International Journal of Game Theory

, Volume 44, Issue 4, pp 903–932 | Cite as

Evolutionary dynamics and equitable core selection in assignment games

Article

Abstract

We study evolutionary dynamics in assignment games where many agents interact anonymously at virtually no cost. The process is decentralized, very little information is available and trade takes place at many different prices simultaneously. We propose a completely uncoupled learning process that selects a subset of the core of the game with a natural equity interpretation. This happens even though agents have no knowledge of other agents’ strategies, payoffs, or the structure of the game, and there is no central authority with such knowledge either. In our model, agents randomly encounter other agents, make bids and offers for potential partnerships and match if the partnerships are profitable. Equity is favored by our dynamics because it is more stable, not because of any ex ante fairness criterion.

Keywords

Assignment games Cooperative games Core Equity Evolutionary game theory Learning Matching markets Stochastic stability 

JEL Classification

C71 C73 C78 D83 

Notes

Acknowledgments

Foremost, we thank Peyton Young for his guidance. He worked with us throughout large parts of this project and provided invaluable guidance. Further, we thank Itai Arieli, Peter Biró, Gabrielle Demange, Sergiu Hart, Gabriel Kreindler, Jonathan Newton, Tom Norman, Tamás Solymosi, Zaifu Yang and anonymous referees for suggesting a number of improvements to earlier versions. We are also grateful for comments by participants at the 23rd International Conference on Game Theory at Stony Brook, the Paris Game Theory Seminar, the AFOSR MUIR 2013 meeting at MIT, the 18th CTN Workshop at the University of Warwick, the Economics Department theory group at the University of York, the Theory Workshop at the Center for the Study of Rationality at the Hebrew University, and the Game Theory Seminar at the Technion. The research was supported by the United States Air Force Office of Scientific Research Grant FA9550-09-1-0538 and the Office of Naval Research Grant N00014-09-1-0751. This paper supersedes the working paper “The evolution of core stability in decentralized matching markets”. Theorem 1 was reported without proof in the conference proceeding (Nax et al. 2013). Heinrich H. Nax acknowledges support by the European Commission through the ERC Advanced Investigator Grant ‘Momentum’ (Grant No. 324247). Bary S. R. Pradelski acknowledges support of the Oxford-Man Institute of Quantitative Finance.

References

  1. Agastya M (1997) Adaptive play in multiplayer bargaining situations. Rev Econ Stud 64:411–426MATHMathSciNetCrossRefGoogle Scholar
  2. Agastya M (1999) Perturbed adaptive dynamics in coalition form games. J Econ Theory 89:207–233MATHMathSciNetCrossRefGoogle Scholar
  3. Arnold T, Schwalbe U (2002) Dynamic coalition formation and the core. J Econ Behav Organ 49:363–380CrossRefGoogle Scholar
  4. Balinski ML (1965) Integer programming: methods, uses, computation. Manage Sci 12:253–313MATHMathSciNetCrossRefGoogle Scholar
  5. Balinski ML, Gale D (1987) On the core of the assignment game. Functional analysis, optimization and mathematical economics 1:274–289Google Scholar
  6. Bayati M, Borgs C, Chayes J, Kanoria Y, Montanari A (2014) Bargaining dynamics in exchange networks. J Econ Theory (forthcoming)Google Scholar
  7. Biró P, Bomhoff M, Golovach PA, Kern W, Paulusma D (2012) Solutions for the stable roommates problem with payments, vol 7551., Lecture notes in computer scienceSpringer, Berlin, pp 69–80Google Scholar
  8. Bush R, Mosteller F (1955) Stochastic models of learning. Wiley, New YorkCrossRefGoogle Scholar
  9. Chen B, Fujishige S, Yang Z (2011) Decentralized market processes to stable job matchings with competitive salaries. KIER working papers 749, Kyoto UniversityGoogle Scholar
  10. Chung K-S (2000) On the existence of stable roommate matching. Games Econ Behav 33:206–230MATHCrossRefGoogle Scholar
  11. Crawford VP, Knoer EM (1981) Job matching with heterogeneous firms and workers. Econometrica 49:437–540MATHCrossRefGoogle Scholar
  12. Davis M, Maschler M (1965) The kernel of a cooperative game. Naval Res Logist Quart 12:223–259MATHMathSciNetCrossRefGoogle Scholar
  13. Demange G, Gale D (1985) The strategy of two-sided matching markets. Econometrica 53:873–988MATHMathSciNetCrossRefGoogle Scholar
  14. Demange G, Gale D, Sotomayor M (1986) Multi-item auctions. J Polit Econ 94:863–872CrossRefGoogle Scholar
  15. Diamantoudi E, Xue L, Miyagawa E (2004) Random paths to stability in the roommate problem. Games Econ Behav 48:18–28MATHMathSciNetCrossRefGoogle Scholar
  16. Driessen TSH (1998) A note on the inclusion of the kernel in the core for the bilateral assignment game. Int J Game Theory 27:301–303MATHMathSciNetCrossRefGoogle Scholar
  17. Driessen TSH (1999) Pairwise-bargained consistency and game theory: the case of a two-sided firm. Fields Inst Commun 23:65–82MathSciNetGoogle Scholar
  18. Estes W (1950) Towards a statistical theory of learning. Psychol Rev 57:94–107CrossRefGoogle Scholar
  19. Foster D, Young HP (1990) Stochastic evolutionary game dynamics. Theoret Popul Biol 38:219–232MATHMathSciNetCrossRefGoogle Scholar
  20. Foster DP, Young HP (2006) Regret testing: learning to play Nash equilibrium without knowing you have an opponent. Theoret Econ 1:341–367Google Scholar
  21. Gale D, Shapley LS (1962) College admissions and stability of marriage. Am Math Mon 69:9–15MATHMathSciNetCrossRefGoogle Scholar
  22. Germano F, Lugosi G (2007) Global Nash convergence of Foster and Young’s regret testing. Games Econ Behav 60:135–154MATHMathSciNetCrossRefGoogle Scholar
  23. Hart S, Mas-Colell A (2003) Uncoupled dynamics do not lead to Nash equilibrium. Am Econ Rev 93:1830–1836CrossRefGoogle Scholar
  24. Hart S, Mas-Colell A (2006) Stochastic uncoupled dynamics and Nash equilibrium. Games Econ Behav 57:286–303MATHMathSciNetCrossRefGoogle Scholar
  25. Heckhausen H (1955) Motivationsanalyse der Anspruchsniveau-Setzung. Psychologische Forschung 25:118–154CrossRefPubMedGoogle Scholar
  26. Herrnstein RJ (1961) Relative and absolute strength of response as a function of frequency of reinforcement. J Exper Anal Behav 4:267–272CrossRefGoogle Scholar
  27. Hoppe F (1931) Erfolg und Mißerfolg. Psychologische Forschung 14:1–62CrossRefGoogle Scholar
  28. Huberman G (1980) The nucleolus and the essential coalitions. Analysis and optimization of systems, vol 28., Lecture notes in control and information systemsSpringer, Berlin, pp 417–422Google Scholar
  29. Inarra E, Larrea C, Molis E (2008) Random paths to P-stability in the roommate problem. Int J Game Theory 36:461–471MATHMathSciNetCrossRefGoogle Scholar
  30. Inarra E, Larrea C, Molis E (2013) Absorbing sets in roommate problem. Games Econ Behav 81:165–178MATHMathSciNetCrossRefGoogle Scholar
  31. Jackson MO, Watts A (2002) The evolution of social and economic networks. J Econ Theory 106:265–295MATHMathSciNetCrossRefGoogle Scholar
  32. Kandori M, Mailath GJ, Rob R (1993) Learning, mutation, and long run equilibria in games. Econometrica 61:29–56MATHMathSciNetCrossRefGoogle Scholar
  33. Kelso AS, Crawford VP (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50:1483–1504MATHCrossRefGoogle Scholar
  34. Klaus B, Klijn F (2007) Paths to stability for matching markets with couples. Games Econ Behav 58:154–171MATHMathSciNetCrossRefGoogle Scholar
  35. Klaus B, Klijn F, Walzl M (2010) Stochastic stability for roommates markets. J Econ Theory 145:2218–2240MATHMathSciNetCrossRefGoogle Scholar
  36. Klaus B, Payot F (2013) Paths to stability in the assignment problem. Working paperGoogle Scholar
  37. Kojima F, Ünver MU (2008) Random paths to pairwise stability in many-to-many matching problems: a study on market equilibrium. Int J Game Theory 36:473–488MATHCrossRefGoogle Scholar
  38. Kuhn HW (1955) The Hungarian method for the assignment problem. Naval Res Logist Quart 2:83–97MathSciNetCrossRefGoogle Scholar
  39. Llerena F, Nunez M (2011) A geometric characterization of the nucleolus of the assignment game. Econ Bull 31:3275–3285Google Scholar
  40. Llerena F, Nunez M, Rafels C (2012) An axiomatization of the nucleolus of the assignment game. Working papers in economics 286, Universitat de BarcelonaGoogle Scholar
  41. Marden JR, Young HP, Arslan G, Shamma J (2009) Payoff-based dynamics for multi-player weakly acyclic games. SIAM J Control Optimiz 48:373–396MATHMathSciNetCrossRefGoogle Scholar
  42. Maschler M, Peleg B, Shapley LS (1979) Geometric properties of the kernel, nucleolus, and related solution concepts. Math Oper Res 4:303–338MATHMathSciNetCrossRefGoogle Scholar
  43. Nash J (1950) The bargaining problem. Econometrica 18:155–162MATHMathSciNetCrossRefGoogle Scholar
  44. Nax HH, Pradelski BSR, Young HP (2013) Decentralized dynamics to optimal and stable states in the assignment game. Proceedings of the 52nd IEEE conference on decision and control, December 10–13, 2013. Florence, Italy, pp 2391–2397Google Scholar
  45. Newton J (2010) Non-cooperative convergence to the core in Nash demand games without random errors or convexity assumptions. Ph.D. thesis, University of CambridgeGoogle Scholar
  46. Newton J (2012) Recontracting and stochastic stability in cooperative games. J Econ Theory 147:364–381MATHCrossRefGoogle Scholar
  47. Newton J, Sawa R (2013) A one-shot deviation principle for stability in matching problems. Economics working papers 2013–09, University of Sydney, School of EconomicsGoogle Scholar
  48. Nunez M (2004) A note on the nucleolus and the kernel of the assignment game. Int J Game Theory 33:55–65MATHMathSciNetCrossRefGoogle Scholar
  49. Pradelski BSR (2014) Decentralized dynamics and fast convergence in the assignment game. Department of Economics. Discussion paper series 700, University of OxfordGoogle Scholar
  50. Pradelski BSR, Young HP (2012) Learning efficient Nash equilibria in distributed systems. Games Econ Behav 75:882–897MATHMathSciNetCrossRefGoogle Scholar
  51. Rochford SC (1984) Symmetrically pairwise-bargained allocations in an assignment market. J Econ Theory 34:262–281MATHMathSciNetCrossRefGoogle Scholar
  52. Roth AE, Vate JHV (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480MATHMathSciNetCrossRefGoogle Scholar
  53. Roth AE, Sotomayor M (1992) Two-sided matching. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 1. North Holland, Amsterdam, pp 485–541CrossRefGoogle Scholar
  54. Roth AE, Erev I (1995) Learning in extensive-form games: experimental data and simple dynamic models in the intermediate term. Games Econ Behav 8:164–212MATHMathSciNetCrossRefGoogle Scholar
  55. Rozen K (2013) Conflict leads to cooperation in Nash Bargaining. J Econ Behav Organ 87:35–42CrossRefGoogle Scholar
  56. Sauermann H, Selten R (1962) Anspruchsanpassungstheorie der Unternehmung. Zeitschrift fuer die gesamte Staatswissenschaft 118:577–597Google Scholar
  57. Sawa R (2011) Coalitional stochastic stability in games, networks and markets. Working Paper, Department of Economics, University of Wisconsin-MadisonGoogle Scholar
  58. Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17:1163–1170MATHMathSciNetCrossRefGoogle Scholar
  59. 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
  60. Selten R (1998) Aspiration adaptation theory. J Math Psychol 42:191–214MATHCrossRefPubMedGoogle Scholar
  61. Shapley LS, Shubik M (1963) The core of an economy with nonconvex preferences. RM-3518. The Rand Corporation, Santa Monica, CAGoogle Scholar
  62. Shapley LS, Shubik M (1966) Quasi-cores in a monetary economy with nonconvex preferences. Econometrica 34:805–827MATHCrossRefGoogle Scholar
  63. Shapley LS, Shubik M (1972) The assignment game 1: the core. Int J Game Theory 1:111–130MathSciNetCrossRefGoogle Scholar
  64. Solymosi T, Raghavan TES (1994) An algorithm for finding the nucleolus of assignment games. Int J Game Theory 23:119–143MATHMathSciNetCrossRefGoogle Scholar
  65. Sotomayor M (2003) Some further remarks on the core structure of the assignment game. Math Soc Sci 46:261–265MATHMathSciNetCrossRefGoogle Scholar
  66. Thorndike E (1898) Animal intelligence: an experimental study of the associative processes in animals. Psychol Rev 8:1874–1949Google Scholar
  67. Tietz A, Weber HJ (1972) On the nature of the bargaining process in the Kresko-game. In: Sauermann H (ed) Contribution to experimental economics. J. C. B. Mohr, Tübingen, pp 305–334Google Scholar
  68. Young HP (1993) The evolution of conventions. Econometrica 61:57–84MATHMathSciNetCrossRefGoogle Scholar
  69. Young HP (2009) Learning by trial and error. Games Econ Behav 65:626–643MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Social SciencesETH ZürichZürichSwitzerland
  2. 2.Oxford-Man Institute, University of OxfordOxfordUK

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