Two major open problems regarding the parallel repetition of games are whether an analogue of Raz's parallel-repetition theorem holds for (a) games with more than two players, and (b) games with quantum players using entanglement. We make progress on both problems: we introduce a class of games we call anchored, and prove exponential-decay parallel repetition theorems for anchored games in the multiplayer and entangled-player settings. We introduce a simple transformation on games called anchoring and show that this transformation turns any game into an anchored game. Together, our parallel repetition theorem and our anchoring transformation provide a simple and efficient hardness-amplification technique in both the classical multiplayer and quantum settings.
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.
The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game G repeated n times in parallel is at most c_G*n^{-1/4}*log(n) for a constant c_G depending on G, provided that the entangled value of G is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.
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 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 obtain quantitative estimates on the decay of the multiplayer optimal value under parallel repetition. In comparison to a previous work of the author in 2025 (arXiv: 2508.09380) which sought to generalize dependency-breaking and anchoring variables from two-player Quantum games, being able to establish quantitative estimates on the decay of the optimal value of a multiplayer game under parallel repetition is of interest to establish under different assumptions. Specifically, independently of the dependency-breaking and anchoring variables that have previously been employed to remove correlations from entangled information shared between Alice and Bob (hence removing dependencies), monotonic concave functions can be used in place of such variables to obtain rates of decay on the optimal value. The game-theoretic setting with two players was first analyzed with monotonic concave functions by Lanzenberger and Maurer. For
qi,xi>0 ∀1≤i≤N where $N > 0 $ is the total number of players we adddress an open question raised in their work regarding potential generalizations of two-player monotonic concave functions, through amplification functions of the form
ΨMult≡Ψ=N−1≤i≤N∏exp[−qixi], which in the multiplayer game-theoretic setting have more intricate combinatorial structures.