Exploring the effects of national and regional popular vote Interstate compact on a toy symmetric version of the Electoral College: an electoral engineering perspective

Miller (2012b) contains an insightful presentation of the Electoral College that he characterizes as “a terrific boon for political science (and public choice) research (and teaching)”.

Trump is the fifth person in US history to become president despite losing the nationwide popular vote (Clinton received about 2.9 million more votes nationwide, a margin of 2.1%).

As pointed out by Neubauer et al. (2012), “(...) for the 56 presidential elections that have taken place since 1780, at least four unpopular elections—nearly 7%—have occurred. Further, the 1960 presidential election has been a matter of discussion among scholars. Some (for instance Edwards 2011) argue that a fair accounting of the popular vote cast in Alabama would make Richard. Nixon -not John. F. Kennedy- the winner of the nationwide popular vote. If we include the 1960 election, fully 8.6% of all American presidential elections have been unpopular”. But those authors also note that “estimates based on the historical record do not necessarily provide good predictions of the likelihood of an unpopular result in a future election because the sample size involved in these estimates is small, and because the political structure has changed dramatically many times in United States history. A more effective way to estimate the likelihood of a future unpopular election is to use the Monte Carlo method: simulate a large number of elections using a suitable randomization approach and then compute the percentage that result in unpopularly elected presidents, using as input data an appropriate set of recent presidential elections”. Neubauer et al. used the principal components approach to generate 20,000 trials and found a 4.9% frequency of unpopular elections. Some other authors have reached similar conclusions. For instance, as a consequence of estimating a probabilistic-voting model of electoral competition, Strömberg (2008) obtains that the probability of an election inversion to be about 4% (see also Chapt. 13 in Morton 2006). Ball and Leuthold (1991) as well as Merrill (1977) also estimate the likelihood of election inversions.

At about the same time, a McClatchy-Marist poll revealed that 52% of registered voters think that the popular vote should be the deciding factor in future elections, and 45% think the Electoral College should remain in place. Three percent of those polled were unsure.

The description below is borrowed from http://​www.​fairvote.​org/​past. Further historical information can be found in Wilderming (1958), Peirce and Longley (1981, Chapt. 6) and Schumaker and Loomis (2002). They discuss debates about changes in the Electoral College that go back to the early nineteenth century.

Interestingly, since both states have adopted the modification, the statewide winners have consistently swept all of the state’s districts as well. Consequently, neither state has ever had to split its electoral votes

We refer our readers to the website: http://​www.​nationalpopularv​ote.​com/​ and Chapt. 6 in Koza et al. (2013) for a more detailed exposition of that proposal.

According to Elliot and Ali (2007), “In the United States of America, an interstate compact is an agreement between two or more states which requires congressional ratification. ... As stated in Article I, Section 10, Clause 3 of the United States Constitution “No State shall, without the Consent of Congress ... enter into any Agreement or Compact with another State”. Koza et al. (2013, Chapt. 5) contains a background on interstate compacts.

See also Amar and Amar (2001).

For instance, Maskin and Sen (2017a), two outstanding social choice theorists, write “Currently, the most promising initiative to replace the Electoral College is the National Popular Vote Interstate Compact”. They point out, however, that “This condition creates a coordinating mechanism—states would move to the new system together, not unilaterally. So far, ten states and the District of Columbia have joined the compact, amounting to 165 electoral votes. All of them are solidly Democratic—probably reflecting the election of 2000, when Al Gore won the popular vote but lost in the Electoral College. In all likelihood, we will have to wait for an election in which the same thing happens to a Republican candidate before any red states sign on. More importantly, even if the compact succeeds (so that the Electoral College is in effect “replaced” ), the election system will remain highly unsatisfactory unless plurality rule—election by less than a majority—is also replaced”.

The media coverage of the fact that Donald Trump, the winner of the 2016 US presidential election, failed to win the popular vote has been very large. Even if it had been a matter of general discussion before (specially after the 2000 election), the 2016 election has resurrected the debate in the United States and beyond. About foreign coverage, let us mention, for instance, that French newspapers have dedicated a lot of space to that election and other peculiarities of the US Electoral College.

For an early analysis of that third feature, see Owen (1975, 2001).

Details are found in table 1.23 in Koza et al. (2013).

Koza et al. (2013) refers to Abbott and Levine (1991) to predict that emerging political trends would lead to more frequent election inversions. They also report some analysis by insiders like, for instance, Maureen Dowd of “How Obama could lose the popular vote and win the election” (Huffington Post, June 6, 2012).

The NPV plan is responsible for lively and partisan appraisals by politicians. Virgin (2017) reports some comments on the NPV plan given on December 7, 2011, by Mitch McConnell (R-KY), then-Senate Minority leader, calling it a ‘dangerous’ and ‘absurd’ scheme Democratic lawmakers and activists were ‘sneaking through’ under cover of metaphorical darkness. “They are well-funded, unfortunately, as they are well-organized, and they are getting close to the finish line”. “We need to kill it in the cradle before it grows up”. Similarly, on November 15, 2016, D. D. Wire reports in the Los Angeles Times the following statement by retiring Senator Barbara Boxer (D-CA): “In my lifetime, I have seen two elections where the winner of the general election did not win the popular vote. The Electoral College is an outdated, undemocratic system that does not reflect our modern society, and it needs to change immediately. Every American should be guaranteed that their vote counts.”

Koza (2016) answers the 24 criticisms of NPV in De Witt and Schwartz (2016), while Koza et al. (2013) develops “answers to 131 myths about the National Popular Vote Plan”.

To quote Miller “The third set of issues pertains to the durability of the interstate compact itself, especially in the face of controversies such as those noted above. The compact provides a nice example of a ‘social contract’ in a cooperative game but also highlights the problem of ‘credible commitment’—whether and how the terms of the ‘social contract’ among states could be enforced promptly and reliably in highly controversial cases. Clearly there would be strong incentives for some states to defect from the compact in precisely the circumstances in which the compact produces a winner different from the Electoral College winner, and such defections would tend to be legitimized in circumstances producing the kinds of controversies noted above”.

In response to the assertion the current system can reject popular-vote winners, they write “But what does that mean ? There are countless formulas for translating popular votes into ‘winners’. What is the right one ? Proponents of the compact may think that they are enforcing rules by majorities, but what they have actually proposed is rule by pluralities: whoever wins the most votes wins the election, even if those votes are not a majority”. This point is also raised by Maskin and Sen (2017a) who write: “More importantly, even if the compact succeeds (so that the Electoral College is in effect “replaced” ), the election system will remain highly unsatisfactory unless plurality rule—election by less than a majority—is also replaced”. Maskin and Sen (2017b) argue further that “There is a risk that the presence of additional major candidates might prevent any one of them from getting 270 votes in the Electoral College. This could be avoided by amending the Electoral College system so that the winner is the candidate who wins the nationwide vote under majority-rule voting. Such a change could be instituted, for example, by revising the National Popular Vote Interstate Compact initiative, in which a state pledges to award its electoral votes to the winner of the national popular vote as long as states totaling at least 270 electoral votes make the same pledge.”

We refer to Miller (2012a, b) for the details of his arguments. Along these lines, he also points out that “There is also the problem that states are not constitutionally required to hold direct popular votes for unified elector slates (so that non-member states might fail, even in principle, to produce statewide presidential popular vote counts), though of course at present all states (including Maine and Nebraska with their district systems) actually do this. The compact requires member states to hold such elections, but obviously it cannot require non-member states to do so. ” This echoes our early discussion of the 1960 US presidential election and the controversies about counting popular votes in Alabama.

Variants of Brams and Kilgour’s MPV plan were discussed before the official NPV plan was proposed formally. Section 6.5 in Koza et al. (2013) describes previous proposals for multi-state electoral legislation. They note on page 282 that “none of the earlier proposals contains a provision making the effective date of the system contingent on the enactment of identical laws in states that collectively possess a majority of the electoral votes”. Section 6.5 develops arguments against these earlier proposals.

Interestingly, Neubauer et al. (2012) also consider that possibility when they ask: “What if states instead took action before they commanded a majority?”

As noted by our referee, both the NMPV and RNPV plans (but specially the former) present most of the practical problems associated with NPV and discussed above.

We are thankful to our referee for calling our attention to that classic book and, in particular, the relevance of Chapt. 3. Schattschneider writes “the practice of prior consultation in order to agree upon a united front is an old one usually described by the word caucus”. In that chapter, Schattschneider pursues the strategic implications of caucuses in a legislative context. In our working paper version, we sketch some of the game-theoretical issues raised by the formation of a caucus. In particular, we introduce the question of other states’ reactions, which is an issue also addressed by Schattschneider under the heading ‘rival caucus’.

We thank our referee for calling our attention to that proposal, described in Koza et al. (2013, Sect. 6.5).

Interestingly, the RMPV alternative also appears in discussions of public mailing lists. For instance, in 2014, on https://​department-lists.​uci.​edu/​pipermail/​law-election, Sean Parnell wrote “Now here’s my question: under this theory, NPV’s inclusion of popular vote totals in non-compact states is basically a courtesy. If they wanted to, the NPV compact would be amended to simply say that member states would collectively award their electors to the candidate who receives the largest number of popular votes in the compact states, and simply ignore states that aren’t members of compacts. Furthermore, while the compact currently says that any state may join the compact, I assume that could be amended to say that a majority of states already in the compact must vote to approve the membership of other states who want to join, or some other limiting feature could be devised.”

In doing so, we follow an important part of the existing literature on the topic.

Therefore, malapportionment issues also are absent. It is not conceptually problematic to consider models integrating all of these effects. The probabilistic models describing richer environments are more complex: the input is not binary, voters and candidates may be strategic and voters may be ex ante biased towards some candidates.

This is properly demonstrated in de Mouzon et al. (2018) for election inversions. They also show however that with a generalized version of IAC with lower (but not null) levels of correlation, the probability of election inversion while small does not vanish to 0.

Some further developments illustrating these points in the simplest conceivable versions of the Electoral College are offered in the ‘long’ version of this work.

The exceptional paper of Weber (1978) should not be forgotten.

The mechanism is not constant.

In the terminology of simple games, the second tier is referred to as a weighted majority game.

Strictly speaking, the range of L is \(\left\{ 0,1,...,K\right\}\), where \(L=0\) means that no interstate compact forms. Note that for MRPV, the cases \(L=0\) and \(L=1\) are both equivalent to the Electoral College, while for MNPV, they are not: \(L=0\) and \(L=1\) are two different mechanisms. Only \(L=0\) corresponds to the Electoral College.

Hereafter, we will refer to the coalition of states identified by their membership to group 1 as an interstate compact. In a game wherein the players are the states, a coalitional deviation refers to a joint deviation from a profile of strategies. To emphasize, the “voting bloc” feature of the interstate compact, we could have instead, as suggested by our referee, chosen the term “bloc”. The term “bloc” or “resolute bloc” appears in Penrose (1946, 1952) to describe a set of players who decide to coordinate their votes. It has to be contrasted with the term ‘block’ which is used by Barberà and Jackson (2006) to define a new probabilistic model, which they call the ‘block model’. Therefore, in their setting the term ‘block’ refers to some exogenous variable while here the terms bloc or interstate compact refers to an endogenous variable.

If \(\sum _{1\le j\le L}n^{j}\) is an even integer (given our assumptions, that will happen when L is an even integer), the rule needs to be supplemented by a tie breaking rule. As already pointed out, that requirement is inconsequential for our analysis. However, to run our simulations, we had to make a choice. To preserve neutrality among candidates, we decided to depart from our deterministic framework by breaking any given tie by a fair coin flip.

On probabilistic models in general, see Gehrlein (2006) and Gehrlein and Lepelley (2011), Niemi and Weisberg (1972) and Straffin (1978).

This choice may seem cumbersome as the utility scale contains only two values. The more recent literature, including Barberà and Jackson (2006), and Beisbart et al. (2005, 2010), considers arbitrary utility scales.

This is equivalent to Weber’s (1978) effectiveness of an electoral mechanism, defined as

Therefore, it is ex ante optimal for any \(\pi\).

We have similar results for \(2\times 10^{k}+1\) voters per state for \(0\le k\le 5\). For \(k\ge 7\), but computing is time-consuming.

When L is odd. When L is even, it is slightly different. We obtain something similar when we break ties by either choosing one candidate or by flipping a coin.

This holds true for any probabilistic model \(\pi .\)

We have similar figures for \(K=101,1001\) and 10001.

We can show that for \(K=101\) and \(K=1001\), the picture remains qualitatively the same for a larger number of states, i.e., \(\Delta _{IC}(K,L)\) is single-dipped with a minimum at \(L=L^{**}(K)\). We speculate that \(L^{**}(K)\) also is a small number compared to K, but we do not have any specific mathematical conjecture to report on its asymptotic behavior.

The longer version contains some further developments on that issue.

These developments are reported in the longer version (see, appendix 3). They suggest an alternative program to perform numerical simulations.

It is conjectured in Le Breton et al. (2016) that \(c_{L}\simeq \sqrt{\frac{6L }{\pi }}\) when L (and therefore also K) is large enough.

We do not have a mathematical proof of this assertion and conjecture that it holds true.

The same shape shows up for all values of K that we have explored, including \(K=101,1001\) and 10001.

The same shape shows up for \(K=101\) and 1001.

This is not surprising. Indeed, the IAC’s average decisiveness curve always is constant since for any simple game, the sum of the IAC’s individual probabilities of being pivotal is equal to 1.

At least, in the toy version that is considered here.

Since this is true when the interstate compact contains a majority of states, it is true when the size of the interstate compact is close to that size. Note also that this argument does not hold when the interstate compact contains a single state since the first effect is totally absent.

In the longer version (see, appendix 6), we report some analytical arguments to illustrate the importance of conditioning.

Precisely, \(0.5+\left( 0.5\right) \times \frac{2}{10^{4}\pi }\).

Precisely \(0.5+\left( 0.5\right) \times \frac{\sqrt{2}}{10^{4}\sqrt{\pi }}\).

The differences to which we alluded before can be made more precise when m is large. We refer to appendix 7 for a proof that, up to a multiplicative constant, the BI share behaves approximatively as \(\frac{1}{\sqrt{m}}\) under IAC and as \(\frac{1}{ m}\) under \(IAC^{*}\).

These figures are available from the authors upon request.

As also noted by Miller (2009) in his analysis of the Electoral College and some of its reforms: “A measure of a priori voting power takes account of the fundamentals of a voting rule but nothing else. Thus the following analysis takes account only of the 2000 population of each state and the District of Columbia, the apportionment of electoral votes based on that population profile, and the requirement that a Presidential candidate receive 270 electoral votes to be elected. It does not take account of other demographic factors, historical voting patterns, differing turnout rates, relative party strength, survey or polling data, etc. This indicates the sense in which a priori voting power analysis is conducted behind a ‘veil of ignorance’ and is blind to empirical contingencies.”

In particular, for the case of the US Electoral College, i.e., on the basis of the current state populations and electoral vote allocations, Neubauer et al. (2012) estimate the consequences of different reforms, including the NMPV plan.

This could be illustrated in a stylistic situation with a biased state (all citizens vote definitely for one candidate) and two swing states (described by the IC model). Evaluations of the states are ex ante evaluations like an insurance contract between two parties. In the hypothetical classroom situation where preferences are such that in half plus one of the states, one candidate wins for sure with a minimal majority and does not get any votes in the other half minus one of the states, uncertainty has disappeared. With such extreme ideological bias, ex ante gains do not exist. We are left with a straightforward ex post dispute. The candidate winning a majority of the states always will prefer the Electoral College while the other always will prefer the popular vote. As noted by Hinich et al. (1972), in such a case election inversion is certain.

For instance, Sect. 9.31 in Koza et al. (2013) entitled “Myth that a Nationwide Vote for President would Favor one Political Party over the Other” examines 14 issues listed under that heading. In his econometric study, Virgin (2017) finds evidence that loyalties to the state and to the party may be competing. Although NPV advances furthest when Democrats control state lawmaking, a state’s status a swing- but not as an over-represented- state weakens the relationship to the point where even Democrats are unlikely to support NPV.

This asymmetric treatment of citizens is listed as one of the three shortcomings of the current system by Koza et al. (2013).

See, for instance, among many others, Brams and Davis (1974) for an early model addressing that question and Strömberg (2008) for a recent analysis of how US presidential candidates should allocate resources across states to maximize the probability of winning the election, based on the estimation of a probabilistic-voting model of political competition under the Electoral College system.

Scholarly works on campaigning games and redistributive politics coincide with common wisdom to conclude that swing states indeed receive more attention at equilibrium.

Under IC, IAC and \(IAC*\), the probability of a tied election is very small in the case of the popular vote if the population of voters is large. If asymmetries among the candidates are introduced, the probability of tied elections is even smaller (see, for instance, Chamberlain and Rothschild 1981).

See, however, Sects. 9.7 and 9.8 in Koza et al. (2013) for discussions of some of these issues.

The spoiler effect is a critical issue. Section 3 in Hinich et al. (1972) contains an interesting analysis of the consequences of having national or regional third parties.

In the current version of the Electoral College, the winner takes all of the electoral votes in a state according to plurality rule. But even without altering the winner-take-all principle, we could conceive of alternative rules: Borda, instant runoff, plurality with runoff and Condorcet, to cite a few.