## Abstract

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 \({1- \epsilon}\) (for \({\epsilon \leq 1/2}\)), the value of the game repeated *n* times in parallel on independent inputs is at most \({(1- \epsilon)^{\Omega(\frac{\epsilon^2 n}{\ell})}}\), 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, \({n=O(\frac{t \ell}{\epsilon})}\) repetitions suffice to reduce the value of a free game from \({1- \epsilon}\) to \({(1- \epsilon)^t}\), and denoting the *input length* of the game by *m*, it follows that \({nm=O(\frac{t \ell m}{\epsilon})}\) 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 \({1- \epsilon}\) to \({(1- \epsilon)^t}\) 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.

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Shaltiel, R. Derandomized Parallel Repetition Theorems for Free Games.
*comput. complex.* **22, **565–594 (2013). https://doi.org/10.1007/s00037-013-0071-y

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### Keywords

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

### Subject classification

- 68Q15
- 68Q17