4. Who should win?

Condorcet Cycles

The question:

In most modern, western countries, we are justly proud of democracy. All citizens have their say in who will lead and administer the country. Every few years, nationwide elections are held – one person one vote! – and the candidate who obtains most votes is elected. Similarly, in board meetings, committees, in businesses, schools, and among friends, we simply take a vote and the majority decides. It is the fairest way.

But is it really? Does our beloved majority rule really reflect the true will of the voters?

The paradox:

Peter, Paul and Mary must decide what to order for their after-dinner drinks. Peter prefers Amaretto to Grappa, and Grappa to Limoncello. Paul prefers Grappa to Limoncello, and Limoncello to Amaretto. Finally, Mary prefers Limoncello to Amaretto, and Amaretto to Grappa.

Peter: Amaretto > Grappa > Limoncello

Paul: Grappa > Limoncello > Amaretto

Mary: Limoncello > Amaretto > Grappa

The three diners decide to go by the majority opinion. A majority prefers Grappa to Limoncello (Peter and Paul), and a majority prefers Amaretto to Grappa (Peter and Mary). Based on these two rounds they can make their decision: order a bottle of Amaretto.

But surprise, surprise: Paul and Mary point out that they would prefer even Limoncello, the lowest ranked option, over Amaretto. How come? Here is the clincher: had the three campers had a third round of voting, between Limoncello and Amaretto, a majority would have preferred Limoncello (Paul and Mary). So let them buy Limoncello and get it over with. But wait a minute. Buy Limoncello, and Peter and Paul will protest just as vigorously. They prefer Grappa to Limoncello. A paradox.

History:

It was the 18th-century French nobleman Marquis de Condorcet (1743-1794), who first identified the problem. One of the foremost intellectuals before and during the French Revolution, the Marquis was a mathematician, an economist, a political scientist, and a defender of human rights.[1]

Condorcet wrote important works that combined mathematics and social issues. Some of his most intriguing texts were his contributions to the theory of voting and elections. In 1785 Condorcet wrote a 200-page pamphlet entitled “Essay on the application of probability analysis to majority decisions”. And that is where he pointed out the paradox.

One way to overcome it was suggested by Condorcet’s contemporay and compatriot, Jean-Charles de Borda (1733 - 1799). He too a nobleman, he too a scientist of note, Borda found his calling not in politics but in the military and distinguished himself in the navy as a formidable maritime engineer.

Borda’s scholarly achievements include important advances in experimental physics and engineering, in geodesy, cartography and other areas. He had also shown interest in the subject of voting and elections and wrote an article entitled “Essay on ballot elections” in 1781. (See below.)

Dénouement:

To understand the mathematical reason underlying the paradox, let’s compare majority rule with weights and measures. If Marc is taller than Nancy, who is taller than Oscar, then Marc is definitely taller than Oscar. In mathematical lingo: the measurements of persons’ heights are transitive. But in elections, ‘majorities prefer A to B, and B to C’ does not imply that ‘A is preferred to C by the majority’. In other words, the Condorcet Paradox arises because majority opinions are not transitive.

***

In Borda’s proposal each voter ranks the candidates according to his or her taste, and then awards them points according to their rank. The candidate at the bottom is given one point, the next-lowest two, the next three, and so on, until the top-ranked candidate is awarded the most points by each voter. Then the total number of points are added up for each candidate, and the one with most points is elected. With many voters awarding their points, it is rare that two or more candidates obtain the exact same number. Hence, there is usually a well-defined Borda-winner which is why Borda-counts are often used in games and TV-shows.

Technical supplement:

Borda’s proposal has problems of its own. First of all, the winner of the Borda count may be nobody’s favorite. Let’s say 30 students mst elect their class president. Their preferences are as follows:

11 students: Paul > Mary > John > Peter

10 students: Peter > Mary > John > Paul

9 students: John > Mary > Peter > Paul

In the traditional voting method, Paul would be elected – though only by what is called a plurality, not a majority). But in a Borda-count, Paul would get 63 points, Peter 69, John 78. Mary, whom nobody really likes, would win with 90 points.

Then, there is the problem that the winner of a Borda-count depends crucially on the number of points awarded at each rank. After all, there is no intrinsic reason why each rank should be rewarded with exactly one additional point. Why not give each voter a certai amount of points, say a hundred, that she can then allocate in any manner, according to the intensity of her feelings towards the candidates? The Borda-winner would vary, depending on the exact system used. In the Peter, Paul, John & Mary example above, for example, if ten points are awarded to the top ranked candidate, six to the second, five to the third, and zero to the last, John would be the winner with195 points, and Mary the runner-up with 180

There is the Bozo-problem. By entering the race, a totally irrelvant candidate could change the outcome. Even though he would be ranked low on every voter’s list, his addition may influence the election’s outcome. Let us assume that 51 electors prefer Ginger to Fred, and 49 prefer Fred to Ginger:

51 electors: Ginger > Fred

49 electors: Fred > Ginger

The Borda count declares Ginger the winner with 151 points, and Fred 149. Now Bozo appears on the scene. Nobody really likes Bozo but his entry persuaded 3 of Fred’s voters to rank Ginger even behind Bozo:

51 electors: Ginger > Fred > Bozo

46 electors: Fred > Ginger > Bozo

3 electors: Fred > Bozo > Ginger

Now Ginger receives 248 points, Fred 249 and Bozo 102. Bozo’s entry caused Fred to win. (See also the chapter on the Independence of Irrelevant Alternatives.).

* * *

Condorcet proposed his own solution. It was as simple as it was impractical.

Every candidate would be paired against every other candidate in a series of showdowns. Each time, the voters express their preferences and the candidate who receives the majority of the votes is considered to be superior to the other. After all pairings have been performed, the candidates are ranked. The person who comes out on top, who beat all other candidates, will be declared the winner.

But things are not quite as simple. First of all, mathematical combinatorics implies that Condorcet’s proposal with N candidates would require N(N-1)/2 showdowns. For twenty candidates, that would mean 190 pairings!

Second, in general, no unambiguous ranking can be drawn up because he results of the pairings are, again, not transitive. Cycles appear, the very cycles that led to the Amaretto-Grappa-Limoncello paradox in the first place.

[1] For more on this subject and about this fascinating personality, see my Numbers Rule (Princeton, 2010).

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