A Strong Parallel Repetition Theorem for Projection Games on Expanders

The parallel repetition theorem states that for any Two Prover Game with value at most 1-€ (for € <; 1/2), the value of the game repeated n times in parallel is at most (1 - €³)^Ω(n/s), where s is the length of the answers of the two provers [24], [17]. For Projection Games, the bound on the value of the game repeated n times in parallel was improved to (1 - €²)^Ω(n) [23] and this bound was shown to be tight [25]. In this paper we study the case where the underlying distribution, according to which the questions for the two provers are generated, is uniform over the edges of a (bipartite) expander graph. We show that if λ is the (normalized) spectral gap of the underlying graph, the value of the repeated game is at most (1 - €²)^{Ω(c(λ)·n/s)}, where c(λ) = poly(λ); and if in addition the game is a projection game, we obtain a bound of (1 - €)^Ω(c(λ)·n), where c(λ) = poly(λ), that is, a strong parallel repetition theorem (when λ is constant). This gives a strong parallel repetition theorem for a large class of two prover games.