Skip to main content
SpringerLink
Log in

Negotiation as a Cooperative Game

  • Reference work entry
  • First Online:
Handbook of Group Decision and Negotiation

  • 1422 Accesses

Abstract

Game theory provides us with a set of important methodologies for the study of group decisions as well as negotiation processes. Cooperative game theory is a subfield of game theory that focuses on interactions in which involved parties have the power to make binding agreements. Many group decision and negotiation processes (such as legal arbitrations) fall into this category, and as such, they have been central in the development of cooperative game theory. Particularly, an area of cooperative game theory, called bargaining theory, focuses on bilateral negotiations as well as negotiation processes where coalition formation is not a central concern. The object of study in bargaining theory is a (bargaining) rule, which provides a solution to each bargaining problem (or in other words, negotiation). Studies on bargaining theory employ the axiomatic method to evaluate bargaining rules. This chapter reviews and summarizes several such studies. After a discussion of the bargaining model, we present the important bargaining rules in the literature (including the Nash bargaining rule), as well as the central axioms that characterize them. Next, we discuss strategic issues related to cooperative bargaining, such as the Nash program, implementation of bargaining rules, and games of manipulating bargaining rules. We conclude with a discussion of the recent literature on ordinal bargaining rules.

This is a revised version of the chapter “Cooperative Game Theory Approaches to Negotiation” which was published in the first edition of this handbook.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 599.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 699.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Cooperative game theory analyzes interactions where agents can make binding agreements and it inquires how cooperative opportunities faced by alternative coalitions of agents shape the final agreement reached. Cooperative games do not specify how the agents interact or the mechanism through which their interaction leads to alternative outcomes of the game (and in this sense, they are different than noncooperative games). Instead, as will be exemplified in this chapter, they present a reduced form representation of all possible agreements that can be reached by some coalition.

  2. 2.

    This set contains all agreements that are physically available to the negotiators, including those that are “unreasonable” according to the negotiators’ preferences.

  3. 3.

    As will be formally introduced later, an agreement is Pareto optimal if there is no alternative agreement that makes an agent better-off without hurting any other agent.

  4. 4.

    We use the following vector inequalities x ≧ y for each i ∈ N, xi ≧ yi; x ≧ y and x ≠ y; and x > y if for each i ∈ N, xi > yi.

  5. 5.

    A stronger assumption called full comprehensiveness additionally requires utility to be freely disposable below d.

  6. 6.

    A decision-maker is risk-neutral if he is indifferent between each lottery and the lottery’s expected (sure) return.

  7. 7.

    This is Pareto optimal since both bargainers prefer accession to rejection. What they disagree on is the tariff rate.

  8. 8.

    A function λi : ℝ→ℝ is positive affine if there is a,b ∈ ℝ with a > 0 such that for each x ∈ ℝ, λi(x) = ax+b.

  9. 9.

    Any (S, d) can be “normalized” into such a problem by choosing \( {\lambda}_i\left({x}_i\right)=\frac{x_i-{d}_i}{N_i\left(S,d\right)-{d}_i} \) for each i ∈ N.

  10. 10.

    Any (S, d) can be “normalized” into such a problem by choosing \( {\lambda}_i\left({x}_i\right)=\frac{x_i-{d}_i}{a_i\left(S,d\right)-{d}_i} \) for each i ∈ N.

  11. 11.

    On problems that are not d-comprehensive, the Egalitarian rule can also violate weak Pareto optimality.

  12. 12.

    For a scale invariant rule, (S1,d1) and (S4,d4) are alternative representations of the same physical problem. (Specifically, E’s payoff function has been multiplied by 2 and thus, still represents the same preferences.) For the Egalitarian rule, however, these two problems (and player E’s) are distinct. Since it seeks to equate absolute payoff gains from disagreement, the Egalitarian rule treats agents’ payoffs to be comparable to each other. As a result, it treats payoff functions as more than mere representations of preferences.

  13. 13.

    This property is weaker than scale invariance because, for an agent i, every translation xi + zi is a positive affine transformation λi(xi) = 1xi + zi.

  14. 14.

    Any (S, d) can be “normalized” into such a problem by choosing λi(xi) = xi−di for each i ∈ N.

  15. 15.

    Thus, as in Nash (1953), each agent demands a payoff. But now, they have to rationalize it as part of a solution proposed by an “acceptable” bargaining rule.

  16. 16.

    This is due to the following fact. Two utility functions represent the same complete and transitive preference relation if and only if one is an increasing transformation of the other.

  17. 17.

    There is no reference on the origin of this rule in Shubik (1982). However, Thomson attributes it to Shapley. Furthermore, Roth (1979) (pp. 72–73) mentions a three-agent ordinal bargaining rule proposed by Shapley and Shubik (1974, Rand Corporation, R-904/4) which, considering the scarcity of ordinal rules in the literature, is most probably the same bargaining rule.

References

  • Abreu D, Pearce D (2015) A dynamic reinterpretation of Nash bargaining with endogenous threats. Econometrica 83(4):1641–1655

    Article  Google Scholar 

  • Anbarci N, Boyd J (2011) Nash demand game and the Kalai-Smorodinsky solution. Games Econ Behav 71:14–22

    Article  Google Scholar 

  • Anbarci N, Sun C (2011) Distributive justice and the Nash bargaining solution. Soc Choice Welf 37:453–470

    Article  Google Scholar 

  • Anbarci N, Sun C (2013) Robustness of intermediate agreements and bargaining solutions. Games Econ Behav 77:367–376

    Article  Google Scholar 

  • Bennett E (1997) Multilateral Bargaining Problems. Games Econ Behav 19:151–179

    Article  Google Scholar 

  • Binmore K, Eguia JX (2017) Bargaining with outside options. In: Schofield N, Caballero G (eds) State, institutions and democracy. Springer, Cham, pp 3–16

    Chapter  Google Scholar 

  • Binmore K, Rubinstein A, Wolinsky A (1986) The Nash bargaining solution in economic modeling. RAND J Econ 17(2):176–189

    Article  Google Scholar 

  • Blackorby C, Bossert W, Donaldson D (1994) Generalized Ginis and Cooperative Bargaining Solutions. Econometrica 62(5):1161–1178

    Article  Google Scholar 

  • Calvo E, Gutiérrez E (1994) Extension of the Perles-Maschler solution to N-person bargaining games. Int J Game Theory 23(4):325–346

    Article  Google Scholar 

  • Calvo E, Peters H (2000) Dynamics and axiomatics of the equal area bargaining solution. Int J Game Theory 29(1):81–92

    Article  Google Scholar 

  • Calvo E, Peters H (2005) Bargaining with ordinal and cardinal players. Games Econ Behav 52(1):20–33

    Article  Google Scholar 

  • Chiappori P-A, Donni O, Komunjer I (2012) Learning from a piece of pie. Rev Econ Stud 79:162–195

    Article  Google Scholar 

  • Chun Y, Thomson W (1990) Bargaining with uncertain disagreement points. Econometrica 58(4):951–959

    Article  Google Scholar 

  • Dagan N, Volij O, Winter E (2002) A characterization of the Nash bargaining solution. Soc Choice Welf 19:811–823

    Article  Google Scholar 

  • De Clippel G, Minelli E (2004) Two-person bargaining with verifiable information. J Math Econ 40(7):799–813

    Article  Google Scholar 

  • Dhillon A, Mertens JF (1999) Relative utilitarianism. Econometrica 67(3):471–498

    Article  Google Scholar 

  • Dubra J (2001) An asymmetric Kalai-Smorodinsky solution. Econ Lett 73(2):131–136

    Article  Google Scholar 

  • Forges F, Serrano R (2013) Cooperative games with incomplete information: some open problems. Int Game Theory Rev 15(02):134009. (17 pages)

    Article  Google Scholar 

  • Gómez JC (2006) Achieving efficiency with manipulative bargainers. Games Econ Behav 57(2):254–263

    Article  Google Scholar 

  • Herrero MJ (1989) The Nash program – non-convex bargaining problems. J Econ Theory 49(2):266–277

    Article  Google Scholar 

  • Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45(7):1623–1630

    Article  Google Scholar 

  • Kalai A, Kalai E (2013) Cooperation in strategic games revisited. Q J Econ 128(2):917–966

    Article  Google Scholar 

  • Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43:513–518

    Article  Google Scholar 

  • Karagözoğlu E, Rachmilevitch S (2018) Implementing egalitarianism in a class of Nash demand games. Theor Decis 85(3–4):495–508

    Article  Google Scholar 

  • Karos D, Muto N, Rachmilevitch S (2018) A generalization of the egalitarian and the Kalai-Smorodinsky bargaining solutions. Int J Game Theory 47(4):1169–1182

    Article  Google Scholar 

  • Kıbrıs Ö (2002) Misrepresentation of utilities in bargaining: pure exchange and public good economies. Games Econ Behav 39:91–110

    Article  Google Scholar 

  • Kıbrıs Ö (2004a) Egalitarianism in ordinal bargaining: the Shapley-Shubik rule. Games Econ Behav 49(1):157–170

    Article  Google Scholar 

  • Kıbrıs Ö (2004b) Ordinal invariance in multicoalitional bargaining. Games Econ Behav 46(1):76–87

    Article  Google Scholar 

  • Kıbrıs Ö (2012) Nash bargaining in ordinal environments. Rev Econ Des 16(4):269–282

    Google Scholar 

  • Kıbrıs Ö, Sertel MR (2007) Bargaining over a finite set of alternatives. Soc Choice Welf 28:421–437

    Article  Google Scholar 

  • Kıbrıs Ö, Tapk İG (2010) Bargaining with nonanonymous disagreement: monotonic rules. Games Econ Behav 68(1):233–241

    Google Scholar 

  • Kıbrıs Ö, Tapk İG (2011) Bargaining with nonanonymous disagreement: decomposable rules. Mathematical Social Sciences 62(3):151–161

    Google Scholar 

  • Kihlstrom RE, Roth AE, Schmeidler D (1981) Risk aversion and Nash’s solution to the bargaining problem. In: Moeschlin O, Pallaschke D (eds) Game theory and mathematical economics. North-Holland, Amsterdam

    Google Scholar 

  • Myerson RB (1977) Two-person bargaining problems and comparable utility. Econometrica 45:1631–1637

    Article  Google Scholar 

  • Myerson RB (1981) Utilitarianism, egalitarianism, and the timing effect in social choice problems. Econometrica 49:883–897

    Article  Google Scholar 

  • Nash JF (1950) The bargaining problem. Econometrica 18(1):155–162

    Article  Google Scholar 

  • Nash JF (1953) Two person cooperative games. Econometrica 21:128–140

    Article  Google Scholar 

  • O’Neill B, Samet D, Wiener Z, Winter E (2004) Bargaining with an agenda. Games Econ Behav 48:139–153

    Article  Google Scholar 

  • Perles MA, Maschler M (1981) A super-additive solution for the Nash bargaining game. Int J Game Theory 10:163–193

    Article  Google Scholar 

  • Peters H (1986) Simultaneity of issues and additivity in bargaining. Econometrica 54(1):153–169

    Article  Google Scholar 

  • Peters H (1992) Axiomatic bargaining game theory. Kluwer Academic, Dordrecht

    Book  Google Scholar 

  • Peters H, Wakker P (1991) Independence of irrelevant alternatives and revealed group preferences. Econometrica 59(6):1787–1801

    Article  Google Scholar 

  • Rachmilevitch S (2011) Disagreement point axioms and the egalitarian bargaining solution. Int J Game Theory 40:63–85

    Article  Google Scholar 

  • Rachmilevitch S (2015) Nash bargaining with (almost) no rationality. Math Soc Sci 76:107–109

    Article  Google Scholar 

  • Raiffa H (1953) Arbitration schemes for generalized two-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II. Princeton University Press, Princeton

    Google Scholar 

  • Roth AE (1979) Axiomatic Models of Bargaining. Springer-Verlag

    Google Scholar 

  • Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50(1):97–109

    Article  Google Scholar 

  • Rubinstein A, Safra Z, Thomson W (1992) On the interpretation of the Nash bargaining solution and its extension to nonexpected utility preferences. Econometrica 60(5):1171–1186

    Article  Google Scholar 

  • Samet D, Safra Z (2005) A family of ordinal solutions to bargaining problems with many players. Games Econ Behav 50(1):89–106

    Article  Google Scholar 

  • Shapley L (1969) Utility comparison and the theory of games. In: La Décision: Agrégation et Dynamique des Ordres de Préférence. Editions du CNRS, Paris, pp 251–263

    Google Scholar 

  • Shapley LS, Shubik M (1974) Rand Corporation Report R-904/4. Game Theory in Economics: Chapter 4, Preferences and Utility

    Google Scholar 

  • Shubik M (1982) Game theory in the social sciences. MIT Press, Cambridge, MA

    Google Scholar 

  • Sobel J (1981) Distortion of utilities and the bargaining problem. Econometrica 49:597–619

    Article  Google Scholar 

  • Sobel J (2001) Manipulation of preferences and relative utilitarianism. Games Econ Behav 37(1):196–215

    Article  Google Scholar 

  • Sprumont Y (2000) A note on Ordinally equivalent Pareto surfaces. J Math Econ 34:27–38

    Article  Google Scholar 

  • Thomson W (1981) Nash’s bargaining solution and utilitarian choice rules. Econometrica 49:535–538

    Article  Google Scholar 

  • Thomson W (1994) Cooperative models of bargaining. In: Aumann R, Hart S (eds) Handbook of game theory with economic applications. North-Holland, Amsterdam

    Google Scholar 

  • Van Damme E (1986) The Nash bargaining solution is optimal. J Econ Theory 38(1):78–100

    Article  Google Scholar 

  • Vidal-Puga J (2015) A non-cooperative approach to the ordinal Shapley-Shubik rule. J Math Econ 61:111–118

    Article  Google Scholar 

  • Yu P (1973) A class of solutions for group decision problems. Manag Sci 19:936–946

    Article  Google Scholar 

  • Zhou L (1997) The Nash bargaining theory with non-convex problems. Econometrica 65(3):681–685

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Arts and Social Sciences, Sabanci University, Istanbul, Turkey

    Özgür Kıbrıs

Authors
  1. Özgür Kıbrıs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Özgür Kıbrıs .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    D. Marc Kilgour

  2. Strathclyde Business School, University of Strathclyde, Glasgow, UK

    Colin Eden

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Kıbrıs, Ö. (2021). Negotiation as a Cooperative Game. In: Kilgour, D.M., Eden, C. (eds) Handbook of Group Decision and Negotiation. Springer, Cham. https://doi.org/10.1007/978-3-030-49629-6_10

Download citation

Publish with us

Policies and ethics

Search

Navigation