Derandomized Parallel Repetition Theorems for Free Games

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Raz's parallel repetition theorem (SIAM J Comput 27(3):763-803, 1998) together with improvements of Holenstein (STOC, pp 411-419, 2007) shows that for any two-prover one-round game with value at most 1-ε (for ε ≤ 1/2), the value of the game repeated n times in parallel on independent inputs is at most, where ℓ is the answer length of the game. For free games (which are games in which the inputs to the two players are uniform and independent), the constant 2 can be replaced with 1 by a result of Barak et al. (APPROX-RANDOM, pp 352-365, 2009). Consequently, n = O(tℓ/ε) repetitions suffice to reduce the value of a free game from 1-ε to (1-ε)t, and denoting the input length of the game by m, it follows that nm = O(tℓm/ε) random bits can be used to prepare n independent inputs for the parallel repetition game. In this paper, we prove a derandomized version of the parallel repetition theorem for free games and show that O(t(m + ℓ)) random bits can be used to generate correlated inputs, such that the value of the parallel repetition game on these inputs has the same behavior. That is, it is possible to reduce the value from 1-ε to (1-ε)t while only multiplying the randomness complexity by O(t) when m = O(ℓ). Our technique uses strong extractors to "derandomize" a lemma of Raz and can be also used to derandomize a parallel repetition theorem of Parnafes et al. (STOC, pp 363-372, 1997) for communication games in the special case that the game is free.

Original languageEnglish
Pages (from-to)565-594
Number of pages30
JournalComputational Complexity
Issue number3
StatePublished - Sep 2013

Bibliographical note

Funding Information:
I am grateful to Avi Wigderson for introducing me to this problem and for many discussions. This research was supported by BSF grant 2004329 and ISF grant 686/07. I thank anonymous referees for helpful comments. A preliminary version of this paper appeared in CCC 2010.


  • 2-prover systems
  • Parallel repetition
  • derandomization
  • randomness extractors

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Mathematics
  • Computational Theory and Mathematics
  • Computational Mathematics


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