balancedness condition holds. Thus, it can be checked in finitely many steps whether a balanced game has a stable core. We say that the game has a super-stable core if each payoff vector that allocates less than v(S) to some coalition S is dominated by some core element and prove that core super-stability is equivalent to vital extendability, requiring that each vital coalition is extendable.

On the Characterizations of Viable Proposals

Sengupta and Sengupta (1994) consider a payoff configuration of a TU game as a viable proposal if it challenges each legitimate contender. Lauwers (2002) prove that the set of viable proposals is nonempty for every game. In the present paper, we prove that the set of viable proposals coincides with the coalition structure core if there exists an undominated proposal; otherwise, it coincides with the set of accessible proposals. This characterization result implies that a proposal is a viable proposal if and only if it is undominated or accessible. Moreover, we prove that the set of viable proposals includes the minimal dominant set, which is another nonempty extension of the coalition structure core introduced by K\'{o}czy and Lauwers (2007). In particular, we prove that the set of viable proposals of a cohesive game coincides with the minimal dominant set.

Correction to: Relationally equal treatment of equals and affine combinations of values for TU games

Correction to: Relationally equal treatment of equals and affine combinations of values for TU games

A generalization of Peleg’s representation theorem on constant-sum weighted majority games

We propose a variant of the nucleolus associated with distorted satisfaction of each coalition in TU games. This solution is referred to as the $$\alpha $$-nucleolus in which $$\alpha $$ is a profile of distortion rates of satisfaction of all the coalitions. We apply the $$\alpha $$-nucleolus to constant-sum weighted majority games. We show that under assumptions of distortions of satisfaction of winning coalitions the $$\alpha $$-nucleolus is the unique normalized homogeneous representation of constant-sum weighted majority games which assigns a zero to each null player. As corollary of this result, we derive the well-known Peleg’s representation theorem.

Some properties of transportation network cooperative games

AbstractA basic question in network analysis concerns the quantification of the importance of each node in terms of network connectivity. To this end, a possible approach consists in using cooperative game theory tools to define a measure of node centrality. In this paper, given a transportation network, a cooperative game model with transferable utility (TU game) is considered. The nodes of the network represent the players in such a game, and the Shapley values of the nodes are used to measure centrality. The model, called Transportation Network cooperative (TNc) game, integrates within the utility function of the TU game the network topology and the demand. Properties of TNc games and their associated utility functions are investigated. Numerical results are reported, to get insights into the obtained properties.

The Alpha-Constant-Sum Games

Given any alpha in [0,1], an alpha-constant-sum game on a finite set of players, N, is a function that assigns a real number to any coalition S (being a subset of the player set N), such that the sum of the worth of the coalition S and the worth of its complementary coalition N\S is alpha times of the worth of the grand coalition N. This class contains the constant-sum games of Khmelnitskaya (2003) (for alpha = 1) and games of threats of Kohlberg and Neyman (2018) (for alpha = 0) as special cases. An alpha-constant-sum game may not be a classical TU cooperative game as it may fail to satisfy the condition that the worth of the empty set is 0, except when alpha = 1. In this paper, we will build a value theory for the class of alpha-constant-sum games, and mainly introduce the alpha-quasi-Shapley value. We characterize this value by classical axiomatizations for TU games. We show that axiomatizations of the equal division value do not work on these classes of alpha-constant-sum games.

Relationally equal treatment of equals and affine combinations of values for TU games

We axiomatize the set of affine combinations between the Shapley value, the equal surplus division value, and the equal division value in cooperative games with transferable utilities. The set is characterized by efficiency, linearity, the balanced contributions property for equal contributors and outsiders, and the differential null player out property. The balanced contributions property for equal contributors and outsiders requires the balance of contributions between two players who contribute the same amount to the grand coalition and whose singleton coalitions earn the same worth. The differential null player out property requires that an elimination of a null player affects the other players identically. These two relational axioms are obtained by investigating Myerson’s (Int J Game Theory 9:169–182, 1980) balanced contributions property and Derks and Haller’s (Int Game Theory Rev 1:301–314, 1999) null player out property, respectively, from the perspective of a principle of Aristotle’s distributive justice, whereby “equals should be treated equally”.

On the Coalitional Rationality and the Egalitarian Nonseparable Contribution

In earlier works we introduced the Inverse Problem, relative to the Shapley Value, then relative to Semivalues. In the explicit representation of the Inverse Set, the solution set of the Inverse Problem, we built a family of games, called the almost null family, in which we determined more recently a game where the Shapley Value and the Egalitarian Allocations are colalitional rational. The Egalitarian Nonseparable Contribution is another value for cooperative transferable utilities games (TU games), showing how to allocate fairly the win of the grand coalition, in case that this has been formed. In the present paper, we solve the similar problem for this new value: given a nonnegative vector representing the Egalitarian Nonseparable Contribution of a TU game, find out a game in which the Egalitarian Nonseparable Contribution is kept the same, but it is colalitional rational. The new game will belong to the family of almost null games in the Inverse Set, relative to the Shapley Value, and it is proved that the threshold of coalitional rationality will be higher than the one for the Shapley Value. The needed previous results are shown in the introduction, the second section is devoted to the main results, while in the last section are discussed remarks and connected problems. Some numerical examples are illustrating the procedure of finding the new game.

Consistency, Anonymity, and the Core on the Domain of Convex Games

We show that neither Peleg's nor Tadenuma's well-known axiomatizations of the core by non-emptiness, individual rationality, super-additivity, and max consistency or complement consistency, respectively, hold when only convex rather than balanced TU games are considered, even if anonymity is required in addition. Moreover, we show that the core and its relative interior are only two solutions that satisfy Peleg's axioms together with anonymity and converse max consistency on the domain of convex games.