The Nucleolus and Kernel of Veto-Rich Transferable Utility Games

Abstract

The nucleolus and kernel of veto-rich transferable utility games Arin, J.; Feltkamp, V. Publication date: 1994 Link to publication Citation for published version (APA):Arin, J., & Feltkamp, V. (1994). The nucleolus and kernel of veto-rich transferable utility games. (CentER Discussion Paper; Vol. 1994-40). CentER. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. May 2, 1994 AbstractThe process of computing the nucleolus of arbitrary transferable utility games is notoriously hard. A number of papers have appeared in which the nucleolus is computed by an algorithm in which either one or a huge number of huge linear programs have to be solved.We show that on the class of veto-rich games, the nucleolus is the unique kernel element. Veto-rich games are games in which one of the players is needed by coalitions in order to obtain a positive payoff. We then provide a fast algorithm which does not use linear programming techniques to compute the nucleolus on these games.Furthermore, we provide several examples of economic situations which belong to the class of veto-rich games and which are treated in the literature.

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

“…Arin and Feltkamp (1997) propose an exponential time algorithm to find the nucleolus of a veto-rich game, i.e., a game (N , v) such that there exists (at least one) veto player being a player i such that v(S) = 0 when i ∈ S. In this paper we modify this algorithm to find, in polynomial time, the nucleolus of restricted games arising from games with a permission structure in which players in a cooperative TU-game belong to a hierarchical structure that is represented by a directed graph. For given D on N , a path between i and j in N is a sequence of distinct nodes…”

“…According to Arin and Feltkamp (1997) the nucleolus assigns positive payoff to every player in N when N is essential. Notice also that r (N ) > r (S) for every S ⊂ N when N is essential.…”

“…Since the top player is always in this set when it belongs to the coalition, and the largest feasible set is the empty set for any coalition not containing the unique top player, it follows that the top player is a veto player, i.e., any coalition not containing the (veto) top player has zero worth in the restricted game. 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.…”

Strongly essential coalitions and the nucleolus of peer group games

“…For veto-rich games (i.e., nonnegative games with a veto player) Arin and Feltkamp (1997) proves that the nucleolus allocation can be calculated as the unique kernel element, and presents an algorithm that sequentially determines (even if the game is not balanced) the unique payo vector satisfying e ciency and the n − 1 kernel conditions corresponding to the veto nonveto pairs. Since peer group games are monotonic (hence balanced) vetorich games, in light of Corollary 1(iii) a specialized variant of that algorithm could be relevant for us.…”

“…Since peer group games are monotonic (hence balanced) vetorich games, in light of Corollary 1(iii) a specialized variant of that algorithm could be relevant for us. Indeed, if we streamline the algorithm of Arin and Feltkamp (1997) in a straightforward way for monotonic and B-restricted veto-rich games, we basically get the specialized algorithm of Kuipers et al (2000, p. 557). Thus, applied to peer group games we would obtain the same O(n 3 ) algorithm with the mentioned imperfection.…”