Transferable Utility Cooperative Games Research Articles

Computational Complexity of a Solution for Directed Graph Cooperative Games

Khmelnitskaya et al. have recently proposed the average covering tree value as a new solution concept for cooperative transferable utility games with directed graph structure. The average covering tree value is defined as the average of marginal contribution vectors corresponding to the specific set of rooted trees, and coincides with the Shapley value when the game has complete communication structure. In this paper, we discuss the computational complexity of the average covering tree value. We show that computation of the average covering tree value is #P-complete even if the characteristic function of the game is {0,1}-valued. We prove this by a reduction from counting the number of all linear extensions of a partial order, which has been shown by Brightwell et al. to be a #P-complete counting problem. The implication of this result is that an efficient algorithm to calculate the average covering tree value is unlikely to exist.

Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with O(4n) constraints whose coefficients belong to {−1,0,1}. We discuss the need of having all these constraints and empirically prove that they can be reduced to O(kmax2n), where kmax is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 players.

Cooperation in an HMMS-type supply chain: A management application of cooperative game theory

We apply cooperative game theory concepts to analyze a Holt-Modigliani-Muth-Simon (HMMS) supply chain. The bullwhip effect in a two-stage supply chain (supplier-manufacturer) in the framework of the HMMS-model with quadratic cost functions is considered. It is assumed that both firms minimize their relevant costs, and two cases are examined: the supplier and the manufacturer minimize their relevant costs in a decentralized and in a centralized (cooperative) way. The question of how to share the savings of the decreased bullwhip effect in the centralized (cooperative) model is answered by the weighted Shapley value, by a transferable utility cooperative game theory tool, where the weights are for the exogenously given “bargaining powers” of the participants of the supply chain.

Aggregate monotonic stable single-valued solutions for cooperative games

This article considers single-valued solutions of transferable utility cooperative games that satisfy core selection and aggregate monotonicity, defined either on the set of all games, G N, or on the set of essential games, E N (those with a non-empty imputation set). The main result is that for an arbitrary set of players, core selection and aggregate monotonicity are compatible with individual rationality, the dummy player property and symmetry for single-valued solutions defined on both G N and E N. This result solves an open question in the literature (see for example Young et al. Water Resour Res 18:463–475, 1982).

Compatibility in Transferable Utility Cooperative Games

In this paper, we introduce the concept of the compatibility of two transferable utility cooperative games. Two games are compatible if there exists at least one allocation which belongs to the core of both games. We look at the class of nonnegative convex games where the value of the grand coalition sums to one. We conclude that for this class of games, two games are compatible if the sum of the values of the two games for all splits of the grand coalition into two is at most one. Here we use this concept of compatibility to justify why coalitions may fall apart when they are faced with two or more issues. We also do an application of this concept of compatibility to the case of EU member countries who are in ``debt crises.'' We show that the future membership of a country in the EU depends on the ``degree of importance of participation'' in a coalition. We include numerical examples as well as graphical illustrations of compatible and incompatible games.

USING THE ASPIRATION CORE TO PREDICT COALITION FORMATION

This work uses the defining principles of the core solution concept to determine not only payoffs but also coalition formation. Given a cooperative transferable utility (TU) game, we propose two noncooperative procedures that in equilibrium deliver a natural and nonempty core extension, the aspiration core, together with the supporting coalitions it implies. As expected, if the cooperative game is balanced, the grand coalition forms. However, if the core is empty, other coalitions arise. Following the aspiration literature, not only partitions but also overlapping coalition configurations are allowed. Our procedures interpret this fact in different ways. The first game allows players to participate with a fraction of their time in more than one coalition, while the second assigns probabilities to the formation of potentially overlapping coalitions. We use the strong Nash and subgame perfect Nash equilibrium concepts.

Sequentially two-leveled egalitarianism for TU games: Characterization and application

A new linear value for cooperative transferable utility games is introduced. The recursive definition of the new value for an n-person game involves a sequential process performed at n−1 stages, applying the value to subgames with a certain size k,1⩽k<n, combining with the rule of two-leveled egalitarianism (additive normalization) in order to guarantee the efficiency property for the new value, sequentially two-leveled egalitarianism, shortly S2EG value, applied to subgames of size k+1. The new value will be characterized in various ways. The S2EG value differs from the Shapley value since, besides efficiency, linearity, and symmetry, it verifies an additional property with respect to so-called scale-dummy player (replacing dummy player property). Consequently, the S2EG value of a game may be determined as the solidarity value of the per-capita game (incorporating the proportional rule due to different levels of efficiency). Various potential representations of the new value are established. In the application to a land corn production economy, it yields allocations, in which the landlord’s interest coincides with striving for a maximum production level. For economies with the linear production function, not only the unique landlord but also all the workers have incentives to increase the scale of the economy.

Inequality decomposition values: the trade-off between marginality and efficiency

This paper presents a general procedure for decomposing income inequality measures by income sources. The methods of decomposition proposed are based on the Shapley value and extensions of the Shapley value of transferable utility cooperative games. In particular, we find that Owen’s value can find an interesting application in this context.We show that the axiomatization by the potential of Hart and Mas-Colell remains valid in the presence of the domain restriction of inequality indices. We also examine the properties of these decomposition rules and perform a comparison with Shorrocks’ decomposition rule properties.

Parameterized fairness axioms on cycle-free graph games

We study cooperative transferable utility games with a communication structure represented by an undirected graph, i.e., a group of players can cooperate only if they are connected on the graph. This type of games is called graph games and the best-known solution for them is the Myerson value, which is characterized by the component efficiency axiom and the fairness axiom. Recently the average tree solution has been proposed on cycle-free graph games, and shown to be characterized by the component efficiency axiom and the component fairness axiom. We propose $${\epsilon}$$ -parameterized fairness axiom on cycle-free graph games that incorporates the preceding fairness axioms, and show the existence and the uniqueness of the solution. We then discuss a relationship between the existing and our proposed solutions by a numerical example.

Cooperative games and cost allocation problems

The objective of this paper is to provide a general view of the literature of applications of transferable utility cooperative games to cost allocation problems. This literature is so large that we concentrate on some relevant contributions in three specific areas: transportation, natural resources and power industry. We stress those applications dealing with costs and with problems arisen outside the academic world.

Characterizing the Shapley value in fixed-route traveling salesman problems with appointments

Starting from her home, a service provider visits several customers, following a predetermined route, and returns home after all customers are visited. The problem is to find a fair allocation of the total cost of this tour among the customers served. A transferable-utility cooperative game can be associated with this cost allocation problem. We introduce a new class of games, which we refer as the fixed-route traveling salesman games with appointments. We characterize the Shapley value in this class using a property which requires that sponsors do not benefit from mergers, or splitting into a set of sponsors.

Algebraic duality theorems for infinite LP problems

In this paper, we consider a primal–dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players.

Cooperation in Service Systems

We consider a number of servers that may improve the efficiency of the system by pooling their service capacities to serve the union of the individual streams of customers. This economies-of-scope phenomenon is due to the reduction in the steady-state mean total number of customers in the system. The question we pose is how the servers should split among themselves the cost of the pooled system. When the individual incoming streams of customers form Poisson processes and individual service times are exponential, we define a transferable utility cooperative game in which the cost of a coalition is the mean number of customers (or jobs) in the pooled system. We show that, despite the characteristic function is neither monotone nor concave, the game and its subgames possess nonempty cores. In other words, for any subset of servers there exist cost-sharing allocations under which no partial subset can take advantage by breaking away and forming a separate coalition. We give an explicit expression for all (infinitely many) nonnegative core cost allocations of this game. Finally, we show that, except for the case where all individual servers have the same cost, there exist infinitely many core allocations with negative entries, and we show how to construct a convex subset of the core where at least one server is being paid to join the grand coalition.

Associated consistency and values for TU games

In the framework of the solution theory for cooperative transferable utility games, Hamiache axiomatized the well-known Shapley value as the unique one-point solution verifying the inessential game property, continuity, and associated consistency. The purpose of this paper is to extend Hamiache’s axiomatization to the class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. For this enlarged class of values, explicit relationships to the Shapley value are exploited in order to axiomatize such values with reference to a slightly adapted inessential game property, continuity, and a similar associated consistency. The latter axiom requires that the solutions of the initial game and its associated game (with the same player set, but a different characteristic function) coincide.

Harsanyi power solutions for graph-restricted games

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions which distribute dividends such that the dividend shares of players in a coalition are based on power measures for nodes in corresponding communication graphs. We provide axiomatic characterizations of the Harsanyi power solutions on the class of cycle-free graph games and on the class of all graph games. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games and equals the position value on the class of cycle-free graph games. The Myerson value is the Harsanyi power solution that is based on the equal power measure. Finally, various applications are discussed.

Axiomatizations of Banzhaf permission values for games with a permission structure

In games with a permission structure it is assumed that players in a cooperative transferable utility game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. We provide axiomatic characterizations of Banzhaf permission values being solutions that are obtained by applying the Banzhaf value to modified TU-games. In these characterizations we use power- and player split neutrality properties. These properties state that splitting a player’s authority and/or contribution over two players does not change the sum of their payoffs.

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.

The aggregate-monotonic core

We introduce the aggregate-monotonic core as the set of allocations of a transferable utility cooperative game attainable by single-valued solutions that satisfy core-selection and aggregate-monotonicity. We provide a necessary and sufficient condition for the coincidence of the core and the aggregate-monotonic core. Finally, we introduce upper and lower aggregate-monotonicity for set-valued solutions, and characterize the aggregate-monotonic core using core-selection and upper and lower aggregate-monotonicity.

Implementing cooperative solution concepts: a generalized bidding approach

This paper provides a framework for implementing and comparing several solution concepts for transferable utility cooperative games. We construct bidding mechanisms where players bid for the role of the proposer. The mechanisms differ in the power awarded to the proposer. The Shapley, consensus and equal surplus values are implemented in subgame perfect equilibrium outcomes as power shifts away from the proposer to the rest of the players. Moreover, an alternative informational structure where these solution concepts can be implemented without imposing any conditions of the transferable utility game is discussed as well.

AVERAGE TREE SOLUTION AND SUBCORE FOR ACYCLIC GRAPH GAMES

In this paper we consider cooperative transferable utility games with limited communication structure, called graph games. Agents are able to cooperate only if they can communicate directly or indirectly with each other. For the class of acyclic graph games the average tree solution has recently been proposed. It was proven that the average tree solution is a core element if the game exhibits super-additivity. We show that the condition of super-additivity can be relaxed to a weaker condition, which admits for a natural interpretation. Moreover, we introduce the concept of subcore, which is a subset of the core, always contains the average tree solution, and therefore is a non-empty refinement of the core.