On the Parallel Repetition of MultiPlayer Games: The NoSignaling Case
Abstract
We consider the natural extension of twoplayer nonlocal games to an arbitrary number of players. An important question for such nonlocal games is their behavior under parallel repetition. For twoplayer nonlocal games, it is known that both the classical and the nonsignaling value of any game converges to zero exponentially fast under parallel repetition, given that the game is nontrivial to start with (i.e., has classical/nonsignaling value <1). Very recent results show similar behavior for the quantum value of a twoplayer 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 nonsignaling value. In this work, we show a parallel repetition theorem for the nonsignaling value of a large class of multiplayer games, for an arbitrary number of players. Our result applies to all multiplayer 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 nonsignaling value smaller than 1, then the nonsignaling value of the $n$fold parallel repetition is exponentially small in $n$. Our parallel repetition theorem for multiplayer games is weaker than the known parallel repetition results for twoplayer games in that the rate at which the nonsignaling value of the game decreases not only depends on the nonsignaling 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 arXiv:
 arXiv:1312.7455
 Bibcode:
 2013arXiv1312.7455B
 Keywords:

 Quantum Physics
 EPrint:
 12 pages, v2: no technical changes, extended related work, added grant acknowledgments