Strong Parallel Repetition Theorem for Free Projection Games

Abstract

The parallel repetition theorem states that for any two provers one round game with value at most 1 − ε (for ε < 1/2), the value of the game repeated n times in parallel is at most (1 − ε ³)^Ω(n/logs) where s is the size of the answers set [Raz98],[Hol07]. For Projection Games the bound on the value of the game repeated n times in parallel was improved to (1 − ε ²)^Ω(n) [Rao08] and was shown to be tight [Raz08]. In this paper we show that if the questions are taken according to a product distribution then the value of the repeated game is at most (1 − ε ²)^Ω(n/logs) and if in addition the game is a Projection Game we obtain a strong parallel repetition theorem, i.e., a bound of (1 − ε)^Ω(n).