The probability of the referendum paradox under maximal culture Gabriele Esposito Vincent Merlin December 2010 Abstract In a two candidate election, a Referendum paradox occurs when the candidates who obtains a majority of votes does not win a majority of seats. Feix et al. (2004) estimated the likelihood of the referendum paradox under the IC and IAC assumption, for N = 3 to 101 jurisdictions. We propose to study the phenomenon for a third a priori assumption, the Maximal Culture Assumption. Using the flexibility of this assumption, we estimate the probabilities for three candidate elections using Monte Carlo simulation. Keywords : voting paradoxes, probability calculations, MC hypothesis. JEL Classification : C9, D72 1 Introduction A referendum paradox (see Nurmi 1999) occurs in a two candidate election when a candidate wins in a majority of districts (or states) while she gets less votes than her opponent in the whole country (or federal union). Such situations have been observed in real political elections, the most famous occurrence being the election of G.W. Bush in 2000. Some results on this question have been obtained by Feix et al (2004) in the simplest case where there are equal EHESS and GREQAM, Centre de la Vieille Charité, 2 rue de la Charité, 13236 Marseille, France. E-mail: esposito@ehess.fr. G. Esposito acknowledges financial support from Marie Curie fellowships for early stage research training. CREM UMR6211, CNRS and Université de Caen, 19 rue Claude Bloch, 14032 Caen cedex, France. E-mail: vincent.merlin@unicaen.fr 1
Table 1: A potential referendum with UK elections Party YouGov Pool BBC Election 20 th April 2010 Seat Calculator Lib Dem 34% 150 Conservative 31% 229 Labour 26 % 242 Others 9% 29 size districts. The results provided by these authors are based on various a priori probabilistic models, including the usual Impartial Culture (IC) and the Impartial Anonymous Culture (IAC) models. Using both analytical and simulation-based arguments, they show, among other results, that the paradox probability under the IAC tends to a limit of around 16,5% when the number of districts goes to infinity. Under IC, the limit probability seems to reach 20.5%. In fact, in a paper that has been completely forgotten until recently 1, Kenneth May (1948) prove that the limit probability for IAC for K districts of n voters is 1 6. However, the paradoxical situations unveiled for two party competition can also appear with first past the post systems with three party or more. At some point during the campaign for 2010 UK parliament, some pools (by YouGov) and seat projections (by BBC) predicted that the Labor party would win more seats with less votes that the Liberal Democrats and the Conservative Party! We can even observe an even stronger paradox on Table 1, as the ordering of the three major parties in terms of votes could have been completly reversed in term of seats 2. Such situations could also appear in countries such as India and Canada, which use the plurality rule in districts to elect their parliament, and count more than two major parties. Thus, the objective of this paper is to estimate the influence of the third party on the a priori likelihood of the referendum paradox, for the election of a parliament using the plurality rule in each constituency. For the sake of simplicity, we will assume, as in Feix at al. (2004), that all the constituencies will have the same size. To perform this task, we will use an a priori probabilistic assumption which is different from the traditional IC and IAC models often used in this literature, the Maximal Culture assumption (MC). MC is characterized by an integer parameter, M, which is the maximal number of voters for each preference type. Then for each preference type, the number of voters with type j is drawn randomly from an uniform 1 We are indebted to Hannu Nurmi and John Roemer, who raised our attention to this forgotten paper. 2 Finally, the conservative party won the 2010 election with 36.1% of the votes and 306 seats. No paradox was observed this time! 2
distribution on [0,M]. Thus, the population is variable from draws to draws. By letting M vary from party to party, we could estimate easily the impact of the growth of the third party on the referendum paradox. In section 2, we will present in details the model, the different definitions of the referendum paradox we can think of in a three party race, and the probabilistic assumption. Section 3 will be devoted to the study of the three party races using Monte Carlo simulations. Section 4 concludes. 2 Assumptions 2.1 Voting rule and paradoxes Consider a parliament with K seats, K 3. The member of the parliaments are elected in each jurisdiction with the plurality rule (also called the first past the post rule), that is, the candidate with more votes will win the seat. We assume throughout the paper that each jurisdiction has the same population n. In each jurisdiction, the three parties, A, B and C, will endorse a candidate. The result in jurisdiction j is given by a vector n j = (n j A,nj B,nj j C ), where ni is the number of votes for party i = A,B,C in this jurisdiction. Due to the geographic repartition of the votes and differences in turnout, we can end up with several paradoxical results, which generalize the definition of the referendum paradox. Case 1 The party which obtained the highest number of votes over the whole country did not obtain the highest number of seats. Case 2 The party which obtained the highest number of votes over the whole country got the smallest number of seats. Case 3 The party which obtained the smallest number of votes over the whole country has more seat than another party. Case 4 The party which obtained the smallest number of votes over the whole country got the highest number of seats. The hypothetical situation described in Table 1 was a combination of Case 2 and 4. I this paper, we will focus the analysis on Case 1, and just briefly comment the results for the three other cases. 3
2.2 Adapting the maximal culture assumption The maximal culture assumption was fist introduced by Gerhlein and Fishburn (1981). It assumes that, for each preference type i, the number of vote is drawn randomly and independently between 0 and a positive integer M, according to an uniform distribution. In this paper, we will first assume that M is not the same one for each party. The maximal number of votes for party i in each jurisdiction will be the same M i, with M A M B M C 0, M A > 0, to define a more flexible version of MC, able to cope with asymmetries among the three parties. To simplify further the simulation, we will assume that the populations in each district are so large that we can consider that the number of voters is continuous. Thus we can normalize the M i s by dividing then by n: m i = M i n, with i m i = 1. Furthermore, if we define the fraction of registered voters that voted for party i in jurisdiction j by: v j i = n j i n, i = A,B,C we can simulate the limit behavior of a Maximal Culture model as n goes to infinity by drawing the v i s according to a uniform distribution on [0,m i ]. Thus, the key parameter is now the the vector m = (m A,m B,m C ), with 3 i m i = 1, v i 0, andm A m B m C, which gives the maximal proportion of the registered voter that can support each party. m = (0.5, 0.5, 0) gives back a two party scenario, and as m C increase, we will study the potential impact of small parties on the magnitude of the paradox. At the extreme point, m = (1/3,1/3,1/3) models an extremely competitive three way race. 3 The probability of the referendum paradox for three candidates Our Monte Carlo simulations provide us with a wide quantity of data about different possibilities of m and the respective probabilities to have a paradox. We report in table 2 the probabilities for the referendum paradox (Case 1) for 3 parties and 3 states under 3. When m C = m B = 0, party A is the unique party and no paradox can exist. The probability progressively rises as m B and m C increase. For the two party case (m C = 0,m A = m B = 0.5), we estimate the probability of the referendum paradox around 15.9%, in between the values 3 Tables for cases 2,3 and 4 and K = 3, 11, 25, 45, 101, and 650 are available upon request, as well as the R program that was used to simulate the data. 4
Table 2: The probabilities for the referendum paradox for 3 parties and 3 states under MC. m C 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.333 m B 0 0 0.04 0.0013 0.0010 0.08 0.0049 0.0054 0.0057 0.12 0.0121 0.0124 0.0122 0.0126 0.16 0.0216 0.0229 0.0231 0.0225 0.0276 0.20 0.0354 0.0385 0.0384 0.0379 0.0415 0.0487 0.24 0.0509 0.0544 0.0555 0.0556 0.0577 0.0626 0.0746 0.28 0.0702 0.0733 0.0761 0.0760 0.0783 0.0833 0.0912 0.1044 0.333 0.0931 0.0969 0.0989 0.1015 0.1003 0.1055 0.1133 0.1281 0.1457 0.36 0.1072 0.1115 0.1156 0.1146 0.1163 0.1214 0.1263 0.1384 0.40 0.1252 0.1282 0.1356 0.1349 0.1332 0.1319 0.44 0.1436 0.1477 0.1467 0.1455 0.48 0.1551 0.1560 0.50 0.1593 proposed by Feix et al (2004) for IAC (12.5%) and IC (16.2%). The probability of the referendum paradox seems slightly less likely in the symmetric three party competition (14.57% for m A = m B = m C = 1 3 ). Figure 1 plots the path of the four paradoxes we considered in section 2 for the extreme case m = (1/3,1/3,1/3) and an increasing number of states. The paths are not monotonous, but seem to be periodic. The fluctuations are of course intrinsic to Monte Carlo simulations, but also depend upon the of the rest of the division of K by 2 and 3, as for example, for K = 6, seat repartition (3,3,0) and (2,2,2) are not considered as paradoxes, whatever the percent of vote parties get. The third kind of paradox broadly seems as likely as the first one, as it is a symmetric case: it considers the party with less votes and check whether it is not last in the parliament. As we could expect, the probability of the biggest (smallest) party being last (first) in the parliament are smaller, but they are not negligible. Anyway, paradoxes 1 and 3 and paradoxes 2 and 4, even if different, converge to 2 limit values. Simulations with 650 states provide a value of 31.45% for paradox 1 and 31.84% for paradox 3, not really higher then the values in figure 1 for just 120 states. Paradox 2 and 4 have respectively a value of 7.79% and 7.57%, still in line with the hypothesis of convergence to a unique limit when the number of states reaches the infinite. Figures 2 show the dynamic of the probability for case 1 with an odd number of jurisdictions when there are two big equal size parties and a third one that increases its size from 0 5
Figure 1: Probability of the referendum paradox for three equal parties with different number of states to one third. When the size of the new party is relatively small, it gives a negative (positive) shock to the probability of paradox when the number of states is odd (even). As the new party only captures its first one seat, it changes the parity of the seats that are left for the two main competitors. But we observe that the probabilities for an even number of seats are higher than probabilities for odd number of seats; Figure 3display this phenomenon, and also shows that this effect stops at some point. Thus, at some point, the probability of the referendum paradox increases a lot as the new party is also able to capture more seats and to win the general election. 4 Conclusion Going back to the example we used in our introduction, the 2010 UK parliamentary elections, we can state that the increasing success of the Lib-Dem party constantly increases also the probability of referendum paradox. When the size in votes is similar among the three parties, our MC simulation with 650 constituencies suggest that the probability of a paradox could be large, as it converges to about 32%. Knowing that in many parliamentary system, the party with more seats is first called to form a government, this paper gives an additional argument against first past the post. From a technical point of view, the MC assumption has proven to be more flexible than IC and IAC to describe asymmetric societies. Furthermore, Adapting 6
Figure 2: Probability of the referendum paradox when a third party increases its size, for an odd number of jurisdictions Figure 3: Probability of the referendum paradox when a third party increases its size 7
the recent literature on Erhart polynomials, (Lepelley, Louichi, Smaoui, 2008, Huang et Chua 2000) to the case of maximal culture could enable us to find exact values for the paradox in scenarios with few sates. References [1] Nurmi I., Voting Paradoxes and How to Deal with them. (1999). [2] Feix M, D Lepelley, V Merlin and JL Rouet (2004) The probability of paradox in a U.S. presidential type election. Economic Theory 23: 227-257. [3] [Fishburn, Peter C.]; Gehrlein, William V.; Constant Scoring Rules for Choosing One Among Many Alternatives ; Quality and Quantity; Vol. 15, No. 2; April, 1981; 203-210; [4] Gehrlein, William V.; Lepelley, Dominique; Condorcet s Paradox under the Maximal Culture Condition ; Economics Letters; Vol. 55, No. 1; 15 August 1997; 85-89; [5] May, Kenneth, (1948), Probabilities of Certain Election Results, The American MAthematical Monthly, vol 55, No. 4. [6] Lepelley D, A Louichi and H Smaoui (2008) On Ehrhart polynomials and probability calculations in voting theory. Social Choice and Welfare 30: 363-383. 8