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Cooperative Game Theory Approaches to Negotiation

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Book cover Handbook of Group Decision and Negotiation

Part of the book series: Advances in Group Decision and Negotiation ((AGDN,volume 4))

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 J. F. Nash (Econometrica, 18(1):155–162, 1950). There, Nash provided a way of modeling negotiation processes and applied an axiomatic methodology to analyze such models. Nash’s approach to modeling negotiation processes is (i) identifying the set of all alternative agreements, (ii) determining the implications of disagreement, (iii) determining how each negotiator values alternative agreements, as well as the disagreement outcome, and (iv) using the obtained payoff functions to reconstruct the negotiations in the payoff space. The feasible payoff set is the set of all payoff profiles resulting from an agreement and the disagreement point is the payoff profile obtained in case of disagreement. This pair is called a bargaining problem in cooperative game theory. The object of study in cooperative bargaining theory is a (bargaining) rule. It maps each bargaining problem to a payoff profile in the feasible payoff set. 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 the chapters by Klamler and Nurmi, this volume.) An axiom is simply a property of a bargaining rule that the researcher argues to be desirable. A typical study on cooperative bargaining theory considers a set of axioms, motivated by a particular application, and identifies the class of bargaining rules that satisfy them. 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. Here, we present the seminal characterizations of the Nash rule, the Kalai-Smorodinsky rule, and the Egalitarian rule. We also discuss some well-known rules such as the Utilitarian rule, the Dictatorial rule, the Equal Area rule, and the Perles-Maschler rule. 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 the chapter by Chatterjee, this volume). In the final part, we present the recent literature on ordinal bargaining rules, that is, rules that do not rely on the assumption that the agents have von Neumann-Morgenstern preferences.

This chapter was partially written while I was visiting the University of Rochester. I would like to thank this institution for its hospitality. I would also like to thank to William Thomson, Marc Kilgour, Arzu Kıbrıs, and İpek Gürsel Tapkı for comments and suggestions. Finally, I gratefully acknowledge the research support of the Turkish Academy of Sciences via a TUBA-GEBIP fellowship.

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Notes

  1. 1.

    Cooperative game theory, pioneered by von Neumann and Morgenstern (1944), 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: \(x\geqq y\) if for each \(i\in N, x_{i}\geqq y_{i}; x\geq y\) if \(x\geqq y\) and \(x\not=y\); and \(x>y\) if for each \(i\in N, x_{i}>y_{i}\).

  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 \(\lambda _{i}:\mathbb{R\rightarrow R}\) is positive affine if there is \(a,b\in \mathbb{R}\) with \(a>0\) such that for each \(x\in \mathbb{R}, \lambda _{i}\left( x\right) =ax+b.\)

  9. 9.

    Any \(\left( S,d\right) \) 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 \(i\in N.\)

  10. 10.

    Kalai and Rosenthal (1978) discuss a variant of this rule where the aspiration payoffs are defined alternatively as \(a_{i}^{\ast }\left(S,d\right) =\arg \max_{x\in S}x_{i}.\)

  11. 11.

    Any \(\left( S,d\right) \) 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 \(i\in N.\)

  12. 12.

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

  13. 13.

    For a scale invariant rule, \(\left( S^{1},d^{1}\right) \) and \(\left(S^{4},d^{4}\right) \) 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.

  14. 14.

    This property is weaker than scale invariance because, for an agent i, every translation \(x_{i}+z_{i}\) is a positive affine transformation \(\lambda _{i}\left( x_{i}\right) =1x_{i}+z_{i}.\)

  15. 15.

    Any \(\left( S,d\right) \) can be “normalized” into such a problem by choosing \(\lambda_{i}\left( x_{i}\right) =x_{i}-d_{i}\) for each \(i\in N.\)

  16. 16.

    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.

  17. 17.

    To implement a cooperative bargaining rule in an equilibrium notion (such as the Nash equilibrium), one constructs a noncooperative game whose equilibria coincides with the rule’s choices on every problem.

  18. 18.

    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.

  19. 19.

    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

  • Anbarcı N, Bigelow JP (1994) The area monotonic solution to the cooperative bargaining problem. Math Soc Sci 28(2):133–142

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Blackorby C, Walter Bossert and David Donaldson (1994) Generalized ginis and co-operative bargaining solutions. Econometrica 62(5):1161–1178

    Article  Google Scholar 

  • Binmore KG (1985) Bargaining and coalitions. In Roth AE (ed) Game theoretic models of bargaining, Cambridge University Press, Cambridge, 269–304

    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 

  • 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 

  • Chatterjee K (2010) Noncooperative bargaining theory. In: Kilgour M, Eden C (eds) Handbook of group decision and negotiation. Springer, Dordrecht

    Google Scholar 

  • Chun Y (1988) Nash solution and timing of bargaining. Econ Lett 28(1):27–31

    Article  Google Scholar 

  • Chun Y Thomson W (1989) Bargaining solutions and relative guarantees. Math Soc Sci 17(3):285–295

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Chun Y, Thomson W (1990b) Egalitarian solutions and uncertain disagreement points. Econ Lett 33(1):29–33

    Article  Google Scholar 

  • Conley J, Wilkie S (2007) The ordinal egalitarian bargaining solution for finite choice sets. mimeo

    Google Scholar 

  • Crawford VP, Varian H (1979) Distortion of preferences and the Nash theory of bargaining. Econ Lett 3(3):203–206

    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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Kalai E, Rosenthal RW (1978) Arbitration of two-party disputes under ignorance. Int J Game Theory 7:65–72

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

    Article  Google Scholar 

  • Kıbrıs Ö, Tapkı IG (2007) Bargaining with nonanonymous disagreement: decomposable rules. Sabancı University Working Paper

    Google Scholar 

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

    Article  Google Scholar 

  • Kıbrıs Ö (2008) Nash bargaining in ordinal environments. Rev Econ Des forthcoming

    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, pp 65–71

    Google Scholar 

  • Klamler C (2010) Fair division. In: Kilgour M, Eden C (eds) Handbook of group decision and negotiation. Springer, Dordrecht

    Google Scholar 

  • Lensberg T (1988) Stability and the Nash solution. J Econ Theory 45(2):330–341

    Article  Google Scholar 

  • Miyagawa E (2002) Subgame-perfect implementation of bargaining solutions. Games Econ Behav 41(2):292–308

    Article  Google Scholar 

  • Moulin H (1984) Implementing the Kalai-Smorodinsky bargaining solution. J Econ Theory 33:32–45

    Article  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 

  • Myerson RB (1984) 2-Person bargaining problems with incomplete information. Econometrica 52(2):461–487

    Article  Google Scholar 

  • Nash J (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 

  • Nurmi H (2010) Voting systems for social choice. In: Kilgour M. Eden C (eds) Handbook of group decision and negotiation. Springer, Dordrecht, pp 167–182

    Google Scholar 

  • Ok E, Zhou L (2000) The Choquet bargaining solutions. Games Econ Behav 33:249–264

    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 

  • Osborne M, Rubinstein A (1990) Bargaining and Markets. Academic New York, NY

    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 

  • Perles MA (1982) Nonexistence of super-additive solutions for 3-person Games. Int J Game Theory 11:151–161

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Peters H, Van Damme E (1991) Characterizing the Nash and Raiffa bargaining solutions by disagreement point properties. Math Oper Res 16(3):447–461

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Peters H (1992) Axiomatic bargaining game theory. Kluwer Academic, New York, NY

    Book  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, NJ, pp 361–387

    Google Scholar 

  • Roemer JE (1998) Theories of distributive justice. Harvard University Press, Cambridge, MA

    Google Scholar 

  • Roth AE (1979) Axiomatic models of bargaining. Springer, Dordrecht

    Book  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 

  • Salonen H (1998) Egalitarian solutions for n-person bargaining games. Math soc sci 35(3):291–306

    Article  Google Scholar 

  • Samet D, Safra Z (2004) An ordinal solution to bargaining problems with many players. Games Econ Behav 46:129–142

    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 

  • 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. Journal of Mathematical Economics 34:27–38

    Article  Google Scholar 

  • Thomson W, Myerson RB (1980) Monotonicity and independence axioms. Int J Game Theory 9:37–49

    Article  Google Scholar 

  • Thomson W (1980) Two characterizations of the Raiffa solution. Econ Lett 6(3):225–231

    Article  Google Scholar 

  • Thomson W (1983) The fair division of a fixed supply among a growing population. Math Oper Res 8(3):319–326

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Thomson W, Lensberg S (1989) The theory of bargaining with a variable number of agents. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Thomson W (1994) Cooperative models of bargaining. In: Aumann R, Hart S (Eds) Handbook of game theory, Chapter 35. North-Holland

    Google Scholar 

  • Thomson W (1996) Bargaining theory: the axiomatic approach, book manuscript. Academic Press

    Google Scholar 

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

    Article  Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton, NJ

    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 

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  1. Faculty of Arts and Social Sciences, Sabanci University, 34956, Istanbul, Turkey

    Özgür Kıbrıs

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  1. Dept. Mathematics, Wilfrid Laurier University, University Ave. W. 75, Waterloo, N2L 3C5, Canada

    D. Marc Kilgour

  2. Strathclyde Business School, University of Strathclyde, Glasgow, G4 0QU, United Kingdom

    Colin Eden

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Kıbrıs, Ö. (2010). Cooperative Game Theory Approaches to Negotiation. In: Kilgour, D., Eden, C. (eds) Handbook of Group Decision and Negotiation. Advances in Group Decision and Negotiation, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9097-3_10

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