A solidarity value forn-person transferable utility games

In this paper, we introduce axiomatically a new value for cooperative TU games satisfying the efficiency, additivity, and symmetry axioms of Shapley (1953) and some new postulate connected with the average marginal contributions of the members of coalitions which can form. Our solution is referred to as the solidarity value. The reason is that its interpretation can be based on the assumption that if a coalition, sayS, forms, then the players who contribute toS more than the average marginal contribution of a member ofS support in some sense their weaker partners inS. Sometimes, it happens that the solidarity value belongs to the core of a game while the Shapley value does not.This research was supported by the KBN Grant 664/2/91 No. 211589101.     

The nucleolus and kernel of veto-rich transferable utility games

The process of computing the nucleolus of arbitrary transferable utility games is notoriously hard. A number of papers have appeared in which the nucleolus is computed by an algorithm in which either one or a huge number of huge linear programs have to be solved. We show that on the class of veto-rich games, the nucleolus is the unique kernel element. Veto-rich games are games in which one of the players is needed by coalitions in order to obtain a non-zero payoff. We then provide a fast algorithm which does not use linear programming techniques to compute the nucleolus of these games. Furthermore, we provide a few examples of economic situations which belong to the class of veto-rich games and which are treated in the literature.     

Polytopes of special classes of the balanced transferable utility games

Under study are the polytopes of (0, 1)-normalized convex and 1-convex (dual simplex) n-person TU-games and monotonic big boss games.We solve the characterization problems of the extreme points of the polytopes of 1-convex games, symmetric convex games, and big boss games symmetric with respect to the coalition of powerless agents. For the remaining polytopes, some subsets of extreme points are described.     

Consistent extensions and subsolutions of the core of multi-choice NTU games

We extend the reduced games introduced by Davis and Maschler (Naval Res Log Q 12:223–259, 1965) and Moulin (J Econ Theory 36:120–148, 1985) to multi-choice non-transferable utility games and define two related properties of consistency. We also show that the core proposed by Hwang and Li (Math Methods Oper Res 61:33–40, 2005) violates these two consistency properties. In order to investigate how seriously it violates these two consistency properties, we provide consistent extensions and consistent subsolutions of the core.     

Consistent extensions and subsolutions of the core for the multichoice transferable-utility games

Abstract

We extend the reduced games introduced by Davis and Maschler (Naval Res. Log. Q. 12:223–259, 1965) and Moulin (J. Econ. Theory. 36:120–148, 1985) to multichoice transferable-utility games. First, we provide an example to illustrate that the core proposed by van den Nouweland et al. (Math Methods Oper. Res. 41:289–311, 1995) violates related consistency properties. Further, we propose the minimal consistent extensions of the core and the maximal consistent subsolutions of the core. We also provide an axiomatization based on related consistency properties and its converse.     

On the core of routing games

A repairman makes a round-trip along a set of customers. He starts in his home location, visits each customer exactly once, and returns home. The cost of his trip has to be shared by the customers. A cooperative cost game, calledrouting game, is associated with this allocation problem, and anO(n ²) algorithm is given which computes a core element of a routing game if the core is non-empty. The non-emptiness of the core depends on the tour which is traversed by the repairman. Several procedures are given to construct tours which guarantee the non-emptiness of the core.     

On the core of network synthesis games

We use polynomial formulations to show that several rational and discrete network synthesis games, including the minimum cost spanning tree game, satisfy the assumptions of Owen's linear production game model. We also discuss computational issues related to finding and recognizing core points for these classes of games.     

On the core of linear production games

An economic production game is treated, in which players pool resources to produce finished goods which can be sold at a given market price. The production process is linear, so that the characteristic function can be obtained by solving linear programs. Duality theory of linear programming is used to obtain equilibrium price vectors and to prove the non-emptiness of the core. Under replication, it is shown that, in the non-degenerate case, the core converges finitely to the set of competitive equilibria; a counter-example shows that this is not true in the degenerate case (i.e., only convergence in the limit can be guaranteed).     

On the core of cooperative queueing games

We consider a class of cooperative games for managing several canonical queueing systems. When cooperating parties invest optimally in common capacity or choose the optimal amount of demand to serve, cooperation leads to “single-attribute” games whose characteristic function is embedded in a one-dimensional function. We show that when and only when the latter function is elastic will all embedded games have a non-empty core, and the core contains a population monotonic allocation. We present sufficient conditions for this property to be satisfied. Our analysis reveals that in most Erlang B and Erlang C queueing systems, the games under our consideration have a non-empty core, but there are exceptions, which we illustrate through a counterexample.     

On the core of information graph games

This paper considers a subclass of minimum cost spanning tree games, called information graph games. It is proved that the core of these games can be described by a set of at most 2n — 1 linear constraints, wheren is the number of players. Furthermore, it is proved that each information graph game has an associated concave information graph game, which has the same core as the original game. Consequently, the set of extreme core allocations of an information graph game is characterized as the set of marginal allocation vectors of its associated concave game. Finally, it is proved that all extreme core allocations of an information graph game are marginal allocation vectors of the game itself, though not all marginal allocation vectors need to be core allocations.     

On non-transferable utility games with coalition structure

We introduce a solution function for Non-transferable Utility (NTU) games when prior coalition structure is given. This solution function generalizes both the Harsanyi solution function forNTU games and the Owen solution forTU games with coalition structure.I would like to thank Sergiu Hart, Bezalel Peleg and Shmuel Zamir for some conversations and constructive remarks on an earlier version of this paper. Part of this research was supported by the Sonderforschungsbereich 303 in the university of Bonn.     

On the core of ordered submodular cost games

A general ordertheoretic linear programming model for the study of matroid-type greedy algorithms is introduced. The primal restrictions are given by so-called weakly increasing submodular functions on antichains. The LP-dual is solved by a Monge-type greedy algorithm. The model offers a direct combinatorial explanation for many integrality results in discrete optimization. In particular, the submodular intersection theorem of Edmonds and Giles is seen to extend to the case with a rooted forest as underlying structure. The core of associated polyhedra is introduced and applications to the existence of the core in cooperative game theory are discussed. Received: November 2, 1995 / Accepted: September 15, 1999?Published online February 23, 2000     

On the core and nucleolus of directed acyclic graph games

We introduce directed acyclic graph (DAG) games, a generalization of standard tree games, to study cost sharing on networks. This structure has not been previously analyzed from a cooperative game theoretic perspective. Every monotonic and subadditive cost game—including monotonic minimum cost spanning tree games—can be modeled as a DAG-game. We provide an efficiently verifiable condition satisfied by a large class of directed acyclic graphs that is sufficient for the balancedness of the associated DAG-game. We introduce a network canonization process and prove various structural results for the core of canonized DAG-games. In particular, we characterize classes of coalitions that have a constant payoff in the core. In addition, we identify a subset of the coalitions that is sufficient to determine the core. This result also guarantees that the nucleolus can be found in polynomial time for a large class of DAG-games.     

On the core of multiple longest traveling salesman games

In this paper we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except to his home location, in such an order that maximizes the total reward. First it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions. Secondly it is shown that MLTS games with five or less players have a nonempty core. However, a six player MLTS game may have an empty core. For the special instance in which the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.     

On bargaining position descriptions in non-transferable utility games

We present a generalization to the Harsanyi solution for non-transferable utility (NTU) games based on non-symmetry among the players. Our notion of non-symmetry is presented by a configuration of weights which correspond to players' relative bargaining power in various coalitions. We show not only that our solution (i.e., the bargaining position solution) generalizes the Harsanyi solution, (and thus also the Shapley value), but also that almost all the non-symmetric generalizations of the Shapley value for transferable utility games known in the literature are in fact bargaining position solutions. We also show that the non-symmetric Nash solution for the bargaining problem is also a special case of our general solution. We use our general representation of non-symmetry to make a detailed comparison of all the recent extensions of the Shapley value using both a direct and an axiomatic approach.     

On cooperative games with large monotonic core

Cooperative games with large core were introduced by Sharkey (Int. J. Game Theory 11:175–182, 1982), and the concept of Population Monotonic Allocation Scheme was defined by Sprumont (Games Econ. Behav. 2:378–394, 1990). Inspired by these two concepts, Moulin (Int. J. Game Theory 19:219–232, 1990) introduced the notion of large monotonic core giving a characterization for three-player games. In this paper we prove that all games with large monotonic core are convex. We give an effective criterion to determine whether a game has a large monotonic core and, as a consequence, we obtain a characterization for the four-player case.     

Stable outcomes of the roommate game with transferable utility

We consider the TU version of Gale and Shapley's roommate game. We find several results that are analogous to known results for the NTU game, such as a characterization of stable outcomes by forbidden minors, a characterization of the extreme points of the core, and a median property of stable outcomes. The TU roommate game is a special case of the TU partitioning game of Kaneko and Wooders. Bondareva and Shapley's balancedness condition for the core of such games is the starting point for our forbidden minors approach. Received: April 1999/Revised version: November 2000