Voting Arithmetic

Voting is simple when there are only two candidates and lots of voters: people vote and one of the candidates will simply get more votes than the other (except in the very unusual case of a tie when a coin is needed). In other words, the candidate who gets a majority of the votes wins; naturally, this is called majority voting. Things start to get complicated when there are more than two candidates and a majority of the votes is required to elect a candidate; in this case, multiple ballots and lots of horse trading can be required until a winner emerges.

The U.S. has continued many practices inherited from England such as the system of common law. One of the most important of these practices is the way elections are run. In the U.K. and the U.S., elections are decided (with some exceptions) by plurality : the candidate who polls the largest number of votes is the winner. Thus if there are 3 candidates and one gets 40% of the votes while the other two get 30%, the one with 40% wins – there is no runoff. This is called plurality voting in the U.S. and relative majority voting in the U.K.

The advantages of plurality voting are that it is easy to tabulate and that it avoids multiple ballots. The glaring disadvantage is that it doesn’t provide for small party representation and leads to two party dominance over the political system – a vote for a third party candidate is a vote thrown away and counts for nothing. As a random example, look at the presidential election in the Great State of Florida in 2000: everything would have been the same if those who voted for Ralph Nader had simply stayed home and not voted at all. As a result, third parties never get very far in the U.S. and if they do pick up some momentum, it is quickly dissipated. In England, it is similar with Labour and the Conservatives being the dominant parties since the 1920’s. Is this handled differently elsewhere? Mystère.

To address this problem of third party representation, countries like Italy and Holland use proportional voting for parliamentary elections. For example, suppose there are 100 seats in parliament, that the 3 parties A, B and C propose lists of candidates and that party A gets 40% of the votes cast and B and C each get 30%; then A is awarded 40 seats in parliament while B and C each get 30 seats. With this system, more than two parties will be represented in parliament; if no one party has a majority of the seats, then a coalition government will be formed.

Another technique used abroad is to have majority voting with a single runoff election. Suppose there are 4 parties (A,B,C,D) with candidates for president; a first election is held and the two top vote getters (say, A and B) face off in a runoff a week or two later. During this time, C and D can enter into a coalition with A or B and their voters will now vote for the coalition partner. So those minority votes in the first round live to fight another day. Also that president, once elected, now represents a coalition which mitigates the kind of extreme Manichaeism of today’s U.S. system. It can lead to some strange bedfellows, though. In the 2002 presidential election in France, to stop the far-right National Front candidate Jean-Marie LePen from winning the second round, the left wing parties had to back their long time right wing nemesis Jacques Chirac.
However, one might criticize and find fault with these countries and their election systems, the fact is that voter participation is far higher than that in the U.S.

For U.S. presidential elections, what with the Electoral College and all that, “it’s complicated.” For Hamilton, Madison and others, the Electoral College would serve as an additional buffer between the masses and the government: one way this was to be achieved was by means of the “faithless elector,” one who does not vote for the candidate he pledged to – this stratagem would overturn a mass vote for a potential despot. This was considered a feature and not a bug; this feature is still in force and some pledged electors do employ it – in the 2016 election, seven electors voted against their pledged candidates, two against Trump and five against Clinton. But, except for faithless electors, how else could the Electoral College stymie the will of the people? Mystère.

That the Electoral College can indeed serve as a buffer between the presidency and the population has been proven by four elections (1876, 1888, 2000, 2016) where the Democratic candidate carried the popular vote but the Republican candidate obtained a majority in the Electoral College; most scandalously, in the 1876 election, in a backroom deal, 20 disputed electoral votes were awarded to the Republican candidate Rutherford B. Hayes to give him a majority of 1 vote in exchange for the end of Reconstruction in the South – “probably the low point in our republic’s history” to cite Gore Vidal.

That the Electoral College can indeed serve as a buffer between the presidency and the population has also been proven by the elections of 1800 and 1824 where no candidate had a majority of the electoral vote; in this case, the Constitution specifies that the election is to be decided in the House of Representatives with each state having one vote. In 1824, the populist candidate, Andrew Jackson, won a plurality both of the popular vote and the electoral vote, but on the first ballot a majority of the state delegations, cajoled by Henry Clay, voted for the establishment candidate John Quincy Adams. In the election of 1800, Jefferson and Burr were the top electoral vote getters with 73 votes each. Jefferson won a majority in the House on the 36th ballot, his victory engineered by Hamilton who disliked Jefferson but loathed Burr – we know how this story will end, unfortunately.

For conspiracy theorists, it is worth pointing out that not only were all four candidates who won the popular vote but lost the electoral vote Democrats but that three of the four were from New York State as was Aaron Burr.

The most obvious shortcoming of the Electoral College system is that it is a form of gerrymandering that gives too much power and representation to rural states at the expense of large urban states; in English terms, it creates “rotten boroughs.” For example, using 2018 figures, California has 55 electoral votes for 39,776,830 people and Wyoming has 3 votes for 573,720; so, if one does the math, 1 vote for president in Wyoming is worth 3.78 votes in California. Backing up, let us “show our work.” When we solved this kind of problem in elementary school, we used the rule, the “product of the means is equal to the product of the extremes”; thus, using the camel case dear to programmers, we start with the proportion

     votesWyHas : itsPopulation :: votesCaShouldHave : itsPopulation

where : is read “is to” and “::” is read “as.” Three of the four terms have known values and so the proportion becomes

     3 : 573,720 :: votesCaShouldHave : 39,776,830

The above rule says that the product of the inner two terms is equal to that of the outer two terms. The term votesCaShouldHave is the unknown so let us call it ; let us apply the rule using * as the symbol for multiplication and let us solve the following equation:

     3 * 39,776,830 = 573,720 * x

which yields the number of electors California would have, were it to have as many electors per person as Wyoming does; this simplifies to

     x = (3 * 39,776,830)/ 573,720 = 207.99

So California would have to have 207.99 electors to make things fair; dividing this figure by 55, we find that 1 vote in Wyoming is worth 207.99/55 = 3.78 votes in California. This is a most undemocratic formula for electing the President. But things can get worse. If the race is thrown into the House, since California has 69.33 times as many people as Wyoming, the ratio jumps to 1.0 to 69.33, making this the most undemocratic way of selecting a President imaginable. For state by state population figures, click  HERE .

One simple way to mitigate the undemocratic nature of the Electoral College would be to eliminate the 2 electoral votes that correspond to each state’s 2 senators. The would change the Wyoming/California ratio and make 1 vote in the Equality State worth only 1.26 votes in the Golden State. With this counting technique, Trump would still have won the 2016 presidential election 236 to 195 (much less of a “massive landslide” than 304 to 227, the official tally) but Al Gore would have won the 2000 race, 228 to 209, even without Florida (as opposed to losing 266 to 271).

To tally the Electoral College vote, most states assign all their votes (via the “faithful” pledged electors) to the plurality winner for president in that state’s presidential tally. Nebraska and Maine, the two exceptions, use the congressional district method which assigns the two votes that correspond to Senate seats to the overall plurality winner and one electoral vote to the plurality winner in each congressional district in the state. By way of example, in an election with 3 candidates, suppose a state has 3 representatives (so 5 electoral votes) and that one candidate obtains 50% of the total vote and the other two 25% each; then if each candidate is the plurality winner in the vote from exactly one congressional district, the top vote-getter is assigned the 2 votes for the state’s senators plus 1 vote for the congressional district he or she has won and the other two candidates receive 1 electoral vote each. This system yields a more representative result but note that gerrymandering will still impact who the winner is in each congressional district. What is intrinsically dangerous about this practice, though, is that, if candidates for more than two parties are running, it can dramatically increase the chances that the presidential election will be thrown into the House of Representatives. In this situation, the Twelfth Amendment ordains that the 3 top electoral vote-getters must be considered for the presidency and so, if this method had been employed generally in the past, the elections of 1860 (Lincoln), 1912 (Wilson) and 1992 (Clinton) could well have given us presidents Stephen Douglas, Teddy Roosevelt and Ross Perot.

The congressional district method is a form of proportional voting. While it could be disastrous in the framework of the Electoral College system for electing a U.S. president, proportional voting itself is successfully implemented in many countries to achieve more equitable outcomes than that furnished by plurality voting and the two party system.

A voting system which is used in many American cities such as Minneapolis, and Oakland and in countries such as Australia and Ireland is known as ranked choice voting or instant-runoff voting. Voters in Maine recently voted for this system to be used in races for seats in the U.S. House of Representatives and in party primaries. Ranked choice voting emulates runoff elections but in a single round of balloting; it is a much more even-handed way to choose a winner than plurality voting. Suppose there are 3 candidates – A, B and C; then, on the ballot, each voter lists the 3 candidates in the order of that voter’s preference. For the first round, the count is made of the number of first place votes each candidate received; if for one candidate that number is a majority, that candidate wins outright. Otherwise, the candidate with the least number of first place votes, say A, is eliminated; now we go into the “second round” with only B and C as candidates and we add to B’s first place total the number of ballots for A that listed B as second choice and similarly for C. Now, except in the case of a tie, either B or C will have a clear majority and will be declared the winner. This will give the same result that staging a runoff between B and C would yield. With 3 candidates, at most 2 rounds are required; if there were 4 candidates, up to 3 rounds could be needed, etc.

Most interestingly, in the Maine 2018 election, in one congressional district, no candidate for the House of Representatives gathered an absolute majority on the first round but a candidate who received fewer first place votes on that first round won on the second round when he caught up and surged ahead because of the number of voters who made him their second choice. (For details, click  HERE ).

Naturally, all this is being challenged in court by the losing side. However, for elections, Section 4 of Article 1 of the U.S. Constitution leaves implementation to the states for them to carry out in the manner they deem fit – subject to Congressional oversight but not to judiciary oversight:

   “The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing (sic) Senators.”

N.B. In this article of the Constitution, the senators are an exception because at that time the senators were chosen by the state legislatures and direct election of senators by popular vote had to wait for 1913 and the 17th Amendment.

At the first legal challenge to it, the new Maine system was upheld vigorously in the United States District Court based in large part on Section 4 of Article 1 above. For the ruling itself, click  HERE . But the story will not likely end so simply.

This kind of voting system is also used by the Academy of Motion Picture Arts and Sciences to select the nominees in each category, but they call it preferential voting. So to determine five directors, they apply the elimination process until only five candidates remain.

With ranked choice voting, in Florida, in that 2000 election, if the Nader voters listed Ralph Nader first, Al Gore (who was strong on the environment) second and George Bush third and if all Pat Buchanan voters listed Buchanan first, Bush second and Gore third, Gore would have carried the day by over 79,000 votes in the third and final round.

However one might criticize and find fault with countries like Australia and Ireland and their election systems, the fact is that voter participation is far higher than that in the U.S. For numbers, click  HERE .

Ranked voting systems are not new and have been a topic of interest to social scientists and mathematicians for a long time now. The French Enlightenment thinker, the Marquis de Condorcet, introduced the notion of the Condorcet Winner of an election – the candidate who would beat all the other candidates in a head-to-head election based on the ballot rankings; he also is the author of Condorcet’s Paradox – that a ranked choice setup might not produce a Condorcet winner. To analyze this situation, the English mathematician Charles Lutwidge Dodgson introduced the Dodgson Method, an algorithm for measuring how far the result for a given election using ranked choice voting is from producing a Condorcet Winner. More recently, the mathematician and economist Kenneth Arrow authored Arrow’s Paradox which shows that there are ways in which ranked voting can sometimes be gamed by using the idea behind Condorcet’s Paradox: for example, it is possible that in certain situations, voters can assure the victory of their most preferred candidate by listing that candidate 2nd and not 1st – the trick is to knock out an opponent one’s favorite would lose to in a head to head election by favoring a weaker opponent who will knock out the feared candidate and who will then be defeated in the final head to head election. For his efforts, Arrow was awarded a Nobel Prize; for his efforts, Condorcet had a street named for him in Paris (click  HERE  ); for his efforts, Charles Lutwidge Dogson had to latinize his first and middle names, then reverse them to form the pen name Lewis Carroll, and then proceed to write Alice in Wonderland and Jabberwocky, all to rescue himself from the obscurity that usually awaits mathematicians. For a detailed but playful presentation on paradoxes and ranked choice voting, click  HERE .

2 thoughts on “Voting Arithmetic

  1. “The congressional district method is a form of proportional voting.” — This statement surely needs to be qualified at least, as the recently elected House shows. Even without gerrymandering, this method distorts majorities for structural reasons.

    1. Point well taken. The paragraph about Maine, Nebraska and congressional district voting has been revised accordingly. The edits might still not capture your comment “for structural reasons” however.

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