Math Goes Pop!

As promised, in this thrilling final installment to the relationship between math and voting (the first two parts can be found here and here), we will look at what many people see as the holy grail of voting systems: Range voting.

The concept of range voting is simple. Given a set of candidates, in a range voting system you simply put a score next to each name that reflects how strongly you support that candidate. Of course, this is quite different from our current voting system, where we only get to vote for one candidate, but more importantly, it differs significantly from other voting systems where you are just asked to rank candidates in order of preference, because a ranking gives no information about the degree to which your support varies from candidate to candidate.

For example, if Anna, Bob, and Charlie are all running for President, you and I may both prefer Anna to Bob, and Bob to Charlie. However, I may LOVE Anna and HATE Charlie, while you may be relatively indifferent, with only a slight preference for one over the other. In a ranked voting system, both our preferences would be recorded as A > B > C. However, using range voting, our preferences may look something more like: A = 99, B = 50, C = 0 for me, and A = 51, B = 50, C = 49 for you.

This example shows that range voting allows us to capture more information about voter preferences than the other voting systems discussed. Therefore, one might heuristically expect that because it captures more information, it leads to better results, i.e. a more accurate representation of the will of the people. The question, of course, is whether this is actually the case.

The answer may depend on your definition of “better,” but by most measures the range voting system comes out on top of the others. One important feature of this system is that it is not subject to the constraints of Arrow’s Impossibility Theorem (discussed in Part 1). In other words, range voting has basically every property you would like a voting system to have. Range voting doesn’t contradict Arrow’s theorem because Arrow’s theorem deals only with voting systems that only rank preferences.

There are many other benefits to range voting as well, and these benefits are no doubt well known by any range voting advocate. The website Rangevoting.org gives a list of reasons why range voting is so great – I won’t list them all here, but I will highlight some of the more interesting ones.

• Range voting encourages honest voting, rather than strategic voting. There is never an incentive for you to score a candidate you support more lower than a candidate you support less.
• Range voting allows for a larger range of political parties to flourish. Because you are not restricted to one vote, people from third parties can feel free to support their candidate without fear of “wasting their vote.” This is also good for independents who may not feel particularly strongly about any major party candidate.
• (Perhaps the cutest result) Range voting maximizes the number of “pleasantly surprised” voters, i.e. the number of voters for whom the winner of the election is better (scored higher) than they thought it would be.

As with any other idea, though, range voting is not without its share of criticism. However, these criticisms pale in comparison to the critiques that can be made about our current voting system. The main critique with range voting has to do with strategic voters, and comes in two forms:

• Why doesn’t this just degenerate into the system we already have? For example, if supporters for one candidate feel strongly enough, they will simply give that candidate the highest score and every other candidate a 0. Doesn’t this benefit dishonest voters, and hurt candidates whose supporters are honest and may not give their candidate the full score, or score everyone else with a 0?

This criticism is a little suspect, because while there are certainly people who may vote in this way, it’s certainly hard to believe that everyone will, or that even a disproportionate number of supporters of one candidate will. It’s more likely that roughly the number of people for each major candidate will feel strongly enough to vote in this way, so that in the end it should all balance out. There certainly are extremes of political belief, but this is true of both the left and the right, with a wider swath of moderates somewhere in the middle.

• Doesn’t the system inflate support for third party candidates? For example, people will be more likely to throw support to a candidate they believe has no chance of winning – this will amplify support for lesser known candidates, and dampen support for well known candidates.

This seems plausible. One way to combat this is to require that a candidate receive a minimum number of scores in order to be viable. For example, we could say that in order for a candidate to be declared the winner, at least 10% of the population must have voted for the candidate. The percentage should be high enough to be significant, but not so high that it’s possible it couldn’t be obtained if enough voters strategically abstain from voting for a particular candidate.

However, it’s still far more likely that a major candidate with a large base of support will win over an independent candidate with a small base of fervent supporters. Overall, range voting will certainly reflect preferences better than the current system, so it’s hard to argue that this is much of a valid criticism when compared to the current system, where your support is for all intents and purposes meaningless unless it is for a major party candidate.

In summary, from a mathematical point of view, there really is no argument: range voting certainly trounces are current voting system, and it looks like it beats everything else as well. The question then becomes: why don’t we use it? I’m not sure there’s a good answer.

Range voting is not used by any democratic nation, but examples of it can be found all over the internet. For a concrete example you are probably familiar with, you need look no further than the Internet Movie Database. On this site, you are free to vote on any movie you like, giving it a rating between 1 and 10. These votes are then compiled into IMDB’s rankings of the top 250 movies of all time, which can be found here. What’s even better, the votes are compiled using a true Bayesian estimate, the formula for which can be found at the bottom of the page. If you have any doubts about the validity of range voting, you need only go view this list and see all the awesome movies on it to conclude that indeed, this system has it going on.

Of course, you may find that this list of movies is terrible, but in this case, don’t worry. It doesn’t mean that range voting doesn’t work, it just means you have bad taste in movies.

In conclusion, our voting system is horribly broken. There are solutions out there, but getting from where we currently are to a new voting system is a problem that goes beyond the scope of a pop culture math blog. For now, we’ll have to deal with things as is, and of course, that includes voting tomorrow, November 4th. So make sure you go out and do it. We can fight the larger fight of voting systems another day.

As you may recall, I have already discussed certain perils associated with different voting systems. However, given all the commotion this election is causing, I thought it may be worthwhile to discuss voting in a bit more detail.

There is plenty of information online regarding the relationship between math and voting, for those with interest enough to seek it out. But perhaps the best centralized internet location on this topic comes from this year’s Mathematics Awareness Month website.

In April of every year, mathaware.org hosts a Mathematics Awareness month, complete with articles and contests related to the year’s theme of forging a bridge between mathematics and what is often times a seemingly disparate discipline. It was no doubt with tremendous foresight that they selected “Mathematics and Voting” for this year’s theme.

A good way to kill a few minutes is with their voting methods simulation. On this page, you can vote for potential presidential candidates using three voting rules. The first voting rule is the standard: vote for a candidate. The second rule allows you to cast a vote for as many candidates as you wish. In the third rule, you are asked to rank your candidates from first to last, and points are then awarded to each candidate according to the Borda Count – essentially, if you are choosing between n candidates, the Borda count awards your first choice n – 1 points, your second choice n – 2 points, and so on, so that your last choice receives 0 points.

I highly recommend you vote for yourself. Once you do so, you can view the results, which are quite interesting in and of themselves.

Beyond that, there are several interesting articles you can peruse at your leisure by following this link. I don’t have the time to discuss every article, but I will highlight a couple here.

The first, in very literally keeping with the theme, is called “Mathematics and Voting,” by Professor Donald G. Saari. This article talks in detail about the problems with voting systems, which include the three given in the simulation discussed above. All the voting systems under consideration fall prey to Arrow’s theorem (discussed in the last post), but even so, it is natural to ask: are some voting systems more broken than others? The short answer is yes.

Professor Saari not only discusses how bad things can become (for instance: given three candidates to choose between there is a roughly 69% chance that the choice of voting rule can alter the outcome of a close election), he discusses how and why some voting rules are “better” than others (by “better” I mean that some voting rules do not allow as many undesirable election outcomes as others). In fact, the aforementioned Borda count allows for far fewer potential election outcomes than most other ranking systems.

Does this suggest that in some sense, the Borda count is better than other voting rules? It is even more interesting to ask this question in light of the simulation election results one can view after voting.

Another paper (which I highlight because it is such a short read) is titled “Yes, It Is Rational to Vote” by Professor Andrew Gelman. He offers an interesting rebuttal to the commonly held view that it is irrational to vote, because the chances of your vote actually determining the outcome of an election are so minimal. His argument is that even if you believe your preferred candidate will bring a marginal increase in the quality of life to the citizenry (equivalent to some small monetary sum, say $50), when we multiply this montary amount by the population of America, the expected value of your vote becomes non negligible. Although the odds of your vote being decisive are extremely small (especially in red state or blue state strongholds), a small benefit compounded by 300 million people then translates into a significant return.

There are plenty of other articles as well, and while some are more technical than others, certainly all of them merit a look.

I would like to close by pointing out that all of the voting systems so far discussed have been based on a ranking system. Given candidates A, B, and C, you are asked to order your preferences, and every voting rule is an attempt to quantitatively translate your preference.

However, just giving a preference does not give the whole story. For example, you and I may both prefer A over B and B over C, but you may love A and hate C, while I feel relatively indifferent among the three candidates. In this case, although we would both say that our rankings are A > B > C, this simple ranking ignores crucial information about the degree of our preferences.

Next time, I will look at the concept of Range voting, which attempts to address this issue. One interesting feature of range voting is that it satisfies all the conditions of a voting system one would like to have, in what may seem like a contradiction to Arrow’s Theorem. It is, however, because of the added information put in to the range voting model that Arrow’s Theorem is not violated. We will discuss this in more detail, and look at perhaps the most common example of range voting, next time.

Many students often ask their teachers, “Why do I have to learn this boring mathematics? Nobody uses mathematics anyhow.” This new feature, entitled Math Gets Around, will attempt to show you that in fact, mathematics will pop up even in the least likely of places. So learn those multiplication tables, chief.

Today, we see how mathematics has weaseled its way into an unlikely place: the realm of politics. This is particularly relevant given the fact that, as some of you may have heard, there is a presidential election in just a few short months.

Among the general population, there will always be dissidents who complain of the failings of our democratic process. Among these dissidents, you may even find those who question the existence of our two party system, and claim that a system with a larger number of parties would be better for everyone involved. But I am here to tell you the shocking truth: from a mathematical standpoint, this is not the case.

Let me explain what I mean in plain terms. Elections are a part of our democracy. In order to ensure that elections are fair, you would like your process to have certain properties. In particular, any reasonable person should agree that any voting system should satisfy the following three properties:

1) the system should not be a dictatorship – in other words, one person’s preferences can’t be imposed on the results of the election. One can call this the dictatorial property.

2) the system should allow for an individual to rank the candidates in any order imaginable; in particular, any candidate on the ballot should be able to win. One can call this the exhaustion property.

3) the system should be non-manipulable, by which I mean that there are no conditions under which a voter could vote in a manner that does not reflect his or her true preferences in order to achieve the long-term goal of having his or her true preferred candidate win the election. One can call this the manipulability property.

Unfortunately, The Gibbard-Satterthwaite theorem tells us that in any voting system with three or more candidates, and at least two voters, no such voting system exists. In other words, any voting system with more than two candidates must either be dictatorial, non-exhaustive, or manipulable.

Since any system which doesn’t satisfy the first two conditions is impractical, the theorem usually amounts to saying that any voting system you will encounter in real life with more than three candidates must be manipulable.

Oh, but come now, Matt, you might say. Manipulable elections? What hogwash! For this, I turn your attention towards none other than the 2003 Governor election in our fine state of California. Here I quote from the Wikipedia entry on “tactical voting:”

One high-profile example of tactical voting was the situation that led to the 2003 California recall. During the primaries, Republicans Richard Riordan (former mayor of Los Angeles) and Bill Simon (a self-financed businessman) were vying for a chance to compete against the unpopular Governor of California, Gray Davis. As California holds open primaries in which anyone can vote for any candidate he or she pleases, Davis supporters were rumored to have voted for Simon because Riordan was perceived as a greater threat to Davis; this combined with a negative advertising campaign by Davis describing Riordan as a “big-city liberal”, and Simon ultimately won the primary despite a last-minute business scandal. However, he lost the election against Davis; discontent soon led to the recall.

Further examples can be found across the globe (click the link above to read in more detail).

Senators Obama and McCain discuss ways to try and outfox mathematics.

Fine, you might say. But what if we don’t want our elections to necessarily pick winners and losers? Elections, at the end of the day, are merely collections of lists of individual preferences. Is there a way that we can use this large pool of individual data to come up with a preference list that works for the entire community, subject of course to some reasonable assumptions? This subject is taken up in Arrow’s Theorem. The assumptions for the voting system under this theorem are as follows:

1) The voting system should not be dictatorial (see above).

2) The aggregate preference list compiled from individual voting preferences should account for everyone’s vote in providing a ranking for the group, and it should do so in a well defined way – in other words, if two collections of preferences are equivalent (say if person A and person B simply swap their voting sheets), then the ranking for the group should be unchanged. This is referred to as the universality property.

3) Say you prefer candidate A to candidate B, and suppose now that candidate C decides to enter the race. You must alter your preferences to reflect this fact; in other words, how to you feel about C relative to A and B? Whatever your feelings are, when C enters the race, it’s natural to impose the restriction that your preferences for A and B can’t change – for example, if A > B, when C enters the race your list of preferences could be A > C > B, C > A > B, or A > B > C, but not C > B > A, because if you prefer B to A when C is in the race, why wouldn’t you prefer B to A when C is ignored? This property, that preferences for a subset of the candidate list should not contradict preferences for the whole list, is called the independence of irrelevant alternatives, or IIA for short.

4) If everybody in the group prefers A to B, then the ranking for the group should also prefer A to B. This is called unanimity, or Pareto efficiency.

Arrow’s theorem tells us that no such ranking system can satisfy all of the properties given above. Sadly, it would seem that from a mathematical standpoint, no voting system can get it quite right.

However, as with most things in life, there is a silver lining. If you feel our system of elections is broken, don’t worry – you can take solace in the fact that any other voting system you can imagine is probably broken too.