Parallel repetition: simplifications and the no-signaling case

TL;DR

This paper simplifies the proof of parallel repetition theorems for certain non-communicating strategies in game theory, extending the results to no-signaling strategies and providing clearer insights into their behavior.

Contribution

The authors provide a simplified proof of Raz's parallel repetition theorem and extend the analysis to no-signaling strategies, broadening understanding of game repetition effects.

Findings

Simplified Raz's parallel repetition proof

Extended results to no-signaling strategies

Established bounds on winning probabilities in repeated games

Abstract

Consider a game where a refereed a referee chooses (x,y) according to a publicly known distribution P_XY, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value "a" and Bob responds with a value "b". Alice and Bob jointly win if a publicly known predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probability that Alice and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most v'^(n/log(s)), where s is the maximal number of possible responses from Alice and Bob in the initial game, and v' is a constant depending only on v. In this work, we simplify Raz's proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication among them.