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.

On 1-convexity and nucleolus of co-insurance games

“…As was already shown in Arin and Feltkamp (1997), for every veto-rich game from the class G + N 0 the core is nonempty and the nucleolus payoff to a veto-player is larger than or equal to that of the other players. From this it easily follows that every vetoremoved game is balanced because the Davis-Maschler reduced game inherits the core property and, moreover, in every nontrivial veto-rich game v 0 ∈ G + N 0 the nucleolus payoff to a veto-player ν 0 (v 0 ) > 0 since in any nontrivial game v 0 ∈ G + N 0 the worth of the grand coalition v 0 (N 0 ) > 0.…”

“…Consider now a game v ∈ G m N together with its associated vetorich game v 0 ∈ G m N 0 . From (11) and already mentioned above statement of Arin and Feltkamp (1997) concerning the nucleolus payoff to a veto-player, it easily follows that v(N) n+1 ≤ ν 0 ≤ v(N). Set a := ν 0 .…”

“…In every veto-rich game from the class G + M with M containing the veto-player 0 the nucleolus coincides with the prenucleolus due to the nonemptiness of the core which was already mentioned above with reference to Arin and Feltkamp (1997). Then, because of the Davis-Maschler consistency of the prenucleolus (Sobolev, 1975), the nucleolus payoffs to the players in the Davis-Maschler reduced game w S 0 ,y ∈ G S 0 constructed in Step 3 of Algorithm 2 are the same as the nucleolus payoffs to the players in S 0 in the game w ∈ G + M .…”

“…Algorithm 2 is closed conceptually to the algorithm for computing the nucleolus for veto-rich games suggested in Arin and Feltkamp (1997).…”

Coalitional games with veto players: Consistency, monotonicity and Nash outcomes

“…Under this resolution of the conflict, the nucleolus of the TU game appears as a candidate to be the Nash outcome of the non-cooperative game. The reason is that in the class of veto balanced games the nucleolus and the prekernel coincide (Arin and Feltkamp, 1997) and the prekernel is the maximal set satisfying the Davis-Maschler reduced game property and the standard solution property. 2 Given a veto balanced game and its associated non-cooperative game we identify the set of allocations to which the outcome of any Nash equilibrium belongs.…”