Equilibrium payoffs and proposal ratios in bargaining models

  • Shunsuke HanatoEmail author
Original Paper


We analyze a bargaining model which is a generalization of the model of Rubinstein (Econometrica 50(1):97–109, 1982) from the viewpoint of the process of how a proposer is decided in each period. In our model, a player’s probability to be a proposer depends on the history of proposers and players divide a pie of size 1. We derive a subgame perfect equilibrium (SPE) and analyze how its SPE payoffs are related to the process. In the bilateral model, there is a unique SPE. In the n-player model, although SPE may not be unique, a Markov perfect equilibrium (MPE) similar to the SPE in the bilateral model exists. In the case where the discount factor is sufficiently large, if the ratio of opportunities to be a proposer converges to some value, players divide the pie according to the ratio of this convergent value under these equilibria. This result implies that although our process has less regularity than a Markov process, the same result as in the model that uses a Markov process holds. In addition to these results, we show that the limit of the SPE (or the MPE) payoffs coincides with the asymmetric Nash bargaining solution weighted by the convergent values of the ratio of the opportunities to be a proposer.


Non-cooperative bargaining Subgame perfect equilibrium Proposal ratio Limit payoff Asymmetric Nash bargaining solution 



I am grateful to Ryo Kawasaki, two anonymous referees, an anonymous associate editor, and the editor for extremely valuable comments. I am also grateful to Keisuke Bando, Toshiyuki Hirai, Shin Kishimoto, Toshiji Miyakawa, Shigeo Muto, Tadashi Sekiguchi, Satoru Takahashi, and participants at Game Theory Workshop 2017 held at The University of Electro-Communications, The 2017 Spring National Conference of The Operations Research Society of Japan held at Okinawaken Shichouson Jichikaikan, East Asian Game Theory Conference 2017 held at The National University of Singapore, and European Meeting on Game Theory 2018 held at University of Bayreuth for helpful comments and suggestions. This work was supported by JSPS KAKENHI Grant Number JP18J20162.


  1. Britz V, Herings P, Predtetchinski A (2010) Non-cooperative support for the asymmetric nash bargaining solution. J Econ Theory 145:1951–1967CrossRefGoogle Scholar
  2. Chae S, Yang J-A (1990) An n-person bargaining process with alternating demands. Seoul J Econ 3(3):255–261Google Scholar
  3. Fershtman C, Seidmann DJ (1993) Deadline effects and inefficient delay in bargaining with endogenous commitment. J Econ Theory 60(2):306–321CrossRefGoogle Scholar
  4. Fudenberg D, Tirole J (1991) Game theory. MIT press, CambridgeGoogle Scholar
  5. Herings PJ-J, Predtetchinski A (2010) One-dimensional bargaining with markov recognition probabilities. J Econ Theory 145(1):189–215CrossRefGoogle Scholar
  6. Kalandrakis T (2004) Equilibria in sequential bargaining games as solutions to systems of equations. Econ Lett 84(3):407–411CrossRefGoogle Scholar
  7. Kalandrakis T (2006) Regularity of pure strategy equilibrium points in a class of bargaining games. Econ Theory 28(2):309–329CrossRefGoogle Scholar
  8. Kultti K, Vartiainen H (2010) Multilateral non-cooperative bargaining in a general utility space. Int J Game Theory 39(4):677–689CrossRefGoogle Scholar
  9. Laruelle A, Valenciano F (2008) Noncooperative foundations of bargaining power in committees and the shapley-shubik index. Games Econ Behav 63(1):341–353CrossRefGoogle Scholar
  10. Mao L (2017) Subgame perfect equilibrium in a bargaining model with deterministic procedures. Theory Decis 82(4):485–500CrossRefGoogle Scholar
  11. Mao L, Zhang T (2017) A minimal sufficient set of procedures in a bargaining model. Econ Lett 152:79–82CrossRefGoogle Scholar
  12. Merlo A, Wilson C (1995) A stochastic model of sequential bargaining with complete information. Econometrica 63(2):371–399CrossRefGoogle Scholar
  13. Merlo A, Wilson C (1998) Efficient delays in a stochastic model of bargaining. Econ Theory 11(1):39–55CrossRefGoogle Scholar
  14. Okada A (1996) A noncooperative coalitional bargaining game with random proposers. Games Econ Behav 16(1):97–108CrossRefGoogle Scholar
  15. Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50(1):97–109CrossRefGoogle Scholar
  16. Shaked A, Sutton J (1984) Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52(6):1351–1364CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Economics, School of EngineeringTokyo Institute of TechnologyTokyoJapan

Personalised recommendations