Elections: Unintuitive Behavior of Seemingly Nice Voting Systems
York College (CUNY)
Jamaica, NY 11451
Over a very long period of time many appealing methods of amalgamating the opinions (votes) of individuals into a societal choice have been proposed. Almost from the start it became apparent that one could find examples of particular patterns of voter ballots where particular methods seemed to find a "peculiar" choice as the winner. These examples are also useful in showing the reach of Arrow's Theorem and Gibbard-Satterthwaite Theorem. More specifically, one shows using examples that a particular election decision method can be "manipulated" in a particular way, or that a particular decision method violates monotonicity. Complicating the situation is that one can seek examples with the smallest number of voters or with the fewest different kinds of preference schedules, or involving the smallest number of alternatives as possible. While these example may not be likely to occur in practice they are of interest not only in their own right but as ways to get students thinking about various aspects of the way mathematics is used in the world, and the tools via which mathematical questions can be attacked. There are a variety of ways one can organize examples. One can look at examples which show how various systems can fail to meet a particular fairness rule (axiom) or one can show for a particular system examples which show the limitations of that system. Here we will do a little of each!
Consider the following 17 vote election:
There is no candidate who gets a majority (there is a tie between A and B), and a tie between A and B for plurality winner. If we use run off or sequential run off, we eliminate candidate C, and in the race between A and B, A would be the winner.
Now suppose that the last two voters decide to make things even better for A by giving him more support. Thus, they interchange their ranking of A and B giving the ranking of the three candidates as:
So replacing the two original ballots with these two new ones, we have the election:
In this election the plurality winner A but A does not have a majority. If we use run off or sequential run off, the candidate with the smallest number of first place votes is B. Thus, B is eliminated. In the resulting election between A and C, C beats A 9 to 8. Thus, by swinging support from their second choice to their first choice, seemingly to give A more support, these voters wind up defeating A! This seems like a "perverse" property of run off and sequential run off.
Here is a way of describing this "perverseness."
If an election decision method would make Z the winner for a particular set of ordinal ballots and one or more voters change their preference with regards to Z so that Z is treated in a more favorable way (without changing the the order in which they prefer the other alternative candidates) then Z should still be the winner.
Plurality voting satisfies (winner) Monotonicity, while run off and sequential run off do not, as shown by the example above.
Condorcet Loser Property
Suppose an alternative Z would lose in every two way race with every other alternative, then a "reasonable" voting rule should not choose Z as the winner.
Here is an election which shows that some methods can select a Condorcet Loser.
Note that B is the Condorcet winner in this election since B can beat A and B can beat C in a two way race. However, the plurality winner is C. C loses to both B and A in two way races. Hence, the plurality method can elect a Condorcet Loser.
Independence of Irrelevant Alternatives
Suppose we think of the votes below as being the preferences of 9 friends at a restaurant and who must make of a choice of one of the three prix fix dinners A, B, or C.
To see what is happening we will construct the pairwise preference matrix:
The row sums in the matrix above are 9 for A, 4 for B, and 8 for C. Thus, the Borda count winner is A, yet the Condorcet winner is C, who beats both A and B in a two-way race.
This example can be used to illustrate the idea of "irrelevant alternatives." If we rank all three candidates A, B, and C using the Borda Count we obtain the ranking:
So based on the Borda Count the choice would be A. However, note that had the waiter informed the group that B is not truly available, then the ballots above imply that C is preferred to A by 4 to 3!
Note that if A is not a choice C is still above B, and if C is not a choice A is still above B.
Thus, we see by this example that the Borda Count violates the "fairness rule" known as the Independence of Irrelevant Alternatives.
For the fun of it, let us look at the Condorcet Ranking. that comes from the example above.
Note that this ranking is independent of irrelevant alternatives. If C is not on the ballot, A still beats B, if B is not on the ballot, C still beats A and if B is not on the ballot C still beats A.
Note: We have been considering elections where all the voters rank all of the candidates and there are no ties in the voters preferences. Many fairness axioms are concerned with the case where votes submit rankings (without indifference) between some proper subset of the alternatives (candidates).
For example, here is an example of such a fairness rule:
A candidate should not be hurt by taking some ballots that do not have X ranked in first place and replacing these ballots by ones where X is ranked at the top but there are no candidates listed below X.
This "axiom" is sometimes called MONO-SUB-PLUMP.
de Swart, and A. Deemen, E. van der Hout, P. Kop, in Theory and Applications of Relational Structures as Knowledge Instruments, Lecture Notes in Computer Science, Vol. 2929, Springer, 2003, pp. 147-195.
Straffin, P., Topics in the Theory of Voting, Birkhäuser, Boston, 1990.
Woodall, D., Monotonicity of single-seat preferential election resultsws, Discrete Applied Mathematics 77 (1997) 81-98.