Give me that which I want, and you shall have this which you want, is the meaning of every such offer (Adam Smith).
Abstract
A new, more fundamental approach is proposed to the classical bargaining problem. The give-and-take feature in the negotiation process is explicitly modelled under the new framework. A compromise set consists of all allocations a player is willing to accept as agreement. We focus on the relationship between the rationality principles (arguments) adopted by players in making mutual concessions and the formation of compromise sets. The bargaining correspondence is then defined as the intersection of players’ compromise sets. We study the non-emptiness, symmetry, efficiency and single-valuedness of the bargaining correspondence, and establish its connection to the Nash solution. Our framework provides a rational foundation to Nash’s axiomatic approach.
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Notes
Myerson (1999) stated that Nash’s bargaining solution was “virtually unanticipated in the literature,” and Binmore (2005) argued that “Nash deserves his Nobel prize more for his bargaining solution than for his equilibrium concept, since his contribution to bargaining theory is entirely original, whereas his equilibrium idea had a number of precursors.”
While it was not explicitly declared when Nash (1950) first proposed his axiomatic approach, the unique existence of the solution (agreement) was formally stated as the first fundamental axiom (assumption) in Nash (1953, p. 136). This assumption is vital in Nash’s framework. Without this assumption, Nash’s axioms, which are properties defined on the assumed solution, would be meaningless, and Nash’s approach would be logically unsound. Binmore (1984) also pointed out that the existence of agreement is implicitly assumed in Nash (1950).
One way to avoid addressing the existence of agreement issue in Nash’s framework is to interpret a bargaining solution as a compromise recommended by an arbitrator rather than a unanimous agreement reached by the parties.
The strategic approach specifies the negotiation process in a multi-stage game, and predicts bargaining outcomes based on a suitable equilibrium concept. In the seminal paper, Rubinstein (1982) proved the existence of subgame perfect Nash equilibrium in a two-person, alternating-offer bargaining game. The existence of stationary no-delay equilibrium in general n-person bargaining games has been established in Banks and Duggan (2000) and Britz et al. (2010, 2014). Recently, Duggan (2017) provided sufficient conditions for the existence of stationary Markov perfect equilibrium in a general class of dynamic bargaining games.
Given \(x,y\in \mathbb {R} ^{n},\) \(x>y\) if \(x_{i}>y_{i}\) for each i, and \(x\ge y\) if \(x_{i}\ge y_{i} \) for each i.
Note that CR itself is a kind of symmetry assumption on the players’ behavior (but obviously much weaker than SYM). Hence we do not completely accomplish Schelling’s goal of abandoning ANY symmetry assumption in game theory. Our view is that this direction is unrealistic.
Compared to the Zeuthen-Harsanyi concession principle, \(\hbox {CIP}_{H}\) is a much weaker condition on mutual concessions, and is compatible with most solution concepts.
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This project was commenced when I was visiting the Department of Economics, University of Rochester. The hospitality of the department is gratefully acknowledged. I am indebted to William Thomson for insightful discussions. I would also like to thank Nejat Anbarci and Shiran Rachmilevitch, conference and seminar participants at the 2014 International Conference on Game Theory (Stony Brook), GRIPS (Tokyo) and University of Queensland as well as an anonymous referee of this journal for their valuable comments and suggestions.
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Sun, Cj. The bargaining correspondence: when Edgeworth meets Nash. Soc Choice Welf 51, 337–359 (2018). https://doi.org/10.1007/s00355-018-1119-3
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DOI: https://doi.org/10.1007/s00355-018-1119-3