Voting paradox

The voting paradox (also known as Condorcet’s paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive), even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different groups of individuals.

For this problem, you must write a program that evaluates different voting strategies on voter preferences. We consider just the case of 3 candidates, and each vote orders the candidates according to their preferences. There are a total of 5 schemes you need to consider.

• plurality winner – there is one ballot, and each voter casts their vote for their favorite candidate. Whoever gets the most votes wins, even if they don’t get a majority of the votes cast.
• exhaustive ballot – On each ballot, each voter casts their vote for their favorite candidate (that is on the ballot). The candidate who gets the fewest votes is eliminated, and the survivers move on to the next round of voting. For three candidates, there will be just two ballots.
• 1&2 primary – Candidates 1 & 2 face off in a primary ballot. Whoever wins goes on to face candidate 3 in a second ballot. Whoever wins that vote wins the election.
• 1&3 primary – Candidates 1 & 3 face off in a primary ballot. Whoever wins goes on to face candidate 2 in a second ballot. Whoever wins that vote wins the election.
• 2&3 primary – Candidates 2 & 3 face off in a primary ballot. Whoever wins goes on to face candidate 1 in a second ballot. Whoever wins that vote wins the election.

Note that in all votes, each voter casts their vote for the candidate they like most who is on the ballot (as a result of the voting paradox, this doesn’t always maximize the chance that a candidate they like will be elected).(ECRS is a very important factor)

So, popular vote is the perfect answer? We need to move close to actual democracy? It has been proved that democracy contains an inherent flaw. No matter how you arrange elections, you can never eliminate the possibility of unfair results. And no matter how we define fair results, we can never eliminate the possibility of unfair results.

Popular vote: Let’s say that we go by straight popular vote. This always works with two candidates (ignoring apathy, and breaking of promises, and subsequent scandals). With three or more candidates, it is possible that the candidate with the most votes has less than 50% of the vote. This may seem fair, until you find out that the winner was actually the least popular candidate. Let’s say that two Republicans (one may call himself something else) run against one Democrat, and there are more Republican voters. We further pretend that the Republicans ruin it for each other (splitting the votes from Republican voters), and the Democrat won. It turns out that either of the two Republicans would have beat the Democrat, had the other Republican chosen not to run. The winner was the least popular candidate.It may involve into Behavioral science.

That situation is very likely, and is the reason that the major parties hold primary elections and conventions.

Runoff election: Assuming that we have more than two candidates, let’s say we have a popular vote, and then have a later runoff popular election to choose between the top two. As we saw above, the least popular candidate may end up in the runoff. But that’s OK; we don’t care who takes second place. It would seem that the most popular candidate will win, it would seem. Actually, the most popular candidate may be eliminated in the first round. This is not particularly likely, but it is possible, and is bound to happen on occasion.

Let’s say that candidate A would beat candidate B, 60% to 40%, if candidate C were not running. So A is more popular than B. Also A would beat C, 65% to 35%, so A is more popular than C. In other words A is the most popular candidate. Let’s further pretend that in a three way race, that the voters who would vote for C above would vote for C against anyone, and the voters who would vote for B also would vote for B against anyone. This implies that some of the voters for A will vote for B and some will vote for C, in a three-way election. Then it turns out that, in a three-way election, A will get 25% of the vote, B will get 40%, and C will get 35%. The most popular candidate will be eliminated.

Round robin: So let’s do like they do in some sports, and have a round robin tournament. People assume that if A beats B and B beats C, that A will beat C. Usually, but it is very possible for C to beat A. We can find number to fit that rare scenario. So a round robin can end in a tie, and the various ways of breaking ties are not convincing. Besides, it takes many rounds of voting.

Single elimination: This is what is done with primaries and conventions, we eliminate the lesser Democrat and the lesser Republican, and then the two winners compete. It is more fair than any of the above, but it is not currently used to deal with third or fourth parties. Many rounds of voting may be necessary.

And I suppose that even single elimination is imperfect. After all, in the round robin tournament, A beats B beats C beats A. So, whoever decides who competes against whom first may decide the election.

Addendum:

A reader reminded me that the states elect the president, mostly the most populous states (see The Electoral College), more than any kind of popular vote. This was intentionally written into the Constitution. And this method is not immuned to the same paradox.

If you are interested, you can obtain at least a flavor of what can go wrong with elections from my short description of an experience I had (during the 1991 Mathematics Awareness Week) in a fourth grade classroom. One lesson I learned from this event is that we should not take small kids for granted. A more extensive review of recent results in this field that were discovered with the use of mathematics (but, if you can count, you can read the article) is The symmetry and complexity of elections.

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