Strongly essential coalitions and the nucleolus of peer group games

Brânzei, R.; Solymosi, Tamás; Tijs, S.H.

doi:10.1007/s00182-005-0213-9

Cited by 27 publications

(22 citation statements)

References 15 publications

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“…For peer group games, Brânzei et al (2005) provide a polynomial time algorithm for computing the nucleolus (Schmeidler 1969) of the restricted game. In van den Brink et al (2010Brink et al ( , 2011 polynomial time algorithms for two subclasses of games with a permission structure, both generalizing peer group situations, are developed.…”

Section: Discussionmentioning

confidence: 99%

Games with a permission structure - A survey on generalizations and applications

Brink

2017

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In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions. In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize existing solutions for TU-games. In this survey, we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures. A central model of games with hierarchies is that of games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.This invited paper is discussed in the comments available at

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Section: Discussionmentioning

confidence: 99%

Games with a permission structure - A survey on generalizations and applications

Brink

2017

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“…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. We denote …”

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

confidence: 91%

“…Brânzei et al (2005) argue that applying the algorithm to the specific case of a peer group game the complexity reduces to a polynomial time algorithm of order O(n 3 ). They show that the algorithm given in their paper to find the nucleolus of a peer group game is a polynomial time algorithm of order O(n 2 ).…”

Section: Complexity Of the Algorithmmentioning

confidence: 99%

“…In Arin and Feltkamp (1997) an exponential time algorithm has been given to compute the nucleolus for veto-rich games. In Brânzei et al (2005) a polynomial time algorithm is given to compute the nucleolus of a peer group game. In this paper we modify the Arin-Feltkamp algorithm to compute the nucleolus of the restricted game induced by more general situations, including the examples mentioned above as special cases.…”

Section: Introductionmentioning

confidence: 99%

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

2010

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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.

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“…As remarked after both definitions, such a cycle in the decomposition cannot occur if we apply only essentiality or only dual essentiality. Strongly essential coalitions discussed by Brânzei et al [3] are not immune to this kind of failure, hence they might not form a characterization set for the nucleolus in a balanced game in which the grand coalition is not vital.…”

Section: C(s)mentioning

confidence: 99%

Characterization sets for the nucleolus in balanced games

Solymosi

Sziklai

2016

Operations Research Letters

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We provide a new modus operandi for the computation of the nucleolus in cooperative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and if the game is monotonic dually saturated coalitions determine both the core and the nucleolus whenever the core is non-empty. We show how these two sets are related to the existing characterization sets. In particular we prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself.

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Strongly essential coalitions and the nucleolus of peer group games

Cited by 27 publications

References 15 publications

Games with a permission structure - A survey on generalizations and applications

Games with a permission structure - A survey on generalizations and applications

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

Characterization sets for the nucleolus in balanced games

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