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TR08-013 | 16th January 2008 00:00
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#### Parallel Repetition in Projection Games and a Concentration Bound

TR08-013
Authors:

Anup Rao
Publication: 28th February 2008 09:39

Downloads: 2048

Keywords:

**Abstract:**
In a two player game, a referee asks two cooperating players (who are

not allowed to communicate) questions sampled from some distribution

and decides whether they win or not based on some predicate of the

questions and their answers. The parallel repetition of the game is

the game in which the referee samples n independent pairs of

questions and sends corresponding questions to the players

simultaneously. If the players cannot win the original game with

probability better than (1-e), what's the best they can do in the

repeated game?

We improve earlier results of Raz and Holenstein, which showed

that the players cannot win all copies in the repeated game with

probability better than (1-e^3)^{\Omega(n/c)} (here c is the

length of the answers in the game), in the following ways:

We prove the bound (1-\e^2)^{Omega(n)} as long as n =

Omega(log(1/e)/e^2) and the game is a ``projection game'', the

type of game most commonly used in hardness of approximation

results. Our bound is independent of the answer length and has a

better dependence on e. By the recent work of Raz,

this bound is essentially tight (we might still hope to get rid of the

restriction on n). A consequence of this bound is that the Unique Games Conjecture of Khot is equivalent to:

[Unique Games Conjecture] For every delta,e >

0, there exists an alphabet size M(e) such that it is NP-hard to

distinguish a Unique Game with alphabet size M for which a 1-e^2

fraction of the constraints can be satisfied from one in which a

1-e^{1-delta} fraction of the constraints can be satisfied.

We prove a concentration bound for parallel repetition (of

general games) showing that for any constant 0<delta <e, the

probability that the players win a (1-e+delta) fraction of the

games in the parallel repetition is at most

exp(-Omega(delta^4 n/c)). An application of this is

in testing Bell Inequalities. Our result implies that the parallel

repetition of the CHSH game can be used to get an experiment that

has a very large classical versus quantum gap.