International Journal of Game Theory

, Volume 45, Issue 1–2, pp 461–496 | Cite as

The complexity of interacting automata

  • Olivier Gossner
  • Penélope Hernández
  • Ron PeretzEmail author
Original Paper


This paper studies the interaction of automata of size m. We characterise statistical properties satisfied by random plays generated by a correlated pair of automata with m states each. We show that in some respect the pair of automata can be identified with a more complex automaton of size comparable to \(m\log m\). We investigate implications of these results on the correlated min–max value of repeated games played by automata.


Complexity Automata De Bruijn sequences Bounded memory 



We are grateful to three anonymous referees and an editor who contributed valuable suggestions that significantly improved the quality of this paper.


  1. Abreu D, Rubinstein A (1988) The structure of nash equilibrium in repeated games with finite automata. Econometrica 56(6):1259–1281CrossRefGoogle Scholar
  2. Aumann RJ (1981) Survey of repeated games. Essays in game theory and mathematical economics in honor of Oskar Morgenstern. Bibliographisches Institut, Mannheim, pp 11–42Google Scholar
  3. Bavly G, Neyman A (2014) Online concealed correlation and bounded rationality. Games Econ Behav 88:71–89CrossRefGoogle Scholar
  4. Ben-Porath E (1993) Repeated games with finite automata. J Econ Theory 59(1):17–32CrossRefGoogle Scholar
  5. Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn. Wiley, New YorkGoogle Scholar
  6. De Bruijn NG (1946) A combinatorial problem. K Ned Akad Wet 49:758–764Google Scholar
  7. Gossner O, Hernández P (2003) On the complexity of coordination. Math Oper Res 28(1):127–140CrossRefGoogle Scholar
  8. Gossner O, Hernández P (2006) Coordination through De Bruijn sequences. Oper Res Lett 34(1):17–21CrossRefGoogle Scholar
  9. Gossner O, Tomala T (2006) Empirical distributions of beliefs under imperfect observation. Math Oper Res 31(1):13–30CrossRefGoogle Scholar
  10. Gossner O, Tomala T (2007) Secret correlation in repeated games with imperfect monitoring. Math Oper Res 32(2):413–424CrossRefGoogle Scholar
  11. Gossner O, Hernandez P, Neyman A (2006) Optimal use of communication resources. Econometrica 74(6):1603–1636CrossRefGoogle Scholar
  12. Kalai E (1990) Bounded rationality and strategic complexity in repeated games. In: Ichiishi T, Neyman A, Tauman Y (eds) Game theory and applications. Economic theory, econometrics, and mathematical economics. Academic Press, San Diego, pp 131–157Google Scholar
  13. Kalai E, Stanford W (1988) Finite rationality and interpersonal complexity in repeated games. Econometrica 56(2):397–410CrossRefGoogle Scholar
  14. Lehrer E (1988) Repeated games with stationary bounded recall strategies. J Econ Theory 46(1):130–144CrossRefGoogle Scholar
  15. Neyman A (1985) Bounded complexity justifies cooperation in the finitely repeated prisoners’ dilemma. Econ Lett 19(3):227–229CrossRefGoogle Scholar
  16. Neyman A (1997) Cooperation, repetition, and automata. Cooperation: game theoretic approaches. NATO ASI series F. Springer, New York, pp 233–255CrossRefGoogle Scholar
  17. Neyman A (1998) Finitely repeated games with finite automata. Math Oper Res 23(3):513–552CrossRefGoogle Scholar
  18. Neyman A (2008) Learning effectiveness and memory size. In: Discussion paper 476, Center for the Study of Rationality, Hebrew University, JerusalemGoogle Scholar
  19. Neyman A, Okada D (1999) Strategic entropy and complexity in repeated games. Games Econ Behav 29(1–2):191–223CrossRefGoogle Scholar
  20. Neyman A, Okada D (2000) Repeated games with bounded entropy. Games Econ Behav 30(2):228–247CrossRefGoogle Scholar
  21. Neyman A, Okada D (2009) Growth of strategy sets, entropy, and nonstationary bounded recall. Games Econ Behav 66(1):404–425CrossRefGoogle Scholar
  22. Neyman A, Spencer J (2010) Complexity and effective prediction. Games Econ Behav 69(1):165–168CrossRefGoogle Scholar
  23. Ornstein D (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4(3):337–352CrossRefGoogle Scholar
  24. Peretz R (2012) The strategic value of recall. Games Econ Behav 74(1):332–351CrossRefGoogle Scholar
  25. Peretz R (2013) Correlation through bounded recall strategies. Int J Game Theory 42(4):867–890CrossRefGoogle Scholar
  26. Shapira A (2007) Symmetric online matching pennies. PhD thesis, Hebrew University, JerusalemGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Olivier Gossner
    • 1
    • 2
  • Penélope Hernández
    • 3
  • Ron Peretz
    • 4
    Email author
  1. 1.École Polytechnique, CNRSParisFrance
  2. 2.London School of EconomicsLondonUK
  3. 3.University of ValenciaValenciaSpain
  4. 4.Bar Ilan University Ramat GanIsrael

Personalised recommendations