Derandomized Parallel Repetition Theorems for Free Games

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.