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  • Cervas, Jonathan, and Bernard Grofman. 2017. Why noncompetitive states are so important for understanding the outcomes of competitive elections: the Electoral College 1868–2016. Public Choice 173(3–4): 251–265. PDF Full Text bibtex | XML
    • --- Published Online: April 2019 [https://doi.org/10.1111/ssqu.12634](https://doi.org/10.1111/ssqu.12634) **To cite this article:** Cervas, Jonathan R., & Bernard Grofman. (2019). "Are Presidential Inversions Inevitable?: Comparing Eight Counterfactual Rules for Electing the US President." _Social Science Quarterly_. 1-21. **Abstract:** We offer a typology of possible reforms to the Electoral College in terms of changes to its two most important structural features: seat allocations that are not directly proportional to population and winner-take-all outcomes at the state level. This typology allows us to classify four major variants of “reform” to the present Electoral College in a parsimonious fashion. Many of the proposals we consider have been suggested by well known figures, some debated in Congress, and they include what we view as most likely to be taken seriously. We evaluate these proposals solely in terms of one simple criterion: ‘Would they be expected to reduce the likelihood of inversions between EC and popular vote outcomes?’. We answer this question by looking at the data on actual presidential election outcomes at the state level over the entire period 1868–2016, and at the congressional district level over the period 1956–2016. We consider the implications for presidential outcomes of these different alternative mechanisms, in comparison to the actual electoral outcome and the popular vote outcome. In addition, we consider the implications of a proposal to increase the size of the U.S. House (Ladewig and Jasinski 2008). Our results show that inversions from the popular vote happen under all proposed alternatives at nearly the same rate as under the current Electoral College rules, with some proposals actually making inversions more frequent. The major difference between the present EC rule and alternative rules is NOT in frequency of inversions, but it is in which particular years the inversions occur. As for the proposal to increase the size of the House, we show that any realistic increase in House size would have made no difference for the 2016 outcome. >

    “The Electoral College is a disaster for democracy.”

    -- Donald Trump (November 6, 2012)

    >

    “The Electoral College is actually genius in that it brings all states, including the smaller ones, into play.”

    -- President‐Elect Donald Trump (November 15, 2016)

    --- # Introduction The Electoral College (EC) we know today is not the one envisioned by the founders. The founders believed its function would be to nominate candidates, from which nominees the House of Representatives would select. Today, electors are pledged in advance to particular candidates and Electors very rarely diverge from their pledged support, and they never have done so in a way that has proved consequential (Longley and Pierce, 1999:23–24). Moreover, outcomes are decided in the first stage of the process, in the EC itself. Only once has the responsibility for choosing the president shifted to the second runoff stage, which involves a congressional vote. [^1] As eminent scholar Robert Dahl (2003) noted, “the elaborate machinery of the electoral college [became] little more than a way of counting votes.” Nonetheless, despite these differences in how the EC now operates, its two most basic features have remained in place: seat allocations that are not fully proportional to population, with allocations based on the combination of congressional seats and U.S. Senate seats in the state; and winner‐take‐all (unit‐rule) outcomes at the state level—though now there are two states, Maine and Nebraska, where the winner‐take‐all feature operates at the level of congressional districts, with only the two “federal” seats allocated on the basis of the state‐wide outcome.[^2] In Federalist, No. 68, Alexander Hamilton opined about the EC, “I … hesitate not to affirm that if the manner of it be not perfect, it is at least excellent.” Both then and now, most agree that the EC is indeed far from perfect.[^3] The claim that it is “excellent” would be met with far more suspicion. In the United States, after each presidential election, especially those where popular and EC vote diverged,[^4] or a third‐party candidacy threatens to undermine the idea that a president should represent a clear majority, proposals to abolish/replace the EC are common. Indeed, Article II, Section 1 (i.e., the EC) is the provision of the U.S. Constitution that has most often had changes proposed to it (Longley and Braun, 1972:42–43; Hardaway, 1994; Longley and Peirce, 1999:133). The academic literature is also full of attacks on the EC (see, e.g., Edwards, 2011; Finkelman, 2002; Dahl, 2003; Bennett, 2006; Abbott and Levine, 1991),[^5] though it does have a few defenders (see, e.g., Best, 1975; Diamond, 1977; Hardaway, 1994; Miller, 2012b; Polsby et al., 2012; Ross, 2012). And yet, since the adoption of the 12th amendment, there have been no further changes to its structure, and attempts to eliminate the EC have proved unavailing.[^6] There are many reasons why reformers have been unsuccessful. First, the winner of the previous election has little incentive to change the rules that elected him (see the Trump quotes above; see also Bowler, Donovan, and Karp, 2006). Second, large states think that they benefit from the EC because the winner‐take‐all rule makes their state more likely to be pivotal (Banzhaf, 1968), while small states think they benefit from the EC because of the two‐seat Senate “bonus.”[^7] Third, public opinion is closely divided (with a strong partisan split).[^8] Fourth, as noted above, the academic and journalistic community has its skeptics about EC reform, with those in opposition to change noting, among other things, that proposed remedies have unknown qualities and are unlikely to cure problems such as a campaign focus on the larger states, and may bring new problems with them, for example, party proliferation, and blackmail potential by minor parties now able to win pledged electors whose vote switches could determine a presidential election outcome (see, e.g., Ross, 2012; DeWitt and Schwartz, 2016; see also Grofman and Feld, 2005). Fifth, there is little innovation by way of unique ideas for reform (Longley and Braun, 1972), and it is far from clear what constituency a reformer would be trying to persuade.[^9] Finally, after an election, attention quickly shifts to other more pressing issues and EC reform goes off the political agenda.[^10] There are many complaints about the EC, such as the claim that virtually all presidential campaign activity is focused on a very limited number of battleground states in a way that affects turnout and interest in politics such as to depress both; and there are always renewed fears about wayward electors,[^11] however, we believe it fair to say that far and away the single most important criticism of the EC is that it does not guarantee the election of the national popular vote winner.[^2] We also recognize that a direct popular vote election for the presidency also has its critics (Best, 1975; Gringer, 2008). Opponents of change to the popular vote note the possibility of a bitterly divided and close election, not unlike those we have experienced much of the past few decades and reminiscent of the late 1800s.[^13] The EC delivers decisive victories,[^14] while a close direct vote might lead to a nationwide recount that might take months or even years to complete, leaving the country in a constitutional crisis.[^15] Another issue is based on the expectation that a national popular vote would dramatically increase the incentives for candidate proliferation. While we recognize that there are many dimensions along which the EC and proposed alternatives to it could be evaluated, both in normative and empirical terms, here we evaluate eight key alternative proposals, and in Appendix C, a proposed change in size of the House of Representatives, solely in terms of one simple criterion: “Would they be expected to reduce the likelihood of inversions between EC and popular vote outcomes?” Although this criterion serves double duty, we eschew the normative standard and focus on the empirical. We address the empirical question by looking at the data on actual presidential election outcomes at the state level[^16] over the entire period 1868–2016, and at the congressional‐district level over the period 1956–2016, taking turnout levels and vote choice as given.[^17] The normative aspect is well‐plowed ground in the previous EC literature and is not repeated here.[^18] There are a number of books and articles comparing the present EC rules to proposed alternatives, but none of which we are aware that both use a time series going back to 1868 and include the 2016 election, and none that empirically evaluate as many alternatives to the present EC rules as are considered here (see, e.g., Longley and Braun, 1972; Hardaway, 1994; Grofman and Feld, 2005; Polsby et al., 2012; Koza et al., 2013; cf. Barthélémy et al. 2014, whom we consider among the most thoughtful and detailed empirical analysis and the one most closely resembling this article's analyses). Moreover, many studies only write about the EC in normative, legal, and theoretical terms, or discuss the prospects for change, and provide no attempt at empirically estimating how a particular change in rule would have affected past voting outcomes (Wilmerding, 1958; Bickel, 1968; Glennon, 1992).[^19] The goal of this article is to examine possible reforms to the EC in terms of the consequence of changes to its two most important structural features: seat allocations that are not directly proportional to population and winner‐take‐all outcomes at the state level. This typology allows for a parsimonious way to classify the reforms that have been or are likely to be taken seriously, including those that have actually previously been debated in Congress. In addition, we examine the implications of a proposal to increase the size of the U.S. House (Ladewig and Jasinski, 2008). Quite to our surprise, our empirical results show that over the full time period, inversions from the popular vote happen under all proposed alternatives at nearly the same rate as under the current EC rules, with some proposals actually making inversions more frequent. The major difference between the present EC rule and alternative rules is not in frequency of inversions, but in which particular years they occur. As for the proposal to increase the size of the House, we show that any realistic increase in House size would have made no difference for the 2016 outcome. We utilize only the Democratic and Republican two‐party vote shares in looking at outcomes under different EC formulae. We make the choice of two‐party vote despite the fact that third‐party candidacies sometimes represent a large proportion of the total vote, such as in 1968, when the leading popular vote recipient, Richard Nixon, won just 43.42 percent of the total votes.[^20] The effects of third‐party candidacies on electoral outcomes is certainly worth further investigation but is beyond the scope of the present essay. Moreover, we expect that most, if not all, of the problems identified with third‐party candidacies would be the same or greater under the alternative versions of the EC we consider here. Of course, we recognize that candidates will adapt strategies to the rules in use, and that a priori rules may affect candidate entry decisions, but we still believe it a worthwhile exercise to examine how the previous voting patterns would have affected outcomes in the 38 presidential elections we review. However, because of such estimation complexities, we must interpret the results, such as shown in Table 2, as ceteris paribus ones.[^21] ### Proposals for Electoral College Reform The EC is often thought of as having two undesirable design features.[^22] The first of these is the allocation of EC seats in each state on a winner‐take‐all basis rather than either allocating candidate votes proportionally on a state‐by‐state basis, or nationally in the form of a direct popular vote. The second design feature is the way in which EC votes are allocated to each state, with objections to the two‐state federal bonus as generating malapportionment, and thus overweighting or underweighting certain states. Many critics of the EC would be satisfied only if both features were eliminated and the EC was replaced with direct popular election of the president; others are prepared to see modifications made in one or both features. While most of the current attention on EC reform has been centered on the state compact to bind electors to vote for the national popular vote winner,[^23] many other more limited proposals for changing the EC have been introduced.[^24] In addition to replacing the present EC either with an election based on winning the national vote (though usually with a runoff rule if the plurality victory margin is not above some threshold), or replacing it with a scheme that makes the EC allocation to the candidates in each state more proportional to each candidate's share of the state‐wide vote, there have been many different alternatives proposed.[^25] We aim for a simple and parsimonious means to classify proposed reforms. We do so by focusing on the two key structural feature of the present EC identified above: seat allocations that are not directly proportional to population and winner‐take‐all outcomes at the state level. However, we do not include in our set the proposals the interstate compact that binds the states to report a slate of electors consistent with the popular vote outcome, even though that proposal has recently attracted a lot of attention, since that is simply the popular vote outcome by another mechanism. We also do not include proposals that require voters to rank‐order candidates because a lack of data on the preference ordering of candidates among the electorate makes it impossible for us to reliably estimate the implications of their use in past elections. While this simple classification lends itself naturally to a 2 × 2 format, there are variants within each element that we wish to take into account, such as keeping the winner‐take‐all feature, but applying it at the level of congressional districts.[^26] Similarly, when we consider ways to make EC results more proportional, we need to distinguish between allocations based on House seat share and allocations based strictly on population.[^27] Additionally, we offer two types of proportional representation, one that allows for fractional shares of EC seats, the other awarding only whole seats. The whole‐number proportionality rule used is the same that is used for apportionment of the House of Representatives, namely, the method of equal proportions.[^28] In the latter, whole electors are allocated, which allows for the continued physical meeting of electors in December at their representative state legislatures.[^29] Though not usually called so, the direct national popular vote is the most pure form of proportional representation. It creates an EC the size of the electorate (with the exception of needing a majority, as a simple plurality would suffice), but since it is also winner‐take‐all, it appears as nonproportional. What we end up with are 10 institutional procedures for aggregating votes, including the current EC and the popular vote. Using this simple classification scheme allows us to capture almost all the reforms that have been or are likely to be taken seriously, including those that have actually previously been debated in Congress. In toto, we offer three versions where the state‐level unit‐rule is maintained and seven variants where the unit‐rule is eliminated or altered. These 10 total electoral rules include four that keep the state‐wide two‐seat bonus (1, 4, 6, 8 in Table 1), six that eliminate the bonus (2, 3, 5, 7, 9, 10), and three that eliminate electors (5, 7, 10). We provide in Appendix B a section a procedure that reallocates the number of Electors based on the ideal size of the U.S. House, namely, one based on the cube root of the population. Table 1 identifies each of the 10 variants with their structural features identified. We provide a more technical description of each of these options in mathematical notation. #### Table 1. Variants of Electoral College Reform
        Two‐Seat Unit Number of
    No. Name Bonus Rule Inversions
    1 EC √ √ 4
    2 EC without two‐seat bonus   √ 3
    3 State‐unit population proportionality   √ 3
    4 Whole‐number proportionality with two‐seat bonus √   2
    5 Whole‐number proportionality without two‐seat bonus     3
    6 Fractional proportionality with two‐seat bonus √   3
    7 Fractional proportionality without two‐seat bonus     1
    8 District rule with two‐seat bonus √ √++District‐level winner‐take‐all rule. 5**The number of inversion is five (31.25 percent) for the 16 elections for which we have district‐level data. In comparison, in the full set of 38 elections, there are at most four inversions (10.5 percent).
    9 District rule without two‐seat bonus   √++District‐level winner‐take‐all rule. 5**The number of inversion is five (31.25 percent) for the 16 elections for which we have district‐level data. In comparison, in the full set of 38 elections, there are at most four inversions (10.5 percent).
    10 Direct popular vote     –
    ###### *Note:* +District‐level winner‐take‐all rule. *The number of inversion is five (31.25 percent) for the 16 elections for which we have district‐level data. In comparison, in the full set of 38 elections, there are at most four inversions (10.5 percent). --- #### Table 2. Concordance of Popular Vote with Winner in 10 Variants of the Electoral College
          Electoral Population Whole‐Number Whole‐Number Fractional Fractional District District
        Electoral College Weighted Proportionality Proportionality Proportionality Proportionality Specific Specific
        College Without Two State Unit with Two Without Two with Two Without Two with Two Without Two
      Popular Vote (%) (%) Seats (%) (%) Seats (%) Seats (%) Seats (%) Seats (%) Seats (%) Seats (%)
    1868 47.337 27.491 28.444 29.086 46.048 46.667 46.999 47.315    
    1872 44.062 18.033 18.493 18.471 42.35 41.781 43.712 44.019    
    1876 51.518 49.864 51.195 51.799 51.22 51.877 51.398 51.871    
    1880 49.949 42.276 40.273 40.326 50.407 51.195 50.857 50.944    
    1884 50.295 54.613 55.077 55.34 50.623 51.692 51.25 51.381    
    1888 50.43 41.895 40.615 41.038 52.369 52 52.16 52.242    
    1892 51.69 61.036 63.202 64.439 52.252 53.09 52.059 53.144    
    1896 47.793 38.926 36.415 36.242 50.336 49.02 50.414 49.636    
    1900 46.832 34.676 33.894 33.655 50.336 50.42 49.983 49.85    
    1904 39.988 27.941 28.238 28.312 44.118 44.56 45.485 45.781    
    1908 45.495 32.298 31.714 32.031 49.482 49.361 49.47 49.549    
    1912 64.344 95.669 96.092 96.04 70.433 70.575 67.456 67.518    
    1916 51.644 51.977 49.655 49.911 57.25 56.782 56.16 55.859    
    1920 36.118 23.917 24.138 24.55 39.171 38.391 41.008 40.818    
    1924 34.785 25.612 25.747 25.776 38.795 38.161 40.596 40.538    
    1928 41.202 16.384 16.322 15.888 42.75 42.529 44.05 44.124    
    1932 59.149 88.889 89.195 88.708 65.348 64.598 63.183 62.976    
    1936 62.459 98.493 99.08 99.054 67.985 68.966 65.99 65.941    
    1940 55 84.557 85.747 85.79 60.64 60.69 59.185 59.125    
    1944 53.774 81.356 82.759 82.449 58.945 58.391 57.494 57.576    
    1948 52.37 62.335 61.839 61.801 55.367 55.172 53.995 54.004    
    1952 44.548 16.761 16.322 17.307 43.879 44.368 45.395 45.696    
    1956 42.248 13.936 13.793 13.829 42.185 42.759 43.547 43.621 22.659 24.083
    1960 50.083 59.032 61.556 61.685 51.024 50.801 50.333 50.545 47.486 47.368
    1964 61.346 90.335 90.826 90.894 61.524 61.697 59.6 59.769 85.688 85.092
    1968 49.594 40.52 42.202 42.573 49.442 49.541 49.291 49.44 41.636 43.578
    1972 38.214 3.16 2.982 3.297 35.13 35.092 37.394 37.531 11.896 13.761
    1976 51.052 55.204 57.11 57.366 50.929 50.917 51.079 51.204 49.907 50.575
    1980 44.695 9.108 8.028 8.136 43.309 42.661 44.69 44.93 26.58 29.587
    1984 40.83 2.416 2.064 2.081 38.662 38.303 40.547 40.738 12.825 14.908
    1988 46.098 20.818 20.642 20.418 45.167 45.183 45.958 46.028 29.926 31.881
    1992 53.455 68.773 69.725 71.517 53.532 54.358 53.197 53.361 60.037 58.945
    1996 54.735 70.446 72.248 72.543 54.647 55.275 54.523 54.72 64.126 64.45
    2000 50.27 49.628 51.606 51.713 50.186 49.771 49.958 50.385 44.981 45.413
    2004 48.756 46.84 48.624 48.764 47.955 47.248 48.439 48.755 41.078 41.514
    2008 53.688 67.658 70.183 69.946 53.717 54.587 53.471 53.775 55.948 55.734
    2012 51.965 61.71 63.761 64.925 50.929 51.606 51.534 51.891 48.885 47.936
    2016 51.112 43.309 43.807 43.692 50.186 50 50.629 51.247 46.097 47.248
    ###### *Note:* Percentages are of the Democratic candidate using the alternative rules. All calculations are of the two‐party vote. Shaded cells are those in which the new rule disagrees with the popular vote. --- Table 2 shows actual popular vote and EC vote shares and also simulated seat share under each of our additional eight EC variants. Cells that are in bold Shaded cells show inversion years. There are a number of interesting results shown in Table 2. First, malapportionment effects, and especially the effects of the two‐state bonus, are not that large. For example, in 2016, Donald Trump would have been elected even had there been no two‐seat bonus. In fact, he would have won in all the different unit‐rule configurations, only losing when a proportional rule such as the popular vote is instituted. Indeed, as can be derived from the differences between Columns 2 and 3 of Table 2, in only three elections in American history has the two‐seat bonus feature of the EC been decisive in reversing an election result. The first time this happened was in 1876 when the two‐seat bonus benefited the Republican candidate,[^30] then again in 1916 when it benefited the Democratic candidate, and finally again in 2000 when it benefited the Republican candidate. However, we observe that the over the last seven elections, the two‐seat bonus has consistently favored the Republican candidate, even when it has not had an impact on election outcome. On the other hand, glancing through Table 2's first two columns reveals several instances where a reversal almost happened. In most of these instances, the Democratic candidate came out on top, for example, in 1960.[^31] Second, while proportionality variants of EC allocations clearly can dramatically change the magnitude of seat outcomes relative to vote outcomes, it is only in the period from 1880 to 1900 that we see repeated evidence of changes in the presidential winner based on choice of a proportional as opposed to a winner‐take‐all rule, though, of course, we also see this in 2000 and 2016. Third, we note that the “reform” that would have the most dramatic effect on recent elections is a winner‐take‐all rule based on district outcomes. In recent elections where a Democratic Party candidate won the election, such a rule would reverse the EC. That such inversions provide net benefits to the Republican Party can be explained by the degree to which Democratic voting strength is inefficiently concentrated in urban districts (Chen and Rodden, 2013), and the degree to which there is greater Republican unified control of state legislatures and governorship than is true for Democrats, giving Republicans a much greater opportunity to engage in successful partisan gerrymandering of congressional district lines.[^32] In 1960, 1976, and 2012, for example, the outcomes would have been reversed (all three times benefiting the Republican candidate) if we allocated based on the results withincongressional districts plus the plurality state winner getting two bonus seats.[^33] In 1976, however, not giving the two‐seat bonus to the state plurality winner reverses yet again back to the actual winner, Jimmy Carter (i.e., benefits the Democratic candidate). Fourth, if we want to understand inversions we must look to when popular vote elections are close. As mathematician Sam Merrill has argued, inversions are essentially a coin‐flip as the popular vote approaches 50 percent (Merrill, 1978). An EC Based on an Expanded U.S. House Now we turn to the last structural variation on the present EC rules that we consider. Ladewig and Jasinski (2008), drawing on ideas in Taagepera (1972), have proposed that the House size should be decennially adjusted to equal the cube root of U.S. population.[^34] The cube root of the U.S. population in 2010 was 676. Using this House size, a congressional seat's average size would be just 458,262 people.[^35] As it turns out, this would have ensured that, had congressional size been increased in 2010 according to this formula, even the smallest state would have received at least one congressional seat based solely on the state population. How would EC malapportionment and outcome effects change if we increased the size of the House to make the allocation rule in that body more nearly proportional to the population of the state using the cube root of population to determine House size? One way in which House size could become determinative is when a reversal actually happens, so the popular vote winner has lost the election. Under this circumstance, as the House size grows to approximate the population size, eventually the popular vote winner will also win the EC. Recalculating the 2016 election for a House size of 676 (but now excluding the two bonus seats for each senator) yields Donald Trump 380 of the 676 electors (56.2 percent). Again, as with the current EC and the version that simply omitted the bonus Senate‐based electors, Trump would still have won the White House even if the House size were 676. In this EC rule, he wins by 84 electors. Although this is a larger number of seats, the EC is also larger. Since Trump won 306 of 538 (56.8 percent) electors in 2016, his percentage would, as expected, slightly decrease under the cube root rule. Trump won a plurality in the majority of states, so the same Trump victory still occurs if we add in the two‐seat “federal bonus.” In 2016, for House size to matter required a House size so huge as to be unrealistic: the effects of increasing House size do not affect the outcome in 2016 for any House size under at least 800 (data omitted for space reasons). In Appendix B, we also include a table that compares the popular vote and EC under present apportionment with those using the cube root rule of assembly size for the entire time period of our study.[^36] In the year 1912, the cube root EC size is actually larger than the actual EC size. In 2000, an increase in the size of the House could have mattered in that, in most House sizes starting above 493, including all of them above 655 (the cube root law value), the popular vote winner, Gore, would also have won the EC, a result previously pointed out by Neubauer and Zeitlin (2003). The years in which a cube root allocation would have changed the outcomes are limited to just 1876 and 2000, which were already reversed, thus reducing inversions by two.[^37] # Discussion A reporter once asked legendary singer/actor Maurice Chevalier: “Mr. Chevalier, how does it feel to have reached the ripe old age of 70?” Without hesitation, Chevalier responded to the reporter: “Old age isn't so bad when you consider the alternative!” Using election results from the period 1868 to 2016, we have constructed a total of 13 counterfactual variants on the EC for the purpose of comparing the actual EC results and popular votes with those from various proposed reforms. Presidential elections have seen four occasions in the modern political party era of American history in which outcomes of the popular vote and the EC vote diverge, with two of these coming within the past two decades. While some may argue that even once is too much, others view the relatively small number of inversions as vindication for the founders (Hardaway, 1994). If we look at modifications to the EC such as eliminating the two‐state bonus, allowing for a more proportional distribution of electors, or switching to House‐district‐based outcomes, we discover that at best they reduce the number of inversions by one, from four to three and, at worst, with the House‐district‐based outcomes, they actually increase the number of inversions. Moreover, with the partial exception of 2016, the years in which inversions occur under alternative EC arrangements are different from those in which they occurred under present EC rules. Thus, changing the rules in the ways identified above seems to serve no useful purpose. It does not eliminate or even substantially reduce the prevalence of inversions; all it does is change the years in which they occur. We also considered the implications of a proposal by Ladewig and Jasinski (2008) to increase the size of the House (and thus of the EC) by picking a House size that was proportional to the cube root of population. Here we found that the election results in 2016 would have been unchanged, though the net effects of this rule over the entire time period do reduce the number of inversions by two. While certainly far from perfect, the EC has proved a robust institution that usually produces clear victories that match the plurality winner. Moreover, the alternatives to it identified above, with the partial exception of a rather large increase in the size of the House of Representatives, have virtually the same flaw in terms of likelihood of creating a reversal between popular vote winner and EC winner, with some even worse. And the two‐seat bonus afforded on the basis of statehood has been shown in this essay to be generally nondeterminative of election outcomes. Reformers should also acknowledge that the EC “wrong winner” is no less legitimate than any legislation passed by senators representing a minority of the population, or Supreme Court decisions that largely are immune from public opinion, and somewhat less affected by electoral tides due to the long length of service on the Court and the absence of a mandatory age‐linked retirement.[^38] As if that were not enough reason to be skeptical about the insistence on majoritarianism in the EC, in the process by which the EC would change through constitutional amendment, senators from the 34 states with the smallest population could vote for a change without the input of a popular majority.[^39] A bill would still need to pass the House regardless of action in the Senate, but extreme gerrymandering also means that a minority of the population can, in effect, carry out a constitutional change. Additionally, when it comes to state ratification, state legislatures are often so severely gerrymandered that a majority of voters fail to elect a majority of legislators, and often seats are so noncompetitive they regularly fail to garner competition at all. Ratification by 38 states is required, and those smallest 38 states amount to just 38.4 percent of the total population.[^40] While it is clear that reformers who believe only in strict majoritarianism are right to criticize the EC, they should first look to reform the other more disproportionate aspects of the U.S. Constitution. Without informed examination, one might assume the EC to be an archaic institution that does more harm than good. The EC is not perfect, a fact that the framers were perfectly aware of. All plausible alternatives, except for the popular vote or something that is its equivalent, do not cure the main problem of inversions. Moreover, many have new and severe problems of their own. For example, changes that would eliminate the state‐level winner‐take‐all and move to district unit‐rule would almost definitely lead to political maneuvering and even more extreme gerrymandering (as would increasing the size of the House). And, in an age of hyperpolarization, with the potential for a close national outcome, the direct popular vote creates problems with respect to a proliferation of election challenges. At best the adaptations result in similar outcomes, and at worse could lead to severe constitutional crises. All in all, it is in our opinion that making changes to the system of electing the president should be looked at with a high degree of skepticism. [^1] : In 1824, John Quincy Adams became the only president to not receive a majority (a requirement for winning in the first stage) of the EC votes. The vote was splintered among multiple factions, with no candidate receiving the necessary plurality. In the election of 1876, where politicking in Congress determined which of several competing slates of electors were to be accorded legitimacy, the outcome of what has been called the “Compromise of 1877” was still recorded as a victory for Rutherford Hayes within the EC. He was awarded 20 disputed EC votes that gave him a one‐elector victory in the EC despite not winning the popular vote. [^2]: Maine adopted this rule in advance of the 1972 presidential election, while Nebraska enacted it starting with the 1992 election. A split has occurred once in each of these states. In 2008, Barack Obama won Nebraska's second congressional district, picking up a Democratic electoral vote in that state for the first time since 1964. In 2016, Donald Trump won Maine's second congressional district. [^3]: Other than the United States, there are no presidential democracies currently using an EC to elect their president. Argentina and Bolivia once had ECs (Matthew Shugart, personal communication, February 2018). Many first‐past‐the‐post elections have a runoff procedure to select a president in a multicandidate contest such that, if no candidate receives a certain percentage of the vote, there will be a second round involving two or more of the candidates with the most votes (Birch, [^2003). All parliamentary democracies choose their executive via an indirect form of election. While the prime minister will normally need to command majority support in the national parliament, a prime minister can sometimes govern with only minority support. Minority governments can be quite common in some countries, for example, Denmark. [^4]: “Diverge,” “reverse,” “wrong winner,” “misfire,” “divided verdict,” “reversal of winners,” “representative inconsistency,” “compound majority paradox,” “referendum paradox,” “majority defeat,” and “inversion” have all been used to describe a situation when the winner of the most votes does not win the presidency (Miller, [^2012a). [^5]: Bickel ([^1968) warned against sudden structural reforms, though he ultimately supported reforming the EC. [^6]: Since the Electoral College process is part of the original design of the U.S. Constitution it would be necessary to pass a Constitutional amendment to change this system …Under the most common method for amending the Constitution, an amendment must be proposed by a two‐thirds majority in both houses of Congress and ratified by three‐fourths of the States. (National Archives and Records Administration) [^7]: Both sides are right (Longley and Peirce, [^1999:153). However, when we look at the likelihood that an individual voter in any given state will be pivotal (e.g., using game‐theoretic indices of pivotality such as the Banzhaf index, Banzhaf, [^1965; or the Shapley–Shubik value, Shapley and Shubik, [^1954; see also Mann and Shapley, [^1962); as far back as Owen, [^1975, it has been recognized that these two effects—greater large state pivotality and small state overrepresentation relative to population tend in opposite directions, making the a priori “power” scores of individual votes to influence EC outcomes much more similar across states than one might think (see Gelman, Silver, and Edlin, [^2012; cf. discussion in Grofman and Feld, [^2005; Strömberg, [^2008). [^8]: In nearly every poll in the Roper Center for Public Opinion Research iPOLL data bank, the public is split about eliminating the EC, especially along partisan lines, albeit with majorities favoring a change to popular vote. After the bitterly fought 2000 election, 41 percent of Republicans would have amended the Constitution while 75 percent of Democratic respondents would have liked to see a change, with an overall support for change of 59 percent and with 3 percent of those polled with no opinion (Cable News Network, USA Today. Methodology: Conducted by Gallup Organization, December 15 to December 17, 2000 [USAIPOCNUS2000‐56]). After the even more bitterly fought 2016 election, Gallup asked again about the EC, this time 49 percent choose the option to amend the Constitution (Gallup Poll 2016 [USGALLUP.120216.R01]. November 28–29, 2016). Again, there was a strong partisan split. Republican support of the current system significantly increased after the election. Gallup found that only 19 percent of Republicans or leaning Republicans favor a system where the winner is the candidate who wins the popular vote (compared to 81 percent of their Democratic counterparts). Aldrich, Reifler, and Munger ([^2014) have modeled the circumstances where we might expect changes in preferences about the desirability of the EC. [^9]: In reviewing the history of the Electoral College, it quickly becomes clear how little anybody has to offer that is new. All the plausible reform ideas, and all the arguments for and against them, have been debated and rehashed for well over a century, in terms that have remained virtually unchanged. (Schwarz, [^2000) [^10]: Google Trends reveals spikes in the popularity of searches of the term “Electoral College” in the months before a presidential election; search numbers quickly diminish to near zero shortly after the election. Some of these data are displayed in Figure [^A1. [^11]: “The people know the candidates of president and vice president; rarely do they know the identity of the electors for whom they actually vote. Such ‘go‐betweens’ are like the appendix in the human body. While it does no good and ordinarily causes no trouble, it continually exposes the body to the danger of political peritonitis” (Henry Cabot Lodge, as cited in Longley and Peirce, [^1999:110). [^12]: For issues of problematicity of the meaning of “popular vote winner,” see Gaines ([^2001). [^13]: The last seven elections have been decided by under 5 percentage points. [^14]: The EC often appears to give the president‐elect a landslide victory even when the popular vote is close. This is a result of the winner‐take‐all rules that translate even small pluralities into 100 percent of the state's electoral slate. By definition, EC victories are always over 50 percent, while popular votes have been as low as 38 percent (in 1860). No president has won with a smaller percentage of the EC than aggregated national popular vote percentages (i.e., Lincoln won 59.4 percent of the EC, but only 39.65 percent of the popular vote). [^15]: It is unclear who would lead the executive branch in a situation where no president is selected before inauguration day. [^16]: The District of Columbia received three seats after the passage of the 23rd Amendment in 1961. [^17]: Of course, we recognize that candidates will adapt strategies to the rules in use, but we still believe it a worthwhile exercise to examine how the previous voting patterns would have affected outcomes under different EC formulae. [^18]: In reading the literature and in presentation of our results at conferences and colloquia, it is clear that many political scientists hold the popular vote principle to be sacrosanct. Nonetheless, it is useful to remind readers that only one state voted for the popular election of the president during the Constitutional Convention, while nine voted against. Popular election of the president was again brought up in Congress as a proposed amendment in 1816, and since then has been proposed in Congress at least 100 times, every time failing. [^19]: Bullock, Gaddie, and Wert ([^2009) examine the potential for a Voting Rights Act based challenge to the EC winner‐take‐all rule that would parallel voting rights challenges to at‐large elections. [^20]: That is, nearly 14 percent of the vote went to candidates who did not finish in the top two. If we had a good way of determining the preferences of voters for these other candidates had only the two mainstream candidates been on the ballot, we might find that Hubert Humphrey would have led in the popular vote among the top two candidates. George Wallace's independent campaign drew support primarily from the South, capturing 46 electoral votes from five southern states. Even had Humphrey won all 9.9 million of Wallace's voters and the 46 EC votes that accompanied them, he still would have lost in the EC. [^21]: In the same tweet in which President Trump said that the EC was “genius,” he also claimed that he would have won the popular vote if, rather than the present EC system, who won the popular vote decided who got elected president. Under that rule for deciding outcomes he said he would have campaigned in populous states that were being conceded to the Democrats under the present winner‐take‐all feature of the EC. But, of course, if he had changed his strategy so, too, would his Democratic opponent have been able to do a better job of motivating turnout among her supporters. Gaines ([^2001:75) has called the popular vote a “nebulous quantity.” [^22]: Longley and Braun ([^1972:18) identify five features, one of which (inversion) flows from the others, moving beyond the first stage, which is highly unlikely without a strong regional third‐party candidate, and the faithless elector, which in our simulations, we do not address. The remaining two are those that we consider in this article. [^23]: S.J. Res 28 1979; National Popular Vote Bill—enacted in 11 states. [^24]: By some estimates, over 700 attempts to change or abolish the EC have been advocated or proposed (Hardaway, [^1994). Most of these proposals are simply rehashing previously failed attempts (Schwarz, [^2000). [^25]: Proposals range from reasonable to absurd. Longley and Braun ([^1972:69) write about a proposal in 1808 by Sen. James Hillhouse (Federalist—CT) that would have had all senators elected to one three‐year term, such that a third of them would retire each year; the president would then be chosen randomly among those retiring senators. Most proposals are written for political expediency (Bowler, Donovan, and Karp, [^2006), such as the Democrats pushing for a direct vote in the 2000s because it is seen as being more favorable to their electoral chances. While the Democrats have won the popular vote in four of the previous five elections, they have only won the EC twice (in 2008 and 2012). [^26]: Winner‐take‐all and two‐seat bonus, winner‐take‐all and no bonus, no winner‐take‐all (proportional) and two‐seat bonus, no winner‐take‐all and no bonus. [^27]: For states that enter the Union after a census has been taken but are still allocated EC seats, we take the population in the subsequent census. [^28]: The U.S. Census has used this method since 1940. For more details, see 2 U.S.C. §2a (1941). [^29]: We recognize that there are many different formulas that can be used to allocate seats, and that the differences might, in the words of Gallagher ([^1991:33), “produce significantly different seat allocations for a given distribution of votes…” Similarly, Gaines and Jenkins ([^2009) observe that when the direct vote is particularly close, choice of apportionment method might be determinative (see, especially, Balinski and Young, [^1982, for a full treatment of divisor methods). Gallagher ([^1991) observes that “each PR method minimizes disproportionality according to its own principles.” For the purposes of this essay, we only look at the alternative results based on the apportionment currently used by the U.S. Census for determining EC seats. [^30]: As noted previously, this was an election that was decided not by the votes of the people but instead by a deal between the Democratic and Republican candidates that involved the federal government ending Reconstruction in the South. [^31]: In such situations, very small permutations in vote shares at the state level can, because of the unit‐rule, take a popular vote and EC convergence and reverse them. In 1960, inconsistencies in popular vote totals and controversial methods for counting the popular vote, especially in Alabama, have led some to argue that Nixon in fact won the popular vote. Nobody, including Nixon himself, believed that the questioned votes would have changed the EC outcome. However, a few votes in specific states would have made Nixon president, and depending on how votes in other places were counted, perhaps made him a reversal president. [^32]: Blatant partisan gerrymandering was made more likely by the Supreme Court's consistent refusal to rein in this practice (McGann et al., [^2016). [^33]: There is an especially large difference when allocating by congressional district in 2012, due in no small part to the aggressive House gerrymandering that took place in the census before the election, mostly to the benefit of Republicans (McGann et al., [^2016). Since benefit from incumbency advantage reduces the vote shares of challengers, ceteris paribus, once incumbents are in place whose election is in part or largely due to gerrymandering, apparent partisan bias in subsequent elections may appear lower (Theodore Arrington, personal communication, February, 2017). [^34]: Taagepera ([^1972) argued that, for optimal communication purposes between representatives and those they represent, an assembly size should be the cube root of the polity's population. He also demonstrated that this model did a rather good job in explaining actual assembly size in the world's democracies, with the United States in the last 100 or so years being one of the most notable exceptions. [^35]: In addition to the work of Ladewig and Jasinski ([^2008), the effects on presidential outcomes under the EC of increasing/varying the size of the House have been studied by other authors (e.g., Neubauer and Zeitlin, [^2003; Barthélémy et al., [^2014; Miller, [^2014). [^36]: In Appendix [^B, we also provide comparisons to cube root results for the EC without the two‐seat bonus, and the whole‐number and fractional proportionality rules, both with and without two‐seat bonus. The district‐based measures cannot be calculated since we cannot know the partisan composition of a House delegation that has never existed, and the popular vote rules would be the same regardless of the House size. [^37]: The effect in 1876 is hard to assess given the log‐rolling involved. Mathematically, based solely on the criteria of popular votes counted on a state‐by‐state basis, a cube‐root‐based apportionment would have resulted in the election of the runner‐up, Samuel J. Tilden. [^38]: Reformers who demand majoritarian winners as normative doctrine of democracy should look no further than the multiparty coalition governments common in proportional representation (PR) systems. Leaders emerge through postelection negotiation, and their party may not even receive a plurality of the votes, though the largest party normally gets first chance to put together a winning majority coalition. [^39]: As of the 2010 census, the lowest population two‐thirds states represented just 30 percent of the total population. Conversely, 34 senators from the 17 smallest states with a total population of 21,031,314 (6.8 percent) could block any amendment. [^40]: Again conversely, 13 state legislatures from the smallest population states could prevent ratification with a total population of 12,562,969 (4.1 percent). [^41]: Population growth and differences in migration patterns throughout a decade lead to suboptimal appropriation, which is most significant in the election preceding a new census. [^42]: For the purposes of this calculation, Washington, DC will still be counted for one House vote in periods after 1960 despite not having a voting member of the House of Representatives. As per the 23rd Amendment, adopted in 1961, District of Columbia is allocated three EC votes regardless of its population. [^43]: Maine adopted this rule in advance of the 1972 presidential election, while Nebraska enacted it starting with the 1992 election. A split has occurred once in each of these states. In 2008, Barack Obama won Nebraska's second congressional district, picking up a Democratic electoral vote in that state for the first time since 1964. In 2016, Donald Trump won Maine's second congressional district. [^44]: Although this plan is more proportional than the state unit‐rule plans, including the current EC, it is not a proportional plan since it still awards electors on a winner‐take‐all basis, except now at the congressional‐district level. Given the potential for partisan gerrymanders, this plan may end up being less proportional than a winner‐take‐all state rule. [^45]: S.J. Res. 12, 90th Congress, first session. [^46]: This proposal can be seen as an attempt to avoid change in the present EC that would be impossible to achieve without a constitutional amendment while still assuring concordance with popular vote outcomes by creating a compact of all the states such that they would report EC results as if the national popular vote winner was the winner in the state. This proposal would only take an act of a Congress, since it sets the size of the U.S. House and thus the percentage of electors who are allocated via population. --- # REFERENCES
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    --- # Appendix A: Electoral College Google Trends #### **Figure A1:** #### Electoral College Term Search Google Trends ![Google Trends](https://wol-prod-cdn.literatumonline.com/cms/attachment/7f15d72f-29e6-4ccd-b4fc-044dba1d77d0/ssqu12634-fig-0001-m.jpg) ###### *Note:* “Numbers represent search interest relative to the highest point on the chart for the given region and time. A value of 100 is the peak popularity for the term. A value of 50 means that the term is half as popular. A score of 0 means there was not enough data for this term” (Google Trends). The term was at its peak search in the days following the 2016 election (an inversion). --- # Appendix B: Cube Root Adjusted Electoral College This appendix provides the simulations for an Electoral College (EC) and a number of alternatives by adjusting the total number of electors such that the U.S. House portion is equal to the cube root of the population. As the size of the House increases, disproportionality is reduced. We provide data for the five alternatives that lend themselves to such analyses. Just like the alternatives in the main text of this article, the cube‐root‐based apportionment adjusted reforms produce a similar number of inversions, with only the most proportional alternatives minimizing the occurrences. Still, every alternative in this set produces at least one reversal, with several years being near misfires. In only one year, 1900, is the popular vote margin greater than 2 percent and produces and inversion. We see this as rather redeeming to the EC as currently constructed because only in very close elections is there a legitimate probability of reversal. We also note that 1900 is the last year where apportionment uses 1890 census populations, and that position makes for maximal disproportionality within a given census period.41 #### Table B1. Cube Root Adjusted Electoral College
    CubeRoot CubeRoot CubeRoot CubeRoot
    Electoral Whole‐Number Whole‐Number Fractional Fractional
    Popular Electoral College Proportionality Proportionality Proportionality Proportionality
    Vote College CubeRoot withTwo WithoutTwo withTwo WithoutTwo
    Year ECNo. (%) (%) (%) Seats(%) Seats(%) Seats(%) Seats(%)
    1868 305 47.337 27.491 28.852 46.900 46.230 47.593 48.055
    1872 315 44.062 18.033 18.73 42.416 42.857 44.183 44.699
    1876 336 51.518 49.864 51.786 51.214 52.381 51.959 52.655
    1880 336 49.949 42.276 40.179 50.971 51.19 50.988 51.125
    1884 366 50.295 54.613 54.918 51.584 52.186 51.271 51.422
    1888 366 50.430 41.895 40.437 53.620 52.459 52.115 52.195
    1892 369 51.69 61.036 64.228 52.954 53.659 53.730 55.628
    1896 396 47.793 38.926 36.616 51.029 49.747 49.739 48.632
    1900 396 46.832 34.676 34.091 50.000 50.505 49.901 49.72
    1904 420 39.988 27.941 28.095 44.314 44.524 45.874 46.327
    1908 424 45.495 32.298 32.075 48.45 49.528 49.565 49.684
    1912 425 64.344 95.669 96.000 69.482 70.353 67.404 67.471
    1916 450 51.644 51.977 50.222 56.960 56.667 55.969 55.546
    1920 450 36.118 23.917 24.444 39.194 38.444 40.952 40.703
    1924 472 34.785 25.612 25.424 38.908 38.983 40.185 40.013
    1928 472 41.202 16.384 15.890 42.958 42.797 43.954 44.015
    1932 472 59.149 88.889 88.771 65.493 65.042 63.128 62.874
    1936 496 62.459 98.493 98.992 68.581 68.548 65.905 65.831
    1940 496 55.000 84.557 85.685 60.304 60.081 59.084 58.995
    1944 507 53.774 81.356 82.446 58.375 58.383 57.574 57.679
    1948 507 52.37 62.335 61.933 54.561 54.635 53.973 53.978
    1952 507 44.548 16.761 17.160 44.776 45.168 45.866 46.293
    1956 531 42.248 13.936 13.936 42.584 42.185 43.733 43.849
    1960 531 50.083 59.032 61.77 50.555 50.847 50.555 50.82
    1964 563 61.346 90.335 91.119 60.902 61.634 59.882 60.104
    1968 563 49.594 40.520 42.629 49.173 49.556 49.555 49.754
    1972 563 38.214 3.160 3.197 35.789 35.524 37.684 37.878
    1976 587 51.052 55.204 57.411 50.943 50.937 51.279 51.434
    1980 587 44.695 9.108 8.007 43.977 43.612 45.050 45.341
    1984 609 40.830 2.416 2.135 39.100 39.737 40.803 41.019
    1988 609 46.098 20.818 20.525 45.992 45.813 46.058 46.138
    1992 609 53.455 68.773 71.757 53.586 53.695 53.501 53.703
    1996 628 54.735 70.446 72.452 54.932 55.255 54.819 55.040
    2000 628 50.270 49.628 51.592 49.726 50.159 50.469 50.923
    2004 655 48.756 46.840 48.855 47.952 48.702 48.701 49.000
    2008 655 53.688 67.658 70.076 53.765 54.198 53.758 54.051
    2012 655 51.965 61.710 65.038 51.651 52.214 52.131 52.523
    2016 676 51.112 43.309 43.787 50.900 51.183 51.212 51.785
    ###### Note: Column 2 (EC No.) is the total number of electors awarded based on the cube root of the population. In 1912, the cube root and actual House size are essentially the same. Since then, the size of the House, which was frozen after this, is smaller than ideal. Shaded cells are those in which the new rule disagrees with the popular vote. # Appendix C: Formal Definition of the 10 Electoral College Variants Being Compared We begin with some notation to elucidate how we will measure these EC alternatives. We are interested in all elections since 1868, . In each year, there is a set of states, , which all receive a proportion of the EC, , as determined by the U.S. Constitution. The equation for the Popular vote C1 is simply, (C1)which translates into the national summed percentage of votes for the Democratic candidate, with the two‐party vote total in the denominator and third‐party votes excluded. This is, of course, the most proportional to the voters, but not necessarily to the population since turnout rates might vary by state (Grofman, Brunell, and Campagna, 1997). It also happens to be the reform that has generated the most demand since it is the only system that can guarantee a plurality winner takes the office. The actual EC C2, assuming unit‐rule for all states, is determined by the following equation, (C2) The first of the alternatives we consider sets an EC vote share equal to the size of the state's delegation in the U.S. House divided by the total number of seats in the House, that is, an EC with the two‐seat Senate bonus removed. We refer to it as an EC without two‐seat bonus.42 The equation is the same as the EC C2, except every is first subtracted by 2. The second sets the EC vote share as identical to the state's share of the national population, with fractional allocations to allow for (nearly) perfect proportionality, that is, an EC that corrects for both House malapportionment and malapportionment due to the two‐seat Senate bonus. We refer to the second as a State‐unit population proportionality C3. Here, instead of the total electors equaling 538, it is set to 1, or 100 percent, and each state gets exactly the percentage of this EC as their census year population, and the winning candidate is the one who wins enough states such that his share of the states' allocations surpasses 50 percent of the population. (C3) In the same way that one might expect campaigns to employ a different strategy than with the EC, a proportionality rule such as a state population allocation might encourage regional candidates or smaller parties to run because it would not be necessary to win a majority of states or votes, since the winner would be the candidate who can attract enough support in a subset of states that is greater than any other candidate. Essentially, even though seats are awarded nearly proportionately, this rule would change the nature of campaigns for the highest office. Any increase in the number of viable candidates who go on to win EC seats would result in the winner of an election winning smaller pluralities. With the winner‐take‐all feature maintained, which results in an unbalanced distribution of votes in some states, we would expect more frequently split popular and electoral votes. For the purposes of this essay, we treat the actual results as if they happened under the alternative rules. The third proposal is to create an EC that allocates its votes in a proportional or more proportional way to the state's share of the present EC, rather than in terms of winner take all. Here there are two main variants, each of which have two minor distinctions. The first major variant uses the current allocation of EC seats, the second allocates electors based on representation in the House of Representatives, that is, with the two‐seat bonus eliminated. For the minor variations, Whole‐number proportionality C4 for both EC and House delegation sizes are given by the following series of equations: where the Priority numbers are ordered and the n‐top priority numbers are allocated to each party. (C4) The second minor variation is the Fractional proportionality C5, in which electors are abolished, and candidates receive their share of the state‐wide vote rounded to the third decimal. This variant has been proposed numerous times and was actually passed by the U.S. Senate in 1950 under what was known as the Lodge‐Gossett Amendment (S.J. Res. 2 of the 81st Congress). The Fractional proportionality (C5) alternative results is an increase in proportionality from Whole‐number proportionality but yet is less proportional than Popular vote because it sets the number of Electors each state gets but relaxes the unit‐rule nature of the election. It failed ratification in the House of Representatives (Koza et al., 2013). The equation is as follows: (C5) The other frequently proposed variant is one in which EC votes are allocated by giving one seat for each House district won, and a two‐seat bonus for the candidate who wins the popular vote in the state. This variation emulates the rules presently practiced in the states of Maine and Nebraska.43 We refer to this as the District rule C6. It has two minor variations, with and without two‐seat bonus.44 It is akin to a plan advocated by Senator Karl E. Mundt (R‐SD), which was opposed by then Senator John F. Kennedy (D‐MA).45 (C6) District rule with two‐seat bonus sets ω to 2, while the District rule without two‐seat bonus instead sets it to 0. In addition, in the subsequent section, we briefly consider an additional type of change, one based on the suggestion in Ladewig and Jasinski (2008) that the House size be decennially adjusted to reflect the cube root of U.S. population. The idea is that increasing the size of the House should increase the proportionality of EC outcomes, and hence make the EC vote look more like the popular vote.46 (C7) Instead of locking the size of the U.S. House at 435, this rule would apportion seats using method of equal proportions (as described in Equation C4), and we replace n with the Cube root House size rounded down to the nearest integer. We can then use the new apportionment to apply to all the alternative EC rules.