Demand Commitment Bargaining: - The Case Of Apex Games -
An apex game is a bargaining situation in which there is one major (apex) player and n “minor” players. The only profitable coalitions contain either the apex player and any one of the minor players or else all of the minor players. The demand commitment
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by
Elaine Bennett and Eric van Damme
Abstract An apex game is a bargaining situation in which there is one major (apex) player and n "minor" players. The only profitable coalitions contain either the apex player and any one of the minor players or else all of the minor players. The demand commitment model is a bargaining procedure, i.e. an extensive form game. This paper investigates the payoffs that result (as subgame perfect outcomes) for apex games when players use the demand commitment bargaining procedure. We show that whenever the apex player has the first move he forms a coalition with a minor player and obtains the fraction (n - l)/n of the coalition's value while his (minor-player) partner obtains the remaining lin. When a minor player has the first move he either forms a coalition with the apex player (and obtains
lin) or else forms a coalition with all of the remaining minor players. When this minorplayer coalition forms there are many subgame perfect payoff distributions. A refinement of su bgame perfection is proposed and is shown to select a unique payoff distri bution (lin for each minor player) for the minor-player coalition.
1
Introduction
In this paper we present a noncooperative model for bargaining in characteristic function games, and explore its implications for the class of apex games. An apex game is a bargaining situation 1 Both
authors are grateful to the Center for Interdisciplinary Research at the University of Bielefeld for its
support and hospitality. They thank Bill Zame for helpful discussions and for suggestions that improved the presentation. The first author i. also grateful to the CentER for Economic Research at Tilburg University for its support and hospitality and to the National Science Foundation grant for its support through grant SES-8706631.
R. Selten (ed.), Game Equilibrium Models III © Springer-Verlag Berlin Heidelberg 1991
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in which there is one major (apex) player and n minor players; only the coalitions consisting of the apex player and anyone of the minor players, and the coalition consisting of all of the minor players are profitable, and these coalitions have equal value. (It is convenient, and involves no loss of generality, to normalize payoffs so that the value of each of these coalitions is n, the number of minor players.) In our model, bargaining takes place by means of a procedure which we call
demand commitment. 2 Each player in turn may set a price (a payoff demand, expressed in utility terms) for his participation in any coalition. Having set his price a player can form a coalition if his partners in the coalition have already named their prices and the coalition can afford these prices. The game terminates as soon as one coalition is formed; players not belonging to the "successful" coalition receive a payoff of zero. Our purpose in this paper is to investigate the effects of the existence of alternative coalitions on bargaining outcomes. The demand commitment model is well-suited to this purpose: In this model, each player must make a
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