Derandomized Parallel Repetition Theorems for Free Games

Raz\´s parallel repetition theorem together with improvements of Holenstein shows that for any two-prover one-round game with value at most 1 - ∈ (for ∈ ≤ 1/2), the value of the game repeated n times in parallel on independent inputs is at most (1-∈)^{Ω(∈2n/ℓ)} 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, Rao, Raz, Rosen and Shaltiel. Consequently, n = O(^tℓ/_∈) repetitions suffice to reduce the value of a free game from 1 - ∈ to (1 - ∈)^t, and denoting the input length of the game by m, if follows that nm = O(^tℓm/_∈) 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. Thus, in terms of randomness complexity, correlated parallel repetition can reduce the value of free games at the "correct rate" when ℓ = O(m). Our technique uses strong extractors to "derandomize" a lemma of, and can be also used to derandomize a parallel repetition theorem of Parnafes, Raz and Wigderson for communication games in the special case that the game is free.