SPEAKER:
Ricky Rosen

TITLE:
A Strong Parallel Repetition Theorem for Projection Games on Expanders

Authors:
Ran Raz and Ricky Rosen

ABSTRACT:
The parallel repetition theorem states that for any Two
Prover Game with value at most 1-\eps (for \eps<1/2),
the value of the game repeated n times in parallel is at most
(1-\eps3)^{\Omega(n/s)}, where s is the length of the
answers of the two provers. For Projection
Games, the bound on the value of the game repeated n times in
parallel was improved to  (1-\eps2)^{\Omega(n)}
and this bound was shown to be tight.

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 \lambda is the (normalized) spectral gap of the
underlying graph, the value of the repeated game is at most
(1-\eps2)^{\Omega(c(\lambda) \cdot n/ s)}, where
c(\lambda) = \poly(\lambda); and if in addition the game is a
projection game, we obtain a bound of (1-\eps)^{\Omega(c(\lambda)
\cdot n)}, where c(\lambda) = \poly(\lambda), that is, a strong
parallel repetition theorem (when \lambda is constant).

This gives a strong parallel repetition theorem for a large class of
two prover games.