The Electoral College : a Majority Efficiency Analysis

Home , Electoral college, Paradox of voting

The Electoral College : A Majority Eﬃciency Analysis.

Vincent R. Merlin,∗ Thomas G. Senn´e†

Preliminary Version The 29th of January 2008

One of the main topics of voting theory is the assessment of paradox proba- bilities in order to compare diﬀerent voting rules. These computations are based upon assumptions on the probability distribution of voters’ preferences. The two principal hypothesis are the Impartial Culture assumption which stipulates that every voter picks his preference among a set of uniformly distributed orderings, and the Impartial Anonymous Culture which states that every voting situation has the same probability to occur. Their main disadvantage is that they are theoretical a priori models and they do not necessarily match the reality. In this paper we propose to calibrate these usual models in order to assess the probability of the referendum paradox, that is, the probability that the popular winner does not obtain a majority of delegates. Indeed, one of the motivation of this paper is to take part of the debate resulting from the criticisms instated by authors such as Gelman et alii [11, 12] and Regenwetter [26]. In this way, we provide a serie of estimations for the referendum paradox probability in the Electoral College according to a continuum of probabilistic models, which more and more match the real data. The second motivation is to use these diﬀerent probability assumptions in order to study the current seat allocation method in comparison with apportionment methods.

Keywords : Voting, IAC, IC, Referendum Paradox, Majority Eﬃciency, Electoral College.

JEL classiﬁcation : D71.

∗corresponding author : CREM, Universit´ede Caen, 14032 Caen Cedex, France (email: [email protected], tel: 02 31 56 62 49) †CREM, Universit´ede Caen, 14032 Caen Cedex, France (email: [email protected])

1 1 Introduction

During Summer 2000, just before the US presidential elections, at APSA’s annual Meeting, numerous political scientists predicted a Democratic (Al Gore) victory by upwards of 6 percentage points. In November, the election outcome was a surprisingly close election. With 48.4% of the popular vote, A. Gore only won 21 States among 51, for a total of 267 electors in the Electoral College, while G. W. Bush got 271 electors with less support from the popular vote. This paradox is known in Social Choice literature as the referendum paradox (RP, see Nurmi [21], Saul [28], Feix, Lepelley, Merlin and Rouet [9]). The U.S. presidential elections displayed the paradox in 1876, 1888 and 2000, and close outcomes (the margin between the top two candidates is less than about 4%) have been observed 11 times since the bipartism was established1 (see Table 1).

Table 1: The eleven closest U.S. presidential elections Year Popular Vote (%) Electoral Vote Margin (%) RP Dem. Rep. Dem. Rep. 1876 51.0 48.0 184 185 3.0 Yes 1880 48.3 48.2 214 155 0.1 No 1884 48.5 48.3 219 182 0.2 No 1888 48.6 47.8 168 233 0.8 Yes 1892 46.1 43.0 277 145 3.1 No 1916 49.2 46.1 277 254 3.1 No 1960 49.8 49.6 303 219 0.2 No 1968 42.7 43.4 191 301 0.7 No 1976 50.1 48.0 297 240 2.1 No 2000 48.4 47.9 267 271 0.5 Yes 2004 48.3 50.7 251 286 2.4 No

This paradox belongs to a class of voting paradoxes characterized by the variability of choice sets in counterintuitive ways when we aggregate differently the same electoral data. Two other famous paradoxes belonging to this class are the Ostrogorski’s paradox (see Ostrogorski [23], Nurmi [21, 22], Laffond and Lainé[19], Saari and Sieberg [27]) and the Anscombe’s paradox (see Ascombe [1], Wagner and Carler [29, 30], Saul [28], Nurmi [21]). The referendum paradox calls into question the institution of the consultative referendum in a representative democracy. In a consultative referendum, the issues which are voted upon in the referendum are finally decided by the parliament. More generally the referendum paradox occurs when, for example, a majority of voters supports an

1The US presidential elections records the paradox once before, in 1824. Three candidates received Electoral College votes but as no presidential candidate received an electoral majority, the election was determined by the House of Representatives. John Quincy Adams won the vote with the support of 13 State delegations though he was not the popular vote winner.

2 issue while the members of parliament, or here the Electoral College, reverses the majority decision. The first motivation of this work is to answer the following question : Is the U.S. presidential election system a good two-tiers voting system ? Indeed, one of the most important question in a federal union is to design a good allocation of the mandates among the union members (here States). There exist several normative criteria to solve this problem. Most of them belong to power index literature (see Penrose [24, 25], Banzhaf [3], Felsenthal and Machover [8]). Re- cently, new ones, based on the total utility maximization, have been proposed in the literature (see Felsenthal and Machover [8], Barberàand Jackson [6], Beis- bart, Bowens and Hartmann [7], Kirsch [17]). Close to this idea, Feix, Lepelley, Merlin and Rouet [9] suggested the concept of majority efficiency : the majority efficieny of a method corresponds to referendum paradox probability usind this model2. Ultimately, we want to maximize the probability that the candidate elected through a two-tiers voting rule reveals the popular choice. Then the following question is : how to compute as fairly as possible the referendum paradox probability ? In fact, in social choice theory, it is usual to set some a priori assumptions on the behavior of the voters. They permit to compare voting systems from a neutral point of view. They are not contingent to history : a voting system may be good for a certain voting pattern, at time t, but the time evolution may make it inadequate in the future. Hence, the probabilistic models used throughout the literature convey some notion of impartiality : they assume that each party is equally likely to win, and one can simulate the votes under different probability models without any reference to a precise political context. The two models often introduced in the Social Choice literature are the Impartial Culture (IC) and the Impartial Anonymous Culture (IAC) assumptions. With IC, all profiles of preference are equally likely. In our case of two candidates, it assumes that each voter selects a party with equal probability. When the number of voters in a State i is sufficiently large, the distribution of the votes follows a normal law. The IAC assumption considers that all voting situations have the same probability of occurrence. In our case, every final distribution of the votes between the two candidates is equally likely to occur. The main advantage of these two models is the notion of veil of ignorance they convey : without data they can assess the frequency numerous of paradoxes or evaluate the influence of voters via a power index, and enable the comparison of different voting rules from a normative point of view. However an important problem has recently been brought out. Electoral data give more information and do not match a priori models. Thus, any recommendation based on a priori voting models may clash with reality. Two main works are at the origin of this critic. In their recent book, Regenwetter, Grofman, Marley, Tsetlin [26] develop conceptual, mathematical, methodolog-

2This concept is equivalent to the concept of Condorcet eﬃciency developed by Fishburn and Gehrlein for the analysis scoring rule. For three candidates and more, the Condorcet eﬃciency is the conditional probability that a voting rule selects the Condorcet winner, given that such a candidate exists. In our case, with two candidates only, a majority winner always exists.

3 ical and empirical foundations of behavioral social choice research. This last notion encompasses two major interconnected paradigms : the development of behavioral choice theory and the evaluation of that theory with empirical data on social choice behavior. In their book, studies of survey data do not at all look like random samples from a uniform distribution, i.e., from an Impartial Cul- ture. This reinforces the view that the Impartial Culture is unrealistic and non descriptive of empirical data on social choice behavior. Moreover, they show the propensity of IC model to overevaluate the frequency of majority cycles, which turn out to be very rare with random sampling. The second work has been made by Gelman, Katz and Bafumi [11] on voting power indices. They showed that the Straffin’s Independence assumption (game theory’s equivalent of Impartial Culture) had to be rejected for the elections of the senators, the representatives and president in the United States; similar conclusions are drawn from the electoral data collected over Europe. This special case of model does not fit any of the empirical data they have examined. They indicate that voting power evaluations are based on an empirically falsified and theoretically unjustified model. For them, a more realistic and reasonable model could be affected by local, regional and national effects. The IAC model is not tested in these two preceeding works but we can think that it is not enough realistic too. Then, the resulting question is : if we still want to use a priori models (because they are relatively easy to implement, can be interpreted normatively and give a neutral point of view for the evaluation of voting rules), which adaptations are possible to make them more realistic ? Theoretical answers have been given by the literature (see Nurmi [20, 21, 22], Feix, Lepelley, Merlin and Rouet [9, 10]). A first approach to consider the preceeding critic is the Gen- eralized Impartial Anonymous Culture (GIAC) model. This kind of modeling results from the Berg’s work [4],[5]. He has shown that IC and IAC belong to the same family of urn models, named the Polya-Eggenberger family. In their recent work, Feix, Lepelley, Merlin, Rouet [10] develop Berg’s reasoning. Let us explain GIAC model. The probability of a voter to vote for a proposal A, p, is itself of probabilistic nature through the introduction of a probability distribution function f(p). The choice of f(p) is the first step for a better description of the electoral behavior, with a possible recourse to real-world data. It justifies the importance of the first part of the paper with graphical and goodness-of-fit tests. The probability of a configuration of n voters with t votes for the Demo- cratic candidate and (n − t) for the Republican candidate is pt(1 − p)n−t, and for a large number of election, it is given by formula (1).

1 f(p)pt(1 − p)n−tdp (1) ˆ0 The key point is that f(p) can be chosen as desired. In fact, our point of view is the following one : there may exist hybrid models, classiﬁed between a priori model (IC or IAC) and unrestricted model, “pure” a posteriori as suggested by Regenwetter et al. [26]. Therefore, this kind of model should remain relatively easy to implement and to interpret, should keep some degree of neutrality, but

4 would consider restrictions based on electoral data study. In this way, we implement a range of probabilistic models taking into consideration stylized facts coming from electoral data but which still respect some notions of impartiality. More precisely, the different assumptions will allow us to introduce possible difference among the States, regarding a possible bias in favor of a candidate and the dispersion of the vote in a given State. However, at the national level, the effects of a bias will be neutral. The same reasoning exists in the work of Feix, Lepelley, Merlin and Rouet [9] where a rescaled and biased IAC model (BRIAC) is suggested and implemented. But in our case, the local parameters will be calibrated with an electoral data study. To connect these different models to the GIAC model, we can say that Expression (1) allows to recover IC 1 and IAC model for specific values of f(p): respectively f(p) = δ(p − 2 ), that 1 is to say f(p) is a Dirac Delta distribution function for which p is equal to ( 2 ) for all elections, and f(p) = 1. Equipped with a range of assumptions, we will compute the majority efficiency according to different voting rules for the Elec- toral College, more precisely according to different methods of seat allocation among the US States. In order to study the goodness of seats division, we turn attention to four particular systems. There exists a host of methods to allocate α seats between union members. The simplest ones come from the ni voting rule α family : mandates received by each federation member are proportional to ni , with ni the population of the State i and α a constant usually such as α ∈ [0; 1]. 1 The first studied case is the actual U.S. system ni + 2 , denoted by USR: State i receives a number of seats in part proportional to its population (shar- ing out the 435 Representatives) but with a premium of two (the two senators) per state. District of Columbia holds three seats. There is so 538 Electors in our simulations, except for 1960 and 1950 (their number is respectively equal 3 1 to 535 and 531 ). The second system is the proportional system ni , denoted by PR: State i receives a number of seats proportional to its population4. The 0 third system is the federal rule system ni , denoted by FR: each State of federation receives the same number of seats. Finally, the fourth allocation rule 1 2 is the square root system ni , denoted by SRR: State i receives a number of seats proportional to the square root of its population. These comparisons will establish whether there exist differences according to the probabilistic models, and whether an optimal voting rules emerges or not. Section 2 presents stylized facts on U.S. presidential electoral data. Section 3 presents IC adaptations, and their implementations. Section 4 presents IAC adaptations, and their implementations. Section 5 is devoted to the conclusion.

3D.C. is not represented in the Electoral College of 1960, and States of Hawaii and Alaska join U.S.A federation in 1959. 4The used rounding oﬀ method is the Hill method.

5 2 Stylized facts and framework. 2.1 Stylized facts Before presenting results of statistical explorations, we will introduce some notations. Let Dit be the number of votes received by the Democratic candidate in the State i for election of year t. Symmetrically, let Rit be the number of votes received by the Republican candidate in the State i for election at the year t. Thus, the score vit, in percent, of the Democratic candidate in the duel with the Republican candidate in the State i for the election at the year t is given by Expression (2).

100.Dit vit = (2) Dit + Rit So, the national result of a presidential election at the date t is expressed by P|N | Dit i v¯.t = P|N | · 100, with N = {1,...,N} the set of States. In our present (Dit+Rit) i case, |N | = 51 corresponds to the 50 States and the District of Columbia forming n 0 o the federation of the United States. T = {1,...,T } and T = 1,...,T are the two used sets of considered elections. T is composed by U.S. presidential 0 elections since 1948, it represents the recent period, whereas T is composed by U.S. presidential elections since 1868. In each State i, there are ρit registered electors at the date t and a population nit. In a first part, we shall study in a descriptive way electoral data. In the second, we shall analytically approach the problem of the agreement of the empirical density to a theoretical density. The chosen data are the State-level popular vote of U.S. presidential election. To obtain these last ones, we built a database from the information diffused in the recent book of Archer & al. [2]. Furthermore, we need to make an usual simplification on the US presidential election system : we make the assumption that the voters have simply to choose between a Republican candidate and a Democratic candidate, and that there is no abstention. In other words, we focus on the Democrat-Republican duel.

2.1.1 Descriptive analysis : State-level voting bias As explained by Regenwetter, Grofman, Marley and Tsetlin in their book [26], the fundamental purpose of a behavioral theory is the development of more realistic models and the statistical evaluation of such models. In our case, the first purpose is to be able to find a probabilistic model which better reflects the reality than the traditional IC and IAC hypothesis, but which still capture particular instance of the veil of ignorance hypothesis. As the development of a new model needs to describe as best as possible the reality of the electoral behavior, the first natural stage of the work is to acquaint with the data in a descriptive manner. The first characteristic to be studied in a probabilistic model is the central tendency, and more exactly here the average. In our case, this indicator allows us to calculate the voting bias of a State in long period. We

6 can deﬁne the bias as the average voting position of a State in comparison with an neutral benchmarck (see Appendix II for explanations on computations). So empirically, 23 States show a bias5 superior to 5 % since 1868, positive or negative, among which 14 are very negative. States in favor of the Republicans are notably Vermont, (-12.91 %), Nebraska (-9.79 %) and Kansas (-9.46 %). If we take into account only the recent period from 1948, the bias falls for the Vermont to -3.16 %. The bias of two other States increase over the recent period, reaching respectively -13.44 % and -10.34 %. So these two States are the most marked Republican. States historically in favor of the democrats are Mississippi (20.07 %), the South Carolina (17.99 %) and Georgia (14.71 %). However in the three cases, the bias shows a strong decline over the recent period: we obtain respectively -1 %, -1.94 % and 2.21 %.

2.1.2 Inference analysis In this part we use framework such that proposed by authors such that Gelman, King and Boscardin ([13]). This paper examines the two-tiers decision procedure of the US presidential elections too but it focuses on power index. It is defined as the probability to be decisive in such an electoral system (see Banzhaf [3]). In order to evaluate Banzhaf power index, the authors use Monte Carlo simulations and thus they need to specify the probability distribution nature. In this way, they compare two kinds of estimated probability that a State is decisive : computed based on frequency of simulations versus an estimate from a fitted beta distribution. The estimates are quite similar, and so they use the estimates based on the beta approximation. We will use this method in order to see if it is effectively realistic to find a probability law which approximates in a very correct way the empirical distribution of the State-level votes. The set 0 of elections used for implementation is T .

Tests of goodness of ﬁt There are two implementation steps of goodness of ﬁt test : the Shapiro-Wilk for Gaussian distribution, testing an IC model, and the Pearson’s chi-square test for the uniform distribution, testing the IAC assumption.

The normality test : Shapiro-Wilk test Why to test the normality of the distribution of vit while the hypothesis IC seems very unrealistic ? In fact, the Impartial Culture model assumes that each voter picks his preference randomly among the possible typical preference according to the uniform prob- 1 ability distribution. In the US presidential case, each voter has a probability 2 1 to cast his vote in favor of Democratic candidate, and a probability 2 to cast his vote for Republican candidate. So, the distribution of vit is a binomial law, and as the number of voters is large in each State, the distribution should tend 1 1 to a normal law, of mean ( ) and standard-deviation equal to σit = √ , with 2 2 nit nit the population in the State i at the election t. However, as we have already

5We consider here the absolute state-level bias (see Appendix II, see [17]).

7 previously seen, the average is rarely equal to 0.5. The interest to test the hypothesis of normality thus is to modify the classical Impartial Culture model. A way to adapt this model is to say that the Bernoulli trial is not any more with a probability of success of 50 % but with a probability pi which corresponds in an empirical way to the estimated average of vit. The results of the tests applied to every State series, are the following : 38 States on 50 follow a distribution which can be approximated by a normal law in an acceptable way, but with a possible bias and rescaling. So, even if we cannot here conclude with certainty that the normal law is the perfect density to make a simulation on all the States, we can see that it is an acceptable distribution most of time.

Pearson’s chi-square test Pearson’s chi-square test is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. We made this test by State with the hypothesis of uniform distribution. The results are : 41 States can be considered as following a truncated uniform law6. However the application of these tests is very delicate because of the low number of observations. It is also due to the nature of the test statistic nature. To conclude on the tests of goodness of ﬁt, the distributions of the various vit do not follow exactly theoretical predeﬁned distribution. But, it is obvi- ous that the normal law, corresponding to a hypothesis of Impartial Culture, and uniform distribution, corresponding to an Impartial Anonymous Culture assumption, are possible probability distribution to implement Monte Carlo simulations. It seems surprising that normal and uniform laws are both acceptable. But rescaling (with a higher standard-deviation for normal distribution and reduction of interval for uniform law) leads to same reconciliation between respective density functions7. Moreover, this kind of test, in particular the Pear- son’s chi-square test, remains delicate to handle in the small sample case. So the results of these tests are delicate to use and graphical analysis remains the best tool. It’s why we apply then the procedure used by Gelman, King and Boscardin [13] to determine the ”proxy law” : QQ-Plots.

QQ-Plots If the empirical distribution coincides with the theoretical law then the graphical points have to follow the ascending diagonal8. Proximity of this diagonal is a proxy of the similitude degree between empirical distribution and testing law. We tested the graph Q-Q for the logistical, uniform, beta, Gaussian and log-normal distribution. For each law, the ﬁrst two moments are beforehand estimated from the samples of every State. For the uniform law, the theoretical law of comparison is a uniform law restricted respectively by the maximum and the minimum observed. The obtained plots have the shape shown in ﬁgure 1.

6 The uniform distribution is truncated by the observed minimum and maximum of vit. 735 States respect the normal and uniform conditions 8In our case, the graphs are not centered on the diagonal. That explains the various positions of this line.

8 Figure 1: Example of QQ-Plots (Texas).

9 What stands out from the reading of graphs is that the closest probability laws for the most part of States is the uniform law and, slightly less, the normal distribution. However no law seems to agree in a strict way for all the States. To conclude with this electoral data study, even if one cannot have certainty on the exact nature of the probability density of the diﬀerent vit, it is visible that both normal law, corresponding to a hypothesis of Impartial Culture, and uniform distribution, corresponding to an Impartial Anonymous Culture assumption, are the two more adequate distributions. Indeed, the results obtained by the goodness-of-ﬁt tests are consolidated by the QQ-plots. Such that distribution can legitimately be advanced in the Monte Carlo simulations presented after- ward.

2.2 The A priori framework. Computer simulations are very often used in the social choice literature in order to determine the frequencies of paradoxes. This method is very useful when a problem cannot be solved analytically. In their Book, Regenwetter and al. [26] explore a Bayesian inference framework led by random sampling for the assessment of the Condorcet efficiency. Their approach contrasts with the commonly studied case of drawing random samples from the Impartial (Anonymous) Cul- ture. Their tool put virtually no constraint on the possible distributions of individual preferences. But, it is maybe interesting to extend models in order to keep the neutrality of the traditional probabilistic model and at the same time take into account restrictions from electoral data. The two classical models in social choice theory are IC and IAC model. As said previously, the Impartial Culture model assumes that each voter picks his preference randomly among the possible preference orderings. Thus, a voting situation is distributed according to normal probability distribution. According to the IAC assumption, the percentage of popular votes for the Democrats follows a uniform law limited between 0 and 1. Before implementing the different a priori models, we have to precise some facts about them. We will consider two-party elections, and assume that all the individual in State i vote. Differences in register voters and/or turnout are neglected. Moreover, under IC and IAC assumptions, in the computation of popular votes for each candidate (Dit and Rit), we transpose directly the simulated duel ratio, vit, without taking into account an eventual calibrated turnout ratio or third candidate share (see [?]). Why these simplifications ? In order to respect the three desirable properties : easy to implement, easy to interpret and respect of neutrality. Therefore, so as to validate extension steps, we have beforehand to check that the study of the seat distribution under IC and IAC in Electoral College, according to the criterion of majority efficiency, leads to well-known or under- standable recommendations, i.e. to check wether “pure” a priori simulations match the computations made in the literature, particularly in the power index area. In order to study the goodness of seat divisions, we turn attention to four particular systems enumered previouly (USR, PR, FR, SRR). At last, in

10 order to manage the rounding problem, we currently use the Hill method (see [15, 18]). Seats in the House of Representatives are allocated by this formula. Let be qe the electoral quotient deﬁned by the formula (3).

P|N | n qe = i=1 i (3) h e Where h denotes the number seats to allocate. If we divide ni by q , we obtained the theoretical number of seats for the State i. But the problem is that this number is not always an integer. Thus, how to round integer to have a total number of integer seats exactly equal to h? In this way, we search a common divisor q, in the same order of qe, so that when it is divided into each State population and resulting quotient are rounded at the geometric mean, the total number of seats will sum h. In the case of the study of the square root rule allocation, the number√ of seats in the State i is no longer given by rounding ni ni 9 of the division q but by q . Finally, we analyze the majority eﬃciency for 6 election conﬁgurations given by post WWII censuses, from 1950 to 200010.

3 IC probabilistic extensions. 3.1 Impartial Culture (IC). This model seems not very realistic because, beyond the fact that every elec- 1 tor has the same probability to vote for each candidate, here 2 , this Culture describes mainly tied elections (a very small standard deviation around an ex- pectation of 50%). The election issue is drawn by a truncated normal law. In our case, the used standard-deviation is equal to σIC = √1 . We can see it 2 nit IC that σit is diﬀerent for each State and for each demographic census. They are 1 no bias, so the mean used in the simulations is 2 . Results are presented in the Table 2 11.

Table 2: The likelihood of the RP according to IC model Census USR PR FR SRR 1950 0.2185 0.2290 0.2458 0.2036 1960 0.2225 0.2354 0.2514 0.2067 1970 0.2244 0.2358 0.2538 0.2131 1980 0.2196 0.2344 0.2457 0.2054 1990 0.2177 0.2331 0.2407 0.1972 2000 0.2181 0.2310 0.2429 0.1991

P|N | √ ni 9 e i=1 So, the electoral quotient is equal to q = h 10In other words, the parameters of the simulations are calibrated on the sample T . 11For each simulation, computations are made from 10000 draws. We present only one simulation per census data.

11 The first observation is the compatibility of our results with the common knowledge : in a two-tiers voting system, number of seats received by a member has to be proportional to the square root of its population (see Penrose [24][25] for the description the square root law). Secondly, USR is systematically the second best choice of distribution; FR seems to be less adapted according to the majority efficiency criteria. Moreover, taking into consideration of differences among populations of States engenders an increase of the paradox probability compared with a theoretical situation where every State has the same population size (see [9]) . Although results seem to check known recommendations, reader can be ques- tioning on the fact that used standard-deviation is too small in regards to the electoral data (see figure 2). It is the subject of the next implementation.

3.2 Rescaled Impartial Culture (RIC). For this second model, the used dispersion, more precisely standard-deviation, no longer depends only on the State population but it is historically calibrated on the period 1948-200412 (see expression (4)). v u |T | |N | u 1 X X nit 2 σ¯ = t (v − v¯ ) (4) .. |T | |N | − 1 P|T | P|N | it .. t=1 i=1 t=1 i=1 nit

wherev ¯.. denotes the long run national weighted average, on the period 1948- 2004 and for all States, of vit.σ ¯.. is equal to 0.08988. With this first extension, we search to take into account that the IC-standard dispersion is too low with regards to the electoral data. Indeed, according to the basic definition of the IC assumption, the standard error depends only on the probability to choose an issue (i.e. 50%) and on the size of the population. However, exogenous factors, such as the evolution of the population (in terms of political tendency and inter- State migrations) and the socioeconomic situation, could influence the voters’ choice and could thus increase the empirical standard error. In other words, rescaling, for IC, is a way to reduce the probability for a State to have a tied election result, i.e. to be pivot in the national election outcome. For Rescaled Impartial Culture (RIC), dispersion is the same for each State and indicates a national dispersion of elections outcomes. Results of Monte Carlo simulations using N (0.5;σ ¯..) are given by Table 3.

12It is a common used period, see for example the work of Gelman et alii [11].

12 Table 3: The likelihood of the RP according to RIC model Census USR PR FR SRR 1950 0.1991 0.1970 0.3107 0.2247 1960 0.1986 0.1957 0.3123 0.2263 1970 0.1974 0.1950 0.3098 0.2231 1980 0.1997 0.1980 0.3092 0.2273 1990 0.1988 0.1961 0.3128 0.2262 2000 0.1994 0.1962 0.3142 0.2274

Firstly, simulated frequencies of paradox indicate there exists a shift between PR, accompanied by the USR, and the two other ones. Equalizing seats between States is clearly the worst solution. Moreover, there is no long run diﬀerence between USR and PR. These observations leads to conclude that rescaling engenders a new recommendation of seats division : the best choice seems to be the proportional, or quasi-proportional, allocation. It meets conclusions of Gel- man et alii [11] : when dispersion of votes does not depend on population size and so, when extreme voting situations (60 % of electors for one candidate for example) are more plausible, SRR no longer holds. In order to assess the rescaling at the national level, we can observe the empirical distribution of simulated vit under RIC (see ﬁgure 2). We can observe the more realistic dispersion of the duel outcome distribution at the national level in comparison with the IC case.

3.3 Biased Rescaled Impartial Culture of type 1 (BRIC1). In his paper, Kirsh (see [17]) builds a model named collective bias model. It stipulates that there can exist a kind of common opinion in the society from which individual voters may deviate but to which they agree in the mean. This kind of bias can exist for different reasons : influence of religious/ideological groups, influence of media, etc... We find here the same idea that some States are “Democratic”, other are “Republican”. We called the bias used in Biased Rescaled Impartial Culture (BRIC1) simulation “relative bias”. It corresponds to the State-level voting position compared with the national average position13. It is computed by the formula (5).

˜ 1 X 1 X X Bi. = (vit − 0.5) − P P nit · (vit − 0.5) (5) |T | nit t∈T i∈N t∈T i∈N t∈T

We use this kind of collective bias because it holds an useful property : its population weighted average is null over the period14. That means that the

13see Appendix II 14However, for a given set of population census. At time t, the P n B . When the i∈N it i.

13 national position is neutral, respecting the a priori framework. In other words, 1 the national mean probability to choose each candidate has to be equal to 2 . In case of BRIC, a diﬀerent bias inﬂuences the individual choice in each State. Therefore, we make the following assumption : in the State i, individual voter 1 ˜ picks his choice with probability 2 + Bi. for the Democratic candidate and 1 ˜ 2 − Bi. for the Republican candidate. With this assumption, the probability to be pivot results from the combination of relative bias and national dispersion. If the bias is small compared with the rescaling, the probability to be pivot is higher than if the bias is high compared with the dispersion. Before to present results, we have to notice that rescaling used here is the same than in RIC model,σ ¯... Results of simulations are presented in Table 4.

Table 4: The likelihood of the RP according to BRIC1 model Census USR PR FR SRR 1950 0.2230 0.2148 0.4137 0.2679 1960 0.2206 0.2139 0.4096 0.2599 1970 0.2152 0.2110 0.3931 0.2510 1980 0.2143 0.2156 0.3580 0.2452 1990 0.2110 0.2102 0.3554 0.2471 2000 0.2123 0.2116 0.3508 0.2444

Results are very close to those presented for RIC model. We can see an gen- eral increase of probability paradoxes, particularly for the FR allocation, which stays the worst repartition method for seats allocation. So, the introduction of State-level common opinions does not disrupt the recommendation : the use of PR, or USR, is the best alternative in the Electoral College. In order to review the national simulated repartition of votes between Demo- cratic and Republican candidates, we can see ﬁgure 2.

3.4 Biased Rescaled Impartial Culture of type 2 (BRIC2). The next step of simulation is to take into account the State-level rescaling. The similarity degree of election outcomes among several elections, or the dispersion, could come from a State-level behavior. We can interpret this extension as a diﬀerentiation between militant States, which have always the same electoral behavior, and the no militant State, which can change more frequently their voting position. Thus, State-level dispersion can be see here as a degree of simulated mean is not null (and only slightly deviate), we unbiased, for each election, our simulations in order to check the neutrality. It is particulary engender by the fact that the bias is a time average whereas population not.

14 election outcome predictability. The standard-deviation is computed for Biased Rescaled Impartial Culture (BRIC2) model according to the Expression (6). v u |T | u 1 X 2 σ¯ = t (v − v¯ ) (6) i. |T | − 1 it i. t=1 1 P|T | ˜ 15 Withv ¯i. = |T | t=1 vit. Bias used is Bi. as in the BRIC1 model . Results are given by the Table 5.

Table 5: The likelihood of the RP according to BRIC2 model Census USR PR FR SRR 1950 0.2222 0.2198 0.3618 0.2620 1960 0.2170 0.2174 0.3669 0.2593 1970 0.2290 0.2183 0.3663 0.2716 1980 0.2271 0.2214 0.3646 0.2618 1990 0.2235 0.2218 0.3693 0.2609 2000 0.2235 0.2212 0.3682 0.2564

The study of the result table leads to similar conclusions in comparison with RIC and BRIC1 models. In order to review the national simulated repartition of votes between Democratic and Republican candidates, we can see ﬁgure 2. Across the four preceding probabilistic assumptions (IC, RIC, BRIC1, BRIC2), we can conclude that, in a priori framework, introduction of relative bias and rescaling leads to one single recommendation : in order the reduce the probability of the referendum paradox, the allocation of seats in the Electoral College should be accomplished according a proportional, or quasi-proportional, allocation.

4 IAC probabilistic extensions.

For this part, we will present the same range of models as in the preceding part. But here, we introduce the notion of anonymity. For discussion on this property, reader can refer to Gerhlein and Fishburn [14] or works of Kuga and Nagatani [16]. Under IC, each preference proﬁle has the same probability to occur. Under IAC, we stipulate now that each anonymous preference proﬁle, or voting situation, has the same probability to occur.

15State-level bias and dispersion parameter are given in Appendix III. Again a factor ε may be use to center when the results of the simulation for each census.

15 Figure 2: Simulated nationalv ¯.t empirical distribution, IC-type assumptions (2000 census data) 1stquantile median 3rdquantile 1stquantile median 3rdquantile 0.4999 0.5000 0.5001 0.4594 0.5000 0.5406

1stquantile mean 3rdquantile 1stquantile meanmedian 3rdquantile 0.4596 0.5000 0.5404 0.4636 0.5000 0.5365

16 4.1 Impartial Anonymous Culture (IAC). According to this model, there is the same probability that an election in a State ends up in 99 % of votes for the Democrats or 99 % for the Republicans. This seems to be little realistic but remains however very useful when we do not precisely know electoral preferences of a State. Thus the election issue is selected according to a uniform law on the interval [0; 1]independently for each state. Table 6 shows results of the simulations.

Table 6: The likelihood of the RP according to IAC model Census USR PR FR SRR 1950 0.1718 0.1690 0.2922 0.1984 1960 0.1731 0.1705 0.2931 0.1970 1970 0.1727 0.1705 0.2907 0.1977 1980 0.1751 0.1715 0.2914 0.1985 1990 0.1767 0.1716 0.3003 0.2069 2000 0.1693 0.1688 0.3024 0.2088

We can see that the standard recommendation under IAC hypothesis, in the case of index power literature, is recovered: the best allocation system of seats is the PR. Nevertheless, there exists no long run diﬀerence between PR and USR. The national empirical distribution ofv ¯.t under IAC (see Figure 3) shows a higher dispersion than IC case. Compared with the IC, the probability of paradox is lower for USR and PR systems, slighlty higher for SRR and by far larger for FR system.

4.2 Rescaled Impartial Anonymous Culture (RIAC). Contrarily to the RIC model, the aim of rescaling for Rescaled Impartial Anony- mous Culture (RIAC) is to reduce the range of possible election outcome. In- deed, it seems unlikely that one of the two candidates receive only 1% of ballots or 99% of them. So, the dispersion is now calibrated by the expression (7).

1 P 1 P ! i∈N Maxi∈N ,t∈T vit − i∈N Mini∈N ,t∈T vit ∆¯ = |N | |N | (7) .. 2

1 ¯ 1 ¯ So, vit is drawn uniformly on the interval 2 − ∆..; 2 + ∆.. . In our case, ∆¯ .. = 0.1557. The dispersion is assumed to be the same for all States. Results are displayed on Table 7.

17 Table 7: The likelihood of the RP according to RIAC model Census USR PR FR SRR 1950 0.1712 0.1674 0.3039 0.2020 1960 0.1707 0.1651 0.3049 0.2004 1970 0.1727 0.1682 0.2934 0.1994 1980 0.1696 0.1657 0.2888 0.1990 1990 0.1672 0.1650 0.2990 0.2005 2000 0.1673 0.1637 0.2991 0.2028

Again, the PR allocation emerges as the best voting rule. We can again see that there is no long run diﬀerence between the USR and the PR systems according to the majority eﬃciency criterion. The empirical national distribution ofv ¯.t under RIAC allows to appreciate the rescaling compared with the IAC assumption (see Figure 3).

4.3 Biased Rescaled Impartial Anonymous Culture of type 1 (BRIAC1). As for BRIC1, we introduce relative bias in these simulations. The used bias is B˜i. deﬁned in BRIC1 paragraph. Contrary to BRIC range assumptions, the bias is here supposed to be inﬂuential no longer on the individual voting decisions but on the the overall decisions of the State. Rescaling is the same as for RIAC. So the duel election result for the State i, v , is drawn uniformly on the interval h i it 1 ˜ ¯ 1 ˜ ¯ 16 2 + Bi. − ∆..; 2 + Bi. + ∆.. . We can study the results in the Table 8.

Table 8: The likelihood of the RP according to BRIAC1 model Census USR PR FR SRR 1950 0.1918 0.1769 0.4234 0.2509 1960 0.1844 0.1741 0.4005 0.2377 1970 0.1842 0.1765 0.3597 0.2243 1980 0.1809 0.1772 0.3246 0.2172 1990 0.1768 0.1746 0.3189 0.2160 2000 0.1781 0.1734 0.3171 0.2140

The PR still stays the best voting rule, going with an overall increase of the referendum paradox frequency. The empirical national distribution ofv ¯.t under BRIAC1 shows that introduction of relative bias does not disrupt the shape of

16An extra ε parameter may be used to center on 0.5 the results of the simulation for each census.

18 the distribution, which stays very similar compared with the RIAC empirical distribution (see Figure 3).

4.4 Biased Rescaled Impartial Anonymous Culture of type 2 (BRIAC2). The rescaling is now speciﬁc to each State. The probability for a State to be pivotal in the Electoral College is the state-level combination relative bias - dispersion. The dispersion is now deﬁned by the Expression (8).

Max v − Min v ∆¯ = t∈T it t∈T it (8) i. 2 The used bias is again B˜ 17. So the duel election result for the State i, i. h i 1 ˜ ¯ 1 ˜ ¯ vit, is drawn uniformly on the interval 2 + Bi. − ∆i.; 2 + Bi. + ∆i. , with an eventual ε parameter per census. Results are presented in Table 9.

Table 9: The likelihood of the RP according to BRIAC2 model Census USR PR FR SRR 1950 0.1966 0.1969 0.3474 0.2254 1960 0.1914 0.1957 0.3582 0.2242 1970 0.2005 0.1990 0.3567 0.2280 1980 0.1886 0.1972 0.3072 0.2094 1990 0.1895 0.1939 0.3025 0.2075 2000 0.1887 0.1952 0.2949 0.2054

The best system is now alternatively the USR or the PR. Basically, both seem indistinguistable. So, the introduction of state-level rescaling does not signiﬁcantly change the IAC point of view. The empirical national distribution under BRIAC2 is given by ﬁgure 3.

5 Concluding remarks.

The main aim of this work was to assess the Electoral College system efficiency according to the majority efficiency criterion. Through this paper, we implement a range of probabilistic model taking into account some stylized facts from electoral data study. Section 2 gives an important information : the two theoretical distributions which seems to be more suitable in order to approxi- mate the different State-level empirical distribution of vit are the normal and the uniform laws. 17State-level bias and dispersion parameter are given in Appendix III

19 Figure 3: Simulated nationalv ¯.t empirical distribution, IAC-type assumptions (2000 Census data) 1stquantile median 3rdquantile 1stquantile median 3rdquantile 0.3813 0.5000 0.6187 0.4646 0.5000 0.5353

1stquantile mean 3rdquantile 1stquantile meanmedian 3rdquantile 0.4644 0.5000 0.5357 0.4655 0.5000 0.5344

20 The main aspect of our work has been to develop a range of probabilistic model. Firstly, we relax the traditional dispersion assumption. According to IC, the dispersion is very low and depends only the number of voters. For IAC, the dispersion is too large. Rescaling was made, in order to match stylized facts from the electoral data. Introduction of bias indicates that voter or election result can be influenced in different ways (media, religion, social and economic environment, ...) inducing a gap between the collective opinion and the neutral voting position. However, the state-level bias can be justifiable but we think that in a long run horizon, the national voting position has to be neutral (each candidate owns the same probability to be elected). So, the different probabilistic models in this paper proposed respect our three conditions : easy to implement, easy to interpret and keeping some degree of neutrality. We can add that these different models can be directly implemented on different social binary choice to which the final decision is taking by a representative college. In our case, we have choose a classical example: U.S. presidential election system. This work, in the majority efficiency framework, has permitted to obtain a important result : if we stipulate that the collective voting dispersion is not link to the population size (i.e. IC case), the best voting rule is the PR. Moreover, the introduction of a small premium by State does not disrupt this recommendation. Firstly, IC an IAC assumptions give the same recommendation compared with the literature of index power : square root allocation of seats for IC and proportional division of Electoral College for IAC. Secondly, introduction of national rescaling (RIC, RIAC) allows to match the two types of probabilistic assumption : the PR, or USR, allocation is the best system according to the majority efficiency criterion. This matching is mainly due to the convergence between national distribution of Democratic-Republican duel results (see Figure 2 and 3) : for RIC, we increase the dispersion, and consequently reduce for a considered State the probability to be pivot in the national result ; for RIAC the rescaling engenders opposite consequences. The matching and the proportional recommendation stand for each extension : introduction of relative collective bias (BRIC1, BRIAC1) and state-level rescaling (BRIC2, BRIAC2). We have to add a remark: there exists no long run difference between the proportional (PR) and the actual U.S. system (USR) of seats allocation. These observations induce several conclusions. Firstly, introduction of anonymity in the preference scheme does not disrupt analysis. Secondly, according to a a priori framework, the standard recommendation of IC seems to be erroneous. The traditional IAC recommendation according to which in a two-tier voting system the Represen- tatives have to be distributed according to a proportional rule seems to be the best alternative. In other words, in a priori framework, use of proportional allocation of seats of Electoral College is the best alternative in order to minimize the probability of occurrence of referendum paradox, and maximize the majority efficiency of the voting system. Introduction of a premium of two seats by State does not disrupt this result.

21 References

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23 A Appendix I : Goodness-of-ﬁt tests A.1 The normality test : Shapiro-Wilk test In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample x1, ..., xN came from a normally distributed population. This test is particularly powerful for short samples. The test statistic is given by (9).

2 PN i=1 aixi W = (9) PN 2 i=1 (xi − x¯) th with xi is the i smallest number in the sample,x ¯ the sample average and the ai constants are given by (10).

mT V −1 (ai, ..., aN ) = 1 (10) (mT V −1V −1m) 2 T where m = (m1, ..., mN ) and m1, ..., mN are the expected values of the order statistics of an identical and independent distributed sample from the standard normal distribution, and V is the covariance matrix of those order statistics. Pearson’s chi-square test Pearson’s chi-square test is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. It tests a null hypothesis that the relative occurrence frequencies of observed events follow a specified frequency distribution. The events are assumed to be independent and have the same distribution, and the outcomes of each event must be mutually exclusive. Chi-square is calculated by finding the difference between each observed and theoretical frequency for each possible outcome, squaring them, dividing each by the theoretical frequency, and taking the sum of the results (see expression (11) ).

N 2 X (Oi − Ei) χ2 = (11) E i=1 i

where Oi, an observed frequency and Ei, an expected (theoretical) frequency, asserted by the null hypothesis.

B Appendix II : Computation of collective bias.

We can deﬁne the State-level absolute bias as the State-level average voting position compared with the neutral voting position (i.e. 0.5). It is computed according the formula (12).

1 X B¯ = (v − 0.5) (12) i. |T | it t∈T

24 In an other hand, we can deﬁned the national collective bias as in expression (13).

¯ 1 X X ¯ B.. = P P nit · Bi. (13) nit i∈N t∈T i∈N t∈T Finally, we deﬁne the relative collective bias as the relative position of a considered State compared with the national average voting position. It can be compute as in expression (14).

B˜i. = B¯i. − B¯.. (14) ˜ P P ˜ One one the most important property of Bi.is that i∈N t∈T nit · Bi. = 0. It signiﬁes that the average relative position on the whole federation is neutral.

C Appendix III : State-level bias and dispersion parameters.

Bias and dispersion parameters used for Monte Carlo simulations are presented in the Table and 11. The sample used to compute parameters are T .

Table 10: Dispersion and bias used in BRI(A)C2 simulations - part 1. State Bei. σi. ∆i. Albama -0.0006 0.1414 0.2733 Alaska -0.0787 0.1906 0.1748 Arizona -0.0634 0.0699 0.1166 Arkansas 0.0327 0.1079 0.2187 California 0.0072 0.0637 0.0932 Colorado -0.0380 0.0725 0.1301 Connecticut 0.0123 0.0868 0.1581 Delaware 0.0120 0.0641 0.1073 District of Columbia 0.3715 0.3953 0.0608 Florida -0.0328 0.0793 0.1567 Georgia 0.0332 0.1417 0.2607 Hawaii 0.0642 0.2468 0.2062 Idaho -0.1194 0.0793 0.1234 Illinois 0.0119 0.0649 0.0962 Indiana -0.0575 0.0544 0.1136 Iowa -0.0081 0.0696 0.1309 Kansas -0.0923 0.0662 0.1210 Kentucky -0.0110 0.0741 0.1440 Louisiana 0.0005 0.0933 0.1749 Maine -0.0096 0.1085 0.1986 Maryland 0.0269 0.0745 0.1379

25 Table 11: Dispersion and bias used in BRI(A)C2 simulations - part 2. State Bei. σi. ∆i. Massachusetts 0.0897 0.0945 0.1799 Michigan 0.0100 0.0671 0.1319 Minnesota 0.0447 0.0544 0.0978 Mississippi 0.0011 0.2028 0.4226 Missouri 0.0111 0.0660 0.1317 Montana -0.0389 0.0686 0.1145 Nebraska -0.1234 0.0656 0.1209 Nevada -0.0406 0.0806 0.1426 New Hampshire -0.0471 0.0927 0.1628 New Jersey -0.0060 0.0872 0.1568 New Mexico -0.0098 0.0668 0.1104 New York 0.0407 0.0887 0.1493 North Carolina -0.0107 0.0842 0.1730 North Dakota -0.0855 0.0762 0.1474 Ohio -0.0150 0.0621 0.1203 Oklahoma -0.0647 0.0940 0.1909 Oregon 0.0039 0.0612 0.1239 Pennsylvania 0.0133 0.0592 0.1267 Rhode Island 0.0979 0.1001 0.1957 South Carolina -0.0083 0.1370 0.2906 South Dakota -0.0573 0.0652 0.1244 Tennessee -0.0112 0.0704 0.1330 Texas -0.0157 0.1026 0.1983 Utah -0.1273 0.0988 0.1641 Vermont -0.0205 0.1209 0.1924 Virginia -0.0420 0.0608 0.1155 Washington 0.0146 0.0623 0.1097 West Virginie 0.0320 0.0773 0.1578 Wisconsin 0.0025 0.0608 0.1207 Wyoming -0.0982 0.0853 0.1398