Correlation, Partitioning and the Probability of Casting a Decisive Vote under the Majority Rule

Transcription

1 TSE 6 February 06 Correlation, Partitioning and the Probability of Casting a Decisive Vote under the Majority Rule Michel Le Breton, Dominique Leelley, Hatem Smaoui

2 Correlation, Partitioning and the Probability of Casting a Decisive Vote under the Majority Rule Michel Le Breton y Dominique Leelley z Hatem Smaoui x February 06 Abstract The main urose of this aer is to estimate the robability of casting a decisive vote under the majority rule for a class of random electorate models encomassing the celebrated Imartial Culture (IC) and Imartial Anonymous Culture (IAC) models. The emhasis is on the imact of correlation across votes on the order of magnitude of this event. Our roof techniques use arguments from robability theory on one hand and combinatorial and algorithmic tools for counting integer oints inside convex olytoes on the other hand. JEL Classi cation umbers: D7, D7. Keywords: Elections, Power Measurement, Voting, Random Electorate. This is the last version of a working aer which has circulated for the rst time in January 0. Early versions have been resented at the Social Choice and Welfare Meeting in Delhi and at the Comutational Social Choice (COMSOC) meeting in Krakow. We thank all articiants for their comments and suggestions. y Toulouse School of Economics, France. address: michel.lebreton@tse-fr.eu. z CEMOI, Université de La Réunion, France. Corresonding author. address: dominique.leelley@univ-reunion.fr. Fax: +33 (0) x CEMOI, Université de La Réunion, France. address: hatem.smaoui@univ-reunion.fr.

3 Introduction The main urose of this aer is to introduce a general model of a random electorate of voters described by their references over two alternatives. Our model will admit, as secial cases, the two most oular models in the literature on ower measurement. The rst one, called Imartial Culture (IC) is the basis of the celebrated Penrose-Banzhaf ower index (Penrose (946), Banzhaf (965)). It assumes that the references of the voters over the two alternatives are indeendent and equally likely: correlation among the references of the voters is totally recluded. The second one, called Imartial Anonymous Culture (IAC) which has been ioneered indeendently in voting theory by Chamberlain and Rothschild (98), Good and Mayer (975), Fishburn and Gehrlein (976) and Kuga and agatani (974) is the basis (as forcefully demonstrated by Stra n (977, 988)) of another celebrated ower index due to Shaley and Shubik (Shaley and Shubik (954), Stra n (977, 988)). The IAC model introduces correlation among voters and the seci c distributional assumtion which is considered imlies that the real random variable de ned as the number of voters suorting the rst alternative is uniform over all feasible integers. From a comutational ersective, this distributional roerty of the IAC model makes it very handy as comared to some other models and robably exlains its success. Further, as noted convincingly by Chamberlain and Rothschild, the IAC model is more attractive than the IC model in the sense that the electoral redictions of the IAC models don t dislay a discontinuity in the neighborhood of the outcome of a tied election. Given a random electorate, the ower of a voter is de ned as the robability of being ivotal i.e. as the robability of being able to change the electoral outcome by his or her vote. Given that we will focus on a symmetric simle game (the ordinary majority game), if the model of random electorate is fully symmetric (i.e. if the references are interchangeable), then all voters will have the same ower denoted P iv(; ). Both the IC and the IAC models are symmetric. For the IC model, this de nes the Penrose-Banzhaf ower index P iv(ic; ) while for the IAC model this de nes the Shaley-Shubik ower index P iv(iac; ). It is well known that P iv(ic; ) is of order equal to. and P iv(iac; ) is The main urose of this aer is to continue the exloration of the imlications of correlation on the asymtotic behavior of the ower index. Precisely, we will consider a general family of models of random electorate and study the asymtotic behavior of P iv(; ) with resect to. Our motivation to do so is to deart from the IAC model Good and Mayer (975) refers to this as the e cacy of a vote.

4 which assumes that the correlation is the same for all airs of voters in the oulation. It is likely that the intensity of the correlation between the votes of i and j will deend uon some characteristics of i and j suggesting that the correlation may vary from one air to another. Most of the aer will however be based on a articular attern of heterogeneity. Precisely, we will assume that the voters are artitioned into grous and that: correlation is ositive and identical for any air of voters belonging to the same grou and null for any air of voters belonging to two di erent grous. We will assume that within each grou the correlation is de ned as in the IAC model. This gives the IC and the IAC models as secial cases: the IC model emerges when all the grous are singletons and the IAC model arises when there is a unique grou which is then the entire oulation. While articular, this model is general enough to cover many situations. We will o er a searate treatment of two olar cases. The rst case is the case where there is a bound on the size of the grous; this bound does not deend uon the size of the oulation. This assumtion is well suited to cature local interactions (within the family or the worklace for instance). The second case is the case where there is a xed number of grous; this means that the size of the grous grows with the size of the oulation. This assumtion is well suited to describe large scale interactions (secial interest grous, geograhical territories, electoral districts, countries if the oulation under scrutiny is multinational,...). After o ering some general results, we roceed to the study of these two cases. The analysis of the two cases uses di erent techniques. When describes the local case, the use of some local versions of the Central Limit Theorem allows to estimate P iv(; ). We show that it is of order and we calculate exlicitly Lim P iv(; ). In contrast, when describes the global! case, our estimation of P iv(; ) is based on di erent mathematical techniques. We address the roblem quite di erently using a combinatorial aroach based on Ehrhart theory and algorithmic tools for comuting the number of integer oints in arametric olytoes. We show that P iv(; ) is of order seci c cases. Related Literature and we calculate exlicitly Lim! P iv(; ) in some The artition random model exlored in this aer has been suggested by Stra n (977) under the name artial homogeneity. He suggests this model as an alternative to the existing IC and IAC models but does not derive any general result. Instead, he roceeds to some numerical calculations of the robability of being ivotal in the Canadian constitutional amendment rocess. Stra n writes: In the Canadian constitution examle, it might be that neither the indeendence assumtion nor the homogeneity assumtion describe the situation very well. British Columbia and Québec, for examle, might reasonably be exected to 3

5 behave indeendently, while the four Atlantic rovinces may have common interests and might reasonably be considered to judge roosed constitutional amendment by a common set of values. The most reasonable thing to do might be to artition the rovinces into subsets whose members are homogenous among themselves, but behave indeendently of the members of other subsets". Chamberlain and Rothschild (98) also consider the case of a artition into two grous and study the asymtotics of the robability of being ivotal under some general conditions: the random draws of the arameter (denoting the robability that any individual votes for the rst alternative) in each of the two grous do not necessarily result from a uniform distribution (a feature shared with Good and Mayer (975)) and the draws are not necessarily indeendent among the two grous. Our model of correlation among voters aims to contribute to the existing studies of the imlications of correlation on ower measurement. Knowing the exact magnitude of the robability of being ivotal is interesting for itself but this information is also essential for the design of the otimal weights of reresentatives, as argued convincingly by Barbera and Jackson (006). They introduce a block model which is quite similar to the model of artitions which is considered here excet for the fact that instead of IAC, they assume erfect correlation within each block/grou. Precisely, they describe it as follows: Each country is made u of some number of blocks of agents, where agents within each block have erfectly correlated references and references across blocks are indeendent. The blocks within a country are of equal size. These assumtions re ect the fact that countries are often made u of some variety of constituencies, within which agents tend to have correlated references. For instance, the farmers in a country might have similar oinions on a wide variety of issues, as will union members, intellectuals, and so forth. The block model is a stylized but useful way to introduce correlation among voters references". They roceed to a searate analysis of the xed-size-block model" and the xed-number-of-blocks model" which arallels exactly our distinction between small and large grous. The block model" was in fact introduced by Penrose (95) in chater 7 of his ioneering monograh. His work is motivated by emirical considerations. He observes that if voters were voting indeendently of each other, then the mean value of the statistics D over an inde nite eriod of years (where D denotes the di erence between the votes of the two sides) would equal unity. This rediction is violated in the case of the twenty-six American Presidential elections that he examined. The mean value is much larger than. He concludes from that that this marked excess over the theoretical value of unity may be interreted as indicating that the voters did not vote as random units but were groued into blocs which voted indeendently. 4

6 The aroximate size of each of a set of blocs taking the lace of individuals is given by the actual mean value of D measured over a eriod of years". The Model of a Random Electorate A random electorate is a trile ( ; X; ) where is a nite set of voters, X is a nite set of alternatives and is a robability distribution on P (the set of functions from to P) where P is the set of linear orders over X. In the case where X consists of two alternatives say 0 and, the set P contains two references (0 is referred to, is referred to 0) which will be coded 0 and and P will be identi ed with the Cartesian roduct f0; g where denotes the cardinality of. The rst oular random electorate model, called Imartial Culture (IC), is de ned by (P ) = for all ro les of references P = (P ; P ; :::; P ) in f0; g. The IC model assumes that the references of the voters are indeendent Bernoulli random variables with a arameter equal to (i.e. the electorate is not biased towards a articular candidate). In contrast, the second oular random electorate model, called Imartial Anonymous Culture (IAC) is de ned as follows. The arameter is drawn in [0; ] from the uniform distribution and, conditional on the draw of, the references of the voters are indeendent Bernoulli references with arameter. The robability of ro le (P ; P ; :::; P ) is therefore (P ) = R 0 k ( ) k d where k is the number of coordinates equal to 0 in P. Using the formula: Z 0 t ( ) t d = (t)!( t)! ( + )! () we obtain that (P ) =. The terminology IAC results from the fact that in the (+)( n k) o IAC model, the events E k P f0; g : # fi : P i = 0g = k for k = 0; ; :::; are equally likely. Since there are k such events, the robability that k voters vote 0 is equal to + for all k = 0; ; :::;. A social choice mechanism is a monotonic maing from f0; g into [0; ] where (P ) denotes the robability of choosing candidate 0 when the ro le of references is P 3. In this aer, we will focus on the standard majority mechanism Maj de ned as follows (with # 0 (P ) denoting the number of 0 in P ): In this binary setting, if (P ) f0; g, a social choice mechanism is de ned alternatively by a simle game (Taylor and Zwicker (999)). 3 We will not make any distinction between references and behavior. There is no room for strategic behavior here: if we interret as a direct revelation game, then voting sincerely according to his/her reference is the unique (weakly) dominant strategy. 5

7 8 < 0 if # 0 (P ) < Maj(P ) = if # 0 (P ) > : if # 0 (P ) = If is odd, the third eventuality never arises and the mechanism is deterministic: the robability of choosing 0 is either 0 or. If is even, the third alternative arises when the electorate is slit into two grous of equal size and the tie is broken by using a fair lottery. The whole aer is about evaluating the robability of being ivotal. Recall that voter i is ivotal if her or his vote is suscetible to change the result of the election. We denote by E k (i) the event de ned by E k (i) = fp f0; g : # 0 (P i ) = kg for k = 0; ; : : : ;, and by P iv(; i; ) the robability that i is ivotal (P i denotes the reduced ro le obtained by removing the vote of i). It is easy to see that P iv(; i; ) = (E (i)) when is odd, and P iv(; i; ) = (E (i)) + (E (i)) when is even. The slight di erence between the even and odd cases lies in the fact that in the odd case, the vote of i is always able to change the result, whereas in the even case, her or his vote creates an equality and the electoral outcome will change with robability, deending on the lottery result. As we consider the standard majority mechanism, if the robability measure is symmetric, then P iv(; i; ) does not deend on i and will be denoted shortly by P iv (; ). robability P iv (; ) has been calculated for the two oular models of random electorate which have just been de ned. For the IC model, P iv (IC; ) = The when is odd and P iv (; ) = when is even. For the IAC model, P iv (IAC; ) = for both cases. Using Stirling s formula,! ', we deduce that when gets large q P iv (IC; ) behaves like ' 0: In this aer, we assume that the electorate is artitioned into K grous ; ; :::; K i.e. [ kk k = and k \ k 0 =? for all k; k 0 such that k 6= k 0. We will denote by P k the size of grou k: K k= k = and without loss of generality we assume that ::: K.We consider the following random electorate model. We assume that the references of any voter i from grou k is the realization of a Bernoulli random variable with arameter k and that conditional on k, the references of any two voters in that grou are indeendent. We assume that the coordinates of the vector ( ; ; :::; K ) are the realizations of K indeendent random variables with a uniform distribution on [0; ]. As P iv(; i; ) is the same for all voters i belonging to the same grou, this robability will be denoted by P iv k (; ) for all members of grou k. We start our calculations by reducing the roblem of comuting P iv k (; ) to a well de ned combinatorial roblem which amounts to count the number of ossible decomositions e 6

8 of a given integer into K integers under some seci c constraints. 4 We will denote by ( M; R ; :::; R k ; :::; R K ) the set of decomositions (x ; : : : ; x K ) of the integer M into K ordered integers (M = x + : : : + x K ) under the constraint that the k th integer x k does not exceed R k, and by ( M; R ; :::; R k ; :::; R K ) the cardinality of this set. To illustrate the use of such decomositions in our calculations, consider a society where the number of voters is odd and multile of 3: = 3K with K odd. The society is therefore divided into K grous of size 3 each (three members). Take K = and suose that i is a member of grou. Voter i will be ivotal if in the rest of the society 3 voters vote 0 and 3 voters vote. How to enumerate the number of ossible decomositions of 3 into integers such that the rst one cannot exceed and the other twenty ones cannot exceed 3? One ossibility is (,,3,3,3,3,3,3,3,3,3,0,...,0) and all subsequent ermutations but we can also vary the choice of integers by taking for instance (,,,,,...,,0,0,0,0,0) where " aears 5 times. 5 Lemma : For all k f; : : : ; Kg,. If is odd, P iv k (; ) = ( ; ; :::; k ; k ; k+ ; :::; K ) k ( Q l6=k. If is even, P iv k (; ) = ( ; ; :::; k ; k ; k+ ; :::; K ) k ( Q l6=k ) l + ) l + (:a) (:b) Proof. Consider the case where is odd. We obtain: X Z k P iv k (; ) = ( ; ;:::; k ; k ; k+ ;:::; K) " Y l6=k By using formula (), we deduce: P iv k (; ) = l x l Z 0 X x k x l l ( l )) l x l d l # ( ; ;:::; k ; k ; k+ ;:::; K) k 0 x k l ( k )) k x k d k Y Y ; ; :::; k ; k ; k+ ; :::; K k l6=k l6=k! = l +! l + 4 ote that there are at most K cells i.e. K non zero integers in the decomosition and there is an uer bound on the entries of each cell. Our roblem is close but not totally equivalent to the roblem of counting comositions of integers with restrictions on the summands and their number as resented in Flajolet and Sedgewick (009). 5 In that case, our rst result (Proosition ) will imly that the total number of decomositions behaves as c4 3 where c is a universal constant. 7

9 The roof in the case where is even roceeds along the same lines. Let us check quickly that the IC and IAC models corresond to two extreme secial cases of this general framework. The IC value is attached to the case where K = i.e. where the artition structure consists of singletons: P iv k (IC; ) = P iv(ic; ) = ; ; :::; ; k ; ; :::; = since ; ; :::; ; 0; ; :::; =. The IAC value is attached to the case where K = i.e. where the artition structure consists of a single set: the set : P iv k (IAC; ) = P iv(iac; ) = ; = since ; = : An alternative aroach to the counting roblem is based on robability. Let X ik denote the Bernoulli random variable describing the reference of voter i in grou k and let S k and bs denote resectively the sums P j k X jk and P K k= Pj k X jk = P K k= S k. With these notations, we can exress the ivot robabilities as follows: P iv k (; ) = bs i = when is odd and P iv k (; ) = bs i = + bs i = when is even This robabilistic aroach will be very useful when we will focus on the asymtotic behavior of P iv k (; ) when tends to in nity. ote that all the random variables X ik are symmetric in the sense that Pr(X ik = 0) = Pr(X ik = ) = since Pr(X ik = 0) = R d =. 0 We have E [X ik ] = 0 + = and V ar [X ik] = E [Xik ] E [X ik] =. But two 4 random variables X ik and X jl are indeendent i k 6= l. If not, we have: Pr(X ik = 0; X jk = 0) = Z 0 d = 3 > 4 The two variables are ositively correlated: Cov(X ik ; X jk ) = 3 of correlation is then equal to. 3 = ; the coe cient 4 8

10 3 The case of Many Small Grous In this section, we will focus on the case where there is an exogenous uer bound S on the size of the grous in the artition ( ; ; :::; K ). This imlies that as gets large, then the number of grous increases. To motivate the general result which will be resented hereafter, it is instructive to consider the case where S =. In any such artition structure, the grous are either singletons or airs. We can think of this artition as describing a society where there are singles and coules but no other family tyes. Consider the case where is even and all the grous are exactly of size. From (:b), we deduce that: We can check that: 6 P iv k (; ) = P iv(; ) = ; ; ; :::; ; ; ; ; :::; ; = b 4 c X k=0! (k!) k! : 3 k k + Indeed, counting how many decomositions of into integers chosen in f0; ; g amounts rst to choose how many airs k we choose among. The number of ossibilities ( is )!. This value of k cannot exceed (k!)(( k)!) 4 To reach the integer, we need The number of ossibilities is ( k)! (. k singletons which can be chosen among k. = ( k) k k)!(k+)! (. k)!(k)! k+ After collecting the terms, we obtain the exression reorted above. Calculating the above sum is not an immediate combinatorial exercise 7 and we will mostly focus on the asymtotic behavior of P iv(; ). We conjecture that:! : 0 Lim () b 4 c X k=0! (k!) k! k C A k + 3 exists. The following Table contains some numerical values of () which suorts this conjecture: 6 bxc denotes the integer art of x. 7 We were not able to derive a closed form value of this sum through the use of combinatorial identities. 9

11 () Table : Values of () Interestingly, the function seems to behave asymtotically as the function de ned as follows: () 0 b 4 c X k=0! (k!) k! C A 3 The following Table contains some numerical values of () which suorts this guess: () Table bis: Values of () We now rove a generalized version of the conjecture. To roceed, we use a robabilistic aroach. We assume that all the grous have a size smaller than S and we will be interested in societies where the set of voters is artitioned into grous of size s where s runs from to S. We will consider societies where gets inde nitely large but such that the roortion of the oulation in each tye of grou (described by its size) remains invariant in the oulation growth rocess. We will denote by s the roortion of voters in a grou of size s. We assume that s = sks where Ks is an integer for all s = ; :::; S and = P S s= sks. The initial society contains K s grous of size s. For any integer R, its R th relica has voters where is de ned as follows: = (R) = R SX K s s In this context, P iv k (; ) is the same for all grous k of size s and will be denoted by P iv (s) ( R ; ), the robability for a voter belonging to a grou of size s to be ivotal. The following roosition gives the asymtotic behavior of P iv (s) ( R ; ) when gets large (equivalently where R gets large). Proosition : Let R be the random electorate de ned above. For all s = ; ; :::; S, Lim P iv(s) ( R ; ) = R! s= q : + + P S 6 l= l l 0

12 Proof. For all R and all i = ; ; :::; (R), we arrange the random variables Xi R describing the individual votes in the R th relica in a triangular array 8 de ned as follows: the rst RK variables describe the vote of voters in grous of size, the next RK variables describe the votes of voters in grous of size and so on. We obtain X X SX (R) V ar( Xi R ) = V ar(xi R ) + RK s s(s )Cov(Xi R ; Xj R ) i= i= s= where Cov(Xi R ; Xj R ) denotes the covariance between Xi R and Xj R when i and j belong to the same grou. We have shown before that: V ar(x R i ) = 4 for all i = ; ; :::; Cov(X R i ; X R j ) = for all i; j = ; :::; if i and j belong to the same grou We obtain: 0v (R) = t SX s sa s= A random variable Xi R is of tye s if R P s l= lkl < i R P s l= lkl. We ack the srk s random variables of tye s into RK s random variables Zks R where Zks R is de ned as follows: Z R ks = r i i=(k Xks )s+ X R is = r krk s This de nes a new triangular array Zks R (indexed by R) where the random ss;krk s variables Zks R are indeendent. Hereafter, we will refer to ZR ks as a random variable of tye s. We note that all random variables are integer valued: the suort of a random variable of tye s is f0; ; :::; sg. Let i (R) be a member of a grou of tye s and for each value of the row index R, consider the random variable S R i S R i = (R) X X R j = SX XK l Z R kl + KX s de ned as follows: Z R ks + W R i j=;j6=i l=;l6=s k= k= 8 A triangular array is a collection of y k ; y k ; :::; yn(k) k of random variables on a robability sace. k

13 where W R i P s j= X R js. The robability that i of tye s is ivotal, P iv (s)( R ; ) is equal to the robability of the event S R i = if is odd and to half the robability of the event S R i = [ S R i = if is even. We note that the san of the random variables Z R kl for l S and k Kl and W R i is equal to. Further, the distribution functions of these random variables belong to a nite set of cardinality at most S, are not degenerate and occur in nitely often (excet ossibly Wi R ) in the sequence Zkl R [ W R ls;kk l i. Let > 0: R If is odd, since E Si R =, we deduce from the local central limit theorem on lattice distributions listed as theorem in Petrov (975) 9 that if R is large enough: Similarly, if is even, since E Si R is large enough: Si R Si R SR i P iv(s) ( R ; ) =, we deduce from Petrov s theorem that if R Pr S R i = Pr S R i = (R) e 8 ( SR i ) e 8 (S R i ) Since e 8 ( SR i ) tends to and (SR i ) tends to when R tends to +, we deduce that if R is large enough: (R) P iv (s)( R ; ) : The random variable S R i = S R X R i introduced in the roof of Proosition counts the number of votes in favor of in the oulation without individual i. Proosition rovides information on the asymtotic behavior of the robability of the event S R i =. To illustrate Proosition, consider the case of an electorate, denoted s R, where all the grous have the same size s. In such case, we deduce from our result that: 9 Chater 7, Section, 89. Petrov s theorem in what follows.

14 s P iv (s) ( R ; ) = P iv( s R; ) ' 3 ( + s) The following Table lists a samle of values of the robability of being ivotal for a samle of values of s. s P iv( s R; ) Table : Probability of being ivotal as a function of s We can also handle mixed situations i.e. random electorates where the sizes of the grous di er across voters. For instance, when the random electorate is such that = 0:, = 0:3; 3 = 0:4 and 4 = 0:, we obtain : P iv( R ) ' 0: We could interret these grous as family grous: singles, coules without children voting, coules with one children voting, and so on. Remark. The roof strategy of Proosition based on a seci c version of the local central limit theorem has exloited the fact that the individuals could be artitioned in a regular way and that in each grou the robability draw of the votes in the grou was not changing with the size of the oulation. From our construction, in each grou of size s, the random number of votes for is described by a multinomial robability law indeendent of with values in the set f0; ; :::; sg. So artitioning er se is not enough to ermit a direct use of that version of the local central limit theorem; we need invariance of the law with resect to. 0 Remark j. We could alternatively look at the robability of being ivotal of a grou of size k where > 0 is xed instead of a grou of size as done until now. Such a grou, acting as a block, is ivotal i : where: S R + S R = (R) X i= 0 We don t mean that it is a necessary condition er se. It is a necessary condition to aeal at the theorem that we have used. We conjecture that additional results could be deduced from other local central limit theorems (like for instance Mc Donald (979)). We thank S. Brams for having raised the question answered in that remark. X R i 3

15 and X R i i(r) is an arbitrary triangular array of Bernoulli random variables of arameter. Let us assume that this triangular array is m(r)-deendent and such that for some > 0 and some constant K: V ar X R i+ + ::: + X R j (j i) K for all i; jand R; V ar X R Lim R! + ::: + X R (R) (R) exists and is nonzero, m(r) + Lim R! (R) Since the Bernoulli variables have moments of any order, we deduce from Berk s theorem that XR +:::+XR (R) (R) (R) = 0: is asymtotically normal with mean 0 and variance v where v V ar(x Lim R+:::+XR n ) (see Aendix ). We deduce that the robability of a grou of relative R! (R) size to be ivotal, denoted P iv(; ); is aroximately 3 equal to: Prob ( ) S R + ' ' n Prob (0; v) o r v This weak version of the ivotality result holds in a much larger class of electorates. The notion of m deendency matches di erent ossibilities. First, we could continue to consider artitions into grous whose size can even increase slowly with. What is essential, as re ected by the other two conditions of Berk s theorem, is to bound in an aroriate way the variance of any ack of random variables and to have the variance of the electorate to behave asymtotically as the size of the electorate. For the sake of illustration of such construction, consider the case where the (R) voters are artitioned into consecutive blocks where each block has a size m = m(r) = b 4 c. We assume that m(r) is even and that within each block the Bernoulli random variables describing the votes are correlated as follows. Z = (Z ; Z ; :::; Z m ; Z m ) be a m dimensional random vector such that (i) the coordinates are indeendent random variables, (ii) the rst m coordinates Z j j = ; :::; m are For a de nition of the notion of m-deendency, see Aendix, footnote 0. 3 ote that here, in contrast to the case where we consider a single voter or even a nite set of voters whose size does not deend uon, the limit robability of the grou of being ivotal does not change if we change the quota j k into the quota j k + C where C is a constant. 4 Let

16 Bernouilli random variables with arameter and (iii) the random variable Zm takes the values 0 and with the robability and the value with the robability where [0; ]. Given Z, the vector X is constructed as follows: Xi R = Z i for all i = ; :::; m if either Z Xm R = m = or Z m = and Maj(Z ; Z ; :::; Z m ) = 0 if either Z m = 0 or Z m = and Maj(Z ; Z ; :::; Z m ) = 0 In words, the m th layer of each block votes indeendently of the other voters with some robability or follows the majority oinion of the other voters with the comlement robability. It is immediate to check that X R m is a Bernouilli random variable with arameter. Let us evaluate how his vote is correlated with the vote of any other voter i. From above we deduce that: Cov(X R m; X R i ) = Prob(X R m = and X R i = ) = Prob(Z m = and X R i = ) + Prob(Z m = ; Maj(Z ; Z ; :::; Z m ) = and Xi R = ) 4 = 4 + Prob( Maj(Z ; Z ; :::; Z m ) = j Xi R = ) So we are left with the evaluation of the conditional robability Prob( Maj(Z ; Z ; :::; Z m ) = j Zi R = ) which is equal to P m m = + m k= m k m m m. If m is large, we deduce from Stirling s formula that: m m ' m 4 r m Collecting q the terms together, we derive Prob( Maj(Z ; Z ; :::; Z m ) = j Zi R = ) ' and therefore: + m 4 Cov(X R m; X R i ) ' r! m 4 ' m for all i = ; :::; m The covariance matrix of each block is almost diagonal. We obtain that in each block V ar X R + ::: + Xm R ' m + ( ) m 4. Conditions (ii) and (iii) of Berk s theorem follow easily. Here v = m(r)+. Proerty (iv) also holds true since for any < ; = + 4 tends 4 (R) to 0 when tends to in nity. In that examle, correlation (which concerns only a unique 5

17 voter) collases as the number of voter grows. We can rovide examles where it is not the case Second, we could even abandon comletely the idea of artitioning of voters into blocs and move instead towards local interaction. Indeed, m deendence (even with m indeendent of ) does not force artition. For instance, let Y ; Y ; :::::: be a sequence of indeendent Bernoulli variables with arameter and let X i be de ned as follows: Y X = Y and X i = i with robability Y i with robability where [0; ]. It is easy to check that the X i are Bernouilli random variables with arameter. ote also that for all i 34 ( ), Cov(X i ; X i ) = and Cov(X 4 i ; X i+m ) = 0 whenever m >. Indeed, X i X i = is the union of the following four disjoint events: either: fx i = Y i = and X i = Y i = g ; or fx i = Y i = and X i = Y i = g or fx i = Y i = and X i = Y i = g or fx i = Y i = and X i = Y i g Using indeendence, we deduce that the robabilities of the rst three events are resectively ( ) ; 4 ( ) and while the robability of the fourth one is equal to 4 4 robability of the event fx i X i = g is equal to +( )+( ) ( ). Therefore the ( ) + = ( ) In this construction, the sequence Y is a sequence of indeendent signals : voter i s reference follows his own signal with some robability and follows the revious signal with robability. This makes every voter correlated with his two adjacent neighbors. There is no way in which we can artition the voters into blocks of size. 4 The Case of Few Large Grous In this section, we consider the olar case of a society divided into a nite (ossibly large) number of grous. This means that as gets larger and larger, the number of voters in each grou gets larger and larger. This eculiarity revents us from alying the same robabilistic aroach as in the receding section To circumvent this di culty, we will tackle the roblem from a di erent combinatorial angle. For xed values of K, the general roblem of comuting the number (M; R ; :::; R k ; :::; R K ) can be hrased as counting the exact 4 When i =, Cov(X ; X ) = 4. 6

18 number of integer solutions of a system of linear inequalities with integer coe cients, where the variables are x k (k = ; :::; K) and the arameters are M and R k (k = ; :::; K), which is equivalent to count the number of integer oints inside a arametric convex olyto. There is a well established mathematical aroach for erforming such a calculation, based on Ehrhart s theory (Ehrhart, 96, 967, 977) and e cient counting algorithms 5. We refer to Leelley et al. (008) and Wilson and Pritchard (007) for more details on the use of these tools in robability calculations under IAC hyothesis in voting theory. Most of the results resented in the following subsections have been obtained by alying (arameterized) Barvinok s algorithm (Barvinok, 994; Barvinok and Pommersheim,999) A Preliminary Result It can be noticed that, when the number of voters in the largest grou reresents more than 50% of the total number of voters, then the robability of casting a decisive vote only deends, in each grou, on the value of. More recisely, we have the following general result (Recall that bxc denotes the integer art of x). Proosition : If k = ; 3; :::; K. +, then P iv (; ) = and P iv k (; ) = + for Proof. Let x k be the value of the k th term in the decomosition of : x + ::: + x k + ::: + x K =. If +, then + 3 +:::+ k +:::+ K. Consequently, for k = ; 3; :::; K, x k can take any integer value between 0 and k (including 0 and k ) and when x ; x 3 ; :::; x K are set, the value of x is given in a unique way by x = The number of ossible decomositions is then given by ( ; ; ; :::; k ; :::; K ) = ( + )( 3 + ):::( K + ) and the result follows from relations (.a) and (.b). 4. The Case of Two Grous x x 3 ::: x K. Let us consider the case where K = i.e. the situation where the voters are artitioned into two grous. This setting has been examined by various authors in the literature including 5 For a general background on Ehrhart theory and on the general roblem of counting integer oints in olytoes, see for examle Beck and Robins (007). 6 For a rigorous descrition of this algorithm and for imlementation details, see Verdoolage et al. (004, 005). 7

19 Beck (975), Kleiner (980), Chamberlain and Rothschild (98) and Le Breton and Leelley (04). In such a case, if is odd, then > as the two integers don t have the same arity. It is easily seen that: and therefore: ; ; = + and ; ; = in accordance with Proosition. 4.3 Three grous of voters P iv (; ) = and P iv (; ) = + In this section, we consider the case where the oulation is divided into three grous of voters i.e. K = 3: 3 and = b +, with b even. Proosition 3:. If b + (in accordance with our reliminary result), P iv (; ) = and P iv (; ) = P iv 3 (; ) = +. If b, P iv (; ) = 4 +4 ( b )+4 4 ( +)+ b ( b +) b 4 ( +)( + b ), P iv (; ) = 4 +4 ( b )+4 4 ( b +)+ b ( b +) 4( +) ( + b ) P iv 3 (; ) = 4 +4 ( b )+4 4 b + b b 4) 4( +)( +)( + b ) Proof. The value of ( b ; ; ; 3 ) is given by the number of integer solutions of the following set of (in)equalities, where x k can be interreted as the number of voters voting for alternative 0 in grou k, k = ; ; 3: 0 x 0 x 0 x 3 3 x + x + x 3 = b Given the last equality, 3 = and the above set of inequalities reduces to: where the arameters satisfy: 0 x 0 x 0 x 3 x + x + x 3 = b 8

20 + 0 and + + A reresentation for the number of integer solutions of this set of inequalities with three variables and three arameters (, and ) can be derived by using the multiarameter version of the Barvinok s algorithm (see Leelley et al. (008)). We obtain: ( b ; ; ; 3 ) = ( b + )( + ) = ( 3 + )( + ) if b + and ( b ; ; ; 3 ) = ( b + b ( + ) 4( + ( ) + ( ))=4 if b Reresentations for ( b ; ; 3 ) and ( b ; ; 3 ) can be derived in a similar way to obtain: if b + and ( b ; ; ; 3 ) = ( b + ) = ( 3 + ) ( b ; ; ; 3 ) = ( b + b ( + ) 4( + ( ) + ( ))=4 if b ; ( b ; ; ; 3 ) = ( b + )( + ) = 3 ( + ) if b + and ( b ; ; ; 3 ) = ( b + b ( + + ) 4( + + ))=4 if b. 9

21 Observe that we recover the results we have mentioned for two grous by taking 3 = 0. From the above results, we can now derive the robability of casting a decisive vote for a voter belonging to each of the three grous. We obtain : if b + and P iv (; ) = ( 3 + )( + ) ( + )( 3 + ) = ( 3 + ) P iv (; ) = ( + ) ( 3 + ) = + P iv 3 (; ) = 3 ( + ) ( + )( + ) 3 = + P iv (; ) = ( b ) ( b + ) + b ( b + ) 4 ( + )( + b ) P iv (; ) = ( b ) ( b + ) + b ( b + ) 4( + ) ( + b ) P iv 3 (; ) = ( b ) b + b b 4) 4( + )( + )( + b ) if b. In order to simlify the above reresentations, let = = b and = = b denote the roortion of voters in the rst and the second grou. Relacing by b and by b and assuming that b is large give, we obtain the following aroximation. Corollary : Let c 3 ( ; ) = ( + ) if 0:50 and c 3 ( ; ) = = if > 0:50. Then for k = ; ; 3, P iv k (; ) ' c 3 ( ; ). We nally obtain that, for large, the robability of casting a decisive vote for a voter belonging to an electorate divided in three grous is aroximately equal to the Shaley- Shubik index multilied by c 3 ( ; ). We give in Table 3 some comuted values of c 3 ( ; ) for various values of and. / / / > 0.50 = = = = = Table 3 : Values of c 3 ( ; ) 0

22 These values show that the robability of casting a decisive vote is maximum when = = =3, i.e. when each of the three grous has the same size. 4.4 The Symmetric Case + We consider here the case with = = ::: = K = b and we assume that = b + K is a multile of K, which imlies that K is odd. In this symmetric case, the value of ( b ; + b + ; b ; :::; + b ) is given as the number of integer solutions of the following set K K K of (in)equalities: + 0 x b K + 0 x b K ::: + 0 x K b K x + x + ::: + x K = b For seci c small values of K, it is fairly easy to obtain close forms for the robability of being ivotal as a function of the arameter. Let us consider the rst values of K. b K = 3: To comute ; + b + ; b ; + b, we make use of the Barvinok s algorithm to obtain: bk ; b K; b K for b = modulo 6. From this result, we derive: P iv(; ) = 9( b + ) 4( b + )( b + 4) = ( b + )( b + 4) for b = modulo 6. otice that this result is consistent with the reresentations given in Proosition 3: relacing and by ( + )=3 in the formulas given in this roosition leads to (). Hence, we get for large: P iv(; ) ' c 3 with c 3 = 9 = :5, in accordance with the result obtained in the receding subsection 4 for = = =3. K = 5: We obtain via Barvinok s algorithm: () ( b ; b + 5 ; b + 5 ; b + 5 ; b + 5 ; b + ) = ( b + )( b + 6)(3 b + 76 b + 98)

23 from which we deduce P iv(; ) = 5( b + )(3 b + 76 b + 98) 9( b + )( b + 6) 3 for b = 4 modulo 0. In this case, the limiting value of the robability of casting a decisive vote is given as: with c 5 = = :995. P iv(; ) ' c 5 K = 7, K = 9 and K = : Although we have been able to obtain the comlete olynomials associated with ( b ; + b + ; b ; :::; + b ) for K = 7; 9;, we only give here K K K the values of c K : and c 7 = = 3:577 c 9 = = 4:076 c = = 4:5: For values of K higher than, the imlementation of the Barvinok s algorithm demands a very long comutation time that revents from obtaining some numerical results. The following roosition describes the asymtotic behavior of ( b ; + b + ; b ; :::; + b ) when K K K gets large. Proosition 4: Let K be an odd number (K 3). Let '(K) = have: lim [!+ K ( b ; + b + ; b ; :::; + b K K K '(K) = K X K ( ) m (K )! m m=0 )]. Then, for each xed value of K, we K m K K : Proof. By de nition, ( b ; + b + ; b ; :::; + b ) is the number of integer solutions of the K K K following arametric linear system: 8 0 x K >< 0 x k for all k = ; :::; K K KX >: x k = k=

24 We know by Ehrhart s theorem (Ehrhart, 967) that this number is a quasi-olynomial of degree K on the variable. Hence, '(K) is equal to the leading coe cient of this quasiolynomial 7. As the additive constants in the second member of the constraints do not a ect this coe cient, '(K) is also the leading coe cient of the quasi-olynomial comuting the number of integer solutions of the system 8 >< 0 x k for all k = ; ; :::; K K KX >: x k = k= The system reresents the dilatation by the factor of the rational (K ) dimensional olytoe Q de ned by: 8>< 0 x k K KX >: x k = k= for all k = ; :::; K By Ehrhart s theorem, and by de nition of '(K), we know that '(K) is equal to the relative volume of Q, which is the (normalized) volume in R K of the full-dimensional olytoe P de ned by: 8 >< 0 x k for all k = ; :::; K K K K >: K X x k k= Let Vol(P) be the volume of P. To comute this volume, we consider some articular subsets of R K. Let and 0 be the K -dimensional simlices de ned by: = fx R K : x k 0 for all k = ; : : : ; K and x + : : : + x K =g 0 = fx R K : x k 0 for all k = ; : : : ; K and x + : : : + x K (K )=Kg It is easy to see that Vol(P) = Vol(A) Vol(B), where: A = fx : x k =K; 8k = ; : : : ; K B = fx 0 : x k =K; 8k = ; : : : ; K g g 7 A degree-d quasi-olynomial on the variable n is a olynomial exression f(n) = P d i=0 c i(n)n i, where the coe cients c i (n) are rational eriodic numbers on n. A rational eriodic number, of eriod q, on the integer variable n is a function U : Z! Q such that U(n) = U(n 0 ) whenever n n 0 mod q. 3

25 We only show how to comute Vol(A), the same method will be alied to obtain Vol(B). For each i in f; : : : ; K emty subset S of f; : : : ; K g let i = fx : x i =Kg. More generally, for each non g, we de ne S by S = \ is i. ote that S = ; for j S j> K. For S such that #S K, let #S = m and let t u be the translation of vector u, where u is the vector of R K de ned by u i = K if i S and u i = 0 if not. It is obvious that t u ( S ) = (m), where (m) = fx R K : x k 0 for all k = ; : : : ; K and x + : : : + x K (K m)=kg. Since translations conserve volumes, and alying the formula giving the volume of a simlex, we obtain: Vol( S ) = Vol((m)) = (K )! K m K K On the other hand, we can write Vol(A) = Vol() Vol([ K i= i). Alying the inclusionexclusion rincile, we get: Vol([ K i= i) = = X X ( ) m K m= K S;jSj=m Vol( S ) X K ( ) m m (K )! m= K m K K Since Vol() = (K )! K, we obtain: Vol(A) = K X K ( ) m (K )! m m=0 K m K K ow, Vol(B) can be comuted in a similar way and we can easily establish that: K 3 X K K m Vol(B) = ( ) m K : (K )! m K m=0 4

26 Finally, the following simle calculus gives the result: Vol(P) = (K )! = (K )! = (K )! = (K )! X K K m ( ) m m K K m=0 K K X K + ( ) m [ + m m= m= K K 3 K K X K K m + ( ) m m K K m=0 X K K m ( ) m m K m=0 K m K X K K m ( ) m K : m K ] K m K K Using the analytical exression obtained in Proosition 3, we can extend the calculation of c K = K K '(K) to larger values of K. The following Table gives the exact value of c K for K = 5 to 49 (K odd). K K c K K c K Table 4 : Exact values of c K otice that the limiting result obtained in this subsection can be easily extended to the case where is even and the oulation is divided into K grous of size. The integer K K can be odd or even and the unique assumtion is that is an even multile of K. Let (K) = lim [ ( ; ; ; :::; )]. With slight modi cations in the roof of!+ K K K K Proosition 3, we obtain: if K is odd, and if K is even. (K) = (K) = K X K ( ) m (K )! m (K )! m=0 m=0 K m K K X K K m ( ) m m K K K As the relation in Proosition 4 is not easy to imlement when K becomes large, we have develoed a robabilistic argument to conjecture that: 5

27 P iv(; ) ' r 6K when K is large. Our conjecture on the asymtotic behavior of c K when K tends to is based uon the following heuristic robabilistic argument. Proceeding as in remark of section 3, the robability of being ivotal for a small grou of size where > 0 is xed can be exressed as the robability of the event : where: S + S = KX X k S k where S k = k= i= X k i and k = K The random variables S ; S ; :::; SK are indeendent and identically distributed. Following the argument used in Proosition 4 of Chamberlain and Rothschild (98), we deduce that for all k = ; :::; K, Sk k converges weakly to the uniform law on the interval [0; ] when k!. Since the S k are indeendent, this imlies that S converges weakly to Z = K P K k= U k where the random variables U k are indeendent and identically distributed, with common distribution the uniform distribution on [0; ]. From the central limit theorem, we deduce that if K is large then: P K k= U k K ' (0; ) K q since is the standard deviation of the uniform variable on [0; ]. We deduce then that the robability of a grou of relative size to be ivotal denoted P iv(; ) is aroximatively equal to Pr Some values of ( q 6K P K k= U k K ) are tabulated below: ' ' Pr r 6K (0; K) q K ::: 49 ::: 99 6K : : 090 3: : : ::: 9: ::: 3:

28 5 Concluding Remarks Table 5 : Aroximate values of c K In this aer, we have studied the imact of correlation across references and votes on the robability of being ivotal under the majority rule and we have shown how increasing correlation reduces this robability. To illustrate our contribution, consider 000 voters divided into K indeendent grous of equal size. In each grou, references are generated according to the IAC assumtion. q When K = 000 we obtain the IC model with a robability of being ivotal equal to = :05; the case with K = corresonds to the usual IAC 500 model, with a robability of being ivotal equal to. The tools o ered in this aer 000 allow to consider all the ossible intermediate situations between these two olar cases. The following Table dislays the robability of being ivotal for various values of K: these robabilities are comuted with the hel of Proosition for large values of K (K > 00) and are deduced from Proosition 4 for small values of K. K P iv() Table 6 : Probability of being ivotal, 000 voters, K grous When the grou sizes are not equal, our results suggest that, for large electorates, the robability of casting a decisive vote does not deend on the size of the grou to which the voter belongs and is only governed by the distribution of the grou sizes. Let us conclude with two remarks. In this aer, we have mostly focused on a seci c attern of correlation that we call the IAC artitioning model. It is imortant to recall that this model is seci c on two grounds. First, it is based on a artition of the individuals such that individuals belonging to two di erent grous in that artition have indeendent references. Second, it has been assumed that in each grou the correlations among the references in the grou were resulting from the IAC model. In this remark, we kee the artitioning assumtion but examine a articular generalization of the existing IAC version. In the IAC setting, the correlation coe cient between the votes of two voters from the same grou is equal to. Let us consider instead the case where the correlation coe cient 3 between the votes of two voters is ositive but arbitrary 8 and denoted by : Cov(X ik ; X jk ), 8 In Aendix, we rove that this construction is always ossible. An alternative construction in the sirit of the de nition of IAC due to Berg (990) is also ossible through the family of Beta densities. 7

29 the covariance between the votes of i and j when they belong to the same grou is then equal to. As before, as long as 6= we obtain: 4 Lim P iv(s) ( R ; ) = R! q + + P S 4 4 l= l l In articular, in the case where is a multile of s and all grous are of size s, we obtain: r Lim P iv(s) ( R ; ) = R! + (s ) We observe that P iv (s) ( R ; ) decreases with s and with. This is consistent with intuition as an increase in s or an increase in leads to more correlation among the votes and less room for ivotality. Our second remark is about another key assumtion, namely the neutrality among the two alternatives. We have assumed that the two alternatives were similar ex ante. One interesting generalization could consist in assuming that there is a artition of the oulation into grous where in each grou the references are as here correlated but also ossibly biased towards one candidate. The bias could of course vary from grou to another. In such a setting a grou could be de ned as a subset of individuals dislaying some homogeneity de ned through a vector of characteristics. We are not aware of an ambitious attemt to generalize the current theory to a setting that would allow for di erences across alternatives. To the best of our knowledge, the only 9 model along these lines is due to Beck (975). He considers a oulation divided into two grous of equal size. In the rst grou, the votes are indeendent and eole vote left with robability. In the second grou, votes are also indeendent and eole vote left with robability. Beck estimates numerically the robability for a voter to be ivotal for several values of the arameter. Modulo a simle adjustment of the roof of Proosition, we obtain an asymtotic exact value of the robability of being ivotal in Beck s model. Precisely, we obtain : Lim P iv(; ) =! ( ) q When =, we obtain the traditional constant = 0: When = 3, we 4 obtain = 0:9 3 and when = 4; we obtain = 0: Moving towards olarization increases drastically the robability of being ivotal! 9 See also Berg (990) for another illustration. 8

30 6 Aendix 6. Berk s Theorem For each k = ;,...let n = n(k) and m = m(k) be seci ed and suose that y; k y; k :::; yn k is an m deendent triangular array of random variables with zero means 0. Assume the following conditions hold. For some > 0 and some constants M and K: (i) E y k i + M for all i and all k. (ii) V ar yi+ k + ::: + yj k (j i) K for all i; j, and k. V ar(y (iii) Lim k+:::+yk n) exists and is nonzero. Denote v the limit. k! n m (iv) Lim + = 0 k! n Then yk +:::+yk n n is asymtotically normal with mean 0 and variance v. 6. Positively Correlated Bernoulli Random Variables In this aendix, we resent a rocedure to generate correlation atterns for vectors of Bernoulli random variables with arameter. Consider a Gaussian vector z (z ; z ; :::; z n ) such that the random variables z i are identically distributed with rst moment equal to 0 and let = ( ij ) i;jn = ij to denote its variance-covariance matrix with i;jn = E(zi ) and ii = for all i = ; :::; n. We derive from z a vector x (x ; x ; :::; x n ) of Bernoulli random variables such that Pr (x i = ) = for all i = ; ; :::; n as follows. Let: if zi 0 x i = 0 if z i < 0 From Stieljes s formula (Guta (963)), we obtain that In articular when = I + Cov(x i ; x j ) = Pr (z i 0 and z j 0) 0 : : : : : : : 4 = arcsin ij : C A where > 0, we obtain = + and ij = for all i 6= j. ote that is ositive de nite and is therefore the variance-covariance matrix of a gaussian vector. We derive ij = + for all i 6= j. In such setting, all the correlation coe cients are equal and san all the values from 0 to when sans the range [0; +[. 0 The triangular array yn(k) k is m deendent if k yk ; y k ; :::; yj k and yj+n k ; yk j++n ; :::; yk j+n+l are indeendent whenever n > m (Billingsley(995)). ote that since arcsin x = x + x x O(x5 ), this imlies that Cov(x i ; x j ) ' ij in the neighborhood of 0. 9