Computing Equilibria in Discounted 2 × 2 Supergames

Abstract

This article examines the subgame perfect pure strategy equilibrium paths and payoff sets of discounted supergames with perfect monitoring. The main contribution is to provide methods for computing and tools for analyzing the equilibrium paths and payoffs in repeated games. We introduce the concept of a first-action feasible path, which simplifies the computation of equilibria. These paths can be composed into a directed multigraph, which is a useful representation for the equilibrium paths. We examine how the payoffs, discount factors and the properties of the multigraph affect the possible payoffs, their Hausdorff dimension, and the complexity of the equilibrium paths. The computational methods are applied to the 12 symmetric strictly ordinal 2 × 2 games. We find that these games can be classified into three groups based on the complexity of the equilibrium paths.

This is a preview of subscription content, access via your institution.

References

  1. Abreu D. (1988) On the theory of infinitely repeated games with discounting. Econometrica 56(2): 383–396

    Article  Google Scholar 

  2. Abreu D., Rubinstein A. (1988) The structure of Nash equilibrium in repeated games with finite automata. Econometrica 56(6): 1259–1281

    Article  Google Scholar 

  3. Abreu D., Pearce D., Stacchetti E. (1986) Optimal cartel equilibria with imperfect monitoring. Journal of Economic Theory 39(1): 251–269

    Article  Google Scholar 

  4. Abreu D., Pearce D., Stacchetti E. (1990) Toward a theory of discounted repeated games with imperfect monitoring. Econometrica 58(5): 1041–1063

    Article  Google Scholar 

  5. Axelrod R. (1984) The evolution of cooperation. Basic, New York

    Google Scholar 

  6. Benoit J. P., Krishna V. (1985) Finitely repeated games. Econometrica 53(4): 905–922

    Article  Google Scholar 

  7. Berg, K. & Kitti, M. (2011). Equilibrium paths in discounted supergames. Working paper.

  8. Brams, S. J. (2003). Negotiation Games: Applying game theory to bargaining and arbitration. Routledge.

  9. Cronshaw M. B. (1997) Algorithms for finding repeated game equilibria. Computational Economics 10: 139–168

    Article  Google Scholar 

  10. Cronshaw M. B., Luenberger D. G. (1994) Strongly symmetric subgame perfect equilibria in infinitely repeated games with perfect monitoring. Games and Economic Behavior 6: 220–237

    Article  Google Scholar 

  11. Edgar G. A., Golds J. (1999) A fractal dimension estimate for a graph-directed iterated function system of non-similarities. Indiana University Mathematics Journal 48(2): 429–447

    Article  Google Scholar 

  12. Edgar G. A., Mauldin R. D. (1992) Multifractal decompositions of digraph recursive fractals. Proceedings of the London Mathematical Society 65: 604–628

    Article  Google Scholar 

  13. Falconer K. J. (1988) The Hausdorff dimension of self-affine fractals. Mathematical Proceedings of the Cambridge Philosophical Society 103: 169–179

    Article  Google Scholar 

  14. Falconer K. J. (1992) The dimension of self-affine fractals ii. Mathematical Proceedings of the Cambridge Philosophical Society 111: 339–350

    Article  Google Scholar 

  15. Fudenberg D., Maskin E. (1986) The folk theorem in repeated games with discounting and incomplete information. Econometrica 54: 533–554

    Article  Google Scholar 

  16. Hauert C. (2001) Fundamental clusters in 2 × 2 spatial games. Proceedings of the Royal Society B 268: 761–769

    Article  Google Scholar 

  17. Judd K., Yeltekin Ş., Conklin J. (2003) Computing supergame equilibria. Econometrica 71: 1239–1254

    Article  Google Scholar 

  18. Kilgour D. M., Fraser N. M. (1988) A taxonomy of all ordinal 2 × 2 games. Theory and Decision 24: 99–117

    Article  Google Scholar 

  19. Kitti, M. (2011). Conditional Markov equilibria in discounted dynamic games. Working paper.

  20. Kitti M. (2011) Conditionally stationary equilibria in discounted dynamic games. Dynamic Games and Applications, 1(4): 514–533

    Article  Google Scholar 

  21. Lehrer E., Pauzner A. (1999) Repeated games with differential time preferences. Econometrica 67: 393–412

    Article  Google Scholar 

  22. Mailath G. J., Samuelson L. (2006) Repeated Games and Reputations: Long-Run Relationships. Oxford University Press, Oxford

    Google Scholar 

  23. Mailath G. J., Obara I., Sekiguchi T. (2002) The maximum efficient equilibrium payoff in the repeated prisoners’ dilemma. Games and Economic Behavior 40: 99–122

    Article  Google Scholar 

  24. Mauldin R. D., Williams S. C. (1988) Hausdorff dimension in graph directed constructions. Transactions of the American Mathematical Society 309(2): 811–829

    Article  Google Scholar 

  25. Maynard Smith J. (1982) Evolution and the theory of games. Cambridge University Press, Cambridge

    Google Scholar 

  26. Ngai S. M., Wang Y. (2001) Hausdorff dimension of self-similar sets with overlaps. Journal of the London Mathematicl Society 63: 655–672

    Article  Google Scholar 

  27. Rapoport A., Guyer M. (1966) A taxonomy of 2 × 2 games. General Systems: Yearbook of the Society for General Systems Research 11: 203–214

    Google Scholar 

  28. Rellick L. M., Edgar G. A., Klapper M. H. (1991) Calculating the Hausdorff dimension of tree structures. Journal of Statistical Physics 64(1): 77–85

    Article  Google Scholar 

  29. Robinson, D. & Goforth, D. (2005). The topology of the 2 × 2 games: A new periodic table. New York: Routledge.

  30. Rubinstein A. (1986) Finite automata play the repeated prisoner’s dilemma. Journal of Economic Theory 39: 83–96

    Article  Google Scholar 

  31. Salonen H., Vartiainen H. (2008) Valuating payoff streams under unequal discount factors. Economics Letters 99(3): 595–598

    Article  Google Scholar 

  32. Tarjan R. (1972) Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2): 146–160

    Article  Google Scholar 

  33. Walliser B. (1988) A simplified taxonomy of 2 × 2 games. Theory and Decision 25: 163–191

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Kimmo Berg.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Berg, K., Kitti, M. Computing Equilibria in Discounted 2 × 2 Supergames. Comput Econ 41, 71–88 (2013). https://doi.org/10.1007/s10614-011-9308-5

Download citation

Keywords

  • Repeated game
  • 2 × 2 game
  • Subgame perfect equilibrium
  • Equilibrium path
  • Payoff set
  • Multigraph