Restricted Bargaining Sets in a Club Economy

The core as a solution concept captures the set of allocations against which no objection exists by any coalition of agents. However, Aumann and Maschler (1961. The Bargaining Set for Cooperative Games. Princeton Univ NJ) emphasized that the objection mechanism does not talk about repercussions from agents and thus introduced the bargaining set. Vind (1992. “Two Characterizations of Bargaining Sets.” Journal of Mathematical Economics 21: 89–97) later introduced the notion of a global bargaining set. In this paper, we consider a club economy proposed by Ellickson et al. (1999. “Clubs and the Market.” Econometrica 67: 1185–217) where club goods are consumed parallel to private goods to capture the social aspects of consumption. We introduce the global bargaining set for our club economy and provide characterizations of it in terms of the size of the (counter-) objecting coalitions thereby extending the works of Schjødt and Sloth (1994. “Bargaining Sets with Small Coalitions.” International Journal of Game Theory 23: 49–55) and Hervés-Estévez and Moreno-García (2015. “On Restricted Bargaining Sets.” International Journal of Game Theory 44: 631–45). In the process, we obtain Schmeidler’s (1972. “A Remark on the Core of an Atomless Economy.” Econometrica 40: 579) and Vind’s (1972. “A Third Remark on the Core of an Atomless Economy.” Econometrica 40: 585) theorem for atomless club economies as simple corollaries. We provide further interpretations of the global bargaining set in terms of approximately robustly efficient states, a notion familiarised by Bhowmik and Kaur (2023. “Competitive Equilibria and Robust Efficiency with Club Goods.” Journal of Mathematical Economics 108: 102876) for club economies.