On the characterizations of viable proposals
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On the characterizations of viable proposals Yi-You Yang1 Ó Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Sengupta and Sengupta (Int Econ Rev 35:347–359, 1994) consider a payoff configuration of a TU game as a viable proposal if it challenges each legitimate contender. Lauwers (Int Econ Rev 43:1369–1371, 2002) prove that the set of viable proposals is nonempty for every game. In the present paper, we prove that the set of viable proposals coincides with the coalition structure core if there exists an undominated proposal; otherwise, it coincides with the set of accessible proposals. This characterization result implies that a proposal is a viable proposal if and only if it is undominated or accessible. Moreover, we prove that the set of viable proposals includes the minimal dominant set, which is another nonempty extension of the coalition structure core introduced by Ko´czy and Lauwers (Games Econ Behav 61:277–298, 2007). In particular, we prove that the set of viable proposals of a cohesive game coincides with the minimal dominant set. Keywords Viable proposal Undominated proposal Accessible proposal Coalition structure core Minimal dominant set
1 Introduction Sengupta and Sengupta (1994) study a disequilibrium process of successive coalition formation in the framework of coalitional games with transferable utility (TU games), and introduce the notion of viable proposal as a solution for TU games. Assume that an initial payoff configuration or a proposal ðx; PÞ is considered as an outcome for the game, where x is an individually rational payoff vector and P is a coalition structure such that the total payoff of each coalition in P coincides with its value. In case the members of some coalition S can gain by forming S, they may The author is very grateful to two anonymous referees for their perceptive comments of the paper. & Yi-You Yang [email protected] 1
Department of Economics, Aletheia University, 32 Chen-Li Street, New Taipei City 25103, Taiwan
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Y.-Y. Yang
deviate to obtain a higher aggregate payoff, thereby leading to an alternative proposal according to their myopic preferences. Sengupta and Sengupta (1994) employ a dominance relation to formulate such a transition from one proposal to another, and interpret a sequence of successively dominating proposals as a process of coalition formation. A proposal ðx; PÞ is called accessible from another proposal ðy; QÞ if it can be reached from ðy; QÞ via a sequence of dominating proposals. In case ðx; PÞ is accessible from any proposal ðy; QÞ with ðy; QÞ 6¼ ðx; PÞ, it is called an accessible proposal. A proposal ðx; PÞ is called a viable proposal if ðx; PÞ is accessible from proposal ðy; QÞ whenever ðy; QÞ is accessible from ðx; PÞ. Lauwers (2002) proves that the set of viable proposals is nonempty for every game. In the present paper, we provide two characterization results for the set of viable proposals. First, we show that every game possesses at least an undominated pr
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