The complexity of interacting automata


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.

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


  1. 1.

    Or, equivalently, in the space of oblivious automata of size at most \({\mathcal {O}}(\log m)\), i.e. those that ignore the actions of the other agent.

  2. 2.

    Here and throughout, \(s\hbox { mod }m\) is defined as the number in [m] which is equal to s modulo m.

  3. 3.

    The function D is not the largest possible. Its concavification \(cav\,D\) is also possible as discussed in Sect. 3.8.3.

  4. 4.

    A perfect matching is a matching of size n.

  5. 5.

    Section 3.8.3 suggests that D can be replaced by \(cav\,D\) which implies that \(n-1\) can be attained whenever the two marginals of Q, \(Q^1\) and \(Q^2\), are uniform distributions supported on sets of size n.

  6. 6.

    Note that \(\mathrm {cor\,min\,max}\,G=\min _{x^1\in \Delta (A^1)}\max _{a^3\in A^3}g(x^1,\cdot ,a^3)\) .


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

    Article  Google 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 11–42

    Google Scholar 

  3. Bavly G, Neyman A (2014) Online concealed correlation and bounded rationality. Games Econ Behav 88:71–89

    Article  Google Scholar 

  4. Ben-Porath E (1993) Repeated games with finite automata. J Econ Theory 59(1):17–32

    Article  Google Scholar 

  5. Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn. Wiley, New York

    Google Scholar 

  6. De Bruijn NG (1946) A combinatorial problem. K Ned Akad Wet 49:758–764

  7. Gossner O, Hernández P (2003) On the complexity of coordination. Math Oper Res 28(1):127–140

    Article  Google Scholar 

  8. Gossner O, Hernández P (2006) Coordination through De Bruijn sequences. Oper Res Lett 34(1):17–21

    Article  Google Scholar 

  9. Gossner O, Tomala T (2006) Empirical distributions of beliefs under imperfect observation. Math Oper Res 31(1):13–30

    Article  Google Scholar 

  10. Gossner O, Tomala T (2007) Secret correlation in repeated games with imperfect monitoring. Math Oper Res 32(2):413–424

    Article  Google Scholar 

  11. Gossner O, Hernandez P, Neyman A (2006) Optimal use of communication resources. Econometrica 74(6):1603–1636

    Article  Google 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 131–157

  13. Kalai E, Stanford W (1988) Finite rationality and interpersonal complexity in repeated games. Econometrica 56(2):397–410

    Article  Google Scholar 

  14. Lehrer E (1988) Repeated games with stationary bounded recall strategies. J Econ Theory 46(1):130–144

    Article  Google Scholar 

  15. Neyman A (1985) Bounded complexity justifies cooperation in the finitely repeated prisoners’ dilemma. Econ Lett 19(3):227–229

    Article  Google Scholar 

  16. Neyman A (1997) Cooperation, repetition, and automata. Cooperation: game theoretic approaches. NATO ASI series F. Springer, New York, pp 233–255

    Google Scholar 

  17. Neyman A (1998) Finitely repeated games with finite automata. Math Oper Res 23(3):513–552

    Article  Google Scholar 

  18. Neyman A (2008) Learning effectiveness and memory size. In: Discussion paper 476, Center for the Study of Rationality, Hebrew University, Jerusalem

  19. Neyman A, Okada D (1999) Strategic entropy and complexity in repeated games. Games Econ Behav 29(1–2):191–223

    Article  Google Scholar 

  20. Neyman A, Okada D (2000) Repeated games with bounded entropy. Games Econ Behav 30(2):228–247

    Article  Google Scholar 

  21. Neyman A, Okada D (2009) Growth of strategy sets, entropy, and nonstationary bounded recall. Games Econ Behav 66(1):404–425

    Article  Google Scholar 

  22. Neyman A, Spencer J (2010) Complexity and effective prediction. Games Econ Behav 69(1):165–168

    Article  Google Scholar 

  23. Ornstein D (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4(3):337–352

    Article  Google Scholar 

  24. Peretz R (2012) The strategic value of recall. Games Econ Behav 74(1):332–351

    Article  Google Scholar 

  25. Peretz R (2013) Correlation through bounded recall strategies. Int J Game Theory 42(4):867–890

    Article  Google Scholar 

  26. Shapira A (2007) Symmetric online matching pennies. PhD thesis, Hebrew University, Jerusalem

Download references


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

Author information



Corresponding author

Correspondence to Ron Peretz.

Additional information

O. Gossner acknowledges financial support from Investissements d’Avenir (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047). P. Hernández acknowledges financial support from the Spanish Economy and Competitiveness Ministry (ECO2013-46550-R) and Generalitat Valenciana (PROMETEOII/2014/054).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gossner, O., Hernández, P. & Peretz, R. The complexity of interacting automata. Int J Game Theory 45, 461–496 (2016).

Download citation


  • Complexity
  • Automata
  • De Bruijn sequences
  • Bounded memory