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.
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.
In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1−α and its repeated version Gn (in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this work the case of multiplayer non-signalling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players we prove a threshold theorem stating that the probability that non-signalling players win more than a fraction 1−α+β of the n games is exponentially small in nβ2, for every 0≤β≤α. For games with incomplete support we derive a similar statement, for a slightly modified form of repetition. The result is proved using a new technique, based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems.
In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1 - α and its repeated version Gn(in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this paper, the case of multiplayer non-signaling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players, we prove a threshold theorem stating that the probability that non-signaling players win more than a fraction 1-α+β of the n games is exponentially small in nβ2for every 0 ≤ β ≤ α. For games with incomplete support, we derive a similar statement for a slightly modified form of repetition. The result is proved using a new technique based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems.
We present a strong parallel repetition theorem for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our result is the first parallel repetition theorem for entangled games involving more than two players. Furthermore, our theorem applies to games where the players are allowed to output (possibly entangled) quantum states as answers. More specifically, let G be a k-player free game, with entangled value val∗(G)=1−ε. We show that the entangled value of the n-fold repetition of G, val∗(G⊗n), is at most (1−ε)Ω(n/k2). In the traditional setting of k=2 players, our parallel repetition theorem is optimal in terms of its dependence on ε and n. For an arbitrary number of players, our result is nearly optimal: for all k, we exhibit a k-player free game G and n>1 such that val∗(G⊗n)≥val∗(G)n/k. Hence, exponent of the repeated game value cannot be improved beyond Ω(n/k). Our parallel repetition theorem improves on the prior results of [Jain, et al. 2014] and [Chailloux, Scarpa 2014] in a number of ways: (1) our theorem applies to a larger class of games (arbitrary number of players, quantum outputs); (2) we demonstrate that strong parallel repetition holds for the entangled value of free games: i.e., the base of the repeated game value is 1−ε, rather than 1−ε2; and (3) there is no dependence of the repeated game value on the input and output alphabets of G. In contrast, it is known that the repeated game value of classical free games must depend on the output size. Thus our results demonstrate a seperation between the behavior of entangled games and classical games.
We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val∗(G)=1−ε, the n-fold repetition of G has entangled value val∗(G⊗n) at most (1−ε3/2)Ω(n/sk4), where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G⊗n)≤(1−ε2)Ω(n/s), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1−ε2)Ω(n/sk2). We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n≥1 such that val∗(G⊗n)≥val∗(G)n/k. Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.
We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.
We investigate the value of parallel repetition of one-round games with any number of players k≥2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are connected in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.