When a plurality is good enough JEFFREY C. O’NEILL Abstract. This paper investigates when a runoff election is desirable and when a plurality result is good enough. A runoff election increases the likelihood that the Condorcet winner will be elected but also entails additional costs. The metric for determining whether a runoff election is desirable will be the probability that the winner of the plurality election would win an ensuing runoff. Models of voter behavior are developed that estimate this probability, which are verified with runoff-election data from United States elections. The models allow governments to make more informed choices in creating rules to decide when to hold runoff elections.
1.
Introduction
In electing one person to an office, the simplest and most common method used in the United States is plurality voting. The candidate with the most votes wins regardless of how many votes that candidate receives. If the winning candidate receives a majority of the votes, then the winner is also the Condorcet winner – the candidate who, if she exists beats all other candidates in pairwise contests. However, if the winner receives less than a majority, she may not be the Condorcet winner, and one could argue that the election result does not reliably represent the will of the electorate. For this purposes of this paper, I will presume that electing the Condorcet winner is a laudable objective, although it is certainly disputable (Riker, 1988; Wright, 1989). The closer the winner is to a majority, the more likely she is the Condorcet winner. In the absence of a majority winner, a runoff election can be used to increase the likelihood that the Condorcet winner wins the election. Without a runoff, the Condorcet winner will win the election only if she is the winner of the first election. Assuming that voter preferences do not change between the first and the runoff election, the Condorcet winner will win the runoff election if she is among the top two of the first election. If the Condorcet winner is not among the top two of the plurality election, a runoff will not elect the Condorcet winner, but it will at least elect the better of the top two candidates. Despite the clear advantages of holding a runoff election when there is no majority winner, the advantages must be weighed against the costs. Holding a second election is expensive, the campaign season is longer, and voters must take the time to go to the polls a second time. In certain situations, a plurality winner is clearly good enough. For example, when the winner of the plurality election is one vote shy of a majority and the runner-up is far behind the plurality winner. In other situations, it is less clear. The relative merits of plurality and runoff elections have been widely explored. Merrill has run simulations suggesting that runoff elections are much more likely to elect the Condorcet winner than plurality elections (1988). Wright and Riker have noted that Merrill’s study did not take into account the fact that, generally, fewer candidates enter plurality elections (1989). Thus, Merrill’s simulations perhaps overstate the benefits of runoff elections. Fishburn and Brams have compared the relative merits of plurality, runoff, and approval voting elections (1981). More recent research has used a geometric analysis to compare plurality and runoff elections (Rivière, 2004). Rather than considering the relative merits of runoff elections and other
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methods, this paper will presume that it has already been decided to use a runoff and investigate the mechanics of runoff elections. The purpose of this paper is to investigate precisely when a plurality winner is good enough. The metric will be the probability that the winner of the plurality election would win a runoff election. When the plurality winner is one vote shy of a majority and the runner-up is far behind, this probability is close to one, and a runoff election is not necessary because it is extremely unlikely to change the result. Where the top two candidates in the plurality election are nearly tied and well below a majority, this probability is close to the lower bound of one-half. A runoff election would then significantly increase the likelihood of electing the Condorcet winner. The probability that a plurality winner would also win a runoff election is computed from a mathematical model of voter behavior. Two mathematical models are proposed. The first is a simple but intuitively reasonable model. The second model is fancier and hopefully more accurate. To test the accuracy of the two models, I collected data from 762 runoff elections in the United States.1 The analysis is based on two assumptions. The first assumption is that the electorate and individual voter preferences do not change between the first and runoff elections. Under this assumption, a candidate could never receive fewer votes in the runoff than she did in the first election, but this clearly can happen in a real runoff election. New information will arise between the two elections that will inevitably change who votes and the voters’ preferences. The second assumption is that the characteristics and ideologies of the candidates cannot be used to calculate the probability that the plurality winner would win a runoff election. Runoff elections may be affected by the race, gender, and incumbency of the candidates (Bullock, 1994, 2001, 2003). Further, comparing the political ideologies of the remaining and eliminated candidates can sometimes be used to predict the winner of a runoff. Allowing such subjective information to factor into the decision would be troublesome, because it could allow for manipulation of the political process. One justification for these two assumptions is that the effects caused by the invalidity of these two assumptions will disappear when aggregating over a large number of election results. Governments have come to different conclusions as how to best trade off the costs and benefits of runoff elections. Most elections in the United States take the extreme positions: either no runoff at all or requiring a majority to avoid a runoff. The State of North Carolina takes an in-between position and requires a runoff if the winner of the plurality election receives less than 40% of the vote (Bullock, 1992: 119). Several countries in the Americas use in-between rules (Colomer, 2004: 105–109). Costa Rica and Nicaragua require the plurality winner to obtain 40% of the vote to avoid a runoff. Ecuador requires the plurality winner to have 40% of the vote and a 10% lead on the runner-up. Argentina requires the plurality winner to have (1) 45% of the vote or (2) 40% of the vote and a 10% lead. The analysis in this paper will allow governments to make more informed choices in deciding when a runoff election is necessary. Another rule for determining when a runoff election is necessary is the doublecomplement rule, proposed by Shugart and Taagepeera (1994). If v1 and v 2 represent the percentage of votes received by the plurality winner and the runner-up, respectively, then a runoff is not necessary when 50 − v 2 > 2(50 − v1 ) . This paper will provide greater insight into the double-complement rule.
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2.
Magnitude and Differential
Two important statistics in calculating the likelihood that the plurality winner would win an ensuing runoff are the magnitude of the plurality winner’s share of the vote and the differential between the plurality winner and the runner-up. For convenience, these statistics will be expressed as a percentage of total votes. The closer the plurality winner is to a majority, the more likely she would win an ensuing runoff. The greater the differential between the plurality winner and the runner-up, the more likely the plurality winner would win an ensuing runoff. The magnitude and differential are not independent as the range of values the differential can take depends on the value of the magnitude. In Figure 1, the allowable values of the magnitude and differential fall within the triangle. Let m be the magnitude and d be the differential. Since m − d ≥ 0 , we know that d ≤ m . The vote totals of the top two candidates must be less than 100 so m + (m − d ) ≤ 100 . Thus, it is clear that d ≥ 2m − 100 . Since a candidate receiving a majority of the vote is presumed to win a runoff election, this paper will focus on the triangular region to the left of the dashed line.
differential
100
50
0
50 magnitude
100
Figure 1. The Magnitude-Differential Space.
Given the values of the magnitude and differential, I would like to compute the probability that the plurality winner would win a runoff election. The boundaries of this region are illustrative. When the differential is zero, along the bottom of the triangle, the probability will be at its lower bound of one-half. When the magnitude is 50, along the right side of triangle, the probability will approach its upper bound of one. Thus, at the lower-right corner of the triangle the probability will be discontinuous. Moving epsilon to the left will give a probability of one-half and moving epsilon up and right will give a probability of one. The values in the rest of the triangle will be between one-half and one.
3.
A Simple Model
In order to compute the probability that the plurality winner would win a runoff election, we need to model how votes would be allocated for the runoff election. The first approach is as follows: (1) the two candidates in the runoff keep all their votes from the first election, and (2) the fraction of the remaining votes that go to the plurality winner is modeled as a uniform distribution between 0 and 1. The justifications for this model are its simplicity and that it is intuitively reasonable. In particular, this model allows one candidate in the runoff to receive all
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of the votes from eliminated candidates, which experience dictates, is not an unusual phenomenon. For this simple model, the probability that the plurality winner would win a runoff can be expressed in closed form and is a function of the magnitude, m, and differential, d. This probability is denoted wS (m, d ) . The share of votes from the first election that need to be assigned to the plurality winner and the runner up in the runoff election is 100 − 2m + d , and the plurality winner needs at least 50 − m to win the runoff. The fraction of votes actually received by the plurality winner is modeled as a uniform random variable as described above and denoted as R . Thus, 50 − m wS (m, d ) = Pr R > , 100 − 2m + d which simplifies to 50 − m + d wS (m, d ) = . 100 − 2m + d It is illustrative to draw lines of equi-probability of wS (m, d ) on the magnitude-differential space, and these are shown in Figure 2 along with a scatter plot of the election data. The line between region I and II corresponds to values of the magnitude and differential with a probability of 0.6. Similarly, the lines between the higher numbered regions correspond to probability values of 0.7, 0.8, and 0.9, respectively. 50
V
differential
40
IV
30
III II
20 10 0
I 0
10
20 30 magnitude
40
50
Figure 2. Five Regions for the Simple Model.
The collected runoff-election data can be used to give a rough estimate of the accuracy of this simple model. For the five regions denoted in Figure 2, I calculated how often the plurality winner actually won the runoff. These averages and the 95% confidence intervals are presented in Table 1. One would expect the average for a region to be approximately the average of the upper and lower boundaries of the region. Thus, the averages for the five regions should be approximately 55%, 65%, 75%, 85%, and 95%. The simple model seems to perform well in regions II, III, and IV but less well in regions I and V. Using this model, one can construct rules for runoff elections. For example, if a runoff election is unnecessary when the probability that the plurality winner would win the runoff is 4
greater than 0.9, then a runoff is required only when m < 50 − d / 8 , which corresponds to values of m and d within regions I–IV. If a probability of 0.6 is sufficient, then a runoff is required only when m < 50 − 2d , which corresponds to values of m and d within region I. Shugart and Taagepeera’s double-complement rule requires a runoff when m < 50 − d , which corresponds to a line bisecting the hypotenuse of the triangle. This rule is equivalent to requiring a runoff election when the probability that the plurality winner would win a runoff election is less than 2/3. Table 1. Region Averages and Confidence Intervals for the Three Models.
Region I II III IV V
4.
Simple Model Fancy Model Average 95% CI Average 95% CI 62% 56–68% 59% 52–66% 66% 59–73% 65% 57–72% 78% 70–85% 76% 68–82% 89% 81–95% 85% 77–91% 89% 78–96% 88% 79–94%
A Fancy Model
One obvious criticism of the simple model is that the two candidates in the runoff are equally likely to receive votes from voters who had initially chosen to vote for other candidates. Suppose that in the first election the plurality winner received 40% of the vote, the runner-up received 20% of the vote, and the remaining votes went to other candidates. This information can be used to create a prior distribution for the relative popularity of the plurality winner and runner-up among the entire electorate. Using this information, one would expect that on average the plurality winner would receive 40% 40% +20% or 2/3 of the votes that initially went to other candidates. A fancy model can be constructed by using a prior distribution that incorporates information from the first election rather than using a uniform distribution. After some experimentation, I chose to use prior distribution based on the standard beta distribution (Evans, 2000:34–42). The probability density function of the standard beta distribution has the form x a −1 (1 − x) b −1 f β ( x) = 0 ≤ x ≤ 1; a, b > 0 , B ( a, b) where B(a, b) is the beta function and a and b are two shape parameters. The mode and variance of a beta distribution can easily be computed from the shape parameters a and b: a −1 mode = a+b−2 ab variance = . 2 (a + b) (a + b + 1) To construct a prior distribution, I will start with values for the mode and variance, and from these values I will compute the shape parameters. This can be accomplished through algebraic manipulations and then finding the roots of a cubic equation.
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First, the mode of the prior distribution is derived from the first election. The mode of the prior distribution should be the most likely estimate of the electorate’s support for the plurality winner with respect to the runner up. Since the plurality winner received m votes and the runner up received m − d votes, the mode of the prior should be m . mode = 2m − d Second, the variance of the prior will depend on the total number of votes received by the plurality winner and the runner up. If this total number is close to 100, then we have a very good estimate of the electorate and the variance should be small. If this total number is close to zero, then our estimate is not very good and the variance should be large. As the variance increases, our prior distribution will converge to a uniform distribution. The variance of a uniform distribution is 1/12, and thus 1/12 is our maximum possible variance. As the variance decreases, our prior distribution will converge to a Dirac delta function, and thus the minimum variance is 0. I set the variance to vary linearly along the full range: 1 2m − d variance = 1 − . 12 100 Three examples of beta distributions for different values of m and d are shown in Figure 3.
f(x)
4 3 2 1 0
0
1
x
Figure 3. Three examples of standard beta distributions for m=45 and d=0 (solid), m=45 and d=15 (dashed), and m=45 and d=30 (dotted).
Finally, it is reasonable to expect that the low probability in the tails of the beta distribution will hamstring the model. To remedy this, I mix the beta distribution with a uniform distribution. Denoting the beta distribution as f β (x) and the uniform distribution as f U (x) , the prior distribution for the fancy model is λ f U ( x) + (1 − λ ) f β ( x) , where λ is the weight of the uniform distribution in the mixture. As will be explained in more detail below, a weight of about 0.8 provides the best fit to the election data. Mixtures of the three distributions from Figure 3 with a weight of 0.8 are shown in Figure 4.
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f(x)
1
0
0
1
x
Figure 4. The three beta distributions from Figure 3 after being mixed with a uniform distribution with weight 0.8.
The fancy model is defined as follows: (1) the two candidates in the runoff keep all their votes from the plurality election, and (2) the fraction of the remaining votes that go to the plurality winner is the prior distribution as defined above. With this model, the probability that the plurality winner would win a runoff election, wF (m, d ) , cannot be expressed in closed form, but it is straightforward to compute numerically. The lines of equi-probability for the five regions of the fancy model are shown in Figure 5. Given the high value of the mixture weight, the similarity with Figure 2 is not surprising. 50
V
40
differential
IV 30
III 20
II
10 0
I 0
10
20 30 magnitude
40
50
Figure 5. Five Regions for the Fancy Model.
As above, the runoff data is used to give a rough estimate of the accuracy of the fancy model. For the five regions denoted in Figure 5, I calculated how often the plurality winner won the runoff. These averages and the 95% confidence intervals are also presented in Table 1. While the confidence intervals are too large to make any strong conclusions, the data suggest that the fancy model is a better fit to the data than the simple model. The fancy model performs the worst for region V. This could be a weakness in the model or indicate an “underdog” effect where voters on the fence are more likely to support the weaker candidate. The models proposed in this paper intentionally avoid attempting to model these kinds of effects.
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5.
Comparing the Two Models
Now that the simple and fancy models are defined, I will compare the models by seeing how well they predict the outcomes of the 762 actual runoff elections. The models can be used to predict two related statistics of the runoff election. First, the models will be compared to see how well they predict the number of votes received by the plurality winner in the runoff election. Second, the models will be compared to see how well they predict whether the plurality winner wins the runoff election. 5.1.
Vote-Based Likelihoods
In the first election, the plurality winner receives m votes, the runner up receives m − d votes, and other candidates receive the remaining 100 − 2m + d votes. The simple and fancy models predict what fraction of these 100 − 2m + d votes go to the plurality winner. The share of votes received by the plurality winner in the runoff election will range from m (when the runner up receives all of the remaining votes) to 100 − m + d (when the plurality winner receives all of the remaining votes). To evaluate the performance of the two proposed models, we can compute the likelihood that the two models produced the outcomes in the 762 collected elections. According to the simple model, each of the outcomes in this range is equally likely, since the fraction of votes that go to the plurality winner is a uniform distribution. For election i , the magnitude and differential of the plurality election are denoted as mi and d i , and the share of the vote received by the plurality winner in the runoff is denoted as xi . The fraction of the remaining votes received by the plurality winner is x i − mi , 100 − 2mi + d i and the likelihood function for the simple model is thus N N x i − mi = ∏ 1 = 1 , l S ( x) = ∏ f U i =1 100 − 2mi + d i i =1 where N is the number of elections, and f U () represents the probability density function of a uniform distribution over [0, 1]. For the fancy model, the fraction of votes received by the plurality winner is a mixture of a uniform distribution and a beta distribution: λf U ( x) + (1 − λ ) f β ( x) . The likelihood function for the fancy model is thus N x i − mi , l F ( x) = ∏ λ + (1 − λ ) f β i =1 100 − 2mi + d i where as above the mixture weight is set to 0.8 (how this was chosen is discussed in the next section). With one complication, it is now straightforward to compute the likelihoods that the simple and fancy models predict the plurality winner’s share of the vote in the runoff election. The simple and fancy models were based on the premises that the electorate does not change between the first and runoff elections, and that voters do not change their mind between the two elections. This is clearly not true in real elections, and as a result, the plurality winner could receive fewer than m votes or greater than 100 − m + d votes in the runoff election. Because
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such events are outside the model they have probability zero and the likelihoods would thus be zero. Of the 762 elections, in 33 the plurality winner received fewer than m votes and in 25 the plurality winner received greater than 100 − m + d votes. For the purposes of computing the vote-based likelihoods, these 58 elections were modified so that the plurality winner received m or 100 − m + d votes, respectively. The likelihoods for the simple and fancy models are shown in Table 2 as vote-based likelihoods. The fancy model is clearly a better predictor of the number votes the plurality winner receives in the runoff election. Table 2. Likelihoods for the Simple and Fancy Models.
5.2.
Simple Model
Vote-Based Likelihoods l S = 1.00
Win-Based Likelihoods log l S = -435.61
Fancy Model Likelihood Ratio
l F = 1.55e+14 l F / l S = 1.55e+14
log l F = -433.69 l F / l S = 6.82
Win-Based Likelihoods
The vote-based likelihoods provide valuable information as to the performance of the simple and fancy models. However, the end goal is not to predict the number of votes received by the plurality winner but to predict whether the plurality winner wins the runoff election. To this end, I also compute the likelihoods that the simple and fancy models predict the winner of the runoff election. Now, xi represents the outcome of the runoff election, which is 1 if the plurality winner wins and 0 otherwise. For the two models presented above, the likelihood function has the form N
l( x) = ∏ w(mi , d i )δ ( x i ) + (1 − w(mi , d i ))(1 − δ ( xi )) i =1
where w(m, d ) corresponds to either wS (m, d ) or wF (m, d ) , and δ () is the Kronecker delta function. The likelihood will necessarily be a very small number as it is the product of numbers less than one. The log-likelihood function will thus be easier to work with and it has the form N
log l( x) = ∑ log(w(mi , d i )δ ( xi ) + (1 − w(mi , d i ))(1 − δ ( xi )) ) . i =1
Above, I set the mixture weight, λ , in the fancy model to 0.8. This weight was chosen to maximize the likelihood that the fancy model predicted the 762 electoral outcomes.2 Now we are in a position to evaluate which of the two models better predicts the outcome of the runoff election. The log likelihoods for the simple and fancy models for the 762 elections are presented in Table 2 as win-based likelihoods. Despite being much more elaborate, the fancy model provides only a relatively small increase in the likelihood of predicting the outcome. 5.3.
Discussion
The comparison between the two models immediately raises the question of whether the added complexity of the fancy model provides any true benefit. Visually, the models are similar to each other, and the fancy model is only marginally better at predicting whether the plurality winner would win a runoff election. The simple model has a simple, closed-form expression. A decision rule created from the simple model involves very basic calculations, e.g., requiring a
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runoff if m < 50 − d . The fancy model can only be computed numerically. A decision rule created from the fancy model requires a mathematically complex computer program. The primary benefit of the fancy model appears to be its validation of the simple model. For example, according the simple model, the double-complement rule requires a runoff if the probability that the plurality winner would win a runoff election is less than about 0.67. From a visual inspection of the fancy model, one could argue that for the double-complement rule, this probability is closer to 0.70. Given that these calculations are necessarily inexact to begin with, the difference does not seem important. Another question is the desirability of one the assumptions underlying the models: that the electorate and their preferences do not change between the two elections. Because of this assumption, in about 8% of the 762 elections, the share of the vote received by the plurality winner in the runoff had probability zero. The high value of the mixture parameter, λ , further suggests that this assumption did limit the performance of the models. The assumption was convenient in that it simplified the modeling, but without it, more accurate models could possibly be constructed. It remains to be seen whether a better simple model or a better fancy model could be constructed.
6.
Conclusions
In conclusion, it is useful to revisit the runoff rules used by different governments. In the United States, most elections have no runoff at all or require a runoff when the plurality winner has less than a majority. The former is underprotective in that no action is taken to increase the likelihood of electing the Condorcet winner. The latter is overprotective in that a runoff is required even when it is highly unlikely to change the result. Other governments use in-between rules that are more effective in balancing the costs and benefits of runoff elections. These inbetween rules are presented graphically in Figure 6, where the shaded regions correspond to values of the magnitude and differential requiring a runoff election and the dashed line indicates the double-complement rule. In Figure 6, we are only interested in the portion of the magnitudedifferential space corresponding to actual elections, which is the portion near the right-angle of the triangle (see the scatter plots of the data in Figure 2 and Figure 5). The rule used by North Carolina, Costa Rica, and Nicaragua provides a better balance than no runoff at all or requiring a majority, but the rule is still underprotective. The non-shaded region below the dashed line indicates values of the magnitude and differential where no runoff is required even though the probability that the plurality winner wins could be close to one-half. The rule used by Ecuador is overprotective. The shaded region above the dashed line indicates cases where a runoff is required even though the probability that the plurality winner wins could be close to one. In contrast, Argentina’s rule is both over and underprotective, albeit to a lesser degree than the previous two rules. These rules are over or underprotective because they don’t account for the discontinuity that occurs at the right-angle of the triangle.
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(a)
(b)
(c)
Figure 6. Shaded regions in the magnitude-differential space indicating when governments require a runoff election: (a) North Carolina, Costa Rica, and Nicaragua; (b) Ecuador; and (c) Argentina. The dashed line indicates the double-complement rule.
In contrast, Shugart and Taagepeera’s double-complement rule does acknowledge the discontinuity. While the double-complement rule is qualitatively reasonable, until now there was no accompanying quantitative analysis. According to the simple model, the double-complement rule requires a runoff if the probability that the plurality winner would win a runoff is less than 2/3. This rule provides a reasonable compromise between ensuring the reliability of the election result and avoiding unnecessary election expenses. A government desiring to place greater emphasis on reliability or conserving its fisc could use the simple model as a guide to creating an appropriate rule. Predicting the winner of a runoff election is necessarily an inexact science, but the models presented in this paper provide valuable insights into deciding when a runoff election is desirable and when a plurality is good enough.
Notes 1. U.S. Congressional elections between 1994 and 2004 account for 117 of the elections. See Federal Elections published every two years by the U.S. Federal Election Commission. State elections in Texas between 1992 and 2004 account for 205 of the elections. See http://elections.sos.state.tx.us/elchist.exe. State elections in Florida between 1978 and 2000 account for 201 of the elections. See http://election.dos.state.fl.us/elections/resultsarchive/. State elections in Oklahoma between 2004 and 1996 account for 56 of the elections. See http://www.state.ok.us/~elections/. State elections in Louisiana between 2004 and 1991 account for 183 of the elections. See http://www.sos.louisiana.gov/. All dates are inclusive. 2. Only increments of 0.1 were considered in the maximization. One might expect that
maximizing the win-based likelihood would necessarily maximize the vote-based likelihood, but this wasn’t the case. Another variation of the fancy model produced a vote-based likelihood of 4.31e+33 and a win-based log likelihood of -436.89, which is even lower than that of the simple model. This could be caused by one of the assumptions underlying the model, which is discussed in more detail below. Alternatively, it could indicate such small differences in the win-based log likelihood are not significant.
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