The extreme core allocations of the assignment game

Although assignment games are hardly ever convex, in this paper a characterization of their set or extreme points of the core is provided, which is also valid for the class of convex games. For each ordering in the player set, a payoff vector is defined where each player receives his marginal contribution to a certain reduced game played by his predecessors. We prove that the whole set of reduced marginal worth vectors, which for convex games coincide with the usual marginal worth vectors, is the set of extreme points of the core of the assignment game.convex games, assignment game, core

The assignment game: core bounds for mixe-pair coalitions

In the assignment game framework, we try to identify those assignment matrices in which no entry can be increased without changing the core of the game. These games will be called buyerseller exact games and satisfy the condition that each mixedpair coalition attains the corresponding matrix entry in the core of the game. For a given assignment game, a unique buyerseller exact assignment game with the same core is proved to exist. In order to identify this matrix and to provide a characterization of those assignment games which are buyerseller exact in terms of the assignment matrix, attainable upper and lower core bounds for the mixedpair coalitions are found. As a consequence, an open question posed in Quint (1991) regarding a canonical representation of a 45olattice by means of the core of an assignment game can now be answered.core, exact games, assignment game

A geometric characterization of the nucleolus of the assignment game

Maschler et al. (1979) provide a geometrical characterization for the intersection of the kernel and the core of a coalitional game, showing that those allocations that lie in both sets are always the midpoint of certain bargaining range between each pair of players. In the case of the assignment game, this means that the kernel can be determined as those core allocations where the maximum amount, that can be transferred without getting outside the core, from one agent to his/her optimally matched partner equals the maximum amount that he/she can receive from this partner, also remaining inside the core (Rochford, 1984). We now prove that the nucleolus of the assignment game can be characterized by requiring this bisection property be satisfied not only for optimally matched pairs but also for optimally matched coalitions.assignment game, core, kernel, nucleolus

Assignment markets with the same core

In the framework of bilateral assignment games, we study the set of matrices associated with assignment markets with the same core. We state conditions on matrix entries that ensure that the related assignment games have the same core. We prove that the set of matrices leading to the same core form a join-semilattice with a nite number of minimal elements and a unique maximum. We provide a characterization of the minimal elements. A sucient condition under which the join-semilattice reduces to a lattice is also given.core, semilattice, assignment game

A geometric chracterization of the nucleolus of the assignment game

core, assignment games, nucleolus, cooperative games, kernel

The Lorenz-maximal core allocations and the kernel in some classes of assignment games

core, assignment game, lorenz domination, kernel

Operations Research Games: A Survey

This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved.Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also face an additional allocation problem in how to distribute these joint costs back to the individual players.This interplay between optimisation and allocation is the main subject of the area of operations research games.It is surveyed on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.cooperative games;operational research

Characterization of the walrasian equilibria of the assignment model

We study the assignment model where a collection of indivisible goods are sold to a set of buyers who want to buy at most one good. We characterize the extreme and interior points of the set of Walrasian equilibrium price vectors for this model. Our characterizations are in terms of demand sets of buyers. Using these characterizations, we also give a unique characterization of the minimum and the maximum Walrasian equilibrium price vectors. Also, necessary and suncient conditions are given under which the interior of the set of Walrasian equilibrium price vectors is non-empty. Several of the results are derived by interpreting Walrasian equilibrium price vectors as potential functions of an appropriate directed graph.

Sequential decisions in allocation problems

In the context of cooperative TU-games, and given an order of players, we consider the problem of distributing the worth of the grand coalition as a sequential decision problem. In each step of the process, upper and lower bounds for the payoff of the players are required related to successive reduced games. Sequentially compatible payoffs are defined as those allocation vectors that meet these recursive bounds. The core of the game is reinterpreted as a set of sequentially compatible payoffs when the Davis-Maschler reduced game is considered (Th.1). Independently of the reduction, the core turns out to be the intersection of the family of the sets of sequentially compatible payoffs corresponding to the different possible orderings (Th.2), so it is in some sense order-independent. Finally, we analyze advantageous properties for the first player.core, reduced game, sequential allocation, tu-game

On the dimension of the core of the assignment game

The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position in the assignment matrix.core, assignment game, core dimension