When Can Limited Randomness Be Used in Repeated Games?

  • Pavel HubáčekEmail author
  • Moni Naor
  • Jonathan Ullman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9347)


The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times.

In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have \(\varOmega (n)\) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits.

When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then \(\varOmega (n)\) random bits remain necessary for equilibria to exist.


Limited randomness Repeated games Nash equilibrium 


  1. 1.
    Aumann, R.J., Hart, S.: Long cheap talk. Econometrica 71(6), 1619–1660 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput. 13(4), 850–864 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Budinich, M., Fortnow, L.: Repeated matching pennies with limited randomness. In: ACM EC 2011, pp. 111–118 (2011)Google Scholar
  4. 4.
    Dodis, Y., Halevi, S., Rabin, T.: A cryptographic solution to a game theoretic problem. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 112–130. Springer, Heidelberg (2000) CrossRefGoogle Scholar
  5. 5.
    Goldreich, O.: The Foundations of Cryptography. Basic Techniques, vol. 1. Cambridge University Press, Cambridge (2001) CrossRefzbMATHGoogle Scholar
  6. 6.
    Halpern, J.Y., Pass, R.: Algorithmic rationality: Game theory with costly computation. J. Econ. Theory (2014)Google Scholar
  7. 7.
    Halprin, R., Naor, M.: Games for extracting randomness. ACM Crossroads 17(2), 44–48 (2010)CrossRefGoogle Scholar
  8. 8.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hubáček, P., Naor, M., Ullman, J.: When can limited randomness be used in repeated games? CoRR abs/1507.01191 (2015).
  10. 10.
    Impagliazzo, R.: Pseudo-random generators for cryptography and for randomized algorithms. Ph.D. thesis, University of California, Berkeley (1992)Google Scholar
  11. 11.
    Impagliazzo, R., Luby, M.: One-way functions are essential for complexity based cryptography (extended abstract). In: FOCS 1989, pp. 230–235 (1989)Google Scholar
  12. 12.
    Kalyanaraman, S., Umans, C.: Algorithms for playing games with limited randomness. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 323–334. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  13. 13.
    Naor, M., Rothblum, G.N.: Learning to impersonate. In: ICML 2006, pp. 649–656 (2006)Google Scholar
  14. 14.
    Neyman, A., Okada, D.: Repeated games with bounded entropy. Games Econ. Behav. 30(2), 228–247 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yao, A.C.: Theory and applications of trapdoor functions (extended abstract). In: FOCS 1982, pp. 80–91 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.Columbia University Department of Computer ScienceNew YorkUSA

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