Evaluating the likelihood of the referendum paradox for mixed voting systems Very preliminary version

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1 Evaluating the likelihood of the referendum paradox for mixed voting systems Very preliminary version Michel Le Breton, Dominique Lepelley, Vincent Merlin March 204 Abstract A referendum paradox (Nurmi, 999) occurs, in a two party competition, each time a party gets a majority of the seats in the parliament while it did not obtain a majority of votes nationwide. This paradox can be viewed as an instance of the Borda paradox, as the voting rules fails to select the Condorcet winner. Feix, Lepelley, Merlin and Rouet (2004), Wilson and Pritchard (2007) and Lepelley, Merlin and Rouet (20) computed the probability of the referendum paradox under the Impatial Culture (IC) assumption and a variant of the Impartial Anonymous Culture (IAC*) assumptions when two parties compete in D equal sized districts. These a priori models for voting are extensively described in Gerhlein (2006). The same paradox may occur for mixed electoral systems. On the top of electing D representatives in districts, the voters also elect L members of the parliament at large. Hence, the parliament is of size D + L. Blais and Massicote (2009) propose an extensive survey of all the mixed electoral systems that are used worldwide. In this paper, assuming that D representatives are elected in equal size jurisdictions, we estimate the probability of the referendum paradox for three different mixed systems : ) when the all the L at large seats are attributed to the party which obtained a majority of votes nationwide 2) when the L at large seats are apportioned according to the proportional rule and 3) when the L seats are apportioned on the basis of the wasted votes, that is, the sums of the votes of the party candidates that were not elected in districts. We perform our estimations under the IC and IAC* hypothesis. As a corollary, we estimate the probability of the referendum paradox as a function of the ratio L/D under the three scenarios. In an electoral design perspective, we are then able to suggest which values for L/D are sufficiently large for the referendum paradox to become negligible, and which mixed system is more able to drastically reduce the likelihood of the paradox. Keywords : referendum paradox, mixed voting rules, probability calculations, IAC hypothesis, IC hypothesis. JEL Classification : C9, D72 Institut Universitaire de France and Toulouse School of Economics, France. michel.lebreton@tse-fr.eu CEMOI, Faculté de Droit et d Economie, Université de La Réunion, 9775 Saint-Denis cedex 9 France. dominique.lepelley@univ-reunion.fr Normandie Université, UCBN, CREM CNRS UMR62. UFR des sciences économiques et de gestion, 9 rue Claude Bloch, 4032 Caen cedex, France. vincent.merlin@unicaen.fr

2 The Model. The Voting Rules Consider a country (elsewhere federation, region, etc.) divided into D districts (elsewhere states, regions, jurisdictions). For the sake of simplicity, we will assume that each region i =,... D has an equal size population of voters n i = n. Hence, the total population size is nd. In each district, the voters will select a representative according with the plurality rule; this case a very common one, used in many democracies such as United Kingdom, United States of America, Canada, India, etc. Throughout the paper, we will also assume that there are only two parties, A and B competing in all the districts in order to obtain a majority at the parliament. Moreover, abstention is not authorized. When we add up the votes across the districts, there is obviously one majority winner or Condorcet winner. The fact that a majority of the D seats does not select the party who has a majority of votes is then called a Referendum Paradox (Nurmi 999). It can also be viewed as an instance of the Borda paradox, that is, an instance where the voting rule fails to pick the majority winner. Our model become less traditional when we assume that, on the top of electing D representatives in single member districts, the voters also elects L representatives at large. We will also assume that a voter cast his vote in the same way at the federal level than in his district. This assumption is of course very restrictive, and could be amended latter on, by assuming that the second vote is just partially correlated with the local vote. Among the many ways that could be used in order to fill the L at large seats, we will restrict ourselves to three cases : Winner takes all (W ). The party which obtains a majority of votes across the districts gets all the L at large seats. The question here is simple. Given that there is a referendum paradox, the extra seats will tend to restore the majority result. An example is the state of Nebraska, for the US presidential election. Out of the 5 electors that it sends to the electoral college, 3 are designated in the representative districts. Next, the candidate who obtains a plurality of votes in Nebraska gets the two extra seats. Clearly, D = 3 and L = 2 is sufficient to restore a majority of electors for the majority winner. Similarly, whenever L D, we are sure that the at large majority winner will be the winner. The issue here is to understand how the likelihood of the referendum paradox decreases as L increases, and to find the minimal bound on L that almost surely eliminates the paradox. Proportional seats (P ). This model seems the most natural one, as it combines a priori the advantages of the plurality rule (giving a large majority of districts to the leading party) and of proportional rule (guarantying some seats for all the parties). Many sophisticated variants are in use worldwide. This idea was also put forward during the 202 French presidential election by many candidates, e.g. François Hollande, François Bayrou, in order to reform the election of the French parliament. We will here stick to the simple case where the at large seats are apportioned according to the Hamilton rule 2 Again, we seek for the lower bound on L that will restore almost surely the election of the majority winner. Compensation seats. These systems are less known, but exist in practice. An extreme case is Notice that a referendum paradox cannot occur if a party wins all the D seats. The reader can check that L D is a sufficient condition for the Condorcet winner to get a majority of seats, whether D is odd or even. 2 In our two party case, this means that theinterger part of division of the total number of votes for a party by the quota nd/l determine its number of seats, and the extra seats will go to the party with the highest remainder. For more about the apportionment methods, see Balinski and Young (983). 2

3 the German electoral system, where the L seats are meant to restore the pure proportionality in D + L 3. A less known case is Trentino Alto Adige, in Italy. This region is divided into six jurisdiction, each one sending one representative with first past the post. However, the votes obtained by the losers are aggregated according to their party affiliation, and the party who got the maximal number of these lost votes gets an extra seat. The idea can be generalized when L >, by again using the Hamilton rule to apportion the wasted votes. Notice that under the Sakorzy presidency in France, the government considered the implementation of this voting rules for regional election, with 80% of the seats elected in districts under first past the post, and 20% of compensatory seats. Again, the main question here is to understand how the likelihood of the referendum paradox evolves as L increases. Notice that we could also study in the same context D districts, and divide L = P + W + C, where P seats are attributed with the proportional rule, W seats given to the majority winner, and C seats are compensation seats. This generalization is however, beyond the scope of this note. In order to evaluate our voting models, we must set some a priori assumption about the behavior of the voters. We will rely on the oldest and simplest models that have been suggested in the literature: the model proposed by May (948), which is a variation of the model known today as the Impartial Anonymous Culture (IAC), and the Impartial Culture assumption (IC), which has been first introduced by Guilbaudt (953 )in social choice litterature..2 The Impartial Culture Assumption In Social Choice Theory, the Impartial Culture describes a voting behavior wherein each voter picks out any of the possible preference type with equal probability independently from the the other. In our case, the probability to select A, p A equals 2. The number of voters who backs up candidate A in district i, n A i is governed by a binomial law: ( ) n P rob(n A i = x) = (p A ) x ( p A ) n x () x As p A = 2, the mean of variable na i is µ A = n 2 and its variance is σ2 A = n 4. Using the Central Limit Theorem, the distribution of a sum of independent variables z N = N i= x i can be approximated by a normal law with mean m = N i= m i and variance σ 2 = N i= σ2 i if N is large enough. Hence, as n tends toward infinity, the distribution of any n A in any district can be approximated by a normal law: f(t) = exp ( (t µ A) 2 ) 2πσA 2σ A Let us denote by P X (n, D) the probability of the referendum paradox for D district of n voters under model X. Hence, (Feix et al 2004) managed to derive some formulas in the case n : ( ) P IC (, 3) = 3 3 π arccos P IC (, 5) Simulations (Feix et al 2004) and approximations (Leppeley et al 20) suggest that P IC (, 0) tends toward 20.5%. 3 For recent debate about the German electoral system and its reforms, see Pukelsheim(202). 3

4 .3 The Impartial Anonymous Culture Assumption and May s Variant One of the main drawback of the IC assumption is to concentrate the probability around the mean of distribution; extreme points, where all the voters have similar preferences, become extremely rare as n increases. An alternative model has been then proposed in Social Choice Theory by, the Impartial Anonymous Culture (IAC). Let ñ i = (n A i, nb i ) be a voting situation, ie the distribution of the n voters between Candidate A and Candidate B in district i. n A i + n B i = n as abstention is not allowed. The IAC assumption will assume that all the voting situations are equally likely: P rob(n A i = x) = n + Hence, in each district, the distribution of n A i has a mean µ A = n 2 and a variance (σ A )2 = (2n+)n 6. We will use throughout the paper the assumption IAC, introduced by May (948): each district selects its voting situation according to the IAC assumption, but the votes across the districts are independent. Let us denote by P X (n, D) the probability of the referendum paradox for D district of n voters under model X. Hence, we obtain (Feix et al 2004, Wilson and Pritchard 2007, Lepelley et al 20): P IAC (n, 3) = n2 + 2n 3 8(n + ) 2 2.5% P IAC (n, 5) = 5(n )(n + 3)(n2 + 22n + 27) 384(n + ) 4 4.3% P IAC (n, 7) = (n )(n + 3)(577n n n n ) 3840(n + ) 6 5.0% (2) The limit probability is /6 as D and n increases (May 948). 2 Estimating the likelihood of the paradox in the winner takes all case 2. The approximation technique under IAC* The method we will use throughout the paper to estimate the likelihood of the referedum paradox for mixed voting rules is presented in Lepelley et al (20). For each state i, define r i = ni n, the proportion of A partisans. We can next assume that r i is drawn from a distribution law f i (r i ) = f(r i ) in each district. If we apply the IAC assumption in each district and assume that n is large, we have: { if r i [0, ]. f(r i ) = 0 otherwise. or, if the define x i = 2r i h(x i ) = { /2 if x i [, ]. 0 otherwise. Assume now that D A districts back A while D B back B. We want to derive the law of z D = D i= x i, the total number of votes for A, with the parliament (D A, D B ). Let us call h A (x) and h B (x) the repartitions of the (relative) difference of votes for A in a state that favors A and B respectively. 4

5 h B (x) = { if x i [, 0]. 0 otherwise. and h A (x) = { if x i [0, ]. 0 otherwise. The respective means are m B = 0.5 and m A = 0.5, while σa 2 = σ2 B = σ2 = /2. If D is sufficiently large, the distribution of z D can be approximated by a normal law with mean m = D i= m i and variance σ 2 = D i= σ2 i. Then for D A districts in favor of A and D B dictricts in favor of B, in this limit, we have with h D D A,D B (z D ) 2π σ exp( (z D m) 2 2σ 2 ) m = D A m A + D B m B and σ 2 = D A σ 2 A + D B σ 2 B The likelihood of the referendum paradox under IAC* for the repartition (D A, D B ), D A < D B, is then: P IAC (, D A, D B ) ( ) D D A D! D A! D B! p D A A pd D A B 2 D 0 0 ( 2π σ exp (x m)2 2σ 2 ) (x m)2 ( 2σ 2 dx 2π σ exp ) dx with m = D A m A + D B m B and σ 2 = Dσ 2 A. We will now detail the computation for D = 0. Assumes that 50 seats backs A, and 5 backs B. The distribution of the votes for A will follow approximately a normal law with m = 0.5N A 0.5N B = 0.5(N A n B ) = 0.5 and. Hence σ 2 = 0 2 = h N 50,5(z D ) 2π 0 48 exp( (z N + 0.5) ) (3) 48 By doing the same exercise for all the repartitions (D A, D B ), we are able to derive the contribution of each case to the likelihood of the referendum paradox. Table displays these values for D A < D B. The value of the paradox is twice the sum, as we have to take into account the symmetric cases with D B < D A. With this technique, we approximate quite well the true value (/6) derived by May (948). We can immediately use the information contained in this table to estimate PIAC W (, D) the probability of the referendum paradox when the national winner gains W at large seats. If W = 2, we can immediately notive that there won t be any referendum paradox anymore for D A = 50, D B = 5, as Candidate A will know obtain 53 seats in the parliament. However, these two extra seats will not be sufficient to ensure him a majority in the parliament if D A 49. Hence, the value of the paradox is given by the sum of the contributions displayed in Table, except the first line. Similarly, for W = 4, we will have to exclude the two first lines, for W = 6, the first three lines, etc. With the convention that a fait coin is tossed in case of an equal number of seats in the parliament to determine which party 5

6 Table : IAC case : paradox contribution for the different values of N A, in the case D = 0, L = 50, proportional N A N B m σ paradox contribution P IAC (, 0).6545 Table 2: The Referendum Paradox under IAC* with W at large seats and D = 0. W PIAC W (, 0)

7 Table 3: IC case : paradox contribution for the different values of D A, in the case D = 0 D A D B m σ 2 paradox contribution P IC (, 0) has the majority, we can estimate all the values of PIAC W (, D) for D = 0. They are displayed in Table 2 We immediately conclude from Table 2 that we need to have have W 9 for the probability of the referendum paradox to fall below %. 2.2 The approximation technique under IC* We can derive approximation results in the IC case too with the wery same method, by just modifying the input values of m A, m B, σ A and σ B. As the values of the n i s follow a binomial law, it suffices to consider that x i, the difference of votes between A and B, is drawn from a normal law N (0, ) if n is large. Hence, we have 2e /2 x 2 h A (x) = if x > 0, 0 otherwise. π with m A = and σa 2 = , and 2e /2 x 2 h B (x) = if x < 0, 0 otherwise. π with m B = and σb 2 = With these new data, we can estimate the likelihhod of P IC (, 0), by estimating the contribution of each repartition (D A, D B ) to the paradox. Now, let PIC W (, D) the probability of the referendum paradox when the national winner gains W at large seats. Table 4 tell us that we need to set W 0 for the likelihood of the referendum paradox to fall below the % threshold. 3 Estimating the likelihood of the paradox for the proportional rule In this section, we assume that on the top of electing D representative in districts, the votes are aggregate at the federal level in order to apportion P extra seats according to the Hamilton rule. Our 7

8 Table 4: The Referendum Paradox under IAC with W at large seats W PIC W (, 0) estimation will again rely upon the technique developed in Lepelley et al (20), used here in a slightly different way. Throughout this section, we will still consider the case D = 0 3. The IAC* case In order to introduce the problem, let us consider the case D = 0 and P = 50. For D A = 50, D B = 5, in case of a referendum paradox, party A needs to capture 26 out of the 50 proportional seats to compensate his defeat at the district level (that is 52% of the proportional seats). Hence, due to the use of Hamilton s apportionment rule, party A needs a proportion q DA,D B = 5% of the vote to escape from paradox. In other words, the paradox persists if party A obtain in between 50% and 5% of the popular vote. Given that have normalized the votes such as x i, the difference between the vote for A and B lies in [, ], Z D range is [ 0, 0]. If party A gets 5% of the votes, the difference between A and B will be 0.02*0 = Hence, if the difference between A and B votes lies in between 0 and the threshold T DA,D B = T 50,5 = 2.02 the paradox will persist, and the contribution of this case is measured by: P P =50 IAC (, 50, 5) ( ) D D A D! D A! D B! p D A A pd D A B 2 D 0 T50,5 0 T50,5 ( 2π σ exp (x m)2 2σ 2 ) (x m)2 ( 2σ 2 dx 2π σ exp ) dx Table 5 indicates that the integration the contribution of the case D A = 50, D B = 5 is now instead of By proceeding in the same way for all the repartitions (D A, D B ), we obtain the value PIAC P =50 (, 0) =

9 Table 5: IAC case : paradox contribution for the different values of N A, in the case D = 0, P = 50, proportional N A N B m σ paradox contribution Threeshold With P = PIAC P (, 0) Table 6: The probability of the referendum paradox under IAC* for a mixed electoral system, with D = 0 and P proportional seat. P PIAC P (, 0) Table 6 indicates how the value of PIAC P (, 0) evolves as P goes from 0 to The key point is that, even with 2000 at large seats, the likelihood of the referendum paradox remains aboe the % threshold. The results are in sharp contrast with the winner takes all case. For example, for P = 0, the likelihood of the paradox barely decreases! The key point is that most of the time, a referendum paradox occurs when the election is tied nationwide. Hence, the repartition of the seats via the Hamilton rule will be extremely close too, failing to give the Condorcet winner enough extra seats to make him victorious. 3.2 The IC case In order to estimate the likelihood of the referendum paradox under IC with P seats distributed according to the Hamilton rule, we need to specify n, the number of voters per district, to calibrate the variance of the distribution of the votes. We will assume here that n = 000. We have then 0000 voters for D = 0, with a relatively low standard deviation σ = Table 7 displays the values of PIC P (000, 0) for some values of P. We immediately observe that the likelihood of the paradox remains extremely important, even for P = 2000! The key point is that the IC assumption create extremely 9

10 Table 7: The probability of the referendum paradox under IC for a mixed electoral system. 0 equal size districts of size 000 and P at large seat. P PIC P (000, 0) split societies. Hence, the repartition of the seat according to the Hamilton rule is also always even between both parties. 4 How to evaluate the likelihood of the referendum paradox for compensation rules? The idea here is to use the use the wasted votes in order to fill L = C seats proportionally. Let N A (resp. N B )be the set of district where A (resp. B) loses. Let us define the wastes votes for A and B w A = r i w B = i N A i N B r i Under IAC, the values of r i for i N A are drawn in f C (r): { 2 if r [0, 0.5]. f C (r) = 0 otherwise. with m C = 0.25 and σ C = /48. If D B is large enough, w A will follow a normal law f DB (w A ) with mean 0.25D B and σ 2 = D B σc 2 : ( (wa 0.25D f DB (w A ) = B ) 2 ) exp 2πDB σc 2 2D B σc 2 Similarly, if D A is large enough, w B will follow a normal law f DB (w B ) with mean 0.25D B and σ 2 = D B σc 2. ( fda (wb) (wb 0.25D = A ) 2 ) exp 2πDA σc 2 2D A σc 2 We can then describe the joint distribution of w A and w B by: With D A < D B, there is a paradox if: f DA,D B (w A, w B ) = f DB (w A )f DA (w B ) w A + D A w B > w B + D B w A (4) w A > w B + D B D A 2 (5) 0

11 Given C compensation seats, A has to reach a quota κ% = g(c, D A, D B ) of the wasted vote to overcome the paradox. The paradox persists if: We hence have to estimate the density between these two constraint. To give an example, consider D A = 50 and D B = 5. Hence, w A w A + w B < κ (6) w A < κ κ w B (7) w A > w B + D B D A 2 w A > w B + 2. (9) With C = 50 compensation seats, party A needs to capture 26 of them, that is κ = 5% of the wasted votes with the Hamilton rule. w A < κ κ w B (0) w A <.0408w B () One can graphically check that the two conditions display a thin cone which does not intersect the center of the distribution f DA,D B (w A, w B ). Though this point needs further investigations, one may conjecture that the likelihood of the referendum paradox for D = 0 districts and C = 50 compensation seats will be extremely low. The results will be even more evident under IC with n large, as the distribution of the wasted votes will be concentrated around the center of the distribution. (8) 5 Preliminary Conclusion In this preliminary note, we revisited the a classical issue, the evaluation of the likelihood of the referendum paradox in two tiers voting systems, when the federation elecst L at large representatives on the top designating D representatives in districts. Though our result needs to be completed, the main conclusion is electing at large seats with a a proportional rule is not an efficient way to reduce the likelihood of the paradox. In contrast, our computation with D = 0 suggest that giving 0% of extra seats to the majority winner would almost surely eliminate the paradox. We also conjecture that apportioning the L at large seats on the basis of wasted vote would be an efficient way to reduce the likelihood of the paradox. References [] Balinski M and P Young (982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Press. [2] Blais B and L Massicote (2009) Mixed electoral systems: a conceptual and empirical survey. Electoral Studies 8: [3] Feix M, D Lepelley, V Merlin and JL Rouet (2004) The probability of paradox in a U.S. presidential type election. Economic Theory 23: [4] Gehrlein W, Condorcet s Paradox. Springer Publishing, 2006,

12 [5] Gehrlein WV and PC Fishburn (976) Condorcet s paradox and anonymous preference profiles, Public Choice 26:-8. [6] Guibauld GT (952) Les théories de l intérêt général et le problème logique de l aggrégation, Economie Appliquée 5: [7] Lepelley D, Merlin V and JL Rouet (20) Three ways to compute accurately the probability of the referendum paradox, Mathematical Social Sciences 62-: [8] Nurmi H (999). Voting Paradoxes, and how to deal with them. Springer. [9] Pukelsheim F, A gentle combination of plurality vote and proportional representation for Bundestag elections. In Felsenthal D.S. and M. Machover Eds (202) Electoral Systems Paradoxes, Assumptions, and Procedures Springer: Berlin, 5-7. [0] Wilson MC, G Pritchard (2007) Probability calculations under the IAC hypothesis. Mathematical Social Sciences 54: