 
Summary: Strong Parallel Repetition Theorem for Free
Projection Games
Boaz Barak # Anup Rao + Ran Raz # Ricky Rosen § Ronen Shaltiel ¶
April 14, 2009
Abstract
The parallel repetition theorem states that for any two provers one round game with value
at most 1  # (for # < 1/2), the value of the game repeated n times in parallel is at most
(1# 3 ) n/ log s) 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# 2 )
# n) [Rao08]
and was shown to be tight [Raz08]. In this paper we show that if the questions are taken accord
ing to a product distribution then the value of the repeated game is at most (1  # 2 )
# n/ log s)
and if in addition the game is a Projection Game we obtain a strong parallel repetition theorem,
i.e., a bound of (1  #)
# n) .
1 Introduction
In a two provers one round game there are two provers and a verifier. The verifier selects randomly
(x, y) # X × Y , a question for each prover, according to some distribution PXY where X is the
questions set of prover 1 and Y is the questions set of prover 2. Each prover knows only the
