Competitive Outcomes and Endogenous Coalition Formation in an n-person Game

Abstract

In this paper we study competitive outcomes and endogenous coalition formation in a cooperative n-person transferable utility (TU) game from the viewpoint of general equilibrium theory. For any given game, we construct a competitive exchange coalition production economy corresponding to the game.First, it is shown that the full core of a TU game is not empty if and only if the completion of the game is balanced. The full core is defined free of any particular coalition structure and the coalitions of the game emerge endogenously from the full core. Second, it is shown that the full core of a completionbalanced general TU game coincides with the set of equilibrium payoff vectors of its corresponding economy and that the coalition structures of the game are endogenously determined by the equilibrium outcomes of the economy. As a consequence, the core of a balanced general TU game coincides with the set of equilibrium payoff vectors of its corresponding economy.

Representation of transferable utility games by coalition production economies

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a b s t r a c tWe prove that, by the method of construction of a coalition production economy due to Sun et al [Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n-person game.

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“…Since an exchange economy is a special case of a coalition production economy, the class of TU games generated by coalition production economies must be larger than the class of TU games generated by exchange economies. Actually, by using a method of construction of a coalition production economy from a TU game due to Sun et al (2008), we prove that any TU game can be generated by a coalition production economy (Theorem 2).…”

“…a TU game. One example was given by Sun et al (2008). They explained the formation of coalitions at a core payoff vector of a TU game as the formation of firms in a market equilibrium for the coalition production economy induced from the given TU game.…”

Autonomous coalitions

“…In the former case, there exists natural generalizations of the core, once it is admitted that players may eventually form either disjoint coalitions, i.e., a partition of the grand coalition, or a collection of coalitions which has the property to be balanced: interpreting the balancing weights as amounts of time and assuming that each player has one unit of time to spend [see for example Peleg and Sudhölter (2003)], this unit can be split among different coalitions of this collection. The c-core (Guesnerie and Oddou 1979;Sun et al 2008) corresponds to the case where players form a partition of the grand coalition, while the d-core (a.k.a. aspiration core) (Bennett 1983;Cross 1967;Albers 1979), which is never empty, corresponds to the more general case where players share their unit of time into different coalitions.…”

Competitive outcomes and the inner core of NTU market games

“…Similarly to the approach of Billera and Bixby (1974), Inoue (2010b) uses coalition production economies as in Sun et al (2008) instead of markets. Inoue (2010b) shows that every compactly generated NTU game can be represented by a coalition production economy.…”