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SIAM Journal on Computing

Parallel Repetition in Projection Games and a Concentration Bound

Anup Rao

Published May 17, 2008• 150 citations
Publisher

Abstract

A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than $(1-\epsilon)$, what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141–172], who showed that the players cannot win all copies in the repeated game with probability better than $(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}$ (here c is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is $(1-\epsilon/2)^{\Omega(\epsilon n)}$ as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant $0<\delta <\epsilon$, the probability that the players win a $(1-\epsilon+\delta)$ fraction of the games in the parallel repetition is at most $\exp\left(-\Omega_{\epsilon}(\delta^3 n/c)\right)$ (here the constant may depend on $\epsilon$); our result has applications to testing Bell inequalities, since it implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap. Our first bound is independent of the answer length and has a better dependence on $\epsilon$. By the recent work of Raz [Proceedings of the $49$th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 369–373], this bound is tight. Our bound gives a generic way to improve the soundness of a probabilistically checkable proof (PCP), in a way that is independent of the answer length of the PCP. Using it, for every k, one can convert any q query PCP with answer length c, size $sc$, and soundness $(1-\epsilon)$ into a two-query PCP with answer length $ck$, size $O(ck (2s)^k)$, and soundness $(1-\epsilon/2q)^{\Omega(\epsilon k/q)}$. Another consequence of our bound is that the unique games conjecture of Khot [Proceedings of the $34$th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] can now be shown to be equivalent to the following a priori weaker conjecture: There is an unbounded increasing function $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that for every $\epsilon> 0$, there exists an alphabet size $M(\epsilon)$ for which it is NP-hard to distinguish a unique game with alphabet size M in which a $(1-\epsilon^2)$ fraction of the constraints can be satisfied from one in which a $(1-\epsilon f(1/\epsilon))$ fraction of the constraints can be satisfied.

DOI
10.1137/080734042

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Parallel repetition: simplifications and the no-signaling case

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$ \newcommand{\vbar}{\bar{v}} $ Consider a game where a referee chooses $(x,y)$ according to a publicly known distribution, 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 $Q(x,y,a,b)$ is satisfied. Assume that the maximum probability that Alice and Bob can win in such a game 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 $\vbar^{\tfrac{n}{\log(s)}}$, where $s$ is the maximal number of possible responses from Alice and Bob in the initial game, and $\vbar<1$ 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 between them.

arXiv:cs/0607139•Theory of Computing•Jan 1, 2009•123 citations

A Counterexample to Strong Parallel Repetition

Ran Raz

The parallel repetition theorem states that, for any two-prover game with value $1-\epsilon$ (for, say, $\epsilon\leq1/2$), the value of the game repeated in parallel n times is at most $(1-\epsilon^c)^{\Omega(n/s)}$, where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803]. Several researchers asked whether this bound could be improved to $(1-\epsilon)^{\Omega(n/s)}$; this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size m, a two-prover game with value $1-1/2m$. We show that the value of the odd cycle game repeated in parallel n times is at least $1-(1/m)\cdot O(\sqrt{n})$. This implies that, for large enough n (say, $n\geq\Omega(m^2)$), the value of the odd cycle game repeated in parallel n times is at least $(1-1/4m^2)^{O(n)}$. Thus the following hold. 1. For parallel repetition of general games, the bounds of $(1-\epsilon^c)^{\Omega(n/s)}$ given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767–775] are of the right form, up to determining the exact value of the constant $c\geq2$. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of $(1-\epsilon^2)^{\Omega(n)}$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1–10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of $1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least $1-\epsilon^2$ satisfiable from instances that are at most $1-(2/\pi)\cdot\epsilon$ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.

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Concentration Bounds from Parallel Repetition Theorems

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This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions: \n \n- A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances). \n \n- Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies. \n \n- A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case.

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Consider a game where a refereed chooses (x,y) according to a publiclyknown distribution PXY, sends x to Alice, and y to Bob. Withoutcommunicating with each other, Alice responds with a value "a" and Bobresponds with a value "b". Alice and Bob jointly win if a publiclyknown predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probabilitythat 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.

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Parallel Repetition for Entangled Games via Conditional Relative Entropy

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quant-pharXiv:2610.21410•Oct 16, 2026•0 citations
A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1−ϵ)(1-\epsilon)(1−ϵ), what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141–172], who showed that the players cannot win all copies in the repeated game with probability better than (1−ϵ/2)Ω(nϵ2/c)(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}(1−ϵ/2)Ω(nϵ2/c) (here c is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is (1−ϵ/2)Ω(ϵn)(1-\epsilon/2)^{\Omega(\epsilon n)}(1−ϵ/2)Ω(ϵn) as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0<δ<ϵ0<\delta <\epsilon0<δ<ϵ, the probability that the players win a (1−ϵ+δ)(1-\epsilon+\delta)(1−ϵ+δ) fraction of the games in the parallel repetition is at most exp⁡(−Ωϵ(δ3n/c))\exp\left(-\Omega_{\epsilon}(\delta^3 n/c)\right)exp(−Ωϵ​(δ3n/c)) (here the constant may depend on ϵ\epsilonϵ); our result has applications to testing Bell inequalities, since it implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap. Our first bound is independent of the answer length and has a better dependence on ϵ\epsilonϵ. By the recent work of Raz [Proceedings of the 494949th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 369–373], this bound is tight. Our bound gives a generic way to improve the soundness of a probabilistically checkable proof (PCP), in a way that is independent of the answer length of the PCP. Using it, for every k, one can convert any q query PCP with answer length c, size scscsc, and soundness (1−ϵ)(1-\epsilon)(1−ϵ) into a two-query PCP with answer length ckckck, size O(ck(2s)k)O(ck (2s)^k)O(ck(2s)k), and soundness (1−ϵ/2q)Ω(ϵk/q)(1-\epsilon/2q)^{\Omega(\epsilon k/q)}(1−ϵ/2q)Ω(ϵk/q). Another consequence of our bound is that the unique games conjecture of Khot [Proceedings of the 343434th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] can now be shown to be equivalent to the following a priori weaker conjecture: There is an unbounded increasing function f:R+→R+f:\mathbb{R}^+\rightarrow \mathbb{R}^+f:R+→R+ such that for every ϵ>0\epsilon> 0ϵ>0, there exists an alphabet size M(ϵ)M(\epsilon)M(ϵ) for which it is NP-hard to distinguish a unique game with alphabet size M in which a (1−ϵ2)(1-\epsilon^2)(1−ϵ2) fraction of the constraints can be satisfied from one in which a (1−ϵf(1/ϵ))(1-\epsilon f(1/\epsilon))(1−ϵf(1/ϵ)) fraction of the constraints can be satisfied.
$ \newcommand{\vbar}{\bar{v}} $ Consider a game where a referee chooses (x,y)(x,y)(x,y) according to a publicly known distribution, sends xxx to Alice, and yyy to Bob. Without communicating with each other, Alice responds with a value aaa and Bob responds with a value bbb. Alice and Bob jointly win if a publicly known predicate Q(x,y,a,b)Q(x,y,a,b)Q(x,y,a,b) is satisfied. Assume that the maximum probability that Alice and Bob can win in such a game is v<1v<1v<1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated nnn times in parallel, then the probability that Alice and Bob win all games simultaneously is at most $\vbar^{\tfrac{n}{\log(s)}}$, where sss is the maximal number of possible responses from Alice and Bob in the initial game, and $\vbar<1$ is a constant depending only on vvv. 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 between them.
The parallel repetition theorem states that, for any two-prover game with value 1−ϵ1-\epsilon1−ϵ (for, say, ϵ≤1/2\epsilon\leq1/2ϵ≤1/2), the value of the game repeated in parallel n times is at most (1−ϵc)Ω(n/s)(1-\epsilon^c)^{\Omega(n/s)}(1−ϵc)Ω(n/s), where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803]. Several researchers asked whether this bound could be improved to (1−ϵ)Ω(n/s)(1-\epsilon)^{\Omega(n/s)}(1−ϵ)Ω(n/s); this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size m, a two-prover game with value 1−1/2m1-1/2m1−1/2m. We show that the value of the odd cycle game repeated in parallel n times is at least 1−(1/m)⋅O(n)1-(1/m)\cdot O(\sqrt{n})1−(1/m)⋅O(n​). This implies that, for large enough n (say, n≥Ω(m2)n\geq\Omega(m^2)n≥Ω(m2)), the value of the odd cycle game repeated in parallel n times is at least (1−1/4m2)O(n)(1-1/4m^2)^{O(n)}(1−1/4m2)O(n). Thus the following hold. 1. For parallel repetition of general games, the bounds of (1−ϵc)Ω(n/s)(1-\epsilon^c)^{\Omega(n/s)}(1−ϵc)Ω(n/s) given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767–775] are of the right form, up to determining the exact value of the constant c≥2c\geq2c≥2. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of (1−ϵ2)Ω(n)(1-\epsilon^2)^{\Omega(n)}(1−ϵ2)Ω(n) given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1–10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of 1−(1/m)⋅Ω~(n)1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})1−(1/m)⋅Ω~(n​) given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least 1−ϵ21-\epsilon^21−ϵ2 satisfiable from instances that are at most 1−(2/π)⋅ϵ1-(2/\pi)\cdot\epsilon1−(2/π)⋅ϵ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.
This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions: \n \n- A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances). \n \n- Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies. \n \n- A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case.
Consider a game where a refereed chooses (x,y) according to a publiclyknown distribution PXY, sends x to Alice, and y to Bob. Withoutcommunicating with each other, Alice responds with a value "a" and Bobresponds with a value "b". Alice and Bob jointly win if a publiclyknown predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probabilitythat 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.
We give a proof of the cubic parallel-repetition bound for finite two-player entangled games using quantum conditional relative entropy. For a game with entangled value 1−ε<11-\varepsilon<11−ε<1 and answer alphabets A,B\mathcal A,\mathcal BA,B, its nnn-fold repeated value is at most exp⁡[−Ω(ε3n/L)],L:=max⁡{1,log⁡(∣A∣∣B∣)}, \exp[-Ω(\varepsilon^3n/L)],\qquad L:=\max\{1,\log(|\mathcal A||\mathcal B|)\}, exp[−Ω(ε3n/L)],L:=max{1,log(∣A∣∣B∣)}, for every n≥1n\ge1n≥1. This recovers Song's cubic rate. We also prove a threshold bound: the probability of winning a fraction at least 1−ε+δ1-\varepsilon+δ1−ε+δ is at most exp⁡[−Ω(δ5n/(ε2L))]\exp[-Ω(δ^5n/(\varepsilon^2L))]exp[−Ω(δ5n/(ε2L))] for 0<δ≤ε0<δ\le\varepsilon0<δ≤ε. Our information charge is the decrease in relative entropy when a question reference is discarded. Our comparison states preserve the original conditional question law. A chain rule bounds the coordinate charges using the cost of conditioning on checked wins and retaining checked answers. Our main lemma converts small charges into nearby pure-state descriptions and local corrections, each depending only on one player's question. Together with Song's sampler, these enable approximate local preparation of the conditioned workspace.