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Negotiation as a Cooperative Game

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Handbook of Group Decision and Negotiation

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Abstract

In this chapter, we present an overview of how negotiation and group decision processes are modeled and analyzed in cooperative game theory. This area of research, typically referred to as cooperative bargaining theory, originated in a seminal paper by Nash (Econometrica 18(1):155–162, 1950). The object of study in cooperative bargaining theory is a (bargaining) rule, which provides a solution to each bargaining problem. Studies on cooperative bargaining theory employ the axiomatic method to evaluate bargaining rules. (A similar methodology is used for social choice and fair division problems, as discussed in chapters “Social Choice” and “Fair Division” of this handbook.) In this chapter, we review and summarize several such studies. In the first part of the chapter, we present the bargaining model of Nash (Econometrica 18(1):155–162, 1950). In the second part, we introduce the main bargaining rules and axioms in the literature. In the third part of the chapter, we discuss strategic issues related to cooperative bargaining, such as the Nash program, implementation of bargaining rules, and games of manipulating bargaining rules (for more on strategic issues, see chapter “Noncooperative Bargaining” of this handbook). In the final part, we present 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.

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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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    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

    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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    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

    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

    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

    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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

Download references

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Authors and Affiliations

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

    Özgür Kıbrıs

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  1. Özgür Kıbrıs
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  1. Wilfrid Laurier University Dept. Mathematics, Waterloo, ON, Canada

    D. Marc Kilgour

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

    Colin Eden

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Kıbrıs, Ö. (2020). Negotiation as a Cooperative Game. In: Kilgour, D., Eden, C. (eds) Handbook of Group Decision and Negotiation. Springer, Cham. https://doi.org/10.1007/978-3-030-12051-1_10-1

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