# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, October 29, 2013, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Felix Weissenberger

## Two-Prover Games for Parallel Repetition

In a **two-prover game**, a verifier chooses questions *(x,y)* according to a publicly known distribution. He sends *x* to Alice and *y* to Bob. They answer with *a(x)* and *b(y)*. The game is won if a publicly known predicate *V(x,y,a(x),b(y))* is satisfied. The value of the game is the winning probability if Alice and Bob play optimally. In the **parallel repetition** of a two-prover game, independent copies of the game are played in parallel, and it is won if and only if all copies are won.

Two-prover games and their parallel repetition are motivated by the error reduction of two prover one round proof systems, which are related to the PCP Theorem and inapproximability.

The best parallel repetition theorem states that for a game with value *1- ε*, for *ε > 0*, and answer alphabet size *2*^{s}, the value of its *k*-fold parallel repetition is at most *e*^{-f(ε,s) · k}, with f(ε,s)∈ Ω (ε^{3}/s). Moreover, there are two independent examples of games which show that *f(ε,s) ∈ O(ε*^{2}) and *f(ε,s) ∈ O(log(s)/s)*.

We present a parallel repetition theorem for **cycle games** where *f(ε,s)* is independent of *s*. In cycle games the questions are drawn from the uniform distribution and the question graph is a cycle.

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