A procedure to compute the nucleolus of the assignment game

The assignment game introduced by Shapley and Shubik (1972) is a model for a two-sided market where there is an exchange of indivisible goods for money and buyers or sellers demand or supply exactly one unit of the goods. We give a procedure to compute the nucleolus of any assignment game, based on the distribution of equal amounts to the agents, until the game is reduced to fewer agents

An efficient algorithm for nucleolus and prekernel computation in some classes of TU-games

We consider classes of TU-games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel. \u

Computing the Least-core and Nucleolus for Threshold Cardinality Matching Games

Cooperative games provide a framework for fair and stable profit allocation in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are such solution concepts that characterize stability of cooperation. In this paper, we study the algorithmic issues on the least-core and nucleolus of threshold cardinality matching games (TCMG). A TCMG is defined on a graph $G=(V,E)$ and a threshold $T$, in which the player set is $V$ and the profit of a coalition $S\subseteq V$ is 1 if the size of a maximum matching in $G[S]$ meets or exceeds $T$, and 0 otherwise. We first show that for a TCMG, the problems of computing least-core value, finding and verifying least-core payoff are all polynomial time solvable. We also provide a general characterization of the least core for a large class of TCMG. Next, based on Gallai-Edmonds Decomposition in matching theory, we give a concise formulation of the nucleolus for a typical case of TCMG which the threshold $T$ equals $1$. When the threshold $T$ is relevant to the input size, we prove that the nucleolus can be obtained in polynomial time in bipartite graphs and graphs with a perfect matching

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

Matching games: the least core and the nucleolus

A matching game is a cooperative game defined by a graph $G=(V,E)$. The player set is $V$ and the value of a coalition $S \subseteq V$ is defined as the size of a maximum matching in the subgraph induced by $S$. We show that the nucleolus of such games can be computed efficiently. The result is based on an alternative characterization of the least core which may be of independent interest. The general case of weighted matching games remains unsolved. \u

The complexity of the nucleolus in compact games

This is the author accepted manuscript. The final version is available from ACM via the DOI in this recordThe nucleolus is a well-known solution concept for coalitional games to fairly distribute the total available worth among the players. The nucleolus is known to be NP-hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. In particular, hardness results have been exhibited over minimum spanning tree games, threshold games, and flow games. However, due to its intricate definition involving reasoning over exponentially many coalitions, a nontrivial upper bound on its complexity was missing in the literature and looked for. This article faces this question and precisely characterizes the complexity of the nucleolus, by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games. The upper bound is established by proposing a variant of the standard linear-programming based algorithm for nucleolus computation and by studying a framework for reasoning about succinctly specified linear programs, which are contributions of interest in their own. The hardness result is based on an elaborate combinatorial reduction, which is conceptually relevant for it provides a "measure" of the computational cost to be paid for guaranteeing voluntary participation to the distribution process. In fact, the pre-nucleolus is known to be efficiently computable over graph games, with this solution concept being defined as the nucleolus but without guaranteeing that each player is granted with it at least the worth she can get alone, that is, without collaborating with the other players. Finally, this article identifies relevant tractable classes of coalitional games, based on the notion of type of a player. Indeed, in most applications where many players are involved, it is often the case that such players do belong in fact to a limited number of classes, which is known in advance and may be exploited for computing the nucleolus in a fast way.Part of E. Malizia’s work was supported by the European Commission through the European Social Fund and by Calabria Regio

Formation of cartels in glove markets and the modiclus

cartels;game theory