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Stochastic Game Theory: Adjustment to Equilibrium Under Noisy Directional Learning

  • Simon P. Anderson


  • Jacob K. Goeree


  • Charles A. Holt


This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the Fokker-Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.

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Paper provided by University of Virginia, Department of Economics in its series Virginia Economics Online Papers with number 327.

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Length: 30 pages
Date of creation: Jul 1999
Date of revision:
Handle: RePEc:vir:virpap:327
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  1. Binmore, Ken & Larry Samuelson, 1994. "Muddling Through: Noisy Equilibrium Selection," Discussion Paper Serie B 275, University of Bonn, Germany.
  2. C. Monica Capra, 1999. "Anomalous Behavior in a Traveler's Dilemma?," American Economic Review, American Economic Association, vol. 89(3), pages 678-690, June.
  3. C. Monica Capra & Charles A. Holt, 1999. "Coordination," Southern Economic Journal, Southern Economic Association, vol. 65(3), pages 630-636, January.
  4. CHEN, Hsiao-Ch. & FRIEDMAN, J.W. & THISSE, Jacques-Francois, 1996. "Boundedly Rational Nash Equilibrium: A Probabilistic Choice Approach," CORE Discussion Papers 1996044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. D. Fudenberg & C. Harris, 2010. "Evolutionary Dynamics with Aggregate Shocks," Levine's Working Paper Archive 496, David K. Levine.
  6. repec:oup:qjecon:v:98:y:1983:i:3:p:525-28 is not listed on IDEAS
  7. Simon P. Anderson & Jacob K. Goeree & Charles A. Holt, 1998. "Rent Seeking with Bounded Rationality: An Analysis of the All-Pay Auction," Journal of Political Economy, University of Chicago Press, vol. 106(4), pages 828-853, August.
  8. Fernando Vega Redondo, 1996. "The evolution of walrasian behavior," Working Papers. Serie AD 1996-05, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
  9. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
  10. Ochs Jack, 1995. "Games with Unique, Mixed Strategy Equilibria: An Experimental Study," Games and Economic Behavior, Elsevier, vol. 10(1), pages 202-217, July.
  11. M. Kandori & G. Mailath & R. Rob, 1999. "Learning, Mutation and Long Run Equilibria in Games," Levine's Working Paper Archive 500, David K. Levine.
  12. Anderson, Simon P. & Goeree, Jacob K. & Holt, Charles A., 1998. "A theoretical analysis of altruism and decision error in public goods games," Journal of Public Economics, Elsevier, vol. 70(2), pages 297-323, November.
  13. Rosenthal, Robert W, 1989. "A Bounded-Rationality Approach to the Study of Noncooperative Games," International Journal of Game Theory, Springer, vol. 18(3), pages 273-91.
  14. Slade, Margaret E, 1994. "What Does an Oligopoly Maximize?," Journal of Industrial Economics, Wiley Blackwell, vol. 42(1), pages 45-61, March.
  15. Binmore, K. & Samuelson, L., 1997. "Evolutionary Drift and Equilibrium Selection," Working papers 9729r, Wisconsin Madison - Social Systems.
  16. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  17. repec:oup:restud:v:66:y:1999:i:2:p:363-93 is not listed on IDEAS
  18. Rhode, Paul & Stegeman, Mark, 2001. "Non-Nash equilibria of Darwinian dynamics with applications to duopoly," International Journal of Industrial Organization, Elsevier, vol. 19(3-4), pages 415-453, March.
  19. Binmore Kenneth G. & Samuelson Larry & Vaughan Richard, 1995. "Musical Chairs: Modeling Noisy Evolution," Games and Economic Behavior, Elsevier, vol. 11(1), pages 1-35, October.
  20. McKelvey Richard D. & Palfrey Thomas R., 1995. "Quantal Response Equilibria for Normal Form Games," Games and Economic Behavior, Elsevier, vol. 10(1), pages 6-38, July.
  21. Goeree, Jacob K. & Holt, Charles A., 2005. "An experimental study of costly coordination," Games and Economic Behavior, Elsevier, vol. 51(2), pages 349-364, May.
  22. Binmore, K. & Samuelson, L. & Vaughan, R., 1993. "Musical Chairs: Modelling Noisy Evolution," Working papers 9324, Wisconsin Madison - Social Systems.
  23. Erev, Ido & Roth, Alvin E, 1998. "Predicting How People Play Games: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria," American Economic Review, American Economic Association, vol. 88(4), pages 848-81, September.
  24. A. Roth & I. Er’ev, 2010. "Learning in Extensive Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Run," Levine's Working Paper Archive 387, David K. Levine.
  25. Binmore, Ken & Samuelson, Larry, 1997. "Muddling Through: Noisy Equilibrium Selection," Journal of Economic Theory, Elsevier, vol. 74(2), pages 235-265, June.
  26. Smith, Vernon L & Walker, James M, 1993. "Monetary Rewards and Decision Cost in Experimental Economics," Economic Inquiry, Western Economic Association International, vol. 31(2), pages 245-61, April.
  27. Selten, Reinhard & Joachim Buchta, 1994. "Experimental Sealed Bid First Price Auctions with Directly Observed Bid Functions," Discussion Paper Serie B 270, University of Bonn, Germany.
  28. Roth, Alvin E. & Erev, Ido, 1995. "Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term," Games and Economic Behavior, Elsevier, vol. 8(1), pages 164-212.
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