Quasi-Cores in Bargaining Sets

Abstract

Abstract:We propose a nonempty-valued subsolution of the Mas-Colell Bargaining Set on the class of TU games satisfying grand coalition zero-monotonicity, a weaker condition than superadditivity, zero-monotouicity and balaneedness. The subsolution is a slight modification of the Shapley-Shubik Quasi-Core. The Zhou Bargaining Set is a refinement of the Mas-Colell Bargaining Set. We also give a nonempty-valued subsolution of the Zhou Bargaining Set on the class of all TU games satisfying grand coalition superadditivity, a weaker condition than superadditivity and ba|ancedness. This subsolution is a modification of the Zhou Bargaining Set which is described by excesses. These results are examined for TU games with coalition structurcs.

The bargaining set and coalition formation

“…Our first theorem extends his result to NTU games similarly to the way Scarf (1967) had extended the core nonemptiness theorem of Bondareva (1963) and Shapley (1967). Shimomura (1997) proves that on the class of TU games, the steady bargaining set and the Zhou bargaining set are nonempty for every coalition structure that can produce the maximal sum of payoffs to all the players. Combining this result with our first theorem, we demonstrate that in a NTU coalitional game, it is not easily predictable which coalitions are likely to form, but the idea of the bargaining set helps explain how coalition structures and payoff allocations are determined.…”

“…We prove Theorem 1 in the appendix. By Theorem 3.2 in Shimomura (1997) and Olive. We denote them by , , and , respectively.…”

“…The Mas-Colell bargaining set of a coalitional game with a coalition structure, discussed in this paper, is the set of payoff profiles feasible for the coalition structure with no justified objection. In a TU game, it contains efficient payoff profiles for at least one coalition structure [Zhou (1994), Shimomura (1997)]. In fact, Mas-Colell (1989, P.138) points out that for many games, his bargaining set "can be quite large" regardless of the requirement of efficiency.…”

The Nucleolus, the Kernel, and the Bargaining Set: An Update

“…The counterobjections defined by weak improvements make Zhou's bargaining set nonempty for general games and the mapping ZB upper hemicontinuous in v given N. Shimomura (1997) defines two modifications of the Zhou bargaining set. In the first, counterobjections are defined by strict improvements, leading to the strict Zhou bargaining set ZB * .…”

“…Shimomura (1997) modifies the Mas-Colell bargaining set and the Zhou bargaining set by defining both objections and counterobjections with strict improvements Izquierdo and Rafels (2018). call them the Mas-Colell bargaining setá la Shimomura and the Zhou bargaining setá la Shimomura, respectively.…”