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

Strongly Essential Coalitions and the Nucleolus of Peer Group Games

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespeci ed collection of size polynomial in the number of players.We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore.As an application, we consider peer group games, and show that they admit at most 2n - 1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2n-1. We propose an algorithm that computes the nucleolus of an n-player peer group game in O(n2) time directly from the data of the underlying peer group situation.game theory;algorithm;cooperative games;kernel estimation;peer games

1-concave basis for TU games

The first stage of research, twenty years ago, on the subclass of 1-convex TU games dealt with its characterization through some regular core structure. Appealing abstract and practical examples of 1-convex games were missing until now. Both drawbacks are solved. On the one hand, a generating set for the cone of 1-concave cost games is introduced with clear affinities to the unanimity games taking into account the complementary transformation on coalitions. The dividends within this new game representation are used to characterize the 1-concavity constraint as well as to investigate the core property of the Shapley value for cost games. We present a simple formula to compute the nucleolus and the τ-value within the class of 1-convex/1-concave games and show that in a 1-convex/1-concave game there is an explicit relation between the nucleolus and the Shapley value. On the other hand, an appealing practical example of 1-concave cost game has cropped up not long ago in Sales’s Ph.D study of Catalan university library consortium for subscription to journals issued by Kluwer publishing house, the so-called library cost game which turn out to be decomposable into the abstract 1-concave cost games of the generating set mentioned above

SUMO regulates p21Cip1 intracellular distribution and with p21Cip1 facilitates multiprotein complex formation in the nucleolus upon DNA damage

We previously showed that p21Cip1 transits through the nucleolus on its way from the nucleus to the cytoplasm and that DNA damage inhibits this transit and induces the formation of p21Cip1-containing intranucleolar bodies (INoBs). Here, we demonstrate that these INoBs also contain SUMO-1 and UBC9, the E2 SUMO-conjugating enzyme. Furthermore, whereas wild type SUMO-1 localized in INoBs, a SUMO-1 mutant, which is unable to conjugate with proteins, does not, suggesting the presence of SUMOylated proteins at INoBs. Moreover, depletion of the SUMO-conjugating enzyme UBC9 or the sumo hydrolase SENP2 changed p21Cip1 intracellular distribution. In addition to SUMO-1 and p21Cip1, cell cycle regulators and DNA damage checkpoint proteins, including Cdk2, Cyclin E, PCNA, p53 and Mdm2, and PML were also detected in INoBs. Importantly, depletion of UBC9 or p21Cip1 impacted INoB biogenesis and the nucleolar accumulation of the cell cycle regulators and DNA damage checkpoint proteins following DNA damage. The impact of p21Cip1 and SUMO-1 on the accumulation of proteins in INoBs extends also to CRM1, a nuclear exportin that is also important for protein translocation from the cytoplasm to the nucleolus. Thus, SUMO and p21Cip1 regulate the transit of proteins through the nucleolus, and that disruption of nucleolar export by DNA damage induces SUMO and p21Cip1 to act as hub proteins to form a multiprotein complex in the nucleolus

The Generalized Nucleolus as a Solution of a Cost Allocation Problem

It has recently become clear that ideas and methods from the theory of cooperative games can be used quite successfully to solve cost allocation problems. Among the extensive literature (see Loughlin, 1977) dealing with this subject we shall concentrate on an article based on work carried out at IIASA (Young et al., 1980). The authors of this article used game theory principles of rationality to solve the problem of sharing the cost of a joint municipal water supply system among the group of Swedish municipalities participating in the project. In Section 1 we introduce the notion of the generalized nucleolus. The nucleolus and all its known mcdifications are special cases of our definition. Section 2 describes a method for calculating the nucleolus that can be readily implemented on a computer, and Section 3 puts forward an analytical criterion for testing the results. In Section 4 we describe some applications of the method to linear-fractional excess functions and convex games, and a number of formulas for three-person games are also given. Section 5 contains numerical results for seven modifications of the nucleolus for the six-person game discussed in Young et al. (1980)

Strongly Essential Coalitions and the Nucleolus of Peer Group Games

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespeci ed collection of size polynomial in the number of players.We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore.As an application, we consider peer group games, and show that they admit at most 2n - 1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2n-1. We propose an algorithm that computes the nucleolus of an n-player peer group game in O(n2) time directly from the data of the underlying peer group situation.

The Least-core and Nucleolus of Path Cooperative Games

Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if it enables a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the CS-core, least-core and nucleolus of path cooperative games. Furthermore, we show that the least-core and nucleolus are polynomially solvable for path cooperative games defined on both directed and undirected network