When a choice is being made between two candidates, the first type of voting system to suggest itself is majority rule: Each voter indicates a preference for one of the two candidates, and the one with the most votes wins. With two candidates, there is no real distinction between a ballot that indicates a voter’s choice for one of the two candidates and what we have called a preference list ballot. The point is that we can, for example, identify a choice for (however indicated) with the list that has over and a choice for with the list that has over .
Majority rule has at least three desirable properties:
The desirability of these three properties, at least in the most common kinds of elections, is easy to see. For example, condition 1 reflects the (grammatically outdated) tenet “one-man, one-vote” that gained prominence from its use in a 1964 Supreme Court majority opinion. Indeed, if any voter had two or more votes, then an exchange of his or her ballot with that of a voter who only had one vote might well change the outcome of an election. If a voting system violated condition 2, then one of the candidates must have an advantage built in by the system. And as for condition 3, any system that is not monotone has the rather bizarre property that a candidate could have enough support to win, but gathering additional support might cause this same candidate to lose.
Explain (in a sentence or two each) why majority rule satisfies each of the three desirable properties.
It is easy to devise voting systems for two candidates in which these properties fail, but each such voting system quickly reveals its undesirability. For example, condition 1 is not satisfied by a dictatorship (in which all ballots except that of the dictator are ignored); condition 2 is not satisfied by imposed rule (in which Candidate wins regardless of who votes for whom); and condition 3 is not satisfied by minority rule (in which the candidate with the fewest votes wins).
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But maybe there are voting systems in the two-candidate case that are superior to majority rule in the sense of satisfying the three properties just listed and some other properties that we might also wish to have satisfied. This, however, turns out not to be the case. In 1952, Kenneth May proved the following, aptly named May’s theorem.
May’s Theorem THEOREM
Among all two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is monotone.
This is an important and elegant result. Thus, mathematical reasoning spares us the trouble of searching for a better voting system for two candidates.
But what if there are three or more candidates? Perhaps we can design a voting system for this situation that, in some way, builds on the success of majority rule in the two-candidate case. In point of fact, there does exist a voting system that arises from precisely this hope, and it is known today as Condorcet's method.
Our description of Condorcet’s method begins with the observation that if we have a sequence of preference list ballots, then—for each pair of candidates—we can determine who the winner would have been had the election involved only these two in a one-on-one contest using majority rule.
To illustrate this notion of a one-on-one contest, consider the following preference list ballots:
| Rank | Number of Voters (3) | ||
|---|---|---|---|
| First | |||
| Second | |||
| Third | |||
In this election, Candidate would defeat Candidate in a one-on-one contest (two votes to one), while would, in turn, defeat in a one-on-one contest, again by a score of 2 to 1. We’ll return to this example in a moment, but we now have at hand all we need to describe Condorcet’s voting system for three or more candidates.
Description of Condorcet’s Method PROCEDURE
With the voting system known as Condorcet’s method, a candidate is a winner precisely when he or she would, on the basis of the ballots cast, defeat every other candidate in a one-on-one contest using majority rule.
Historically, the voting system we are calling Condorcet’s method dates back at least to Ramon Llull in the 13th century (see Spotlight 9.1). It was rediscovered and popularized in the 18th century by the Marquis de Condorcet (1743–1794).
Let’s look at a couple of examples of elections using Condorcet’s method.
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The Historical Record Spotlight 9.1
The following letter was written by Friedrich Pukelsheim of the University of Augsburg, Germany. He is imagining what Ramon Llull (1232–1316) might say if he were alive today.
Dear Editors:
It is my distinct pleasure to respond “from the beyond” to your kind invitation to set the historical record straight. I was born in 1232 on the Island of Mallorca in the Mediterranean Sea, which in your times is known as a popular tourist place. In my days it was a strong political center of that part of the world, with a population that was a mix of Christians, Jews, and Muslims. It was my dream to persuade people of the virtues of Christian belief by relying, not on force, but on reason.
Unfortunately, people did not find it easy to follow my arguments, so I was more than pleased to discover some down-to-earth applications, including an election system. My idea was to oppose every pair of candidates, one on one, and ask the electors whom of the two they would prefer—very much like a medieval jousting tournament. But how to combine the results from all the duels into a winner of the election? I first proposed electing the candidate who won the most duels, then later suggested a system of successive eliminations.
I wrote three papers on the topic, the second of which I “smuggled” into my novel Blanquerna in 1283. More than a century after my death, in 1428, the young German scholar Nicolaus Cusanus (1401–1464) journeyed to Paris to read my works in libraries there. He even copied out the third of my electoral writings, which I had completed on 1 July 1299 in Paris, and his manuscript is the only copy handed down to your days. Reading my papers, Cusanus was inspired to invent his own electoral system. Did he not understand mine, or just find it inadequate? Who knows?
While I had been concerned with electing Church officials, Cusanus sought a system to elect the Holy Roman Emperor. In his system, each elector assigns each candidate a rank score, with the lowest candidate getting a score of 1, the second lowest a score of 2, and the best candidate the highest score possible, that is, 10 when there are 10 candidates. The scores are totaled for each candidate and the candidate with the highest score wins. If you are a soccer player or a hockey player, you will have a good sense for one difference between our systems: Whereas I count victories, Cusanus adds up goals. Cusanus applauded himself for having invented an absolutely ingenious and novel electoral system.
Also, I advocated open voting, whereas Cusanus favored a secret ballot. He was concerned that voters might sell their votes, or that the candidates might pressure the voters. Well, that certainly happened all of the time in elections for worldly authorities! But for election to clerical office, I thought it good enough if electors took an oath to vote for the most worthy candidate and submitted themselves to the social control that comes with an open election.
Cusanus was famous in his times, as I was in mine, but fame indeed is transitory. Sure enough, my electoral system was reinvented by the Marquis de Condorcet (1743–1794), and Cusanus’s system was proposed afresh by the Chevalier de Borda (1733–1799)—neither of whom, I am sure, wasted a thought on the possibility that “their” systems might already be on record. But, as my works had fallen into oblivion as had those of Cusanus, neither Condorcet nor Borda should be blamed for failing to acknowledge our priority.
My first electoral paper—actually the one that is longest and most detailed, written around 1280—was rediscovered only in 2000, filed away in the Vatican Library. How would you feel if your work attracts fresh attention after more than 700 years? Actually, I am utterly pleased that mine has resurfaced at last! The text was excavated by a mathematician interested in voting systems, Friedrich Pukelsheim of the University of Augsburg, Germany. Since the text is handwritten in Latin, handling it became an interdisciplinary project that brought together experts on medieval manuscripts, Church Latin and theology, and even computer scientists. As a result, my electoral writings are now on the Internet (in the original and in translations into English and German) at www.uni-augsburg.de/llull/.
Looking back on my lack of success in preaching peace among Christians, Jews, and Muslims, and all the writing and copying by hand of my works, I hope you can appreciate how highly I value the printed book (such as this one) and, even more, instant communication worldwide over the Internet. May that ease of communication help facilitate the religious peace that I so dearly sought.
Yours truly,
Ramon Llull (1232-1316)
Left Choir Chapel
San Francisco Cathedral
Palma de Mallorca
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EXAMPLE 1 Condorcet’s Method
The mathematics department is hiring a new faculty member and the five-person hiring committee has interviewed four candidates: Adam, Beth, Carol, and Dan. They have decided to use Condorcet’s method on their five ballots (reproduced in the table below). Let’s see who gets the offer.
| Number of Voters (5) | |||||
|---|---|---|---|---|---|
| Rank | Voter 1 | Voter 2 | Voter 3 | Voter 4 | Voter 5 |
| First | Adam | Dan | Carol | Adam | Beth |
| Second | Beth | Carol | Beth | Carol | Dan |
| Third | Carol | Beth | Dan | Dan | Carol |
| Fourth | Dan | Adam | Adam | Beth | Adam |
To find a winner using Condorcet’s method, we begin by choosing an ordering of the candidates, which we’ll take to be alphabetical for this example. Thus, we first pit Adam against each of the others in a one-on-one contest based on the ballots. In an Adam-versus-Beth election, based on these ballots, Voters 1 and 4 would vote for Adam; and Voters 2, 3, and 5 would vote for Beth. Hence, Beth would win this one- on-one contest against Adam, so we know Adam is not going to be the winner using Condorcet’s method. But Beth still has a chance, so we move on to see how Beth would fare in a one-on-one contest against Carol (knowing already that Beth would defeat Adam). But here it is easy to see that Beth would lose to Carol (with Voters 2, 3, and 4 voting for Carol).
Hence, Beth is not a winner with Condorcet’s method, but Carol has not yet been eliminated (and neither has Dan, but remember that we have chosen to check out the candidates in alphabetical order). First pitting Carol against Adam, we see that Carol wins 3 to 2. And pitting Carol against Dan, we see that Carol again wins by this same 3 to 2 score. This shows that Carol is the winner using Condorcet’s method!
By the way, if it seems like we never gave Dan a chance, notice that our determination that Carol is the Condorcet winner means that we already know that she beats Dan one on one. So Dan can’t be a winner in this election using Condorcet’s method.
Suppose that the department makes an offer to Carol, as the above vote suggests, but she refuses the offer. The department decides to again use Condorcet’s method with the same ballots, but with Carol’s name erased. Who gets the next offer?
EXAMPLE 2 Condorcet’s Method
Suppose we have four candidates (, , , and , with these initials chosen for a soon-to-be-revealed reason) and the following sequence of preference list ballots, where the heading of “6” indicates that 6 of the 15 voters hold the ballot with over over over , the heading of “5” indicates that 5 of the 15 voters hold the ballot with over over over , and so on.
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| Number of Voters (15) | ||||
|---|---|---|---|---|
| Rank | 6 | 5 | 3 | 1 |
| First | ||||
| Second | ||||
| Third | ||||
| Fourth | ||||
We claim that is the winner in this election if we use Condorcet’s method. Let’s check the one-on-one scores for each possible pair of opponents:
versus : is over on of the ballots, while the reverse is true on of the ballots. Thus, defeats by a score of 8 to 7.
versus : is over on of the ballots, while the reverse is true on 3 of the ballots. Thus, defeats by a score of 12 to 3.
versus : is over on of the ballots, while the reverse is true on 1 of the ballots. Thus, defeats by a score of 14 to 1.
This shows that is the winner using Condorcet’s method.
Like majority rule, Condorcet’s method satisfies some very desirable properties, as we’ll see later in this section. But it also has a tragic flaw, and this flaw is called Condorcet’s voting paradox.
Condorcet’s Voting Paradox THEOREM
With three or more candidates, there are elections in which Condorcet’s method yields no winners. In particular, the following ballots (often called the “Condorcet voting paradox ballots”) constitute an election in which Condorcet’s method yields no winner.
| Rank | Number of Voters (3) | ||
|---|---|---|---|
| First | |||
| Second | |||
| Third | |||
The Condorcet voting paradox ballots given above are the same ones we used earlier in illustrating the notion of a one-on-one contest. We pointed out then that defeats one on one and defeats one on one. The additional observation needed is that we also have defeating one on one. Thus, cannot be a winner using Condorcet’s method (he or she loses to ), cannot be a winner (he or she loses to ), and cannot be a winner (he or she loses to ). We will revisit Condorcet’s voting paradox in Section 9.3.
Notice that because of our assumption that the number of voters is odd, Condorcet’s method yields either no winner or a unique winner (see Exercise 35).
It is tempting at this point to suggest modifying Condorcet’s method as we have presented it by declaring all the candidates to be tied for the win if there is no candidate who defeats each of the others one on one. The drawback to this modification is that a number of the desirable properties possessed by Condorcet’s method then evaporate. We’ll explore this in the upcoming exercises.