We consider the natural extension of two-player nonlocal games to an arbitrary number of players. An important question for such nonlocal games is their behavior under parallel repetition. For two-player nonlocal games, it is known that both the classical and the non-signaling value of any game converges to zero exponentially fast under parallel repetition, given that the game is non-trivial to start with (i.e., has classical/non-signaling value <1). Very recent results show similar behavior for the quantum value of a two-player game under parallel repetition. For nonlocal games with three or more players, very little is known up to present on their behavior under parallel repetition; this is true for the classical, the quantum and the non-signaling value. In this work, we show a parallel repetition theorem for the non-signaling value of a large class of multi-player games, for an arbitrary number of players. Our result applies to all multi-player games for which all possible combinations of questions have positive probability; this class in particular includes all free games, in which the questions to the players are chosen independently. Specifically, we prove that if the original game has a non-signaling value smaller than 1, then the non-signaling value of the n-fold parallel repetition is exponentially small in n. Our parallel repetition theorem for multi-player games is weaker than the known parallel repetition results for two-player games in that the rate at which the non-signaling value of the game decreases not only depends on the non-signaling value of the original game (and the number of possible responses), but on the complete description of the game. Nevertheless, we feel that our result is a first step towards a better understanding of the parallel repetition of nonlocal games with more than two players.
Additional Metadata
MSC Quantum computation (msc 81P68)
THEME Information (theme 2)
Editor A. Harrow
Persistent URL dx.doi.org/10.4230/LIPIcs.TQC.2014.24
Project Quantum Cryptography , Networks
Conference Conference on Theory of Quantum Computation, Communication and Cryptography
Grant This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/024.002.003 - Networks
Citation
Buhrman, H.M, Fehr, S, & Schaffner, C. (2014). On the Parallel Repetition of Multi-Player Games: The No-Signaling Case. In A Harrow (Ed.), . doi:10.4230/LIPIcs.TQC.2014.24