Notes on Modeling: Session IV

Prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

1. We have seen that voting and elections is a complex environment varying with regard to the number alternatives to choose from, whether a single candidate is to selected or a ranking (with or without ties) is to be produced, what kind of ballot is used, and the way that the ballots are acted upon (election decision method).

There are many appealing election decision methods. In the period since the seminal work of Kenneth Arrow, it is typical to look at what fairness properties (axioms) these different decision methods obey so that one can see the pros and cons of selecting different alternative methods.

As an example of a different fairness rule from any yet discussed consider:

Majority:

Suppose that more than half of the voters have the same set M of candidates (but the order they are listed can vary from one voter to another) at the "top" of their ballots, then at least one of M's members should be selected.

(The wording of this allows for the case that we are selecting more than a single winner in the election.)

Example:

Consider this example (due to Douglas Woodall) which features ballot truncation:

Plurality Elects C, A is the Condorcet winner. Thus, we see that the plurality method need not elect the Condorcet winner when there is one (we have seen this many times in the past) but also, plurality does not obey the "Majority axiom." This is because Majority would require that B or A win since this set is is at the top of more than half of all all the voters ballots. You may want to think about what would happen if these voters had not truncated their ballots but put the missing candidate from each ballot type below the candidates already listed.

It is also now understood that with information about the voting system to be used and knowledge about what other voters are likely to do, that single or groups of voters may try to produce ballots other than "sincere" ballots in order to get an outcome that is viewed as being superior. Some people think that because of this, poll information just prior to an election not be made available to the public.

There have been many threads of interest as a consequence of Arrow's work.

a. How likely is it that "Condorcet paradox" will occur in actual elections? Can one find examples where it appears to have occurred? (Answer: yes.)

b. If a group of voters has some common core of values, might there be methods which would obey the fairness axioms one would like and which are decisive? Results in this direction are theorems of Sen and Black. These give sufficient conditions under which a set of ballots will have a Condorcet winner. Black's Theorem says that if a collection of ballots is single peaked, then there will always be a Condorcet Winner. However, the maximal number of single peaked ballots with n alternatives is 2n-1 while voters, even if indifference is not allowed, might produce n! ballots. Thus, it seems not very likely that one can be certain there will be a Condorcet winner in real elections because the ballots produced are indeed single peaked.

c. Another line or research concerns election decision methods which when there is a Condorcet winner chooses the Condorcet winner and which chooses some winner in all cases. A simple to describe example of such a method is the "sequential run off" based on the Borda Count. Thus, compute the Borda Count of all alternatives, eliminate the candidate with lowest Borda Count. Remove this candidate from the ballot set and repeat the process. The winner must be the Condorcet winner if there is one.

d. There is a growing list of axioms to consider when one tries to find an attractive voting decision method. To get some idea of the richness of ideas that are being considered here, you may want to download some of the papers of Douglas Woodall which can be found in the author list for Voting Matters. What is referred as STV (Single Transferable Vote) gives rise to the same method as what we have been calling sequential run-off (IRV) when there is only one seat to be filled. However, when the term STV is used it is nearly always in the context of proportional representation issues, which come to the fore when in the election one is trying to select more than one candidate from a field of many alternatives. The two papers on the web site above:

http://www.votingmatters.org.uk/AUTHOR.HTM

by Droop and Hare are 19th century contributions to elections and voting theory.

2. We will now move to other aspects of voting. Here the idea is that in some types of voting situations, the "players" are not of equal status. In this case the people who are voting ('players") can cast votes of unequal strength. Thus, in a legislative body involving the European Union, it might not seem reasonable to have Germany and France cast the same number of votes as Liechtenstein since Germany and France have more people and greater economic power than Liechtenstein. Systems of this type can sometimes be implemented as weighted voting systems. Here, there is some quota Q to "pass a bill." The different players cast votes of a certain weight, and when the weights of the votes cast be a "coalition" of players is greater than or equal to Q the "bill" passes. Many of the legislative bodies within the European Union operate in this way. Another voting system of this kind is the American Electoral College. Here there are 51 players, one for each state and the District of Columbia. The player for DC casts three votes. Each other player casts a vote equal to the sum of the number of senators and members of the House of Representatives that state has. Thus, California casts many more votes than Wyoming, which seems reasonable since California has many more people than Wyoming. We will see that unfortunately, the analysis of weighted voting games has "mysteries" similar to those we encountered when we looked at voting decision methods.

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