Condorcet Jury Theorem: The dependent case

Transcription

1 Condorcet Jury Theorem: The dependent case Bezalel Peleg and Shmuel Zamir Center for the Study of Rationality The Hebrew University of Jerusalem. Abstract We provide an extension of the Condorcet Theorem. Our model includes both the Nitzan-Paroush framework of unequal competencies and Ladha s model of correlated voting by the jurors. We assume that the jurors behave informatively ; that is, they do not make a strategic use of their information in voting. Formally, we consider a sequence of binary random variables X = X,X,...,X n,...) with range in {,} and a joint probability distribution P. The pair X,P) is said to satisfy the Condorcet Jury Theorem CJT ) if lim n P Σ n i= X i > n ) =. For a general dependent) distribution P we provide necessary as well as sufficient conditions for the CJT that establishes the validity of the CJT for a domain that strictly and naturally) includes the domain of independent jurors. In particular we provide a full characterization of the exchangeable distributions that satisfy the CJT. We exhibit a large family of distributions P with liminf n nn ) Σn i= Σ j icovx i,x j ) > that satisfy the CJT. We do that by interlacing carefully selected pairs X,P) and X,P ). We then proceed to project the distributions P on the planes p,y), and p,y ) where p = liminf n p n, y = liminf n EX n p n ), y = liminf n E X n p n, p n = p + p,...+ p n )/n, and X n = X + X,... + X n )/n. We determine all feasible points in each of these planes. Quite surprisingly, many important results on the possibility of the CJT are obtained by analyzing various regions of the feasible set in these planes. In the space P of all probability distributions on S p = {,}, let P be the set of all probability distributions in P that satisfy the CJT and let P = P\P be the set of all probability distributions in P that do not satisfy the CJT. We prove that both P and P are convex sets and that P is dense in P in the weak topology). Using an appropriate separation theorem we then provide an affine functional that separates these two sets. We thank Marco Scarsini and Yosi Rinott for drawing our attention to de Finetti s theorem.

2 Introduction The simplest way to present our problem is by quoting Condorcet s classic result see Young 997)): Theorem. CJT Condorcet 785) Let n voters n odd) choose between two alternatives that have equal likelihood of being correct a priori. Assume that voters make their judgements independently and that each has the same probability p of being correct < p < ). Then, the probability that the group makes the correct judgement using simple majority rule is n [n!/h!n h)!]p h p) n h h=n+)/ which approaches as n becomes large. We generalize Condorcet s model by presenting it as a game with incomplete information in the following way: Let I = {,,...,n} be a set of jurors and let D be the defendant. There are two states of nature: g in which D is guilty, and z in which D is innocent. Thus the set of states of nature is S = {g,z}. Each juror has an action set A with two actions: A = {c,a}. The action c is to convict D. The action a is to acquit D. Before the voting, each juror i gets a private random signal t i T i := {tg i,ti z }. In the terminology of games with incomplete information, T i is the type set of juror i. The interpretation is that juror i of type tg i thinks that D is guilty while juror i of type ti z thinks that D is innocent. The signals of the jurors may be dependent and may also depend on the the state of nature. In our model the jurors act informatively not strategically ); that is, the strategy of juror i is σ i : T i A given by σ i tg i) = c and σ i tz i ) = a. The definition of informative voting is due to Austen-Smith and Banks 996), who question the validity of the CJT in a strategic framework. Informative voting was, and is still, assumed in the vast majority of the literature on the CJT, mainly because it is implied by the original Condorcet assumptions. More precisely, assume, as Condorcet did, that Pg) = Pz) = / and that each juror is more likely to receive the correct signal that is, Ptg i g) = Pti z z) = p > /); then the strategy of voting informatively maximizes the probability of voting correctly, among all four pure voting strategies. Following Austen-Smith and Banks, strategic voting and Nash Equilibrium were studied by Wit 998), Myerson 998), and recently by Laslier and Weibull 8), who discuss the assumption on preferences and beliefs under which sincere voting is a Nash equilibrium in a general deterministic majoritarian voting rule. As we said before, in this work we do assume informative voting and leave strategic considerations and equilibrium concepts for the next phase of our research. The action taken by a finite society of jurors {,...,n} i.e. the jury verdict) is determined by a simple majority with some tie-breaking rule, e.g., by coin tossing). We are interested in the probability that the finite) jury will reach the correct decision. Again in the style of games

3 with incomplete information let Ω n = S T,..., T n be the set of states of the world. A state of the world consists of the state of nature and a list of the types of all jurors. Denote by p n) the probability distribution on Ω n. This is a joint probability distribution on the state of nature and the signals of the jurors. For each juror i let the random variable X i : S T i {,} be the indicator of his correct voting, i.e., X i g,tg i) = X iz,tz i) = and X i g,tz) i = X i z,tg) i =. The probability distribution p n) on Ω n induces a joint probability distribution on the the vector X = X,...,X n ), which we denote also by p n). If n is odd, then the probability that the jury reaches a correct decision is ) n p n) X i > n i=. Figure illustrates our construction in the case n =. In this example, according to p ) the state of nature is chosen with unequal probabilities for the two states: Pg) = /4 and Pz) = 3/4 and then the types of the two jurors are chosen according to a joint probability distribution that depends on the state of nature. t g t g 4 /4 g t z 4 Nature t g 3/4 z t g t z t z 4 4 t z Figure The probability distribution p ). Guided by Condorcet, we are looking for limit theorems as the the size of the jury increases. Formally, as n goes to infinity we obtain an increasing sequence of worlds, Ω n ) n=, such that for all n, the projection of Ω n+ on Ω n is the whole Ω n. The corresponding sequence of probability distributions is p n) ) n= and we assume that for every n, the marginal distribution of p n+) on Ω n is p n). It follows from the Kolmogorov extension theorem see Loeve 963), p. 93) that this defines a unique probability measure P on the projective, or inverse) limit Ω = lim n Ω n = S T... T n... such that, for all n, the marginal distribution of P on Ω n is p n). 3

4 In this paper we address the the following problem: Which probability measures P derived in this manner satisfy the Condorcet Jury Theorem CJT ); that is, Which probability measures P satisfy lim Σ P n n i= X i > n ) =. As far as we know, the only existing result on this general problem is that of Berend and Paroush 998), which deals only with independent jurors. Rather than working with the space Ω and its probability measure P, it will be more convenient to work with the infinite sequence of binary random variables X = X,X,...,X n,...) the indicators of correct voting ) and the induced probability measure on it, which we shall denote also by P. Since the pair X,P) is uniquely determined by Ω,P), in considering all pairs X,P) we cover all pairs Ω,P). We provide a full characterization of the exchangeable sequences X =X,X,...,X n,...) that satisfy the CJT. For a general dependent) distribution P, not necessarily exchangeable, we provide necessary as well as sufficient conditions for the CJT. We exhibit a large family of distributions P with liminf n nn ) Σn i= Σ j icovx i,x j ) > that satisfy the CJT. In the space P of all probability distributions on S p = {,}, let P be the set of all probability distributions in P that satisfy the CJT and let P = P\P be the set of all probability distributions in P that do not satisfy the CJT. We prove that both P and P are convex sets and that P is dense in P in the weak topology). Using an appropriate separation theorem we then provide an affine functional that separates these two sets. Sufficient conditions Let X = X,X,...,X n,...) be a sequence of binary random variables with range in {,} and with joint probability distribution P. The sequence X is said to satisfy the Condorcet Jury Theorem CJT) if lim Σ P n n i= X i > n ) = ) We shall investigate necessary as well as sufficient conditions for CJT. Given a sequence of random binary variables X = X,X,...,X n,...) with joint distribution P denote p i = EX i ), VarX i ) = EX i p i ) and CovX i,x j ) = E[X i p i )X j p j )], for i j, where E denotes, as usual, the expectation operator. Also let p n = p + p,...+ p n )/n and X n = X + X,...+X n )/n. Our first result provides a sufficient condition for CJT : Theorem. Assume that Σ n i= p i > n for all n > N and EX n p lim n ) n p n =, ) ) 4

5 or equivalently assume that p n lim = ; 3) n EXn p n ) then the CJT is satisfied. Proof. P Σ n i= X i n ) = P Σ n i= X i n ) = P Σ n i= p i Σ n i= X i Σ n i= p i n ) P Σ n i= p i Σ n i= X i Σ n i= p i n ) By Chebyshev s inequality assuming Σ n i= p i > n ) we have P Σ n i= p i Σ n i= X i Σ n i= p i n ) E Σ n i= X i Σ n i= p ) i Σ n i= p i n ) = EX n p n ) p n ) As this last term tends to zero by ), the CJT ) then follows. Corollary 3. If Σ n i= Σ j icovx i,x j ) for n > N in particular if CovX i,x j ) for all i j) and lim n npn ) =, then the CJT is satisfied. Proof. Since the variance of a binary random variable X with mean p is p p) /4 we have for n > N, EX n p n ) = n E Σn i= X i p i )) Therefore if lim n npn ) =, then = n Σ n i= VarX i )+Σ n i= Σ j icovx i,x j ) ) 4n lim n EX n p n ) p n ) lim n 4np n ) = Remark 4. Note that under the condition of corollary 3, namely, for bounded random variable with all covariances being non-positive, the weak) law of large numbers LLN) holds for arbitrarily dependent variables see, e.g., Feller 957) volume I, exercise 9, p. 6). This is not implied by corollary 3 since, as we shall see later, the CJT, strictly speaking, is not a law of large numbers. In particular, CJT does not imply LLN and LLN does not imply CJT. 5

6 Remark 5. When X,X,...,X n,... are independent, then under mild conditions lim n npn ) = is a necessary and sufficient condition for CJT see Berend and Paroush 998)). Remark 6. The sequence X = X,X,...,X n,...) of i.i.d. variables with PX i = ) = PX i = ) = / satisfies the LLN but does not satisfy the CJT, since it does not satisfy Berend and Paroush s necessary and sufficient condition; therefore LLN does not imply CJT. Given a sequence X = X,X,...,X n,...) of binary random variables with a joint probability distribution P, we define the following parameters of X,P): p p y y y y We first observe the following: Remark 7. If p > / and y = then the CJT is satisfied. := lim inf n n 4) := lim sup p n 5) n := lim inf n p n n ) 6) := lim supex n p n ) n 7) := liminf n p n n 8) := limsupe X n p n n 9) Proof. As EX n p n ), if y = then lim n EX n p n ) =. Since p > /, there exists n such that p n > /+ p)/ for all n > n. The result then follows by Theorem. Necessary conditions using the L -norm Given a sequence X = X,X,...,X n,...) of binary random variables with a joint probability distribution P, if y >, then we cannot use Theorem to conclude CJT. To derive necessary conditions for the CJT, we first have: Proposition 8. If the CJT holds then p. Proof. Define a sequence of events B n ) n= by B n = {ω X n ω) / }. Since the CJT holds, lim n P Σ n i= X i > n ) = and hence limn PB n ) =. Since p n = EX n )) PΩ\B n), taking the liminf, the right-hand side tends to zero and we obtain that liminf n p n = p. 6

7 We shall first consider a stronger violation of Theorem than y > ; namely, assume that y >. We shall prove that in this case, there is a range of distributions P for which the CJT is false. First we notice that for x, x x. Hence E X n p n EX n p n ) for all n and thus y > implies that y > We are now ready to state our first impossibility theorem, which can be readily translated into a necessary condition. Theorem 9. Given a sequence X = X,X,...,X n,...) of binary random variables with joint probability distribution P, if p < + y, then the X,P) violates the CJT. Proof. If y =, then the CJT is violated by Proposition 8. Assume then that y > and choose ỹ such that < ỹ < y and t := ỹ + p >. First we notice that, since EX n p n ) =, we have E max, p n X n ) = E max,x n p n ), and thus, since y >, we have E max, p n X n ) > ỹ for n > n. ) If Ω,P) is the probability space on which the sequence X is defined, for n > n define the events B n = {ω p n X n ω) max, ỹ t)} ) By ) and ), PB n ) > q > for some q and Choose now a subsequence n k ) k= p n X n ω) ỹ t for ω B n, n > n. ) such that By ) and 3), for all ω B nk we have, p nk < ỹ + t = p+t, k =,,... 3) X nk ω) p nk ỹ +t <, and thus PX nk > ) q <, which implies that P violates the CJT. Corollary. If liminf n p n and liminf n E X n p n >, then P violates the CJT. 7

8 3 Necessary conditions using the L -norm Let X = X,X,...,X n,...) be a sequence of binary random variables with a joint probability distribution P. In this section we take a closer look at the relationship between the parameters y and y see 7) and 9)). We first notice that y > if and only if y >. Next we notice that p n for n > n implies that X n p n for n > n. Thus, by corollary, if y > and the CJT is satisfied then max,x n p n ) for n > n. Finally we observe the following lemma, whose proof is straightforward: Lemma. If limsup n P{ω p n X n ω) p n /} >, then the CJT is violated. We now use the previous discussion to prove the following theorem: Theorem. If i) liminf n p n > and ii) liminf n PX n > p n /) =, then y y. Proof. Condition i) implies that p n > for all n > N for some N. This implies that lim n PX n p n < ) =. Also, ii) implies that lim n Pp n X n ) =, and thus and lim P n X n p n ) = 4) Define the events B n = {ω X nω) p n }; then by 4) lim inf X n p n n ) dp = y 5) B n lim inf X n p n n dp = y. 6) B n Since any u [, ] satisfies u u, it follows from 5) and 6) that y y. Corollary 3. Let p = liminf n p n and y = liminf n EX n p n ). Then if p < + y then P does not satisfy the CJT. Proof. Assume that p < +y. If y =, then CJT is not satisfied by Proposition 8. Hence we may thus assume that y >, which also implies that y >. Thus, if liminf n p n then CJT fails by Corollary. Assume then that liminf n p n >. By Lemma we may also assume that liminf n P X n > p n / ) = and thus by Theorem we have y y and hence p < + y + y and the CJT fails by Theorem 9. 8

9 4 Dual Conditions A careful reading of Sections and 3 reveals that it is possible to obtain dual results to Theorems 9 and and Corollary 3 by replacing liminf by limsup. More precisely, for a sequence X = X,X,...,X n,...) of binary random variables with joint probability distribution P, we let p = limsup n p n and y = limsup n E X n p n, and we have: Theorem 4. If p < + y, then the X,P) violates the CJT. Proof. As we saw in the proof of Corollary 3, we may assume that liminf n p n and hence also p = limsup p n liminf p n n n, and hence y >. Choose ỹ such that < ỹ < y and t = ỹ + p >. Let X n k ) k= be a subsequence of X such that lim k E X n k p nk = y. As in ) we get E max, p nk X nk ) > ỹ Define the events B nk ) k= by By 7) and 8), PB nk ) > q for some q > and for k > k. 7) B nk = {ω p nk X nk ω) ỹ t}. 8) p nk X nk ω) ỹ t for ω B n k and k > k. 9) Now lim sup p n = p < ỹ n + t. ) Thus, for n sufficiently large p n < ỹ + t. Hence, for k sufficiently large and all ω B n k, X nk ω) p nk ỹ +t <. ) Therefore PX nk > ) q < for sufficiently large k in violation of the CJT. Similarly we have the dual results to those of Theorem and Corollary 3: Theorem 5. If i) liminf n p n > and ii) liminf n PX n > p n /) =, then y y. Corollary 6. If p < + y then P does not satisfy the CJT. The proofs, which are similar respectively to the proofs of Theorem and Corollary 3, are omitted. 9

10 5 Existence of distributions satisfying the CJT In this section we address the issue of the existence of distributions that satisfy the CJT. For that, let us first clarify the relationship between the CJT and law of large numbers which, at first sight, look rather similar. Recall that an infinite sequence of binary random variables X = X,X,...,X n,...) with a joint probability distribution P satisfies the weak) law of large numbers LLN) if in our notations): ε >, lim n P X n p n < ε ) = ) while it satisfies the Condorcet Jury Theorem CJT ) if: lim P X n > ) = 3) n Since by Proposition 8, the condition p is necessary for the validity of the CJT, let us check the relationship between the LLN and the CJT in this region. Our first observation is: Proposition 7. For a sequence X = X,X,...,X n,...) with probability distribution P satisfying p >, if the LLN holds then the CJT also holds. Proof. Let p = /+3δ for some δ > and let N be such that p n > /+δ for all n > N ; then for all n > N we have P X n > ) P X n ) + δ P X n p n < δ ) Since the last expression tends to as n, the first expression does too, and hence the CJT holds. Remark 8. The statement of Proposition 7 does not hold for p =. Indeed, the sequence X = X,X,...,X n,...) of i.i.d. variables with PX i = ) = PX i = ) = / satisfies the LLN but does not satisfy the CJT see Remark 6). Unfortunately, Proposition 7 is of little use to us. This is due to the following fact: Proposition 9. If the random variables of the sequence X = X,X,...,X n,...) are uniformly bounded then the condition is a necessary condition for LLN to hold. lim n E X n p n ) = The proof is elementary and can be found, e.g., in Uspensky 937), page 85. For the sake of completeness it is provided here in the Appendix. It follows thus from Proposition 9 that LLN cannot hold when y > and thus we cannot use Proposition 7 to establish distributions in this region that satisfy the CJT. Nevertheless, we shall exhibit a rather large family of distributions P with y > and p > /) for which the CJT holds. Our main result is the following:

11 Theorem. Let t [, ]. If F is a distribution with parameters p,y ), then there exists a distribution H with parameters p = t +t p and ỹ = ty that satisfy the CJT. Proof. To illustrate the idea of the proof we first prove somewhat informally) the case t = /. Let X = X,X,...,X n,...) be a sequence of binary random variables with a joint probability distribution F. Let G be the distribution of the sequence Y = Y,Y,...,Y n,...), where EY n = for all n that is, Y = Y =...Y n =... and PY i = ) = i). Consider now the following interlacing of the two sequences X and Y : Z = Y,Y,X,Y 3,X,Y 4,X 3,...,Y n,x n,y n+,x n...), and let the probability distribution H of Z be the product distribution H = F G. It is verified by straightforward computation that the parameters of the distribution H are in accordance with the theorem for t =, namely, p = + p and ỹ = y. Finally, as each initial segment of voters in Z contains a majority of Y i s thus with all values ), the distribution H satisfies the CJT, completing the proof for t =. The proof for a general t [, /) follows the same lines: We construct the sequence Z so that any finite initial segment of n variables, includes about, but not more than the initial tn segment of the X sequence, and the rest is filled with the constant Y i variables. This will imply that the CJT is satisfied. Formally, for any real x let x be the largest integer less than or equal to x and let x be smallest integer greater than or equal to x. Note that for any n and any t we have tn + t)n = n; thus, one and only one of the following holds: i) tn < tn+) or ii) t)n < t)n+) From the given sequence X and the above-defined sequence Y of constant variables) we define now the sequence Z = Z,Z,...,Z n,...) as follows: Z = Y and for any n, let Z n = X tn+) if i) holds and Z n = Y t)n+) if ii) holds. This inductive construction guarantees that for all n, the sequence contains tn X i coordinates and t)n Y i coordinates. The probability distribution H is the product distribution F G. The fact that Z, H) satisfies the CJT follows from: t)n t)n > tn tn, and finally p = t +t p and ỹ = ty is verified by straightforward computation. Remark. The interlacing of the two sequences X and Y described in the proof of Theorem may be defined for any t [,]. We were specifically interested in t [,/] since this guarantees the CJT.

12 6 Feasibility considerations The conditions developed so far for a sequence X = X,X,...,X n,...) with joint probability distribution P to satisfy the CJT involved only the parameters p, p,y,y,y, and y. In this section we pursue our characterization in the space of these parameters. We shall look at the distributions in two different spaces: the space of points p,y ), which we call the L space, and the space p,y), which we call the L space. 6. Feasibility and characterization in L With the pair X,P) we associate the point p,y ) in the Euclidian plane R. It follows immediately that p. We claim that y p p) holds for all distributions P. To see that, we first observe that E X i p i = p i p i ); hence E X n p n = n n n E X i p i ) i= n E i p i i= X The function n i= p i p i ) is strictly) concave. Hence E X n p n Finally, let p = lim k p nk ; then n i= ) n n p i p i ). i= n p i p i ) p n p n ). 4) y = liminf E X n p n n liminf E X n k p nk lim p nk p nk ) = p p). k k The second inequality is due to 4). Thus, if u,w) denote a point in R, then any feasible pair p,y ) is in the region FE = {u,w) u, w u u)}. 5) We shall now prove that all points in this region are feasible; that is, any point in FE is attainable as a pair p,y ) of some distribution P. Then we shall indicate the sub-region of FE where the CJT may hold. We first observe that any point u,w ) FE on the parabola w = u u), for u, is feasible. In fact such u,w ) is attainable by the sequence X = X,X,...,X n,...) with identical variables X i, X = X =... = X n..., and EX = u clearly p = u, and y = u u ) follows from the dependence and from E X i p i = p i p i ) = u u )). Let again u,w ) be a point on the parabola, which is thus attainable. Assume that they are the parameters p,y ) of the pair X,F). Let Y,G) be the pair of constant variables) described in the proof of Theorem and let t [,]. By Remark the t- interlacing of X,F) and Y,G) can be constructed to yield a distribution with parameters p = t p + t) and ỹ = ty see the proof of Theorem ). Thus, the line segment defined by ũ = tu + t) and w = tw for t, connecting u,w ) to,),

13 w = y* / u, w ) w = u u) w = u) w u FE / M u = p Figure The feasible set FE. consists of attainable pairs contained in FE. Since any point u,w) in FE lies on such a line segment, we conclude that every point in FE is attainable. We shall refer to FE as the feasible set, which is shown in Figure. We now attempt to characterize the points of the feasible set according to whether the CJT is satisfied or not. For that we first define: Definition. The strong CJT set, denoted by scjt, is the set of all points u,w) FE such that any pair X,P) with parameters p = u and y = w satisfies the CJT. The weak CJT set, denoted by wcjt, is the set of all points u,w) FE for which there exists a pair X,P) with parameters p = u and y = w that satisfies the CJT. We denote scjt = FE \scjt and wcjt = FE \wcjt. For example, as we shall see later see Proposition 4),,) scjt and /,) wcjt. By Theorem 9, if u < / + /w, then u,w) wcjt. Next we observe that if u,w ) is on the parabola w=u u) and M is the midpoint of the segment [u,w ),,)], then by the proof of Theorem, the segment [M,,)] wcjt see Figure ). To find the upper boundary of the union of all these segments, that is, the locus of the mid points M in Figure, we eliminate u,w ) from the equations w = u u ), and u,w) = /u,w )+/,), and obtain w = u ) u) 6) This is a parabola with maximum /4 at u = 3/4. The slope of the tangent at u = / is ; that is, the tangent of the parabola at that point is the line w = u defining the 3

14 region wcjt. Finally, a careful examination of the proof of Theorem, reveals that for every u,w ) on the parabola w = u u), the line segment [u,w ),M] is in scjt see Figure ). Our analysis so far leads to the conclusions summarized in Figure 3 describing the feasibility and and regions of CJT possibility for all pairs X, P). / w = y * w = u u) w = u /4 wcjt scjt w = u ) u ) / wcjt 3/4 u = p Figure 3 Regions of possibility of CJT in L. Figure 3 is not complete in the sense that the regions wcjt and scjt are not disjoint, as it may mistakenly appear in the figure. More precisely, we complete Definition by defining: Definition 3. The mixed CJT set, denoted by mcjt, is the set of all points u,w) FE for which there exists a pair X,P) with parameters p = u and y = w that satisfies the CJT, and a pair ˆX, ˆP) with parameters ˆp = u and ŷ = w for which the CJT is violated. Then the regions scjt, wcjt, and mcjt are disjoint and form a partition of the feasible set of all distributions FE FE = wcjt scjt mcjt 7) To complete the characterization we have to find the regions of this partition, and for that it remains to identify the region mcjt since, by definition, wcjt \mcjt scjt and scjt\mcjt wcjt. Proposition 4. All three regions scjt, wcjt, and mcjt are not empty. Proof. As can be seen from Figure 3, the region wcjt is clearly not empty; it contains for example the points, ) and /, /). As we remarked already following Definition ), the region scjt contains the point,). This point corresponds to a unique pair X,P), in which X i = for all i with probability, that trivially satisfies the CJT). The 4

15 region mcjt contains the point /,). To see this recall that the Berend and Paroush s necessary and sufficient condition for CJT in the independent case see Remark 5) is lim npn n ) = 8) First consider the pair X, P) in which X i ) i= are i.i.d with P X i = ) = / and P X i = )=/. Clearly np n ) = for all n and hence condition 8) is not satisfied, implying that the CJT is not satisfied. Now consider X,P) in which X =,,,,, ) with probability. This pair corresponds to the point /,) since { + X n = p n = n if n is even + n if n is odd, and hence p = / and y =. Finally this sequence satisfies the CJT as X n > with probability one for all n. 6. Feasibility and characterization in L Replacing y = liminf n E X n p n by the parameter y = liminf n EX n p n ), we obtain results in the space of points p, y) similar to those obtained in the previous section in the space p,y ). Given a sequence of binary random variables X with its joint distribution P, we first observe that for any i j, Therefore, CovX i,x j ) = EX i X j ) p i p j minp i, p j ) p i p j. EX n p n ) = n { n i= j i n { n i= j i CovX i,x j )+ n i= [minp i, p j ) p i p j ]+ p i p i ) n i= } p i p i ) } 9). 3) We claim that the maximum of the last expression 3), under the condition n i= p i = p n, is p n p n ). This is attained when p = = p n = p n. To see that this is indeed the maximum, assume to the contrary that the maximum is attained at p = p,, p n ) with p i p j for some i and j. Without loss of generality assume that: p p p n with p < p j and p = p l for l < j. Let < ε < p j p )/ and define p = p,, p n) by p = p + ε, p j = p j ε, and p l = p l for l / {, j}. A tedious, but straightforward, computation shows that the expression 3) is higher for p than for p, in contradiction to the assumption that it is maximized at p. We conclude that EX n p n ) p n p n ). 5

16 Let now p nk ) k= be a subsequence converging to p; then y = lim inf n lim inf k EX n p n ) liminf EX n k p nk ) k p n k p nk ) = p p). We state this as a theorem: Theorem 5. For every pair X, P), the corresponding parameters p, y) satisfy y p p). Next we have the analogue of Theorem, proved in the same way. Theorem 6. Let t [, ]. If F is a distribution with parameters p,y), then there exists a distribution H with parameters p = t +t p and ỹ = t y that satisfy the CJT. We can now construct Figure 4, which is the analogue of Figure in the L space p,y). The feasible set in this space is FE = {u,w) u, w u u)} 3) w = y /4 u, w ) w = u u) w = u u w FE / M + u u = p Figure 4 The feasible set FE. The geometric locus of the midpoints M in Figure 4 is derived from ) u = u +, ) w = 4 w, and 3) w = u u ) and is given by w = u ) u). This yields Figure 5, which is the analogue of Figure 3. Note, however, that unlike in Figure 3, the straight line w = u is not tangential to the small parabola w = u ) u) at,). 6

17 w = y /4 w = u u) w = u / /8 wcjt /6 / scjt wcjt 3/4 w = u /) u) u = p Figure 5 Regions of possibility of CJT in L. The next step toward determining the region mcjt in the L space Figure 5) is the following: Proposition 7. For any u,w) {u,w) < u < ; w u u)}, there is a pair Z,H) such that: i) EZ i ) = u, i. ii) liminf n EZ n u) = w. iii) The distribution H does not satisfy the CJT. Proof. For < u <, let X,F ) be given by X = X =... = X n =... and EX i ) = u; let Y,F ) be a sequence of of i.i.d. random variables Y i ) i= with expectation u. For < t let Z t,h t ) be the pair in which Z t i = tx i + t)y i for i =,,... and H t is the product distribution H t = F F that is, the X and the Y sequences are independent). Note first that EZi t ) = u for all i and lim n EZt n u) = lim n u u) t) +tu u) n ) = tu u), and therefore the pair Z t,h t ) corresponds to the point u,w) in the L space, where w = tu u) ranges in,u u)) as < t. Finally, Z t,h t ) does not satisfy the CJT since for all n, PrZ t n > ) PrZt = Zt =... = ) = t u) <. 7

18 As this argument does not apply for t = it remains to prove that, except for, ), to any point u,) on the x axis corresponds a distribution that does not satisfy the CJT. For u /, the sequence Y,F ) of of i.i.d. random variables Y i ) i= with expectation u does not satisfy the CJT, as follows from the result of Berend and Paroush 998). For / < u < such a sequence of i.i.d. random variables does satisfy the CJT and we need the following more subtle construction: Given the two sequences X,F ) and Y,F ) defined above we construct a sequence Z = Z i ) i= consisting of alternating blocks of X i-s and Y i -s, with the probability distribution on Z being that induced by the product probability H = F F. Clearly EZ i ) = u for all i, in particular p n = u for all n and p = u. We denote by B l the set of indices of the l-th block and its cardinality by b l. Thus nl) = Σ l j= b j is the index of Z i at the end of the l-th block. Therefore B l+ = {nl)+,...,nl)+b l+ )} and nl+) = nl)+b l+. Define the block size b l inductively by:. b =, and for k =,,...,. b k = kσ k j= b j and b k+ = b k. Finally we define the sequence Z = Z i ) i= to consist of X i-s in the odd blocks and Y i -s in the even blocks, that is, { Xi if i B Z i = k for some k =,,... Y i if i B k for some k =,,... Denote by n x l) and n y l) the number of X coordinates and Y coordinates respectively in the sequence Z at the end of the l-th block and by nl) = n x l)+n y l) the number of coordinates at the end of the l-th block of Z. It follows from and in the definition of b l ) that for k =,,..., n x k ) = n y k )+ 3) n x k) and hence also n xk) n y k) k nk) k 33) It follows from 3) that at the end of each odd-numbered block k, there is a majority of X i coordinates that with probability u) will all have the value. Therefore, Pr Z nk ) < ) u) for k =,,..., and hence lim inf Pr Z n > ) u < ; n 8

19 that is, Z,H) does not satisfy the CJT. It remains to show that y = liminf n EZ n p n ) =. To do so, we show that the subsequence of { EZ n p n ) ) } of the even-numbered blocks converges to, namely, Indeed, lim EZ nk) p nk) ) =. k n= corresponding to the end EZ nk) p nk) ) nx k) = E nk) X u)+ nk) Σn yk) i= Y i u)). Since the Y i -s are i.i.d. and independent of X we have EZ nk) p nk) ) = n xk) n k) u u)+ n yk) n u u), k) and by property 33) we get finally: ) lim EZ nk) p nk) ) lim u u)+ u u) =, k k k nk) concluding the proof of the proposition. Proposition 7 asserts that for every point u,w) in Figure 5, except for the point,), there is a distribution with these parameters that does not satisfy the CJT. This was known from our previous results in the regions wcjt and scjt. As for the region wcjt, Proposition 7 and Theorem 6 yield the following conclusions, presented in Figure 6. Corollary 8.. The region below the small parabola in Figure 5, with the exception of the point,), is in mcjt, that is, {p,y) p < ; and y } p ) p) mcjt.. The point p,y) =,) is the only point in scjt. It corresponds to a single sequence with X = = X n = with FX i = ) =. 9

20 w = y /4 w = u u) w = u / / 3/4 w = u /) u) /8 wcjt scjt /6 scjt mcjt u = p Figure 6 mcjt and scjt in the L space. 7 Exchangeable variables In this section we fully characterize the distributions of sequences X = X,X,...,X n,...) of exchangeable variables that satisfy the CJT. We first recall: Definition 9. A sequence of random variables X = X,X,...,X n,...) is exchangeable if for every n and every permutation k,...,k n ) of,...,n), the finite sequence X k,...,x kn ) has same n-dimensional probability distribution as X,...,X n ). We shall make use of the following characterization theorem due to de Finetti see, e.g., Feller 966), Vol. II, page 5). Theorem 3. A sequence of binary random variables X = X,X,...,X n,...) is exchangeable if and only if there is a probability distribution F on [,] such that for every n: PrX = = X k =, X k+ =... = X n = ) = PrX + +X n = k) = θ k θ) n k df 34) ) n k θ k θ) n k df 35) The usefulness of de Finetti for our purposes is that it enables an easy projection of the distribution into our L space: Theorem 3. Let X = X,X,...,X n,...) be a sequence of exchangeable binary random variables and let F be the corresponding distribution function in de Finetti s theorem. Then, y = lim n EX n u) = VF), 36)

21 where Proof. We have and for i j, u = θdf and VF) = u = EX i ) = PrX i = ) = θ u) df. xdf ; VX i ) = u u) So, CovX i,x j ) == PrX i = X j = ) u = x df u = VF). EX n u) ) = E n Σn X i u) which implies equation 36). = n Σn VX i)+ n Σ i jcovx i,x j ) = nu u) n + We can now state the characterization theorem: nn ) n VF), Theorem 3. A sequence X = X,X,...,X n,...) of binary exchangeable random variables with a corresponding distribution Fθ) satisfies the CJT if and only if ) Pr < θ =, 37) that is, if and only if the support of F is in the open semi-interval /,]. Proof. The only if part follows from the fact that any sequence X = X,X,...,X n,...) of binary i.i.d. random variables with expectation EX i ) = θ /, violates the CJT by the Berend and Paroush s necessary condition). To prove that a sequence satisfying condition 37) also satisfies the CJT, note that for < ε < /, Pr X n > ) Pr θ ) + ε Pr X n > θ ) + ε. 38)

22 For the second term in 38) we have: Pr X n > θ ) + ε = Σ k> n Pr X + +X k = k θ ) + ε 39) ) n = Σ k> n θ k θ) n k df 4) k +ε [ ] n = Σ k> n )θ k θ) n k df 4) k := +ε Now, using Chebyshev s inequality we have: S n θ) = Pr X n > ) θ Pr X n > ) + ε θ S n θ)df 4) +ε 43) VX n θ) θ θ) θ = ε) nθ 44) ε) Since the last expression in 44) converges to uniformly on [/+ε,] as n, taking the limit n of 4) and using 44) we have: lim X Pr n > θ ) + ε df = Pr θ ) + ε. 45) n From 38) and 45) we have that for and fixed ε >, lim Pr X n > ) [ Pr θ )] n + ε. 46) Since 46) must hold for all / > ε >, and since Pr < θ ) =, we conclude that lim Pr X n > ) =, 47) n +ε i.e., the sequence X = X,X,...,X n,...) satisfies the CJT. To draw the consequences of Theorem 3 we prove first the following: Proposition 33. Any distribution F of a variable θ in [/, ] satisfies VF) u ) u), 48) where u = EF), and equality holds in 48) only for F in which Prθ = ) = u) and Prθ = ) = u. 49)

23 Proof. We want to show that or, equivalently, / θ dfθ) u u ) u), 5) / θ dfθ) 3 u+. 5) Replacing u = / θdfθ) and = / dfθ), inequality 5) is equivalent to / θ 3 θ + )dfθ) := / gθ)dfθ). 5) The parabola gθ) is convex and satisfies g/) = g) = and gθ) < for all / < θ <, which proves 5). Furthermore, equality to in 5) is obtained only when F is such that Pr/ < θ < ) =, and combined with u = EF) this implies 49). The next proposition provides a sort of an inverse to proposition 33. Proposition 34. For any pair u,w) where / < u and w < u /) u), there is a distribution Fθ) on /, ] such that EF) = u and VF) = w. Proof. For u =, the only point in this region is when w = and for this point,) the claim is trivially true with the distribution Prθ = ) = ), and so it suffices to consider only the case u <. Given u,w), for any y satisfying / < y u < define the distribution F y for which Prθ = y) = u)/ y) and Prθ = ) = u y)/ y). This distribution satisfies EF y ) = u and it remains to show that we can choose y so that VF y ) = w. Indeed, VF y ) = u y y + u y y u. For a given u < this is a continuous function of y satisfying: lim y u VF y ) = and lim y / VF y ) = u /) u). Therefore, for w < u /) u), there is a value y for which VF y ) = w. The geometric expression of Theorem 3 can now be stated as follows: In the L plane of p,y) let A = { p,y) < p ; and y < p ) p) } {,)} This is the region strictly below the small parabola in Figure 6, excluding /,) and adding,). 3

24 Theorem 35.. Any exchangeable sequence of binary random variables that satisfy the CJT corresponds to p,y) A.. To any p,y) A there exists an exchangeable sequence of binary random variables with parameters p, y) that satisfy the CJT. Proof. The statements of the theorems are trivially true for the point,), as it corresponds to the unique distribution: PrX =... = X n...) =, which is both exchangeable and satisfies the CJT. For all other points in A, Part. follows de Finetti s Theorem 3, Theorem 3 and Proposition 33. Part. follows de Finetti s Theorem 3, Theorem 3 and Proposition 34. Remark 36. Note that Theorem 6 and part ii) of Theorem 35 each establish the existence of distributions satisfying the CJT in the interior of the region below the small parabola in Figure 6. The two theorems exhibit examples of such a distribution: while the proof of part ii) of Theorem 35) provides to each point p,y) in this region a corresponding exchangeable distribution that satisfies the CJT, Theorem 6), provides other distributions satisfying the CJT that are clearly not exchangeable. 8 General interlacing We now generalize the main construction of the proof of Theorem. This may be useful in advancing our investigations. Definition 37. Let X = X,X,...,X n,...) be a sequence of binary random variables with joint probability distribution F and let Y = Y,Y,...,Y n,...) be another sequence of binary random variables with joint distribution G. For t [,], the t-interlacing of X,F) and Y,G) is the pair Z,H) := X,F) t Y,G) where for n =,,..., { X tn if tn > tn ) Z n = Y t)n if, 53) t)n > t)n ) and H = F G is the product probability distribution of F and G. The following lemma is a direct consequence of Definition 37. Lemma 38. If X,F) and Y,G) satisfy the CJT then for any t [,] the pair Z,H) = X,F) t Y,G) also satisfies the CJT. 4

25 Proof. We may assume that t, ). Note that { ω Z n ω) > } { ω X tn ω) > } {ω Y t)n ω) > } By our construction and the fact that both X, F) and Y, G) satisfy the CJT, lim F X tn > ) = and lim G Y t)n > ) =. n n As H Z n > ) F X tn > ) G Y t)n > ), the proof follows. Corollary 39. The region wcjt is star-convex in the L space. Hence, in particular, it is path-connected in this space. Proof. Let u,w) be a point in wcjt in the L space. Then, there exists a pair X,F) which satisfies CJT, where X is the sequence of binary random variables with joint probability distribution F satisfying p = u and y = w. By Remark, Lemma 38, and the proof of Theorem, the line segment [u,w),,)] is contained in wcjt, proving that wcjt is star-convex. Corollary 4. The region wcjt is path-connected in the L space. Proof. In the L space a point u,w) corresponds to p = u and y = w. By the same arguments as before, the arc of the parabola w = u)/ u )) w connecting u,w) to,) see Figure 4) is contained in wcjt, and thus wcjt is path-connected. 9 CJT in the space of all probability distributions. We look at the space S p = {,} already introduced in the proof of Proposition 7 on page 7. We consider S p both as a measurable product space and as a topological product space. Let P be the space of all probability distributions on S p. P is a compact metric space in the weak topology. Let P P be the set of all probability distributions in P that satisfy the CJT and let P = P\P be the set of all probability distributions in P that do not satisfy the CJT. Lemma 4. If P and P are two distributions in P, and if P P then for any < t <, the distribution P 3 = tp + t)p is also in P. 5

26 Proof. For n =,,, let B n = { n x = x,x, ) S p i > n i=x } 54) Since P P, there exists a subsequence B nk ) k= and ε > such that P B nk ) ε for k =,,. Then P 3 B nk ) = tp B nk )+ t)p B nk ) t + t) ε) = ε t), implying that P 3 P. Corollary 4. The set P is dense in P in the weak topology) and is convex. We proceed now to separate P from P in the space P. We first observe that P is convex by its definition and P is convex by Lemma 4. In order to separate P from P by some linear functional we first define the mapping T : P R N where N = {,,...} in the following way: For P P let TP) = χ dp, B χ B dp,..., ) χ Bn dp,... where the sets B n are defined in 54) and χ Bn is the indicator function of the set B n, that is, for x S p and n =,,..., { if x Bn χ Bn x) =. if x B n The mapping T is affine and continuous when R N is endowed with the product topology. Clearly, TP) l. Also TP ) and TP ) are convex and if z TP ) and z TP ) then liminf k z,k z,k ), where z i = z i,,z i,...) for i =,. Let B : l R be any Banach limit on l see Dunford and Schwartz 958), p. 73); then B is a continuous linear functional on l and as Bz) liminf k z k for every z l, we have Bz ) Bz ) whenever z TP ) and z TP ). Thus the composition B T, which is also an affine function, satisfies P P and P P = B TP ) B TP ). We can now improve upon the foregoing separation result between TP ) and TP ) as follows: Proposition 43. For any y TP ) there exists ψ l such that i) ψz ) ψz ) for every z TP ) and z TP ); ii) ψz) > ψy) for all z TP ). 6

27 Proof. Given y TP ), let C = y TP ). Then, C is a convex subset of l and there exists an ε > such that liminf k z k < ε for every z C. Let { } D = z l liminf z k ε ; k then D is convex and C D = φ. Furthermore, D is an interior point. Hence by Dunford and Schwartz 958) p. 47, there exists ψ l, ψ such that ψd) ψc) for all d D and c C. Since l + = {x l x } is a cone contained in D, and ψ is bounded from below on D and hence on l + ) it follows that ψ that is ψx) for all x ). Also, as C and IntD), there exists δ > such that ψd) δ for all d D and ψd) δ, ). Hence ψc) δ for all c C, that is, ψy x) δ for all x TP ). Finally, the cone C = {z l liminf k z k } is contained in D; hence ψ is nonnegative on C. As TP ) TP ) C, the functional ψ separates weakly between TP ) and TP ). Remark 44. The mapping ψ T, which is defined on P, may not be continuous in the weak topology on P. Nevertheless, it may be useful in distinguishing the elements of P from those of P. References Austen-Smith, D. and J.S. Banks 996), Information aggregation, rationality, and the Condorcet Jury Theorem. The American Political Science Review 9: Berend, D. and J. Paroush 998), When is Condorcet Jury Theorem valid? Social Choice and Welfare 5: Condorcet, Marquis de 785), Essai sur l application de l analyse à la probabilité des décisions rendues à la pluralité des voix. De l Imprimerie Royale, Paris. Feller, W. 957, third edition), An Introduction to Probability Theory and Its Applications Volume I, John Wiley & Sons, New York. Feller, W. 966, second corrected printing), An Introduction to Probability Theory and Its Applications, Volume II, John Wiley & Sons, New York. Dunford, N. and J.T. Schwartz 958), Linear Operators, Part I, Interscience Publishers, New York. Ladha, K.K. 995), Information pooling through majority rule: Condorcet s Jury Theorem with correlated votes. Journal of Economic Behavior and Organization 6: Laslier, J.-F. and J. Weibull 8), Committee decisions: optimality and equilibrium. Mimeo. Loève, M. 963, third edition), Probability Theory, D. Van Norstrand Company, Inc., Princeton, New Jersey. 7

28 Myerson, R.B. 998), Extended Poisson games and the Condorcet Jury Theorem. Games and Economic Behavior 5: 3. Myerson, R.B. 998), Population uncertainty and Poisson games. International Journal of Game Theory 7: Nitzan, S. and J. Paroush 985), Collective Decision Making: An Economic Outlook. Cambridge University Press, Cambridge. Shapley, L.S. and B. Grofman 984), Optimizing group judgemental accuracy in the presence of interdependencies. Public Choice 43: Uspensky, J.V. 937), Introduction to Mathematical Probability, McGraw-Hill, New York. Wit, J. 998), Rational Choice and the Condorcet Jury Theorem. Games and Economic Behavior : Young, P.H. 997), Group choice and individual judgements. In Perspectives on Public Choice, Mueller, D.C. ed.), Cambridge University Press, Cambridge, pp. 8-. Then Appendix 9. Proof of Proposition 9 Let C be the uniform bound of the variables, i.e., X i C for all i in our case C = ). Let Q n = P X n p n ) ε) and hence P X n p n ) > ε) = Qn. E X n p n ) C Q n )+ε Q n C Q n )+ε If LLN holds, then lim n Q n = and thus the limit of the left-hand side is less than or equal to ε. Since this holds for all ε >, the limit of the left-hand side is zero. 8