The B-Nucleolus of TU-Games

Reijnierse, Hans; Potters, J.A.M.

doi:10.1006/game.1997.0629

Cited by 28 publications

(25 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This means that instead of having to solve a big LP with 2 n constraints, the CG algorithm solves less than 500 small relaxed LPs and 500 constraint generation problems. The small numbers of iterations used is in fact aligns with the results in [31] where the authors show that there exists a set of 2(n − 1) coalitions that determines the nucleolus of an n-players game. Columns 8 and 9 show that the computation times required to solve these relaxed LPs and the CG problems are relatively small.…”

Section: Large Weighted Voting Gamessupporting

confidence: 76%

See 1 more Smart Citation

Finding the nucleoli of large cooperative games

Nguyen

Thomas

2016

European Journal of Operational Research

View full text Add to dashboard Cite

Highlights• We propose a new nested-LPs-based approach for finding the nucleolus• We resolve subtle issues in handling multiple optimal solutions in each LP• The approach involves fewer and smaller LPs compared to existing methods• The approach proves to work well in several large combinatorial games AbstractThe nucleolus is one of the most important solution concepts in cooperative game theory as a result of its attractive properties -it always exists (if the imputation is non-empty), is unique, and is always in the core (if the core is non-empty). However, computing the nucleolus is very challenging because it involves the lexicographical minimization of an exponentially large number of excess values. We present a method for computing the nucleoli of large games, including some structured games with more than 50 players, using nested linear programs (LP). Although different variations of the nested LP formulation have been documented in the literature, they have not been used for large games because of the large size and number of LPs involved. In addition, subtle issues such as how to deal with multiple optimal solutions and with tight constraint sets need to be resolved in each LP in order to formulate and solve the subsequent ones. Unfortunately, this technical issue has been largely overlooked in the literature. We treat these issues rigorously and provide a new nested LP formulation that is smaller in terms of the number of large LPs and their sizes. We provide numerical tests for several games, including the general flow games, the coalitional skill games and the weighted voting games, with up to 100 players.

show abstract

Section: Large Weighted Voting Gamessupporting

confidence: 76%

“…In fact, Reijnierse and Potters [31] show that there exist a set of 2(n−1) coalitions that determines the nucleolus. That means we can still find an optimal solution by including only a subset of constraints.…”

Section: Finding the Nucleoli Of Large Games 41 Constraint Generatiomentioning

confidence: 99%

Finding the nucleoli of large cooperative games

Nguyen

Thomas

2016

European Journal of Operational Research

View full text Add to dashboard Cite

show abstract

“…In fact, in any n player game there are at most (2n − 2) coalitions which actually determine the nucleolus, see Brune (1983) and Reynierse and Potters (1998). Although, as noticed by Brânzei et al (2005), identifying these coalitions is no less laborious as computing the nucleolus itself, in the following we state some facts for games with non-empty Core which will appear to be useful later on.…”

Section: E(s X) = V(s) − X(s)mentioning

confidence: 92%

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

2010

View full text Add to dashboard Cite

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.

show abstract

“…It seemingly depends on all coalitional values, but a closer look reveals the inherent high redundancy. Indeed, as Brune (1983), and more recently Reijnierse and Potters (1998) have proved: in any n-player game there are at most 2n − 2 coalitions which actually determine the nucleolus. Unfortunately, the identi cation of these coalitions is no less laborious as computing the nucleolus itself.…”

Section: Introductionmentioning

confidence: 97%

Strongly essential coalitions and the nucleolus of peer group games

2005

View full text Add to dashboard Cite

Most of the known e cient 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 2 n−1 . We propose an algorithm that computes the nucleolus of an n-player peer group game in O(n 2 ) time directly from the data of the underlying peer group situation. JEL classi cation: C71

show abstract

The B-Nucleolus of TU-Games

Cited by 28 publications

References 10 publications

Finding the nucleoli of large cooperative games

Finding the nucleoli of large cooperative games

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Strongly essential coalitions and the nucleolus of peer group games

Contact Info

Product

Resources

About