Skip to main content
SpringerLink
Log in

When is tit-for-tat unbeatable?

  • Published:
International Journal of Game Theory Aims and scope Submit manuscript

Abstract

We characterize the class of symmetric two-player games in which tit-for-tat cannot be beaten even by very sophisticated opponents in a repeated game. It turns out to be the class of exact potential games. More generally, there is a class of simple imitation rules that includes tit-for-tat but also imitate-the-best and imitate-if-better. Every decision rule in this class is essentially unbeatable in exact potential games. Our results apply to many interesting games including all symmetric 2\(\times \)2 games, and standard examples of Cournot duopoly, price competition, public goods games, common pool resource games, and minimum effort coordination games.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In this paper we shall call the tit-for-tat player, or more generally, the imitator “he” and all possible opponents “she”.

  2. In Axelrod’s tournament on the prisoners’ dilemma, tit-for-tat submitted by Anatol Rapoport also prescribed “cooperate” as initial action (see Axelrod 1980a, b). While the prisoners’ dilemma has a well-defined cooperative action, not all games possess such an action. We therefore consider a definition of tit-for-tat without restrictions on the initial action.

  3. In the context of the prisoner’s dilemma, the class includes strategies like two-tits-for-tat, “always D”, and grimm trigger but not tit-for-two-tats or Pavlov (Nowak and Sigmund 1993).

  4. Given the symmetry of \((X,\pi )\), the second equation plays the role usually played by the quantifier “for all players“ in the definition of potential games.

References

  • Alós-Ferrer C, Ania AB (2005) The evolutionary stability of perfectly competitive behavior. Econ Theory 26:497–516

    Article  Google Scholar 

  • Apesteguia J, Huck S, Oechssler J (2007) Imitation—theory and experimental evidence. J Econ Theory 136:217–235

    Article  Google Scholar 

  • Apesteguia J, Huck S, Oechssler J, Weidenholzer S (2010) Imitation and the evolution of Walrasian behavior: theoretically fragile but behaviorally robust. J Econ Theory 145:1603–1617

    Article  Google Scholar 

  • Axelrod R (1980a) Effective choice in prisoner’s dilemma. J Confl Resolut 24:3–25

    Article  Google Scholar 

  • Axelrod R (1980b) More effective choice in prisoner’s dilemma. J Confl Resolut 24:379–403

    Article  Google Scholar 

  • Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211:1390–1396

    Article  Google Scholar 

  • Balder E (1997) Remarks on Nash equilibria for games with additively coupled payoffs. Econ Theory 9:161–167

    Article  Google Scholar 

  • Brânzei R, Mallozzi L, Tijs S (2003) Supermodular games and potential games. J Math Econ 39:39–49

    Article  Google Scholar 

  • Bryant J (1983) A simple rational expectations Keynes-type coordination model. Q J Econ 98:525–528

    Article  Google Scholar 

  • Duersch P, Kolb A, Oechssler J, Schipper BC (2010) Rage against the machines: how subjects play against learning algorithms. Econ Theory 43:407–430

    Article  Google Scholar 

  • Duersch P, Oechssler J, Schipper BC (2012a) Unbeatable imitation. Games Econ Behav 76(1):88–96

    Article  Google Scholar 

  • Duersch P, Oechssler J, Schipper BC (2012b) Pure strategy equilibria in symmetric two-player zero-sum games. Int J Game Theory 41(3):553–564

    Article  Google Scholar 

  • Huck S, Normann H-T, Oechssler J (1999) Learning in Cournot oligopoly—an experiment. Econ J 109: C80–C95

    Google Scholar 

  • Milgrom P, Roberts J (1990) Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58:1255–1277

    Article  Google Scholar 

  • Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14:124–143

    Article  Google Scholar 

  • Nowak M, Sigmund K (1993) A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364:56–58

    Article  Google Scholar 

  • Peleg B (1998) Almost all equilibria in dominant strategies are coalition-proof. Econ Lett 60:157–162

    Article  Google Scholar 

  • Offerman T, Potters J, Sonnemans J (2002) Imitation and belief learning in an oligopoly experiment. Rev Econ Stud 69:973–997

    Article  Google Scholar 

  • Osborne M (2004) An introduction to game theory. Oxford University Press, Oxford

    Google Scholar 

  • Press WH, Dyson FJ (2012) Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent. Proceedings of the National Academy of Sciences, early edition www.pnas.org/cgi/doi/10.1073/pnas.1206569109

  • Rapoport A, Chammah AM (1965) Prisoner’s dilemma: a study of conflict and cooperation. Michigan University Press, Ann Arbor

    Google Scholar 

  • Sandholm W (2010) Population games and evolutionary dynamics. MIT Press, Cambridge

    Google Scholar 

  • Schipper BC (2009) Imitators and optimizers in Cournot oligopoly. J Econ Dyn Control 33:1981–1990

    Article  Google Scholar 

  • Schipper BC (2003) Submodularity and the evolution of Walrasian behavior. Int J Game Theory 32:471–477

    Google Scholar 

  • Selten R, Hammerstein P (1984) Gaps in Harley’s argument on evolutionarily stable learning rules and in the logic of “tit for tat”. Behav Brain Sci 7:115–116

    Article  Google Scholar 

  • Topkis DM (1998) Supermodularity and complementarity. Princeton University Press, Princeton

    Google Scholar 

  • Van Huyck J, Battalio R, Beil R (1990) Tacit coordination games, strategic uncertainty and coordination failure. Am Econ Rev 80:234–248

    Google Scholar 

  • Vega-Redondo F (1997) The evolution of Walrasian behavior. Econometrica 65:375–384

    Article  Google Scholar 

  • Walker JM, Gardner R, Ostrom E (1990) Rent dissipation in a limited-access common-pool resource: experimental evidence. J Environ Econ Manag 19:203–211

    Article  Google Scholar 

Download references

Acknowledgments

We are extremely thankful to two anonymous referees who provided very valuable input. This paper subsumes the earlier working paper “Once beaten, never again: Imitation in two-player potential games”. Some of the material was previously circulated in a companion paper “Unbeatable Imitation”. We thank Carlos Alós-Ferrer, Chen Bo, Drew Fudenberg, Alexander Matros, Klaus Ritzberger, Karl Schlag, and John Stachurski for interesting discussions. Seminar audiences at Australian National University, Melbourne University, Monash University, UC Davis, UC San Diego, the Universities of Heidelberg, Konstanz, Vienna, and Zürich, the University of Queensland, the University of Oregon, Calpoly, at the International Conference on Game Theory in Stony Brook, 2009, the Midwestern Economic Theory Conference in Evanston 2010, and at the Econometric Society World Congress 2010 in Shanghai contributed helpful comments.

Author information

Authors and Affiliations

  1. Department of Economics, University of Heidelberg, Heidelberg, Germany

    Peter Duersch & Jörg Oechssler

  2. Department of Economics, University of California, Davis, USA

    Burkhard C. Schipper

Authors
  1. Peter Duersch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jörg Oechssler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Burkhard C. Schipper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jörg Oechssler.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duersch, P., Oechssler, J. & Schipper, B.C. When is tit-for-tat unbeatable?. Int J Game Theory 43, 25–36 (2014). https://doi.org/10.1007/s00182-013-0370-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00182-013-0370-1

Keywords

JEL-Classifications

Search

Navigation