Less then Perfect

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451



web page:


The fact that there are elections in which there is no Condorcet winner may seem to be nothing more than a quaint curiosity. Unfortunately, the situation is not this benign.

You should verify for yourself that the election below has no Condorcet winner. B beats A in a two-way race, A beats C in a two-way race, and C beats B in a two-way race.

In voting bodies such as the United States Senate, the United States House of Representatives and other similar legislative bodies votes are taken between pairs of options that take the form of votes between pairs of alternatives, usually one of the alternatives being the status quo (do nothing). The order in which voting takes place is often specified via a "rules committee" or by "rules of order." One sometimes speaks of an "agenda" that governs the order in which the votes are taken.

Consider the voting situation above, where A is the status quo, and B is a proposed change which has been amended to take form C. Hence, the legislators have to form opinions of these three alternatives. Suppose the preferences of the 31 legislators is as above. I claim that what emerges as the "law of the land" depends merely on the order in which the votes are taken!!!

Suppose that a vote is taken between A and C and the winner is voted on against B. Since A beats C, A wins, and when A and B are voted on, B becomes the law of the land.

Suppose a vote is taken between A and B and the winner is voted on against C. Since B beats A, when B and C are voted on, C becomes the law of the land.

Finally, suppose B and C are voted on first, and the winner is voted on against A. Since C beats B, when C and A are voted on, A becomes the law of the land.

Thus, the legislative results depend only on the order in which votes are taken, not on the content of the actions being considered! Of course, this unpleasant situation is a result of the sharply divided opinions of the legislators, which takes the form of the ballots produced above.

Another important issue for the fairness of election method systems is raised by the examples below. Seemingly appealing and intuitively "fair" procedures such as Sequential Run-off (single transferable vote or STV; also sometimes called IRV, instant Run-off voting) do not obey the monotonicity principle. Monotonicity is the principle that if voters push a candidate higher on their ballots, this should not harm the candidate whose name was moved higher.

Example (S. Brams, Professor of Government at NYU)

The number of votes for the different preference schedules above, reading from left to right are: 7, 6, 5, and 3. votes.

Since no candidate has a majority (11 votes), the lowest vote getter D is eliminated. In the next round C gets the votes that went to D in the previous round but still no candidate has a majority. Hence, B is eliminated. In the final round vote between A and C the winner is A, by 13 to 8. Now consider what happens if the three voters who ranked D first change their ranking by interchanging A and D, leaving the positions relative to the other candidates the same. The new election is shown below:

The number of votes for the different preference schedules above, reading from left to right are: 7, 6, 5, and 3. votes.

Now, in the first round C is eliminated. In the election that results between A and B, B wins 11 to 10! Thus, increased support for A results in B's winning instead of A. This feature of the Single Transferable Vote (sequential run-off) is rather disconcerting!

Here is another example.

Example: (H. Moulin; Moulin until recently a professor at the University of Texas in Austin, but now returned to Europe, where he trained as a mathematician.)

This example shows that the run-off method does not obey
the Monotonicity Principle.

Consider first the election L:

votes for the preference schedules from left to right: 6 4 2 5

Since C gets the fewest first-place votes, alternative C is eliminated and in the run-off phase of the decision process, A beats B by 11 votes to C. Hence, A is the winner.

Now consider election M, which differs from L in that the preference schedule which was voted for by two voters has been altered so that now alternative A is preferred to alternative B. Note C is still rated last by these two voters.

Election M:

votes for the preference schedules from left to right:: 6 4 2 5

In this election, B gets the fewest first-place votes and is, therefore, eliminated. In the run-off phase of the election between A and C, C beats A by 9 to 8. Hence, the additional support that A received costs A the election!

Yet another example:

Here are two elections, "before" and "after" some of the voters have changed their preferences.

Compute the winner of the elections using a variety of methods and see what you discover.

Warning: If an election method behaves well in particular cases, will this guarantee that it will behave well in all examples?