Non-signalling parallel repetition using de Finetti reductions
aa r X i v : . [ qu a n t - ph ] N ov Non-signalling parallel repetition using de Finetti reductions
Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick Institute for Theoretical Physics, ETH-Z¨urich, CH-8093, Z¨urich, Switzerland Department of Computing and Mathematical Sciences,California Institute of Technology, Pasadena, CA, USA
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 G n (in which n games are played together, in parallel), can the players use strategies that are substantially betterthan ones in which each game is played independently? This question is relevant in physics for thestudy of correlations and plays an important role in computer science in the context of complexityand 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-zeroprobability to be asked) with any number of players we prove a threshold theorem stating thatthe probability that non-signalling players win more than a fraction 1 − α + β of the n games isexponentially small in nβ , for every 0 ≤ β ≤ α . For games with incomplete support we derivea similar statement, for a slightly modified form of repetition. The result is proved using a newtechnique, based on a recent de Finetti theorem, which allows us to avoid central technical difficultiesthat arise in standard proofs of parallel repetition theorems. I. INTRODUCTION
1. Multiplayer games and parallel repetition
Multiplayer games are relevant in many areas of both theoretical physics and theoretical computerscience. In physics, multiplayer games give an intuitive way to study the role and implications ofentanglement and correlations, e.g., in the setting of Bell inequalities [1, 2]. In computer science suchgames arise in the context of complexity theory [3–5] and cryptography [6, 7].In a game G , a referee asks each of the cooperating players a question chosen according to a givenprobability distribution. The players then need to supply answers which fulfil a pre-defined requirementaccording to which the referee accepts or rejects the answers. In order to do so, they can agree on astrategy beforehand, but once the game begins communication between the players is not allowed. Ifthe referee accepts their answers the players win. The goal of the players is, of course, to maximisetheir winning probability in the game.According to the field of interest, one can analyse any game under different restrictions on the players(in addition to not being allowed to communicate). In classical computer science the players are usuallyassumed to have only classical resources, or strategies. That is, they can use only local operations andshared randomness. In contrast, one can also consider quantum strategies: before the game starts theplayers create a multipartite quantum state that can be shared among them. When the game beginseach player locally measures their own part of the state and base the answer on their measurementresult. It is well-known that sharing quantum entanglement can significantly increase the winningprobability in some games [2, 8].Another, more general, type of strategies are those where the players can use any type of correlationsthat do not allow them to communicate, also called non-signalling correlations. That is, the only restriction on the players is that they are not allowed to communicate (as will be defined formallylater).Considering the non-signalling case is interesting for several reasons. A first reason is to minimisethe set of assumptions to the mere necessary one. Indeed, if the players are allowed to communicate bysending signals they can win any game. Minimising the set of assumptions can be useful in cryptographywhen one wishes to get the strongest result possible, i.e., one where the attack strategies of maliciousparties are only restricted minimally (as in [9–11] for example). In theoretical physics, non-signallingcorrelations enable the study of generalised theories possibly beyond quantum theory. It is also impor-tant to mention that, due to their linearity, the non-signalling constraints are often easier to analysethan the quantum or the classical constraints. Therefore, even if additional constraints hold, focusingon the non-signalling ones serves as a way to get first insights into a given problem.One of the most interesting questions regarding multiplayer games is the question of parallel repe-tition. Given a game G with optimal winning probability 1 − α (using either classical, quantum, ornon-signalling strategies), we are interested in analysing the winning probability in the repeated game G n . In G n the referee gives the players n independent tuples of questions at once, to which the playersshould reply. The most natural winning criterion is that the players answer a certain fraction 1 − α + β of the n game instances correctly, and one can then ask what is the probability that the players succeedas a function of β , as the number of repetitions n increases.The players can always use the trivial independent and identically distributed (i.i.d.) strategy: theyjust answer each of the n questions independently according to the optimal one-game strategy. In thiscase the fraction of successful answers is highly concentrated around 1 − α (alternatively, the probabilityto win all games simultaneously is (1 − α ) n ). But can they do significantly better?If correlated strategies for G n are not substantially better than independent ones, even in an asymp-totic manner, we learn that “one cannot fight independence with correlations”. As long as the questionsare asked, and the answers are verified, in an independent way, creating correlations between the dif-ferent answers using a correlated strategy cannot help much. The resulting threshold theorem can thenbe used, for example, when considering a series of Bell violation experiments performed in parallel, orfor hardness amplification in complexity theory and security amplification in classical, quantum andnon-signalling cryptography.
2. Related work
Raz was the first to show in [12] an exponential parallel repetition theorem for classical two-playergames. That is, he showed that if the classical optimal winning probability in a game G is smallerthan 1, then the probability to win all the games in the repeated game G n , using a classical strategy,decreases exponentially with the number of repetitions n . Raz’s result was then improved and adaptedto the non-signalling case by Holenstein [13]. Another improvement was made by Rao in [14], wherea threshold theorem for the classical two-player case was proven: Rao showed that the probability towin more than a fraction 1 − α + β of the games for any β >
3. de Finetti theorems in the context of parallel repetition
The main difficulty in proving a parallel repetition result comes from the, almost arbitrary, correla-tions between the different questions-answers pairs in the players’ strategy for G n : as the players getall the n tuples of questions together they can answer them in a correlated way. In most of the knownparallel repetition results (e.g., [12–15]) the main idea of the proof is to bound the winning probabilityfor some of the questions, conditioned on winning the previous questions. However, as the strategyitself introduces correlations between the different tuples of questions, a large amount of technical workis devoted to dealing with the effect of conditioning on the event of winning the previous questions.When considering the correlations in a strategy for the repeated game there is one type of symmetrywhich one can take advantage of, but which is usually virtually ignored – permutation invariance. Asthe game G n itself is invariant under joint permutation of the tuples of questions and answers, wecan restrict our attention to permutation-invariant strategies without loss of generality. Permutation-invariant strategies are strategies which are indifferent to the ordering of the questions given by thereferee. That is, the probability of answering a specific set of tuples of questions correctly does notdepend on the ordering of the tuples.Once we restrict our attention to permutation-invariant strategies, de Finetti theorems seem like anatural tool to leverage for the analysis. A de Finetti theorem is any type of theorem which relatespermutation-invariant states to a more structured state, having the form of a convex combination ofi.i.d. states, called a de Finetti state. The specific relation between the states depends on the typeof theorem. The first de Finetti theorem [19] established that the collection of infinitely exchangeablesequences, in other words those distributions on infinite strings that are invariant under all permutations,exactly coincides with the collection of all convex combinations of i.i.d distributions. Subsequent resultsestablish quantitative bounds on the distance of any permutation-invariant state, or subsystems thereof,from some de Finetti state or an approximation of a de Finetti state [20–25]. A different form ofstatement, also called a de Finetti reduction, relates any permutation-invariant state to an explicit deFinetti state by an inequality relation [26, 27]. The common feature of all de Finetti theorems is thatthey enable a substantially simplified analysis of information-processing tasks by exploiting permutationinvariance symmetry. Indeed, quantum de Finetti theorems play a significant role in many quantuminformation problems such as quantum cryptography [26, 28], tomography [29], channel capacities [30]and complexity [25].In the context of games and strategies, de Finetti theorems suggest one may be able to reduce theanalysis of general permutation-invariant strategies to the analysis of a de Finetti strategy, i.e., a convexcombination of i.i.d. strategies. As the behaviour of i.i.d. strategies is trivial under parallel repetition, areduction of this type could simplify the analysis of parallel repetition theorems and threshold theorems.Yet, de Finetti theorems were not used in the past in this context, and for a good reason. The manyversions of quantum de Finetti theorems (e.g., [23, 26]) could not have been used as they depend on thedimension of the underlying quantum strategies, while in the quantum multiplayer game setting onedoes not wish to restrict the dimension. Non-signalling de Finetti theorems, as in [31, 32], were also notapplicable for non-signalling parallel repetition theorems, as they restrict almost completely the typeof allowed correlations in the strategies for the repeated game by assuming very strict non-signallingconstraints between the different repetitions, i.e., between the different questions-answers pairs.In this work we use the recent de Finetti theorem of [27], which imposes no assumptions at allregarding the structure of the strategies (apart from permutation invariance), and is therefore applicablein the context of parallel repetition. This allows us to devise a proof technique which is completelydifferent from the known proofs of parallel repetition results. In particular, at least in the non-signallingcase presented here, the conditioning problem described at the beginning of this section disappearscompletely and the number of players does not play a role in the proof structure. A. Results and contributions
The main result presented in this work is a threshold theorem (also called a concentration bound)for the n -fold repetition of any m -player complete-support game, in which the players are allowed toshare any non-signalling strategy. A game is said to have complete-support if all possible combinationsof questions to the players have some non-zero probability of being asked. Denote by w ns the optimalnon-signalling winning probability in a game G . We prove the following theorem. Theorem 1.
For any complete-support game G with w ns = 1 − α there exist C ( G, n ) and C ( G ) , where C ( G, n ) is polynomial in the number of repetitions n , such that for every < β ≤ α and large enough n , the probability that non-signalling players win more than a fraction − α + β of the n questions inthe repeated game G n is at most C ( G, n ) exp (cid:2) −C ( G ) nβ (cid:3) . That is, for sufficiently many repetitions the probability to win more than a fraction 1 − α + β of the n games is exponentially small. The constant C ( G, n ) is such that C ( G, n ) < m |Q||A| ( n + 1) |Q||A|− where m is the number of players, and |Q| and |A| are the number of possible questions and answers,respectively, in G . C ( G ) is a finite constant that can be computed by solving the polynomial-size linearprogram given in Equation (5). A sufficient condition on the number of repetitions for the bound inthe theorem to hold is n = Ω (cid:0) |Q||A| C β ln ( |Q||A| C β ) (cid:1) . We refer to Equation (12), and the choice ofconstants made around Equation (25), for a more precise bound.There are two main differences between the exponential bound given in the threshold theorem of [15](Theorem 15 therein) and the bound we give here. First, while our bound suffers from the polynomialdependency on the number of repetitions in C ( G, n ) (which is inherent to the use of a de Finetti Depending on the context, a state can be a probability distribution, a quantum density operator or a conditionalprobability distribution. reduction), there is no such dependency in [15]. As the number of repetitions goes to infinity, however,the exponential factor quickly dominates. Both our constant C ( G ) and the constant µ in Theorem 15from [15] depend on the size of the game through a certain linear program (see the proof of Lemma 27and the discussion that follows it in this paper, and the proof of Proposition 18 in [15]), making adirect comparison difficult. Another point of comparison between the bounds is the dependency on β :we obtain the optimal (as follows from optimal formulations of the Chernoff bound) dependency β ,as compared to β in [15]. As far as we are aware, this is the first threshold theorem where optimaldependency on β is achieved (see also [14]).Theorem 1 applies to complete-support games. The result is extended in two different directions.First, based on ideas from [33], we show in Appendix A 1 that when considering two-player games without complete-support Theorem 1 still holds. Second, for general multiplayer games we consider inAppendix A 2 a small modification of the repetition procedure. Instead of the usual parallel repetitionprocedure, in which n tuples of questions are chosen according to the game distribution Q , we changethe distribution of questions in the repeated game by sometimes (with small positive probability η )asking the players a tuple of questions q which does not appear in the original game G . We callsuch questions “dummy questions”; for these questions any answer from the players is accepted. Theremaining questions, for which Q ( q ) >
0, are called the “real questions” and the modified game isdenoted by ˜ G n . We prove the following threshold theorem: Theorem 2.
For any game G with w ns = 1 − α there exist C ( G, n ) and C ( G ) , where C ( G, n ) ispolynomial in the number of repetitions n , such that for every < β ≤ α and large enough n , theprobability that non-signalling players win more than a fraction − α + β of the real questions in themodified repeated game ˜ G n is at most C ( G, n ) exp (cid:2) −C ( G ) nβ (cid:3) . The constants C ( G, n ) and C ( G ) have the same form as in Theorem 1, but they now depend alsoon the perturbation η of the original questions distribution. For more details on the definition of ˜ G n and the proof of Theorem 2 see Appendix A 2.A similar modification was previously considered in both classical [34] and quantum [35] parallelrepetition theorems, where the repetitions in which dummy questions are selected were called “confusionrounds”. For many applications this modification is harmless, especially as the success probability of“honest” players is not affected by it. However, it is important to note that Theorem 2 only holds forthe modified form of repetition of the original game.In addition to the bounds themselves our, perhaps most important, contribution in this work is the,arguably simpler, proof technique. While most of the known parallel repetition results build on theproof technique of [12] we give a completely different proof, with ideas based on de Finetti theoremsand tomography (as explained in the next section). Our proof technique allows us to avoid the usualdifficulties which arise in proofs of parallel repetition theorems, such as conditioning on some of thequestions and answers or considering an arbitrary number of players. In this sense our proof can beseen as more natural than previous proofs, and therefore more likely to be extendable to the classicaland quantum multiplayer cases as well. B. Proof idea and techniques
The goal of this section is to give the reader an intuitive understanding of the proof idea and tech-niques. The formal and more technical implementations of these ideas are given in the following sections.Nevertheless, the following two definitions are needed.
Definition 3 (Multiplayer game) . An m -player game G = ( Q , A , Q, R ) is defined by a set of possibletuples of questions Q together with a probability distribution Q : Q → [0 ,
1] (according to which thereferee choses the questions) over it, a set of possible tuples of answers A and a winning condition R : Q × A → { , } . An m -tuple of questions q = ( q , q , . . . , q m ) ∈ Q describes the questions given tothe different players. Similarly an m -tuple of answers a = ( a , a , . . . , a m ) ∈ A describes the answersgiven by the different players. Definition 4 (Strategy) . A strategy for an m -player game G = ( Q , A , Q, R ) is a conditional probabilitydistribution O A | Q : A × Q → [0 , P a O A | Q ( a | q ) = 1 for all q ∈ Q . Similarly, a strategy for arepeated game G n is a conditional probability distribution denoted by P ~A | ~Q : A n × Q n → [0 , G are denoted by O A | Q and strategies for the repeatedgame G n are denoted by P ~A | ~Q .
1. Permutation invariance and de Finetti theorems
The first trivial, but crucial, observation made is that when considering strategies for the repeatedgame, one can concentrate without loss of generality on permutation-invariant strategies. Permutation-invariant strategies are indifferent to the ordering of the tuples of questions given by the referee. That is,the referee can ask the players to answer q , q , q or q , q , q (each q i is an m -tuple); in both cases thewinning probability will be the same if the players are using a permutation-invariant strategy. Note thatthe permutation changes only the order of the tuples of questions. In particular, the players themselvesare not being permuted and the questions of all players are permuted in exactly the same way (seeDefinition 21 and Lemma 22 for the formal argument).Considering only permutation-invariant strategies allows us to use the de Finetti theorem of [27]which relates any permutation-invariant strategy to a de Finetti strategy. The exact statement of thede Finetti theorem will only be relevant later. For now, using just the intuition of de Finetti theorems,one can think of any permutation-invariant strategy as being a convex combination of i.i.d. strategies.That is , P ~A | ~Q ≈ Z O ⊗ nA | Q dO A | Q (1)where dO A | Q is some measure on the space of one-game strategies and O ⊗ nA | Q is a product of n identicalstrategies O A | Q .Unfortunately, the convex combination itself (meaning, the measure dO A | Q ) is unknown. Moreover,even though we assume that the strategy P ~A | ~Q does not allow the m players to communicate, i.e., it isnon-signalling, the convex combination might still include signalling parts, i.e., signalling O A | Q . Indeed,in general, a convex combination of signalling strategies can still be non-signalling.For the non-signalling parts of the convex combination one can easily prove a strong parallel repetitionor threshold theorem. These parts are just i.i.d. non-signalling strategies. The only thing which is left toprove is therefore that the signalling part of the convex combination of Equation (1) has an exponentiallysmall weight . We find this question interesting by itself, and of course, the same question can be askedin the classical and quantum case – given a classical or quantum strategy P ~A | ~Q , what is the weight ofthe non-classical or non-quantum i.i.d. parts in the convex combination?
2. Bounding the signalling part
As the convex combination itself in Equation (1) is unknown, one cannot just calculate the weightof the signalling part. We therefore take a more operational approach, following ideas from quantumtomography [29].Consider a particular (unknown) part O ⊗ nA | Q of the convex combination and divide the n copies of thestrategy O A | Q into two groups – a test group consisting of n/ n copies, and a game group of n/ EST A | Q of the strategy O A | Q ,which will then help us in proving our claims.More specifically, we are interested in knowing whether O A | Q is signalling or not (if it is non-signallingthen its winning probability in G is obviously bounded by the optimal non-signalling winning probability1 − α ). For this we define a signalling measure and an operational (and hypothetical) signalling test.Given questions and answers which are distributed according to the n/ A | Q and Q , thesignalling test will create an estimation O EST A | Q and calculate its signalling value. If the signalling valueis above a certain threshold the test will accept, and otherwise it will reject.In order to bound the weight of the signalling part in Equation (1) one can bound the probability thatthe signalling test accepts. To prove that the acceptance probability is exponentially small we use acombination of two lemmas, which we call the weak and the strong lemma. These lemmas are based ona special guessing game that we construct and on applications of the de Finetti theorem. Both lemmas We emphasise once again that this is not a quantitive statement that we claim to be correct. This is just useful as anintuitive way of understanding the proof idea. As mentioned above, this statement does not hold for an arbitrary decomposition of a non-signalling strategy as aconvex combination of other strategies. We will crucially use the fact that each term in the convex combination has ani.i.d. structure. together show that if the probability of the test accepting is too high, then the original strategy P ~A | ~Q must have been signalling – a contradiction.
3. From intuition to practice
In practice, the de Finetti theorem [27] is an inequality relation between any permutation-invariantstrategy and a given de Finetti strategy (see Lemma 23) which does not imply Equation (1). Asa consequence, considering the test copies and game copies as above does not directly make sense.Nevertheless, we can follow similar ideas by considering the questions-answers pairs in a specific instanceof the repeated game. That is, every time the game is played using a strategy P ~A | ~Q , we divide the data,the questions and answers, of this specific run into two groups – test data and game data, consistingof n/ II. PRELIMINARIES
Throughout the proof many constants and parameters are used. For convenience, apart from in-troducing them when necessary, we list all of them together with their role in Table I in the end ofSection V.We use the letters q, r and s to denote tuples of questions and a and b to denote tuples of answers.In the following we define the notation using only q and a . A. Games and strategies
In this work we consider a general m -player game G = ( Q , A , Q, R ) as defined in Definition 3 inthe previous section. A strategy for a game G is described by a conditional probability distributionO A | Q : A × Q → [0 ,
1] as defined in Definition 4. For the joint questions-answers distribution we useO AQ = Q × O A | Q . Definition 5 (Winning probability) . The winning probability of a strategy O A | Q in game G =( Q , A , Q, R ) is given by w (cid:0) O A | Q (cid:1) = P q,a Q ( q ) R ( q, a )O A | Q ( a | q ).We use the following definition to measure the distance between two one-game strategies. Definition 6 (Distance measure) . The distance between K A | Q and R A | Q is defined as (cid:12)(cid:12) K A | Q − R A | Q (cid:12)(cid:12) = E q ∈Q X a ∈ A (cid:12)(cid:12) K A | Q ( a | q ) − R A | Q ( a | q ) (cid:12)(cid:12) where the m -tuples of questions q ∈ Q are distributed according to Q defined by the game G .Note that this is not the standard definition – instead of a maximisation over the tuples of questionsas in the usual definition of the trace distance we consider the average over the tuples according tothe game distribution. Therefore, the distance between the strategies depends on the specific game G considered.In the repeated game G n the referee asks each of the players n questions, all at once. The questionsare chosen according to the distribution Q ⊗ n , i.e., independently using Q . The answers are then checkedindependently according to the winning condition R . A strategy for the repeated game G n is denotedby P ~A | ~Q : A n × Q n → [0 ,
1] and the joint questions-answers distribution is then P ~A ~Q = Q ⊗ n × P ~A | ~Q .When the distributions are clear from the context we sometimes omit the subscripts and write just Oand P.When considering many questions-answers pairs in the repeated game we denote all the questionsand answers as vectors ~q, ~a . We use a subscript index as in ~q j to denote the j ’th tuple of questions givento the players. We denote by O ⊗ nA | Q a product of n identical strategies O A | Q . That is, O ⊗ nA | Q is definedaccording to O ⊗ nA | Q ( ~a | ~q ) = Q nj =1 O A | Q ( a j | q j ) for all ~a, ~q .For any m -tuple of questions q = ( q , . . . , q m ) ∈ Q and any i ∈ [ m ] = { , · · · , m } we denoteby q i , using a superscript index, the question given to the i ’th player by the referee, and by q ¯ i =( q , . . . q i − , q i +1 , · · · , q m ) the ( m − i . Similarly, for asubset I ⊂ [ m ], q I denotes the questions given to all the players i ∈ I and q ¯ I denotes the complementaryset of questions, i.e., the questions given to all the players i / ∈ I . An analogous notation is used for theanswers. Similarly, when considering many questions-answers paris, ~q i denotes all the questions givento the i ’th player, and so on.A tuple of questions q = ( q , . . . q i − , q i , q i +1 , · · · , q m ) can then be also written as ( q i , q ¯ i ) where it isunderstood which player gets which question. Therefore in this notation Q ( q i , q ¯ i ) = Q ( q ) and similarlyO( a i , a ¯ i | q i , q ¯ i ) = O( a | q ). Moreover, Q ( q i | q ¯ i ) = Q ( q i ,q ¯ i ) P ri Q ( r i ,q ¯ i ) denotes the probability that the i ’th playerreceives question q i given that the other players receive q ¯ i .In the following we prove Theorem 1, which applies to games with complete-support. A game hascomplete-support if every possible combination of questions to the players has some non-zero probabilityaccording to the question distribution Q . Formally, Definition 7 (Complete-support game) . An m -player game has complete-support if for every possiblecombination of questions to the players q , . . . , q m (i.e., q , . . . , q m such that for all i ∈ [ m ] there exist s ¯ i for which Q (cid:0) ( s , . . . , s i − , q i , s i +1 , . . . , s m ) (cid:1) > Q ( q ) > B. Estimated strategies
The specific questions and answers in one run of the repeated game ~q, ~a are also called the data ofthe game. As mentioned in the previous section, the data ~q, ~a is divided into two disjoint sets whichwe call the test data and the game data. We denote the n/ ~q t , ~a t respectively and the n/ ~q g , ~a g respectively. Using thisnotation ~q is the concatenation of ~q t and ~q g and ~a is the concatenation of ~a t and ~a g . Note that althoughwe denote here the test questions as appearing before the game questions, they are indistinguishablefrom one another, as they are chosen according to the exact same distribution Q . Had this not beenthe case, the permutation invariance symmetry would have been broken.Given the test data ~q t , ~a t we create an estimation O EST1 A | Q of a one-game strategy in the following way.For every tuple of questions q ∈ Q and answers a ∈ A let f qa be the frequency of the tuple of answers a in ~a t restricted to the indices j ∈ [ n/
2] for which ~q tj = q (if q did not appear at all set f qa = 0). Thendefine O EST1 A | Q such that O EST1 A | Q ( a | q ) = f qa .Similarly O EST2 A | Q is created in the same way, using the game data ~q g , ~a g (see Figure 1). Note thatsince P ~A | ~Q might be signalling between the different n tuples of questions, both O EST1 A | Q and O EST2 A | Q candepend also on the other questions which are not considered in the estimation process.To evaluate the accuracy of the estimation process described above we will use the following lemma –an application of Sanov’s theorem (see, e.g., [36] Section 11.4) to our scenario. Lemma 8.
Let δ ( l, ǫ ) = ( l + 1) |A|·|Q|− e − lǫ / . Then for every i.i.d. strategy O ⊗ lA | Q , Pr ~a,~q ∼ O ⊗ lAQ h | O EST A | Q − O A | Q | > ǫ i ≤ δ ( l, ǫ ) where O EST A | Q is estimated as above from the data ~a, ~q . distributedaccording toP ~A ~Q ~a : ~a t ~a g ~q : ~q t ~q g defines O EST1 A | Q defines O EST2 A | Q FIG. 1. Division to test and game data
C. Linear programs
Linear programs (see, e.g., [37]) are a useful tool when considering non-signalling games, as the non-signalling constraints are linear. The following results regarding the sensitivity of linear programs willbe of use for us.
Lemma 9 (Sensitivity analysis of linear programs, [37] Section 10.4) . Let max { c T x | Ax ≤ b } be aprimal linear program and min { b T y | A T y = c, y ≥ } its dual. Denote the optimal value of the programsby w and the optimal dual solution by y ⋆ . Then the optimal value of the perturbed program w e = max { c T x | Ax ≤ b + e } for some perturbation e is bounded by w e ≤ w + e T y ⋆ . Lemma 10 (Dual optimal solution bound, [37] Section 10.4) . Let A be an r × r -matrix and let ∆ besuch that for each non-singular submatrix B of A all entries of B − are at most ∆ in absolute value. Let c be a row vector of dimension r and let y ⋆ be the optimal dual solution of min { b T y | A T y = c, y ≥ } .Then κ = r X j =1 | y ⋆j | ≤ r ∆ r X j =1 | c j | . III. DETECTING SIGNALLINGA. The non-signalling constraints
We start by defining a non-signalling strategy. To simplify notation we define it using one-gamestrategies O A | Q . The definition is identical for the strategies P ~A | ~Q . Definition 11 (Non-signalling strategy) . An m -player strategy O A | Q is called non-signalling if for anyset of players I ⊂ [ m ], ∀ a ¯ I , q ¯ I , q I , r I O A | Q ( ◦ , a ¯ I | q I , q ¯ I ) = O A | Q ( ◦ , a ¯ I | r I , q ¯ I )where ◦ denotes a marginal, e.g., O A | Q ( ◦ , a ¯ I | q I , q ¯ I ) = P a i | i ∈ I O A | Q ( a | q I , q ¯ I ).Alternatively, one can define a non-signalling strategy using a set of linearly independent non-signalling constraints from which all the constraints in the above definition follow. Lemma 12 (Lemma 2.7 in [38]) . An m -player strategy O A | Q is non-signalling if and only if for anyplayer i ∈ [ m ] , ∀ a ¯ i , q ¯ i , q i , r i O A | Q ( ◦ , a ¯ i | q i , q ¯ i ) = O A | Q ( ◦ , a ¯ i | r i , q ¯ i ) . (2)From Equation (2) it is clear that for every i and q ¯ i the marginal states O A | Q ( ◦ , a ¯ i | q i , q ¯ i ) are all equiv-alent and independent of q i . Therefore another equivalent formulation of the non-signalling constraintsis given by ∀ a ¯ i , q ¯ i , q i O A | Q ( ◦ , a ¯ i | q i , q ¯ i ) = X r i Q ( r i | q ¯ i )O A | Q ( ◦ , a ¯ i | r i , q ¯ i ) . Here we defined the marginal, which is independent of r i , as an average over the different O A | Q ( ◦ , a ¯ i | r i , q ¯ i ),where the average is taken according to the distribution of the game question Q . It is easy to verifythat this condition is equivalent to Equation (2).We can now write the optimisation problem of finding the optimal winning probability in a complete-support game G using a non-signalling strategy as the following linear program over the variables O( a | q ):max X q,a Q ( q ) R ( q, a )O( a | q ) (3a)s.t. Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) = 0 ∀ i, q i , q ¯ i , a ¯ i (3b) X a O( a | q ) = 1 ∀ q (3c)O( a | q ) ≥ ∀ a, q (3d)The objective function, Equation (3a), is exactly the winning probability of the game using strat-egy O( a | q ) as defined in Definition 5. Equations (3c) and (3d) are the normalisation and positivityconstraints on the strategy O( a | q ).In Equation (3b) all the non-signalling constraints are listed, up to a factor of Q ( q ) which does notchange the constraints when considering complete-support games, but will be important later in thefollowing section. We note that the only place in the proof where the complete-support property of thegame is used is for writing down the linear program above. In Appendix A we explain the implications ofthe linear program (3) to games with incomplete-support. In particular, in Appendix A 1 we show howto modify program (3) for the case of two-player games with incomplete-support such that our resultstill holds. In Appendix A 2 we show how one can slightly modify the parallel repetition procedure toderive a general (although modified) threshold theorem for any game.Next, one can relax the linear program (3) to the following:max X q,a Q ( q ) R ( q, a )O( a | q )s.t. Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ∀ i, q i , q ¯ i , a ¯ i (4a) X a O( a | q ) = 1 ∀ q O( a | q ) ≥ ∀ a, q To see that the relaxation of the non-signalling constraints (3b) to the constraints (4a) does not changethe program, i.e., does not change the value of the optimal solution, assume there exists i, q i , q ¯ i , a ¯ i forwhich Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) < . That is, O( ◦ , a ¯ i | q i , q ¯ i ) is smaller than the average P r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ), and therefore there mustbe some s i for which O( ◦ , a ¯ i | s i , q ¯ i ) is larger than the average. Meaning, Q ( s i , q ¯ i ) " O( ◦ , a ¯ i | s i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) > , but this contradicts the constraints in (4a).0The dual program of the primal (4) is given below.min X q z ( q )s.t. z ( q ) + X i y i ( q, a ¯ i ) Q ( q ) − X i X r | r ¯ i = q ¯ i y i ( r, a ¯ i ) Q ( r ) Q ( r i | q ¯ i ) ≥ Q ( q ) R ( q, a ) ∀ a, q (5a) y i ( q, a ¯ i ) ≥ ∀ i, q, a ¯ i B. Signalling measure
Given a general strategy O A | Q we would like to measure the amount of signalling from every player i ∈ [ m ] to all the other players together. Following the linear program (4), we quantify signalling usingDefinition 13 below.In the definition we derive all the relevant conditional and marginal distributions from O AQ . Con-cretely we use the following notation: O( ◦ , b ¯ i | s i , s ¯ i ) = P b i O( b i , b ¯ i | s i , s ¯ i ) as before, O( ◦ , b ¯ i , ◦ , s ¯ i ) = P b i ,s i O( b i , b ¯ i , s i , s ¯ i ), andO( ◦ , s i | b ¯ i , s ¯ i ) = X b i O( b i , s i | b ¯ i , s ¯ i ) = X b i O( b i , b ¯ i , s i , s ¯ i )O( ◦ , b ¯ i , ◦ , s ¯ i ) . Definition 13 (Signalling measure) . The signalling of strategy O A | Q in direction i → ¯ i using outputs b ¯ i and inputs s i , s ¯ i is defined asSig ( i,b ¯ i ,s i ,s ¯ i ) (O) = Q ( s i , s ¯ i ) " O( ◦ , b ¯ i | s i , s ¯ i ) − X r i Q ( r i | s ¯ i )O( ◦ , b ¯ i | r i , s ¯ i ) (6)= O( ◦ , b ¯ i , ◦ , s ¯ i ) h O( ◦ , s i | b ¯ i , s ¯ i ) − Q ( s i | s ¯ i ) i . (7)That is, we have a signalling measure for every ( i, b ¯ i , s i , s ¯ i ). If Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) > i, b ¯ i , s i , s ¯ i ). A negative signalling value, Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) <
0, is notrelevant due to the inequality in Equation (4a).The two forms of Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) given in equations (6) and (7) are equivalent according to Bayes’rule and they will be useful in different places in the proof. Equation (7) for example allows us toquantify the amount by which input s i is more or less probable given b ¯ i , compared to the prior Q ( s i | s ¯ i ).The following lemma shows that our measure of signalling is continuous. That is, if two strategiesare close to one another according to Definition 6 then their signalling values are also close. The proofis given in Appendix B. Lemma 14 (Continuity of Sig) . Let O and O be two one-game strategies such that (cid:12)(cid:12) O − O (cid:12)(cid:12) ≤ ǫ .Then ∀ i, b ¯ i , s i , s ¯ i (cid:12)(cid:12) Sig ( i,b ¯ i ,s i ,s ¯ i ) (O ) − Sig ( i,b ¯ i ,s i ,s ¯ i ) (O ) (cid:12)(cid:12) ≤ ǫ . C. Signalling tests
In the following we will need an operational way of testing whether a one-game strategy O A | Q issignalling. This can be done by using many copies of O A | Q – given data ~q, ~a which is distributedaccording to many independent copies of O AQ it is possible to create an estimation of O A | Q , O EST1 A | Q ,and then evaluate Sig ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) O EST1 (cid:1) .To formulate this process we first define an indicator function which will be used in the test. Moreprecisely, for every tuple ( i, b ¯ i , s i , s ¯ i ) we define a function T ( i,b ¯ i ,s i ,s ¯ i ) : Q t × A t → { , } :1 Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) 0 ζ − ǫ ζ Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) ≥ ζ − ǫ FIG. 2. The different forms of signalling: every i and every b ¯ i , s i , s ¯ i define a line as in the figure. The value ofSig ( i,b ¯ i ,s i ,s ¯ i ) (O) tells us exactly where we are on the line. T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = ( ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) O EST1 (cid:1) ≥ ζ − ǫ EST1 is estimated from ~q t , ~a t and ζ, ǫ > ζ ≥ ǫ and ǫ ≤ min q Q ( q ).See Figure 2 for a visualisation of the different signalling forms ( i, b ¯ i , s i , s ¯ i ) and the signalling valuesconsidered in the test.The following observation will be crucial later on: Remark 15.
According to Definition 13, in order to evaluate
Sig ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) O EST1 (cid:1) there is no need toknow O EST1 completely; only the marginals O EST1 ( ◦ , b ¯ i | r i , s ¯ i ) for all r i are needed. For every ( i, b ¯ i , s i , s ¯ i ) we can now consider a signalling test. Given a strategy P ~A | ~Q for the repeatedgame G n we sample n tuples of questions ~q using the game distribution Q ⊗ n and use them to get n tuples of answers ~a which are distributed according to P ~A | ~Q . Finally, if T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 the testaccepts, and otherwise rejects . Note that if a question s does not appear in the test data ~q t the test T ( i,b ¯ i ,s i ,s ¯ i ) rejects by definition.The following lemma shows that the test is reliable when applied to an i.i.d. strategy O ⊗ nA | Q . That is,if Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) ≥ ζ the test will detect it with high probability, i.e. the test will accept with highprobability, and if O A | Q is non-signalling then the test will reject with high probability. The proof canbe found in Appendix B. Lemma 16 (Reliable signalling test) . Assume the players share an i.i.d. strategy O ⊗ nA | Q . For every ( i, b ¯ i , s i , s ¯ i ) ,1. If Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) ≥ ζ then Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > − δ .2. If Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) = 0 then Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 0 i > − δ .where δ = δ (cid:0) n , ǫ (cid:1) = (cid:0) n + 1 (cid:1) |A|·|Q|− e − nǫ / . Given a specific signalling test T ( i,b ¯ i ,s i ,s ¯ i ) we define two relevant sets of one-game strategies: σ ( i,b ¯ i ,s i ,s ¯ i ) = n O (cid:12)(cid:12) ∀ ¯O s.t. | O − ¯O | ≤ ǫ , ¯O is ζ signalling or more in ( i, b ¯ i , s i , s ¯ i ) o (9)Σ ( i,b ¯ i ,s i ,s ¯ i ) = n O (cid:12)(cid:12) ∃ ¯O s.t. | O − ¯O | ≤ ǫ ∧ Pr ~a,~q ∼ ¯O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > δ o (10)The following two lemmas allow us to address these sets also according to the signalling values of therelevant strategies (see also Figure 3). Lemma 17.
For all O / ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) , Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) < ζ + 2 ǫ . Lemma 18.
Let ν > be any parameter such that ν < ζ − ǫ . Then ∀ O ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) , Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) > ν . Lemma 17 follows right away from Lemma 14 and the definition of σ ( i,b ¯ i ,s i ,s ¯ i ) . Lemma 18 is provenin Appendix B. As P ~A | ~Q can be signalling between the different n tuples of questions-answers one has to input all the questions beforegetting the test answers. Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) 0 ν ζ ζ + 2 ǫ Sig of O ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) Sig of O / ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) constant gapFIG. 3. Visualization of the signalling values which are relevant for Lemma 18 and the sets σ ( i,b ¯ i ,s i ,s ¯ i ) , Σ ( i,b ¯ i ,s i ,s ¯ i ) . IV. USING DE FINETTI STRATEGIES
In this section we start analysing the relation between the test questions-answers ~q t , ~a t and the gamequestions-answers ~q g , ~a g in one instance of the repeated game G n using a strategy P ~A | ~Q . More precisely,we denote the one-game strategy which is estimated from ~q g , ~a g by O EST2 A | Q , and we are interested inknowing what is the probability that O EST2 A | Q ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) or O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) given the result of T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ).We first do this for any i.i.d. strategy and then extend the results to any permutation-invariantstrategy using a de Finetti reduction [27]. A. de Finetti strategies
As mentioned in Section I, de Finetti strategies are strategies that can be written as a convex com-bination of i.i.d. strategies. Formally,
Definition 19 (de Finetti strategy) . A de Finetti strategy τ ~A | ~Q is a strategy of the form τ ~A | ~Q = Z O ⊗ nA | Q dO A | Q , where dO A | Q is some measure on the space of one-game strategies.In the following lemma we are interested in the relation between the test questions-answers ~q t , ~a t andthe game questions-answers ~q g , ~a g in one instance of the repeated game G n . For i.i.d. strategies (andtherefore also for de Finetti strategies) this is simple: ~q t , ~a t and ~q g , ~a g are independent of each otherand conditioning on a property of one of them does not affect the other. Lemma 20.
For a de Finetti strategy τ ~A | ~Q and every ( i, b ¯ i , s i , s ¯ i ) Pr ~a,~q ∼ τ ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 ∧ O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ δ Pr ~a,~q ∼ τ ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 ∧ O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ δ The proof of this lemma (given in Appendix C) follows from Sanov’s theorem stated in Lemma 8.Intuitively, if T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 then O EST1 is signalling and therefore so should O
EST2 be, and viceversa.
B. de Finetti reductions
Of course, considering just de Finetti strategies is not interesting by itself. Luckily, we can now use ade Finetti reduction to extend the results of the previous section to any permutation-invariant strategy,where the permutation is performed on the questions-answers pairs (we do not permute the players).As the repeated game G n is by itself permutation invariant we can restrict the strategies of the playersto be permutation invariant without loss of generality.3 Definition 21 (Permutation invariance) . Given a strategy P ~A | ~Q and a permutation π of the questionsand answers we denote by P ~A | ~Q ◦ π the strategy which is defined by ∀ ~a, ~q (cid:16) P ~A | ~Q ◦ π (cid:17) ( ~a | ~q ) = P ~A | ~Q ( π ( ~a ) | π ( ~q )) . P ~A | ~Q is permutation invariant if for any permutation π , P ~A | ~Q = P ~A | ~Q ◦ π .The following lemma shows that we can restrict our analysis to permutation-invariant strategieswithout loss of generality. Lemma 22.
For every strategy P ~A | ~Q for the repeated game G n there exists a permutation-invariantstrategy ˜P ~A | ~Q such that w (cid:16) P ~A | ~Q (cid:17) = w (cid:16) ˜P ~A | ~Q (cid:17) .Proof. Given P ~A | ~Q define its permutation-invariant version to be˜P ~A | ~Q = 1 n ! X π P ~A | ~Q ◦ π . The winning probability of the game is linear in the strategy, therefore we have w (cid:16) ˜P ~A | ~Q (cid:17) = w n ! X π P ~A | ~Q ◦ π ! = 1 n ! X π w (cid:16) P ~A | ~Q ◦ π (cid:17) . (11)Since the tuples of questions in the repeated game are chosen in an i.i.d. manner and the winningcondition is checked for each tuple separately, the winning probability is indifferent to the orderingof the questions-answers pairs. As π permutes the tuples of questions and answers together we have w (cid:16) P ~A | ~Q ◦ π (cid:17) = w (cid:16) P ~A | ~Q (cid:17) .Combining this with Equation (11) we get w (cid:16) ˜P ~A | ~Q (cid:17) = w (cid:16) P ~A | ~Q (cid:17) . Lemma 23 (de Finetti reduction for conditional probability distributions [27]) . Let c = ( n +1) |Q| ( |A|− .There exists a de Finetti strategy τ ~A | ~Q such that for every permutation-invariant strategy P ~A | ~Q ∀ ~a, ~q P ~A | ~Q ( ~a | ~q ) ≤ c · τ ~A | ~Q ( ~a | ~q ) . The de Finetti strategy τ ~A | ~Q is constructed explicitly in [27] but the specific construction is notrelevant for our purposes. In some special cases the constant c in Lemma 23 can also be made smallerby taking into account symmetries of the game G itself. For further details see [27].We now use the de Finetti reduction to show that the properties proven in Lemma 20 for the de Finettistrategy also hold true for permutation-invariant strategies, although with slightly weaker parameters.Concretely, the bound of 2 δ in Lemma 20 is replaced by 2 cδ in the following lemma. Nevertheless, thebound still decreases exponentially fast with the number of repetitions . Lemma 24 (Reduction) . For every permutation-invariant strategy P ~A | ~Q and every ( i, b ¯ i , s i , s ¯ i ) Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 ∧ O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ cδ .2. Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 ∧ O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ cδ . One would have liked to apply a similar argument to the winning probability of the repeated game right away. Thatis, w (P ~A | ~Q ) ≤ cw ( τ ~A | ~Q ). This claim is indeed correct, but not useful. A look at the explicit construction of τ ~A | ~Q itselfin [27] will reveal that it is a signalling strategy, hence no non-trivial bound on w ( τ ~A | ~Q ) holds a priori. For a furtherdiscussion see Section VI B. Proof.
We prove both of the claims together. Denote the relevant event by E ( ~a, ~q ) and note that forboth events we can write Pr ~a,~q ∼ P ~A~Q [ E ( ~a, ~q ) = 1] = X ~a,~q | E ( ~a,~q )=1 P ~A ~Q ( ~a, ~q ) . From Lemma 23 we get P ~A ~Q ( ~a, ~q ) ≤ c · τ ~A ~Q ( ~a, ~q ) and thereforePr ~a,~q ∼ P ~A~Q [ E ( ~a, ~q ) = 1] = X ~a,~q | E ( ~a,~q )=1 P ~A ~Q ( ~a, ~q ) ≤ c · X ~a,~q | E ( ~a,~q )=1 τ ~A ~Q ( ~a, ~q ) = c · Pr ~a,~q ∼ τ ~A~Q [ E ( ~a, ~q ) = 1] . Combining this with Lemma 20 proves the lemma.
V. THRESHOLD THEOREM
In this section we prove our threshold theorem, Theorem 1. Before going into the details of the proof,let us explain the high-level idea.First, to see the connection between what was done so far and a threshold theorem note that thewinning probability of O
EST2 A | Q in the game G , w (O EST2 A | Q ), is exactly the fraction of coordinates in whichthe game data ~q g , ~a g satisfies the winning condition R . Therefore, in order to prove a threshold theoremit is sufficient to prove an upper bound on w (O EST2 A | Q ) which holds with high probability.To do so we use the following sequence of lemmas. The first two lemmas bound the proba-bility that the estimate O EST2 A | Q is significantly signalling in any direction ( i, b ¯ i , s i , s ¯ i ) for whichPr ~a,~q ∼ P ~A~Q (cid:2) T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 (cid:3) = 0. Lemma 25, which we also call the weak lemma, establishesthat even conditioned on the test T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) detecting signalling the distribution O EST2 A | Q itselfcannot be signalling with very high probability. The proof of the lemma is based on a reduction toa certain guessing game which is used to derive a contradiction between the conclusion that O EST2 A | Q would be signalling and the assumption that the overall distribution P ~A | ~Q is not. Lemma 26, calledthe strong lemma, amplifies the conclusion of the weak lemma to show that O EST2 A | Q cannot display toomuch signalling, even only with small probability. The amplification is obtained by using the propertiesof permutation-invariant strategies which were proven in Lemma 24 in the previous section.Having shown that with high probability O EST2 A | Q cannot be too signalling, Lemma 27 derives an upperbound on the winning probability w (O EST2 A | Q ). Intuitively, if the strategy O EST2 A | Q does not display strongsignalling in any direction it should not lead to a large advantage over strictly non-signalling strategies inthe game G . The quantitative argument is based on performing a sensitivity analysis of the appropriatelinear program. The three lemmas are brought together in Lemma 28, from which Theorem 1 follows.We are now ready to prove the following lemmas and the threshold theorem. Lemma 25 (Weak lemma) . Let n be such that n ln( n ) > | Q || A | ln(2 /ǫ ) ǫ , (12) and P ~A | ~Q a non-signalling strategy for G n . For any ( i, b ¯ i , s i , s ¯ i ) denote by P ~A ~Q |T =1 the probability dis-tribution P ~A ~Q conditioned on the event T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 , whenever such a conditional probabilitydistribution is defined. Then, Pr ~a g , ~q g ∼ P ~A~Q |T =1 h O EST2 A | Q ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i < − cδ . (13) In the words of the explanation given in Section I B, this is where we prove that the signalling weight is exponentiallysmall. Proof.
For every signalling test T ( i,b ¯ i ,s i ,s ¯ i ) we construct a guessing game. Our goal is to derive acontradiction by showing that if Equation (13) is not true, then the guessing game can be won withprobability higher than the optimal non-signalling winning probability.The guessing game is defined as follows. A referee gives the players n/ m -tuples of gamequestions ~q g where each tuple is distributed according to the questions distribution Q of the originalgame G . Players ¯ i are then allowed to communicate and their goal is to guess and output an index j ∈ [ n/
2] such that ~q gj = ( s i , s ¯ i ) (if there is no such index the players lose automatically).If the players share a non-signalling strategy P ~A | ~Q then the marginals of players ¯ i are the same forall q i . Therefore, their outputs a ¯ i do not give them any information about the question that the i ’thplayer got from the referee (even when players ¯ i are allowed to communicate among themselves, but notwith player i ). The best non-signalling strategy is therefore to choose, uniformly at random, an index j for which ~q g ¯ ij = s ¯ i if it exists. The winning probability is then given by W ns = Q ( s i | s ¯ i ) < ~A | ~Q for whichPr ~a g , ~q g ∼ P ~A~Q |T =1 h O EST2 A | Q ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≥ − cδ (14)then they can win the above guessing game with probability higher than the optimal non-signallingwinning probability W ns .The general idea is as follows. The players use the questions given by the referee as the game questions ~q g and choose, using shared randomness, the inputs for the test questions ~q t . They input the questionsinto P ~A | ~Q . Players ¯ i then check if T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 – they can do this since they are allowed tocommunicate among themselves and they know all the inputs for the test questions of player i (as theywere chosen using shared randomness which is available to all the players). Recalling Remark 15, theyhave all the information they need.The players proceed according to the following conditions:1. If T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 they use the non-signalling strategy described above. That is, they choosea random index j ∈ [ n/
2] such that ~q g ¯ ij = s ¯ i .2. If T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 they choose a random index j ∈ [ n/
2] such that ~q g ¯ ij = s ¯ i and ~a g ¯ ij = b ¯ i ifit exists (otherwise they use the non-signalling strategy described above).Let us show that, as long as Pr ~a,~q ∼ P ~A~Q (cid:2) T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:1) = 1 i = 0, this strategy achieves a winningprobability which is higher than W ns . If T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 then the winning probability is W ns .However, if T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 then ~q g , ~a g can be seen as data which is distributed according to n/ EST2 , which is with high probability in Σ ( i,b ¯ i ,s i ,s ¯ i ) according to Equation (14). FromLemma 18 this implies Pr ~a g , ~q g ∼ P ~A~Q |T =1 h Sig ( i,b ¯ i ,s i ,s ¯ i ) (O EST2 ) > ν i ≥ − cδ , (15)where ν > ν < ζ − ǫ (recall Lemma 18).Using the definition of Sig in Equation (7) we know that if indeed Sig ( i,b ¯ i ,s i ,s ¯ i ) (O EST2 ) > ν thenO EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) > EST2 ( ◦ , s i | b ¯ i , s ¯ i ) > ν O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) + Q ( s i | s ¯ i ) (16)= ν O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) + W ns . That is, by choosing an index for which a ¯ i = b ¯ i players ¯ i increase the winning probability.On the other hand, if Sig ( i,b ¯ i ,s i ,s ¯ i ) (O EST2 ) ≤ ν , which can happen with probability 2 cδ , then theplayers might decrease their winning probability. In the worst case the winning probability is 0.Therefore, if T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 we get the following winning probability W |T =1 ≥ (1 − cδ ) (cid:18) ν O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) + W ns (cid:19) + 2 cδ · W > Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 0 i W ns + Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i (1 − cδ ) (cid:18) ν O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) + W ns (cid:19) = W ns − Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i cδW ns + Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i (1 − cδ ) ν O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) . Finally,
W > W ns for ν > cδ − cδ W ns ≥ cδ − cδ W ns · O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) . (18)Using W ns · O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) ≤ cδ ≤ ( n + 1) | Q || A | e − nǫ / (see Table I), we see that as long as n/ ln( n ) > | Q || A | ǫ − ln(2 /ǫ ) the quantity 2 cδW ns O EST2 ( ◦ , b ¯ i , ◦ , s ¯ i ) / (1 − cδ ) is strictly less than ǫ .Assuming ζ ≥ ǫ , there is a choice of ν that satisfies both (18) and the earlier condition that ν <ζ − ǫ .The bound given in Equation (13) is weak for two reasons. First, the game data ~q g , ~a g is distributedaccording to the conditional distribution P ~A ~Q |T =1 and not according to P ~A ~Q itself. Second, it only tellsus that Pr ~a g , ~q g ∼ P ~A~Q |T =1 (cid:2) O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) (cid:3) ≥ cδ , i.e., the probability that O EST2 A | Q has a small valueof signalling is higher than 2 cδ . We show how the statement in the weak lemma can be amplified usingthe de Finetti reduction from Lemma 24. Lemma 26 (Strong lemma) . Let P ~A | ~Q be a permutation-invariant non-signalling strategy for G n . Thenfor any ( i, b ¯ i , s i , s ¯ i ) such that Q ( s i , s ¯ i ) = 0 and Q ( s i | s ¯ i ) = 1 , Pr ~a,~q ∼ P ~A~Q h O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ cδ . Proof.
From Lemma 24 part 1 we getPr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > cδ ⇒ Pr ~a,~q ∼ P ~A~Q |T =1 h O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ cδ and this can be rewritten asPr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > cδ ⇒ Pr ~a,~q ∼ P ~A~Q |T =1 h O EST2 A | Q ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≥ − cδ . According to Lemma 25, this impliesPr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > cδ ⇒ P ~A | ~Q is signalling . Therefore it must be that Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i ≤ cδ (19)or alternatively, Pr ~a,~q ∼ P ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 0 i ≥ − cδ (20)Next, combining Lemma 24 part 2 with Equation (20) we getPr ~a,~q ∼ P AQ |T =0 h O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ cδ . Using Equation (19) we get Pr ~a,~q ∼ P ~A~Q h O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ cδ . ~A | ~Q is a permutation-invariant non-signalling strategy then the probabilitythat O EST2 A | Q is in a given set σ ( i,b ¯ i ,s i ,s ¯ i ) is exponentially small. In the next lemma we use this propertyto get a bound on the winning probability of O EST2 A | Q in the game G . Lemma 27.
Let κ = P dj =1 | y ⋆j | where d is the number of signalling tests and y ⋆ is an optimal solutionof the dual program (5) . Let O EST2 A | Q be a strategy such that for all ( i, b ¯ i , s i , s ¯ i ) , O EST2 A | Q / ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) .Then w (O EST2 A | Q ) ≤ − α + ( ζ + 2 ǫ ) κ .Proof. According to Lemma 17, if O
EST2 A | Q / ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) for every σ ( i,b ¯ i ,s i ,s ¯ i ) thenSig ( i,a ¯ i ,q i ,q ¯ i ) (O EST2 ) < ζ + 2 ǫ (21)for every i and every b ¯ i , s i , s ¯ i . That is, O EST2 A | Q is not “too signalling” in any direction. This can be usedto bound the winning probability of O EST2 A | Q in the game G .The following linear program describes the optimal winning probability of a strategy O A | Q whichfulfils Equation (21):max X q,a Q ( q ) R ( q, a )O( a | q )s.t. Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ζ + 2 ǫ ∀ i, q i , q ¯ i , a ¯ i X a O( a | q ) = 1 ∀ q O( a | q ) ≥ ∀ a, q (22)As O EST2 A | Q is a strategy we have P a O EST2 ( a | q ) = 1 for all q . Hence O EST2 A | Q satisfies all the constraintsof the above program and therefore its winning probability in G is bounded by the optimal value of theprogram. Program (22) can be seen as a perturbation of the linear program (4), we can therefore boundits optimal value by using known tools for sensitivity analysis of linear programs, stated in Lemmas 9and 10.Denote by y ⋆ an optimal solution of the dual program (5) and let κ = P dj =1 | y ⋆j | where d is thenumber of signalling tests. That is, κ is the sum of all the dual variables which are associated to thenon-signalling constraints.According to Lemma 9 the perturbed winning probability is then bounded by w e ≤ − α + ( ζ + 2 ǫ ) κ. To get κ in the above lemma, one can use any of the following:1. Given a description of a game one can easily get κ by solving the dual program (5) .2. If the game involves only 2 players, then following [33] one can get κ ≤ d where d is the numberof different signalling tests ( d < m |Q||A| ).3. Otherwise, the general bound of Lemma 10 can be used. In our case the bound reads κ ≤ |A| |Q| ∆,where ∆ depends only on the game .Finally we are ready to prove the last lemma: Lemma 28 (Main lemma) . Let w ( G ) = 1 − α be the optimal winning probability of a non-signallingstrategy in G . Let < β ≤ α be some constant and n the number of repetitions such that Equation (18) is satisfied. Then for any non-signalling strategy P ~A | ~Q of the repeated game, Pr ~a,~q ∼ P ~A~Q h w (O EST2 A | Q ) > − α + β i ≤ cdδ . We are only interested in the value of y ⋆ as z ⋆ will not affect the bound. Solving the linear program is anyhow usually necessary for knowing the optimal non-signalling value 1 − α . A similar bound was also used in [15]. Symbol Meaning First appears Fulfils m (g) − α (g) optimal NS winning probability in Gκ (g) bound on an optimal dual solution y ⋆ P dj =1 | y ⋆j | W ns (g) optimal NS winning probability in guessing game max q,i Q ( q i | q ¯ i ) n (t) β (t) deviation in the threshold theorem ǫ confidence interval of the test Equation (8) ǫ ≤ min q Q ( q ) ζ signalling threshold of the test Equation (8) 7 ǫ ≤ ζ ≤ ζ + 2 ǫ ≤ βκ δ confidence level of the test Lemma 16 δ = ( n/ |A|·|Q|− e − nǫ / ν signalling threshold Lemma 18 cδ − cδ W ns < ν < ζ − ǫc de Finetti constant Lemma 23 c = ( n + 1) |Q| ( |A|− d d < m |Q||A| TABLE I. Constants, parameters and their relations. (g) next to the symbol denotes that this is a constantwhich depends on the considered game and (t) denotes a parameter of the threshold theorem. All other constantsshould be chosen such that all the requirements in the last column of the table are fulfilled.
Proof.
Let ζ, ǫ > ζ + 2 ǫ ≤ βκ , ǫ ≤ min q Q ( q ) and 7 ǫ ≤ ζ ≤ s appear at least once in the game data then according to the definition ofO EST2 we have P b O EST2 ( b | s ) = 1 for all s . We can therefore apply Lemma 27 in combination withLemma 26 and getPr ~a,~q ∼ P ~A~Q " w (O EST2 A | Q ) > − α + β (cid:12)(cid:12) X b O EST2 ( b | s ) = 1 ∀ s ≤ Pr ~a,~q ∼ P ~A~Q h ∃ σ ( i,b ¯ i ,s i ,s ¯ i ) s.t. O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ d · cδ . The probability that some tuple of questions does not appear in the game data is upper bounded by |Q| (cid:16) − min s Q ( s ) (cid:17) n/ ≤ |Q| e − min s Q ( s ) n/ ≤ |Q| e − ǫn/ ≤ dδ and therefore all together we havePr ~a,~q ∼ P ~A~Q h w (O EST2 A | Q ) > − α + β i ≤ cdδ . Our threshold theorem, Theorem 1, follows from Lemma 28:
Proof of Theorem 1.
Let f g , f t and f denote the winning frequency in the game data, test data andthe entire data respectively (i.e., the fraction of coordinates in which the players win the game).Form Lemma 28 we know that Pr ~a,~q ∼ P ~A~Q [ f g > − α + β ] ≤ cdδ , as the winning frequency in thegame questions is exactly w (O EST2 A | Q ). As the game data and test data are symmetric (i.e., there isno difference between them except for the name we gave them), the same result also holds for f t ,Pr ~a,~q ∼ P ~A~Q [ f t > − α + β ] ≤ cdδ .Finally, as the winning frequency in the entire data is given by f = ( f t + f g ) we havePr ~a,~q ∼ P ~A~Q [ f > − α + β ] ≤ Pr ~a,~q ∼ P ~A~Q (cid:2)(cid:0) f t > − α + β (cid:1) ∨ ( f g > − α + β ) (cid:3) ≤ cdδ . (23)The relations between all the constants and parameters of the theorem and the proofs are listed inTable I. Note that for any game and choice of parameters the bound 10 cdδ is exponentially decreasingwith the number of repetitions n .To get a better feeling of the result, without trying to optimise it, one can make the following choices.Let ǫ = β κ , ζ = 8 ǫ and ν = ǫ (assuming min q Q ( q ) > β κ ). Using these choices, our proof holds for n β such that n ln( n ) > |Q||A| ln(20 κ/β )( β/ κ ) (24)with the following constants in Theorem 1: C ( G, n ) = 10 m |Q||A| ( n + 1) |Q||A|− , C ( G ) = (20 κ ) − . (25)The theorem then readsPr ~a,~q ∼ P ~A~Q [ f > − α + β ] ≤ m |Q||A| ( n + 1) |Q||A|− e − n ( β κ ) . (26)A different choice of parameters can improve the dependency of the constants on the game G . VI. CONCLUSIONS AND OPEN QUESTIONSA. Current work and possible extensions
In this work a threshold theorem for multiplayer non-signalling games was proven. The thresholdtheorem given in Theorem 1 is applicable to any multiplayer complete-support game and for everytwo-player game (not necessarily with complete-support, as proven in Appendix A 1). Hence, all casesfor which parallel repetition was already known prior to our work [13, 15] are covered by our proof. Formultiplayer-games with incomplete-support we considered a small modification of the parallel repetitionprocedure which results in Theorem 2. We believe a similar modification can be considered to extendthe result of [15].In both theorems it might be possible to improve the dependency of the result on the parametersof the considered game, i.e., improve the constants C ( G, n ) and C ( G ). The polynomial dependencyof C ( G, n ) on the number of repetitions, on the other hand, is inherent to the use of the de Finettitheorem. Moreover, further investigation of the dual program (5) could lead to an explicit bound on C ( G ). This could then be used to extend Theorem 1 to games with incomplete-support, as done fortwo-player games.The most important contribution of this work is a new proof technique for parallel repetition theorems,based on ideas of de Finetti theorems and tomography. de Finetti theorems seem like a natural toolfor parallel repetition theorems, yet, this is the first time that such a result is proven using a de Finettitheorem.Apart from allowing a different point of view on parallel repetition questions, and the study ofcorrelations in general, the new proof technique has several advantages over the previous proofs.For instance, note that in the standard proofs of parallel repetition theorems, i.e., proofs followingthe approach of [12] such as [13–15], most of the difficulties arise due to the effect of conditioning on theevent of winning some of the game repetitions. As this event is one that depends on the structure ofthe game and we have no control over it, it can introduce arbitrary correlations between the questionsused in different repetitions of the game, a major source of difficulty for the remainder of the argument.In our proof we also need to analyse the effect of conditioning on a certain event, the event of the non-signalling test accepting, and this is done in Lemma 25, the weak lemma. However, the key advantageof our approach is that the test has a very specific structure, and in particular conditioning on the testpassing can be done locally by the players in a way that respects the non-signalling constraints. As aresult it is almost trivial to deal with the conditioning in the remainder of the proof. This shift fromconditioning on an uncontrolled event, success in the game, to a highly controlled one, a non-signallingtest that we design ourselves, is a key simplification that we expect to play an important role in anyextension of our method to other scenario such as classical or quantum strategies. More specifically, byfinding appropriate “non-classicality” and “non-quantumness” measures which can replace our signallingmeasure in Definition 13 one may be able to adapt the proof to the multiplayer classical and quantumcases as well. The results of Sections III and IV should follow easily for most “non-classicality” and“non-quantumness” measures of one-game strategies. The main difficulty, however, is finding a measurefor which Lemma 25 can be proven.0 B. What parallel repetition tells us about de Finetti theorems
In the light of the de Finetti reduction stated in Lemma 23, it is tempting to try and prove a parallelrepetition theorem by claiming that for every permutation-invariant strategy P ~A | ~Q , w (cid:16) P ~A | ~Q (cid:17) ≤ c · w (cid:16) τ ~A | ~Q (cid:17) . (27)This claim is correct but, unfortunately, not very useful as τ ~A | ~Q itself is signalling according to theexplicit construction given in [27], hence, no non-trivial bound holds on w (cid:0) τ ~A | ~Q (cid:1) .One might hope that this is just a technical problem; perhaps a different de Finetti reduction canbe proven, where both P ~A | ~Q and τ ~A | ~Q can be taken to be non-signalling (or analogously, quantum orclassical). Such a de Finetti reduction, if it existed, would have implied a strong parallel repetitiontheorem (up to the polynomial factor c ) for any game right away using Equation (27). This howeverwill stand in contradiction to known impossibility results, such as the result of [39].We therefore learn an interesting fact about de Finetti reductions by considering parallel repetitiontheorems: in order to prove a general de Finetti reduction as in Lemma 23, the de Finetti strategy musthave some signalling parts. Fortunately, as shown by our result, this does not render a proof for thenon-signalling case impossible. ACKNOWLEDGMENTS
The authors would like to thank Anthony Leverrier, Yeong-Cherng Liang, and David Sutter for helpfuldiscussions, and Christian Schaffner for pointing out a mistake in a preliminary version of this work.RAF and RR acknowledge support from the European Research Council (ERC) via grant No. 258932,the European Commission STReP project “RAQUEL”, the Swiss National Science Foundation (SNSF)via the National Centre of Competence in Research “QSIT” and the CHIST- ERA project “DIQIP”.TV acknowledges support from the Simons Institute in Berkeley, the Ministry of Education, Singaporeunder the Tier 3 grant MOE2012-T3-1-009 and the Perimeter Institute in Waterloo, Canada. [1] J. S. Bell et al. , Physics , 195 (1964).[2] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Physical Review Letters , 880 (1969).[3] U. Feige, S. Goldwasser, L. Lov´asz, S. Safra, and M. Szegedy, J. ACM , 268 (1996).[4] U. Feige and L. Lov´asz, in Proceedings of the 24th Annual ACM Symposium on Theory of Computing(STOC) (1992) pp. 733–744.[5] I. Dinur, J. ACM (2007), 10.1145/1236457.1236459.[6] M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, in Proceedings of the 20th Annual ACM Symposiumon Theory of Computing (STOC) (1988) pp. 113–131.[7] Y. Tauman Kalai, R. Raz, and R. D. Rothblum, in
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Appendix A: Extending the result to general games
Before we show how to extend the threshold theorem to games with incomplete-support, let us explainwhy the proof given for Theorem 1 holds only for complete-support games.As mentioned in the main text, the reason lies in the linear program (3), and more specifically, inthe non-signalling constraints given in Equation (3b). Indeed, if for some q we have Q ( q ) = 0 then therelevant constraint in Equation (3b) is vacuous. It is therefore clear that in this case the constraints givenin Equation (3b) are in fact relaxations of the standard non-signalling constraints given in Equation (2).For some games, this relaxation of the non-signalling constraints is strict. For example , considera game of 3 players where the questions are uniformly distributed over Q = { (0 , , , (0 , , , (1 , , } and the winning condition is given by the following predicate: R ( q, a ) = q = (0 , ,
1) and a = a q = (0 , ,
0) and a = a q = (1 , ,
0) and a = a (as can be shown by solving a linearprogram). However, in the linear program (3) there are no non-trivial constraints (i.e., all the constraintsin Equation (3b) are of the form 0 = 0). Hence, the optimal solution of the program (3) is 1, which isstrictly larger than . Thus even though the non-signalling conditions are enforced over all “relevant”questions, this does not suffice to guarantee that there exists a strategy achieving the resulting optimumsuccess probability 1 and that can be extended to a non-signalling strategy defined on all questions.For games with incomplete-support in which the optimal value of program (3) is not trivial (i.e., it issmaller than 1), it follows that our proof can be applied as is to derive a non-trivial threshold theorem.Irrespectively of whether this is the case or not one might also elect to work with the weaker definition ofnon-signalling strategies that is implied by the constraints in (3b), where the behaviour of the strategyis not required to be well-defined for questions which do not appear in the game. In this case the linearprogram (3) exactly describes the optimal winning probability of such strategies and Theorem 1 holdswithout any modification.In other cases, on the other hand, we have to slightly modify the linear program in order to derive acorrect threshold theorem. In the following sections we show how to do this. This example was communicated to us by Christian Schaffner.
1. Two-player games
For two-player games we consider the following modification of the linear program (3).max X q,a Q ( q ) R ( q, a )O( a | q )s.t. Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) = 0 ∀ i, a ¯ i , ∀ q i , q ¯ i s.t. Q ( q ) = 0 η " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) = 0 ∀ i, a ¯ i , ∀ q i , q ¯ i s.t. Q ( q ) = 0 X a O( a | q ) = 1 ∀ q O( a | q ) ≥ ∀ a, q (A1)where η > X q,a Q ( q ) R ( q, a )O( a | q )s.t. Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ∀ i, a ¯ i , ∀ q i , q ¯ i s.t. Q ( q ) = 0 (A2a) η " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ∀ i, a ¯ i , ∀ q i , q ¯ i s.t. Q ( q ) = 0 (A2b) X a O( a | q ) ≤ ∀ q O( a | q ) ≥ ∀ a, q Moreover, following [33] it can also be shown that the dual variables y ⋆ which are associated withthe primal constraints of Equations (A2a) and (A2b) are all upper bounded by 1, independently of thevalue of η . This implies that κ = P dj =1 | y ⋆j | ≤ d is also independent of η (where d is now the totalnumber of constraints in Equations (A2a) and (A2b) together).When applying our proof using the linear program (A2) we get the following perturbed linear programin Lemma 27 (instead of the one given in Equation (22)):max X q,a Q ( q ) R ( q, a )O( a | q )s.t. Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ζ + 2 ǫ ∀ i, a ¯ i , ∀ q i , q ¯ i s.t. Q ( q ) = 0(A3a) η " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ζ + 2 ǫ ∀ i, a ¯ i , ∀ q i , q ¯ i s.t. Q ( q ) = 0(A3b) X a O( a | q ) ≤ ∀ q (A3c)O( a | q ) ≥ ∀ a, q EST2 A | Q fulfils the constraints of Equation (A3a) as in the proof in the maintext. Moreover, it fulfils Equation (A3c) by definition (see Section II B). Therefore, in order to ensurethat the winning probability of O EST2 A | Q is bounded by the optimal value of the program (A3) we onlyneed to choose η ≤ ζ + 2 ǫ such that the constraints of Equation (A3b) will hold as well.To see that this is possible, recall that the values of ζ and ǫ are chosen such that ζ + 2 ǫ ≤ βκ . As both β and κ are independent of η we can just choose η ≤ ζ + 2 ǫ . The rest of the proof then follows in thesame way as in the main text and Theorem 1 is derived (without any dependence on η ).
2. General games
As the technique of the previous section is relevant only for two-player games , the aim of this sectionis to explain how our proof can be adapted to derive a useful result for multiplayer games which donot have complete-support, as stated in Theorem 2. To do so we slightly modify the parallel repetitionprocedure.Instead of considering the usual parallel repetition, in which n tuples of questions are chosen accordingto the game distribution Q , we change the distribution of questions in the repeated game by sometimes(with small positive probability) asking the players a tuple of questions q for which Q ( q ) = 0. Wecall such questions “dummy questions”; for these questions any answer from the players is accepted.The remaining questions, for which Q ( q ) >
0, are called the “real questions”. We denote the modifiedrepeated game by ˜ G n .It is important to note that the standard definition of the non-signalling constraints implies thata non-signalling strategy should have a well-defined behaviour for all possible inputs. As the refereeignores the players’ answers to the additional questions, the specific behaviour of the strategy on dummyquestions is irrelevant. Therefore, if the optimal non-signalling winning probability in G is 1, then thewinning probability in both G n and ˜ G n is also 1: our modification does not harm the success probabilityof “honest” players.To prove Theorem 2 we proceed in two steps: we make a small change in the linear program (4) andthen apply our proof using the modified program. a. Changing the linear program As a first step we define ˜ Q to be a complete-support version of Q in the following way .Let I( q ) be the indicator function such that I( q ) = 1 if q is a dummy question, i.e., if Q ( q ) = 0, and 1otherwise. Denote by D the number of dummy questions D = |{ q | I( q ) = 1 }| .Let η > q and d ∈ { , } :P ˜ QD ( q, d ) = η D if I( q ) = 1 and d = 1 Q ( q )(1 − η ) if I( q ) = 0 and d = 00 otherwiseThen ˜ Q ( q ) = P d ∈{ , } P ˜ QD ( q, d ) and we haveP ˜ Q | D =0 ( q ) = P ˜ QD ( q, P q P ˜ QD ( q,
0) = P ˜ QD ( q, − η = Q ( q ) . That is, when conditioning on the event of a question not being a dummy question we retrieve Q from ˜ Q . To be more precise, it holds for any game where κ can be bounded by a constant independent of the questionsdistribution Q . In [34, 35] a subset of indices in which dummy, or “confusion”, questions are asked is chosen. We choose to make asmall modification in the questions distribution instead, such that permutation invariance is maintained. Q to write the non-signalling constraints (but keep Q in the objective function):max X q,a Q ( q ) R ( q, a )O( a | q )s.t. ˜ Q ( q i , q ¯ i ) " O( ◦ , a ¯ i | q i , q ¯ i ) − X r i ˜ Q ( r i | q ¯ i )O( ◦ , a ¯ i | r i , q ¯ i ) ≤ ∀ i, q i , q ¯ i , a ¯ i X a O( a | q ) = 1 ∀ q O( a | q ) ≥ ∀ a, q (A4)This linear program replaces the program (4). The distance measure in Definition 6 and the signallingmeasure in Definition 13 should now be defined with respect to ˜ Q as well. b. Deriving Theorem 2 Following the proof of Theorem 1 with the above changes we get the following statement in the mainLemma, Lemma 28: Pr ~a,~q ∼ P ~A~ ˜ Q h w (O EST2 A | Q ) > − α + β i ≤ cdδ . (A5)where the data ~a, ~q is now distributed according to P ~A ~ ˜ Q = ˜ Q ⊗ n × P ~A | ~Q and the parameter δ nowdepends on the change we did in the question distribution ˜ Q (through κ which depends on the solutionof the dual program of program (A4), and thus has an implicit dependence on η ).As the objective function of program (A4) is given using Q and not ˜ Q , w (O EST2 A | Q ) in Equation (A5)is the winning probability with respect to the original question distribution Q . It is therefore equal tothe winning frequency in the real questions (i.e. it does not take the indices where dummy questionswere asked into account). Hence, it leads to the desired statement:Pr ~a,~q ∼ P ~A~ ˜ Q [ f > − α + β ] ≤ cdδ , where f is the winning frequency in the real questions. This proves Theorem 2.The parameter η can be optimised in different ways, depending on the application. If one is interestedin the bound itself and is not concerned by the modification of the repeated game the precise valueof η should be chosen in order to optimise the constants C ( G, n ) and C ( G ) appearing in the bound.Alternatively, if one does not wish to change the game by too much, small values for η will ensure that˜ G n is relatively close to G n (due to the definition of ˜ Q above). A smaller η will lead to a smaller fractionof dummy questions, but could result in worse constants C ( G ). Appendix B: Proofs of Section III
In this section we present all the proofs which are relevant to the signalling measures and signallingtests.The first proof is a proof of Lemma 14 which shows that the signalling measure given in Definition 13is continuous. We repeat Lemma 14 here:
Lemma 14.
Let O and O be two one-game strategies such that (cid:12)(cid:12) O − O (cid:12)(cid:12) ≤ ǫ . Then ∀ i, a ¯ i , q i , q ¯ i (cid:12)(cid:12) Sig ( i,a ¯ i ,q i ,q ¯ i ) (O ) − Sig ( i,a ¯ i ,q i ,q ¯ i ) (O ) (cid:12)(cid:12) ≤ ǫ . Proof.
We prove a stronger result from which the lemma follows. We prove ∀ i X a ¯ i ,q (cid:12)(cid:12) Sig ( i,a ¯ i ,q i ,q ¯ i ) (O ) − Sig ( i,a ¯ i ,q i ,q ¯ i ) (O ) (cid:12)(cid:12) ≤ ǫ . (cid:12)(cid:12) O − O (cid:12)(cid:12) = E q X a (cid:12)(cid:12) O ( a | q ) − O ( a | q ) (cid:12)(cid:12) ≥ E q X a ¯ i (cid:12)(cid:12)(cid:12) X a i (cid:16) O ( a i , a ¯ i | q ) − O ( a i , a ¯ i | q ) (cid:17) (cid:12)(cid:12)(cid:12) = E q X a ¯ i (cid:12)(cid:12) O ( ◦ , a ¯ i | q ) − O ( ◦ , a ¯ i | q ) (cid:12)(cid:12) = X a ¯ i ,q Q ( q ) (cid:12)(cid:12) O ( ◦ , a ¯ i | q ) − O ( ◦ , a ¯ i | q ) (cid:12)(cid:12) , therefore if (cid:12)(cid:12) O − O (cid:12)(cid:12) ≤ ǫ then X a ¯ i ,q Q ( q ) (cid:12)(cid:12) O ( ◦ , a ¯ i | q ) − O ( ◦ , a ¯ i | q ) (cid:12)(cid:12) ≤ ǫ . (B1)Next, using Equation (6) X a ¯ i ,q (cid:12)(cid:12) Sig ( i,a ¯ i ,q i ,q ¯ i ) (O ) − Sig ( i,a ¯ i ,q i ,q ¯ i ) (O ) (cid:12)(cid:12) = X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − X r i Q ( r i | q ¯ i )O ( ◦ , a ¯ i | r i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) + X r i Q ( r i | q ¯ i )O ( ◦ , a ¯ i | r i , q ¯ i ) (cid:12)(cid:12)(cid:12) = X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) + X r i Q ( r i | q ¯ i ) (cid:16) O ( ◦ , a ¯ i | r i , q ¯ i ) − O ( ◦ , a ¯ i | r i , q ¯ i ) (cid:17) (cid:12)(cid:12)(cid:12) ≤ X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) (cid:12)(cid:12)(cid:12) + X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) X r i Q ( r i | q ¯ i ) (cid:16) O ( ◦ , a ¯ i | r i , q ¯ i ) − O ( ◦ , a ¯ i | r i , q ¯ i ) (cid:17) (cid:12)(cid:12)(cid:12) ≤ X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) (cid:12)(cid:12)(cid:12) + X a ¯ i ,q X r i Q ( r i | q ¯ i ) Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | r i , q ¯ i ) − O ( ◦ , a ¯ i | r i , q ¯ i ) (cid:12)(cid:12)(cid:12) = X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) (cid:12)(cid:12)(cid:12) + X a ¯ i ,q ¯ i X r i Q ( r i | q ¯ i ) Q ( q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | r i , q ¯ i ) − O ( ◦ , a ¯ i | r i , q ¯ i ) (cid:12)(cid:12)(cid:12) = X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) (cid:12)(cid:12)(cid:12) + X a ¯ i ,q Q ( q i , q ¯ i ) (cid:12)(cid:12)(cid:12) O ( ◦ , a ¯ i | q i , q ¯ i ) − O ( ◦ , a ¯ i | q i , q ¯ i ) (cid:12)(cid:12)(cid:12) ≤ ǫ where the last inequality follows from Equation (B1).Next we give the proof of Lemma 16: Lemma 16.
Assume the players share an i.i.d. strategy O ⊗ nA | Q and let ζ, ǫ > be the the parametersdefined as in Equation (8) . For every ( i, b ¯ i , s i , s ¯ i ) ,1. If Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) ≥ ζ then Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > − δ (B2)
2. If
Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) = 0 then Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 0 i > − δ (B3) where δ = δ (cid:0) n , ǫ (cid:1) = (cid:0) n + 1 (cid:1) |A|·|Q|− e − nǫ / . Proof.
For the first part of the lemma assume that Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) ≥ ζ . ThenPr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 0 i = Pr ~a,~q ∼ O ⊗ nAQ h Sig ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) O EST1 (cid:1) < ζ − ǫ i ≤ Pr ~a,~q ∼ O ⊗ nAQ (cid:2) | O EST1 − O | > ǫ (cid:3) ≤ δ where the first inequality is due to Lemma 14 and the second due to Lemma 8. This implies Equa-tion (B2). Equation (B3) can be proven in an analogous way.The last proof of this section is the proof of Lemma 18: Lemma 18.
Let ν > be any parameter such that ν < ζ − ǫ . Then for every ( i, b ¯ i , s i , s ¯ i ) , ∀ O ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) , Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) > ν . Proof.
Assume by contradiction that there exists O ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) such that Sig ( i,b ¯ i ,s i ,s ¯ i ) (O) ≤ ν . SinceO ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) there exists ¯O such that | O − ¯O | ≤ ǫ andPr ~a,~q ∼ ¯O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i > δ . (B4)Using Lemma 14 we get Sig ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) ¯O (cid:1) ≤ ν + 2 ǫ .From Lemma 8 we know that Pr ~a,~q ∼ ¯O ⊗ nAQ (cid:2) | ¯O EST1 − ¯O | > ǫ (cid:3) ≤ δ and therefore, using Lemma 14again, Pr ~a,~q ∼ ¯O ⊗ nAQ h Sig ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) ¯O EST1 (cid:1) > ν + 4 ǫ i ≤ δ . Since ν < ζ − ǫ this impliesPr ~a,~q ∼ ¯O ⊗ nAQ h Sig ( i,b ¯ i ,s i ,s ¯ i ) (cid:0) ¯O EST1 (cid:1) > ζ − ǫ i ≤ δ and therefore, according to the definition of the test,Pr ~a,~q ∼ ¯O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) (cid:16) ~q t , ~a t (cid:17) = 1 i ≤ δ , which contradicts Equation (B4). Appendix C: Proofs of Section IV
In this section we prove the relevant properties of the de Finetti strategy. We prove Lemma 20:
Lemma 20.
For a de Finetti strategy τ ~A | ~Q and every ( i, b ¯ i , s i , s ¯ i ) Pr ~a,~q ∼ τ ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 ∧ O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ δ Pr ~a,~q ∼ τ ~A~Q h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 ∧ O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ δ Proof.
Since a de Finetti strategy is a convex combination of i.i.d. strategies, it is sufficient to prove thisfor i.i.d. strategies O ⊗ nA | Q and the lemma will follow. We start by proving the first part of the lemma.If Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 i ≤ δ then we are done. Consider therefore states O A | Q such thatPr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 i > δ . For such statesPr ~a,~q ∼ O ⊗ nAQ h O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ Pr ~a,~q ∼ O ⊗ nAQ h | O EST2 A | Q − O A | Q | > ǫ i ≤ δ ( i,b ¯ i ,s i ,s ¯ i ) and the second from Lemma 8.All together we get Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 1 ∧ O EST2 A | Q / ∈ Σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ δ as required for thefirst part of the lemma.We now proceed to the second part of the lemma.If Pr ~a,~q ∼ O ⊗ nAQ h O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤ δ then we are done. Consider therefore states O A | Q such thatPr ~a,~q ∼ O ⊗ nAQ h O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i > δ . Using Lemma 8 we know that there exists a state O
EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) such that | O EST2 A | Q − O A | Q | ≤ ǫ and according to the definition of σ ( i,b ¯ i ,s i ,s ¯ i ) this implies that O A | Q is ζ signalling or more. Therefore,according to Lemma 16, Pr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 i ≤ δ . All together we getPr ~a,~q ∼ O ⊗ nAQ h T ( i,b ¯ i ,s i ,s ¯ i ) ( ~q t , ~a t ) = 0 ∧ O EST2 A | Q ∈ σ ( i,b ¯ i ,s i ,s ¯ i ) i ≤2