Economic Theory Bulletin

, Volume 6, Issue 2, pp 141–155 | Cite as

Anchoring on Utopia: a generalization of the Kalai–Smorodinsky solution

  • Carlos Alós-Ferrer
  • Jaume García-Segarra
  • Miguel Ginés-Vilar
Research Article


Many bargaining solutions anchor on disagreement, allocating gains with respect to the worst-case scenario. We propose here a solution anchoring on utopia (the ideal, maximal aspirations for all agents), but yielding feasible allocations for any number of agents. The negotiated aspirations solution proposes the best allocation in the direction of utopia starting at an endogenous reference point which depends on both the utopia point and bargaining power. The Kalai–Smorodinsky solution becomes a particular case if (and only if) the reference point lies on the line from utopia to disagreement. We provide a characterization for the two-agent case relying only on standard axioms or natural restrictions thereof: strong Pareto optimality, scale invariance, restricted monotonicity, and restricted concavity. A characterization for the general (n-agent) case is obtained by relaxing Pareto optimality and adding the (standard) axiom of restricted contraction independence, plus the minimal condition that utopia should be selected if available.


n-Person bargaining Utopia point Axiomatic approach Kalai–Smorodinsky solution 

JEL Classification

C71 C78 



We are grateful to J. Vte. Guinot, Carmen Herrero, M. Carmen Marco, Juan D. Moreno-Ternero, Hervé Moulin, Hans Peters, Hannu Salonen, William Thomson, two anonymous referees and an associate editor for their useful comments. Financial support from projects ECO2015-68469 Ministerio de Educación, PREDOC/2007/28 Fundación Bancaja, and P1.15-1B2015-48 and E-2011-27 (Pla de Promoció de la Investigació) of the Universitat Jaume I are gratefully acknowledged.


  1. Alós-Ferrer, C., García-Segarra, J., Ginés-Vilar, M.: Super-additivity and concavity are equivalent for Pareto optimal \(n\)-agent bargaining solutions. Econ. Lett. 157, 50–52 (2017)CrossRefGoogle Scholar
  2. Anbarci, N.: Reference functions and balanced concessions in bargaining. Can. J. Econ. 28(3), 675–682 (1995)CrossRefGoogle Scholar
  3. Balakrishnan, P.V.S., Gómez, J.C., Vohra, R.V.: The tempered aspirations solution for bargaining problems with a reference point. Math. Soc. Sci. 62(3), 144–150 (2011)CrossRefGoogle Scholar
  4. Chun, Y.: The equal-loss principle for bargaining problems. Econ. Lett. 26(2), 103–106 (1988)CrossRefGoogle Scholar
  5. Chun, Y.: The separability principle in bargaining. Econ. Theory 1(1), 227–235 (2005)CrossRefGoogle Scholar
  6. Chun, Y., Peters, H.J.H.: The lexicographic equal-loss solution. Math. Soc. Sci. 22(2), 151–161 (1991)CrossRefGoogle Scholar
  7. Conley, J.P., Wilkie, S.: An extension of the Nash bargaining solution to nonconvex problems. Games Econ. Behav. 13(1), 26–38 (1996)CrossRefGoogle Scholar
  8. Dubra, J.: An asymmetric Kalai-Smorodinsky solution. Econ. Lett. 73(2), 131–136 (2001)CrossRefGoogle Scholar
  9. Freimer, M., Yu, P.L.: Some new results on compromise solutions for group decision problems. Manag. Sci. 22(6), 688–693 (1976)CrossRefGoogle Scholar
  10. García-Segarra J.: Efficiency, Monotonicity, and Uncertainty in Bargaining Problems: an Axiomatic Approach. PhD thesis, Departament of Economics. Universitat Jaume I de Castelló (2015)Google Scholar
  11. García-Segarra, J., Ginés-Vilar, M.: The impossibility of Paretian monotonic solutions: a strengthening of Roth’s result. Oper. Res. Lett. 43(5), 476–478 (2015)CrossRefGoogle Scholar
  12. Gupta, S., Livne, Z.A.: Resolving a conflict situation with a reference outcome: an axiomatic model. Manag. Sci. 34(11), 1303–1314 (1988)CrossRefGoogle Scholar
  13. Harless, P.: Endowment additivity and the weighted proportional rules for adjudicating conflicting claims. Econ. Theory 63(3), 755–781 (2017)CrossRefGoogle Scholar
  14. Herrero, C., Marco, M.C.: Rational equal-loss solutions for bargaining problems. Math. Soc. Sci. 26(3), 273–286 (1993)CrossRefGoogle Scholar
  15. Hougaard, J.L., Tvede, M.: Nonconvex \(n\)-person bargaining: efficient maxmin solutions. Econ. Theory 21(1), 81–95 (2003)CrossRefGoogle Scholar
  16. Kalai, E.: Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45(7), 1623–1630 (1977)CrossRefGoogle Scholar
  17. Kalai, E., Smorodinsky, M.: Other solutions to Nash’s bargaining problem. Econometrica 43(3), 513–518 (1975)CrossRefGoogle Scholar
  18. Karagözoglu E, Keskin K, Özcan-Tok E (2015) Between anchors and aspirations: a new family of bargaining solutions, Working Paper, Bilkent UniversityGoogle Scholar
  19. Myerson, R.B.: Utilitarianism, egalitarianism, and the timing effect in social choice problems. Econometrica 49(4), 883–897 (1981)CrossRefGoogle Scholar
  20. Nash, J.F.: The bargaining problem. Econometrica 18(2), 155–162 (1950)CrossRefGoogle Scholar
  21. Perles, M.A., Maschler, M.: The super-additive solution for the Nash bargaining game. Int. J. Game Theory 10, 163–193 (1981)CrossRefGoogle Scholar
  22. Peters, H.J.H.: A note on additive utility and bargaining. Econ. Lett. 17(3), 219–222 (1985)CrossRefGoogle Scholar
  23. Peters, H.J.H.: Simultaneity of issues and additivity in bargaining. Econometrica 54(1), 153–169 (1986)CrossRefGoogle Scholar
  24. Peters, H.J.H., Tijs, S.H.: Individually monotonic bargaining solutions of \(n\)-person bargaining games. Methods Oper. Res. 51, 377–384 (1984)Google Scholar
  25. Peters, H.J.H., Tijs, S.H.: Characterization of all individually monotonic bargaining solutions. Int. J. Game Theory 14(4), 219–228 (1985)CrossRefGoogle Scholar
  26. Qin, C.Z., Shi, S., Tan, G.: Nash bargaining for log-convex problems. Econ. Theory 58(3), 413–440 (2015)CrossRefGoogle Scholar
  27. Rosenthal, R.W.: An arbitration model for normal-form games. Math. Oper. Res. 1(1), 82–88 (1976)CrossRefGoogle Scholar
  28. Roth, A.E.: Independence of irrelevant alternatives, and solutions to Nash’s bargaining problem. J. Econ. Theory 16(2), 247–251 (1977)CrossRefGoogle Scholar
  29. Salonen, H.: A solution for two-person bargaining problems. Soc. Choice Welf. 2(2), 139–146 (1985)CrossRefGoogle Scholar
  30. Salonen, H.: Partially monotonic bargaining solutions. Soc. Choice Welf. 4(1), 1–8 (1987)CrossRefGoogle Scholar
  31. Thomson, W.: A class of solutions to bargaining problems. J. Econ. Theory 25(3), 431–441 (1981)CrossRefGoogle Scholar
  32. Thomson W.: Cooperative models of bargaining. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, vol. 2, chap 35, pp. 1237–1284. Elsevier, Amsterdam (1994)Google Scholar
  33. Thomson, W.: Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math. Soc. Sci. 45(3), 249–297 (2003)CrossRefGoogle Scholar
  34. Thomson, W.: Bargaining and the Theory of Cooperative Games: John Nash and Beyond. Edward Elgar Publishing, Cheltenham (2010)CrossRefGoogle Scholar
  35. Thomson, W.: For claims problems, compromising between the proportional and constrained equal awards rules. Econ. Theory 60(3), 495–520 (2015a)CrossRefGoogle Scholar
  36. Thomson, W.: Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update. Math. Soc. Sci. 74, 41–59 (2015b)CrossRefGoogle Scholar
  37. Yu, P.L.: A class of solutions for group decision problems. Manag. Sci. 19(8), 936–946 (1973)CrossRefGoogle Scholar

Copyright information

© Society for the Advancement of Economic Theory 2017

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CologneCologneGermany
  2. 2.Department of EconomicsUniversitat Jaume I de CastellóCastellóSpain

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