Advertisement

Journal of Evolutionary Economics

, Volume 25, Issue 5, pp 1011–1026 | Cite as

Equity dynamics in bargaining without information exchange

  • Heinrich H. NaxEmail author
Regular Article

Abstract

In this paper, completely uncoupled dynamics for n-player bargaining are proposed that mirror key behavioral elements of early bargaining and aspiration adjustment models (Zeuthen, 1930; Sauermann and Selten, 118:577–597 1962). Individual adjustment dynamics are based on directional learning adjustments, solely driven by histories of own realized payoffs. Bargaining this way, all possible splits have positive probability in the stationary distribution of the process, but players will split the pie almost equally most of the time. The expected waiting time for almost equal splits to be played is quadratic.

Keywords

Bargaining Cooperative game theory Equity Evolutionary game theory (Completely uncoupled) learning 

JEL Classifications

C71 C73 C78 D83 

Notes

Acknowledgments

Much of this work is part of the author’s D.Phil. (University of Oxford, 2011; supported by the Economic and Social Research Council [grant number PTA-031-2005-00118] and Postdoc under the Office of Naval Research Grant [grant number N00014-09-1-0751]. I also acknowledge current support by the European Commission through the ERC Advanced Investigator Grant ‘Momentum’ [grant number 324247]. I am particularly thankful to Peyton Young and Steffen Issleib, with whom I worked on this topic, and to Françoise Forges, Chris Wallace and Francis Dennig for comments on earlier versions of this model. The proof technique used is due to previous, joint unpublished work. I would also like to thank anonymous referees and a helpful editor for valuable comments on earlier versions of the paper.

References

  1. Adams JS (1965) Inequity in social exchange. In: Berkowitz L (ed) Advances in experimental social psychology 2, pp 267–299Google Scholar
  2. Alexander J, Skyrms B (1999) Bargaining with neighbors: is justice contagious? J Philos 96:588–598CrossRefGoogle Scholar
  3. Babichenko Y (2010) Uncoupled automata and pure Nash equilibria. Int J Game Theory 39:483–502MathSciNetCrossRefGoogle Scholar
  4. Babichenko Y (2012) Completely uncoupled dynamics and Nash equilibria. Games and Economic Behavior 76(1):1–14MathSciNetCrossRefGoogle Scholar
  5. Bayer R-C, Renner E, Sausgruber R (2013) Confusion and learning in the voluntary contributions game. Exp Econ 16:478–496CrossRefGoogle Scholar
  6. Binmore KG, Piccione M, Samuelson L (1998) Evolutionary stability in alternating-offers bargaining games. J Econ Theory 80:257–291MathSciNetCrossRefGoogle Scholar
  7. Binmore KG, Samuelson L, Young HP (2003) Equilibrium selection in bargaining models. Games and Economic Behavior 45:296–328MathSciNetCrossRefGoogle Scholar
  8. Brems H (1976) From the years of high theory: Frederik Zeuthen (1888–1959). History of Political Economy 8:400–411CrossRefGoogle Scholar
  9. Burton-Chellew M, Nax HH, West S (2015) Learning, not pro-sociality, explains the decline in cooperation in public goods games. Proc R Soc B Biol Sci 282 (1801):20142678CrossRefGoogle Scholar
  10. Bush R, Mosteller F (1955) Stochastic models of learning. NY, WileyCrossRefGoogle Scholar
  11. Cross JG (1983) A theory of adaptive economic behavior. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  12. Crössmann H, Tietz R (1983) Market behavior based on aspiration levels. In: Tietz R (ed) Lecture notes in economics and mathematical systems 213, Berlin, 1982, pp 170–185Google Scholar
  13. Ding J, Nicklisch A (2013) On the impulse in impulse learning. MPI Collective Goods Preprint 13(/02)Google Scholar
  14. Ellingsen T (1997) The evolution of bargaining behavior. Q J Econ 112:581–602CrossRefGoogle Scholar
  15. Erev I, Roth AE (1998) Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria. Am Econ Rev 88:848–881Google Scholar
  16. Estes W (1950) Towards a statistical theory of learning. Psychol Rev 57:94–107CrossRefGoogle Scholar
  17. Foster D, Young HP (1990) Stochastic evolutionary game dynamics. Theor Popul Biol 38:219–232MathSciNetCrossRefGoogle Scholar
  18. Foster D, Young HP (2006) Regret testing: learning to play Nash equilibrium without knowing you have an opponent. Theor Econ 1:341–367Google Scholar
  19. Gale J, Binmore K, Samuelson L (1995) Learning to be imperfect: the ultimatum game. Games and Economic Behavior 8:56–90MathSciNetCrossRefGoogle Scholar
  20. Germano F, Lugosi G (2007) Global Nash convergence of Foster and Young’s regret testing. Games and Economic Behavior 60:135–154MathSciNetCrossRefGoogle Scholar
  21. Grosskopf B (2003) Reinforcement and directional learning in the ultimatum game with responder competition. Exp Econ 6:141–158CrossRefGoogle Scholar
  22. Güth W, Schmittberger R, Schwarze B (1982) An experimental analysis of ultimatum bargaining. J Econ Behav Organ 3(4):367–388CrossRefGoogle Scholar
  23. Harley CB (1981) Learning the evolutionarily stable strategy. J Theor Biol 89:611–633CrossRefPubMedADSGoogle Scholar
  24. Harsanyi JC (1956) Approaches to the bargaining problem before and after the theory of games: a critical discussion of Zeuthen’s, Hicks’, and Nash’s theories. Econometrica 24:144–157MathSciNetCrossRefGoogle Scholar
  25. Hart S, Mas-Colell A (2003) Uncoupled dynamics do not lead to Nash equilibrium. Am Econ Rev 93:1830–1836CrossRefGoogle Scholar
  26. Hart S, Mas-Colell A (2006) Stochastic uncoupled dynamics and Nash equilibrium. Games and Economic Behavior 57:286–303MathSciNetCrossRefGoogle Scholar
  27. Heckhausen H (1955) Motivationsanalyse der Anspruchsniveau-Setzung. Psychol Forsch 25:118–154CrossRefPubMedGoogle Scholar
  28. Herrnstein RJ (1961) Relative and absolute strength of response as a function of frequency of reinforcement. J Exp Anal Behav 4:267–272PubMedCentralCrossRefPubMedGoogle Scholar
  29. Hoppe F (1931) Erfolg und Mißerfolg. Psychol Forsch 14:1–62CrossRefGoogle Scholar
  30. John R, Raith MG (1999) Strategic step-by-step negotiation. J Econ 70:127–154CrossRefGoogle Scholar
  31. Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623–1630MathSciNetCrossRefGoogle Scholar
  32. Karandikar R, Mookherjee D, Ray D, Vega-Redondo F (1998) Evolving aspirations and cooperation. J Econ Theory 80:292–331MathSciNetCrossRefGoogle Scholar
  33. Konrad KA, Morath F (2014) Bargaining with Incomplete Information: Evolutionary Stability in Finite Populations, Working Paper of the Max Planck Institute for Tax Law and Public Finance No. 2014–16Google Scholar
  34. Luce RD, Raiffa H (1957) Games and decisions: introduction and critical survey. NY, WileyGoogle Scholar
  35. Marden JR, Young HP, Arslan G, Shamma JS (2009) Payoff-based dynamics for multiplayer weakly acyclic games. SIAM. J Control Optim 48(1):373–396MathSciNetCrossRefGoogle Scholar
  36. Marden JR, Young HP, Pao LY (2014) Achieving Pareto optimality through distributed learning. SIAM J Control Optim 52(5):2753–2770MathSciNetCrossRefGoogle Scholar
  37. Maynard Smith J (1974) The theory of games and the evolution of animal conflicts. J Theor Biol 47(1):209–221MathSciNetCrossRefGoogle Scholar
  38. Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246(5427):15–18CrossRefGoogle Scholar
  39. Nash J (1950) The Bargaining Problem. Econometrica 18:155–162MathSciNetCrossRefGoogle Scholar
  40. Nax HH (2011) Evolutionary cooperative games, D.Phil. thesis, University of OxfordGoogle Scholar
  41. Nax HH, Perc M (2015) Directional learning and the provisioning of public goods. Sci Rep 5:8010PubMedCentralCrossRefPubMedADSGoogle Scholar
  42. Nax HH, Pradelski BSR (2015) Evolutionary dynamics and equitable core selection in assignment games. International Journal of Game Theory, forthcomingGoogle Scholar
  43. Nax HH, Pradelski BSR, Young HP (2013) Decentralized dynamics to optimal and stable states in the assignment game. IEEE Proceedings 52(CDC):2398–2405Google Scholar
  44. Nowak M, Page KM, Sigmund K (2000) Fairness versus reason in the ultimatum game. Science 289(5485):1773–1775Google Scholar
  45. Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428(6983):646–650CrossRefPubMedADSGoogle Scholar
  46. Pradelski BSR, Young HP (2012) Learning efficient Nash equilibria in distributed systems. Games and Economic Behavior 75:882–897MathSciNetCrossRefGoogle Scholar
  47. Raiffa H (1953) Arbitration schemes for generalized two-person games. In: Kuhn H, Tucker A, Dresher M (eds) Contributions to the theory of games, vol. 2. Princeton University Press, NJ, pp 361–387Google Scholar
  48. Roth AE (1995) Bargaining experiments. In: Kagel J, Roth AE (eds) Handbook of experimental economics. Princeton University Press, NJ, pp 253–348Google Scholar
  49. Roth AE, Erev I (1995) Learning in extensive form games: experimental data and simple dynamic models in the intermediate term. Games and Economics Behavior 8:164–212MathSciNetCrossRefGoogle Scholar
  50. Rubinstein (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97–109Google Scholar
  51. Saez-Marti M, Weibull JW (1999) Clever agents in Young’s evolutionary bargaining model. J Econ Theory 86:268–279MathSciNetCrossRefGoogle Scholar
  52. Sandholm W (2010) Population Games and Evolutionary Dynamics. MIT Press, CambridgeGoogle Scholar
  53. Sauermann H, Selten R (1962) Anspruchsanpassungstheorie der Unternehmung. Zeitschrift für die Gesamte Staatswissenschaft 118:577–597Google Scholar
  54. Schaffer ME (1988) Evolutionary stable strategies for a finite population and a variable contest size. J Theor Biol 132(4):469–478MathSciNetCrossRefPubMedGoogle Scholar
  55. Schelling TC (1956) An essay on bargaining. Am Econ Rev 46(3):281–306Google Scholar
  56. Selten R, Buchta J (1998) Experimental sealed bid first price auction with directly observed bid functions. In: Budescu D, Zwick IER (eds) Games and human behavior, essays in honor of Amnon RapoportGoogle Scholar
  57. Selten R, Stoecker R (1986) End behavior in sequences of finite prisoner’s dilemma supergames: a learning theory approach. J Econ Behav Organ 7:47–70CrossRefGoogle Scholar
  58. Suppes P, Atkinson AR (1959) Markov learning models for multiperson situations. Stanford University Press, StanfordGoogle Scholar
  59. Thorndike E (1898) Animal intelligence: an experimental study of the associative processes in animals. Psychol Rev:8Google Scholar
  60. Tietz R (1975) An experimental analysis of wage bargaining behavior. Zeitschrift für die gesamte Staatswissenschaft 131:44–91Google Scholar
  61. Tietz R, Bartos O (1983) Balancing of aspiration levels as fairness principle in negotiations. In: Tietz R (ed) Lecture Notes in Economics and Mathematical Systems, 213, pp 52–66Google Scholar
  62. Tietz R, Weber H (1972) On the nature of the bargaining process in the Kresko-game. In: Sauermann H (ed) Contributions to experimental economics, Vol. 3, pp 305–334Google Scholar
  63. Tietz R, Weber H (1978) Decision behavior in multi-variable negotiations. In: Sauermann H (ed) Contributions to experimental economics , Vol.7, pp 60–87Google Scholar
  64. Tietz R, Weber H, Vidmajer U, Wentzel C (1978) On aspiration forming behavior in repetitive negotiations. In: Sauermann H (ed) Contributions to experimental economics, Vol. 7, pp. 88–102Google Scholar
  65. Weber H (1976) On the theory of adaptation of aspiration levels in a bilateral decision setting. Zeitschrift für die gesamte Staatswissenschaft 132:582–591Google Scholar
  66. Weibull JW (1995) Evolutionary game theory. MIT Press, MAGoogle Scholar
  67. Young HP (1993) The evolution of conventions. Econometrica 61:57–84MathSciNetCrossRefGoogle Scholar
  68. Young HP (2004) Strategic learning and its limits. Oxford University Press, London, UKCrossRefGoogle Scholar
  69. Young HP (2009) Learning by trial and error. Games and Economic Behavior 65:626–643MathSciNetCrossRefGoogle Scholar
  70. Zeuthen F (1930) Problems of monopoly and economic warfare. Routledge, London, UKGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Social SciencesETH ZürichZürichSwitzerland

Personalised recommendations