Mind Your Decisions

Game Theory and Voting

7 October 2008

Read more Game Theory (new article every Tuesday)

Voting is a prized civic duty. But it is also a high stakes competition with clear winners and losers. In this light, voting can be viewed as a contest of strategy. If you want to make your vote count, it is useful to learn the rules of the game.

History suggests that some votes don’t “count.” In at least five U.S. presidential elections, the most popular candidate ended up losing (more on this below in point 5). The results depended on a variety of factors, including strategic voting. The lesson? We need to be on guard. Under current rules, votes can be gamed.

How can we make elections fair? Eventually we might adopt a different voting system, one less susceptible to strategy. But until we get there, we can level the playing field by telling people about voting theory. Doing so would lesson the impact of gaming. We would return voting to its original purpose: a test of popular will.

In that spirit, I share some ideas from voting and game theory. There are too many to do full justice so I will focus on 5 important ones:

1. The pivotal vote: one reason the Vice Presidential candidate really matters

A pivotal (or decisive) vote is a tie-breaking vote. An example: if your friends are deadlocked on a vote of where to eat, and they ask you to make the call, then you hold the pivotal vote. Because the pivotal vote tips a decision, it holds much power.

There’s one voting tie deemed so important that the U.S. Constitution weighs in. The question is about what to do when the Senate is tied on a decision. The constitution grants the pivotal vote to the Vice President, which turns out to be a big power.

The first Vice President John Adams cast 29 tie-breaking votes, more than anyone since. His votes were pivotal in deciding the location of the nation’s capital and preventing war with Great Britain. More recently, Al Gore cast 4 tie-breaking votes and Dick Cheney 8. (Read a summary of the others).

Although ties are infrequent in the 100 person Senate, they occur on the most divisive and often controversial issues. The lesson is that we must demand a competent Vice President.

I’m sure that message is weighing on the minds of people who have decided to vote. But there are many people who don’t think voting pays. Why do they say that, and does it make sense? These questions motivate the next two topics.

2. The payoff to voting, or why many economists don’t vote

In a game-theory sense, your vote matters only when it is pivotal. The proof follows from a thought experiment. If the election was hypothetically decided by two or more votes, then you could have safely abstained from voting without affecting the majority rule. In other words, your vote was not needed.

How often will your vote be pivotal? A mathematical approach is to calculate the odds that all the other voters will be tied. The approach treats each voter as having some probability of voting for one candidate or the other. The odds of a tie are maximized when each voter is equally likely to vote for one candidate or the other. Here are some estimates from this methodology. At 1,000 voters, the optimistic odds of a tie, making you pivotal, are less than 3 percent. At 100 million voters, the optimistic odds are less than 0.01 percent (roughly 1 in 10,000).

In fact, the true odds are lower because candidates are not equally favored. Small preferences among voters can lead to margins of victory that make your vote irrelevant. The odds can be estimated in an empirical approach that examines at the history of elections. This exercise was done by economists Casey Mulligan and Charles Hunter, and here are their results as summarized in the New York Times:

Even in the closest elections, it is almost never the case that a single vote is pivotal. Of the more than 40,000 elections for state legislator that Mulligan and Hunter analyzed, comprising nearly 1 billion votes, only 7 elections were decided by a single vote, with 2 others tied. Of the more than 16,000 Congressional elections, in which many more people vote, only one election in the past 100 years - a 1910 race in Buffalo - was decided by a single vote. (source)

The conclusion is that your vote is very, very unlikely to affect the outcome. An economic argument extends the logic to say “voting doesn’t pay.” This is because voting has little expected benefit but costs time and effort. This view holds voting in the same light as buying a lottery ticket: a losing bet.

But people still buy lottery tickets, and they still vote on the order of millions. Might there be a reason?

It turns out there might be one that depends both on strategy and on logic.

3. The paradox of the bright line: what if no one votes?

Mathematical calculation and empirical studies can miss the importance of strategy. Whether your vote counts ultimately depends on what others do. It’s a question of strategy and best responses.

Imagine what would happen if a lot of voters decided “voting doesn’t pay.” Voting turnouts would dwindle and your chance of being pivotal would skyrocket. In the extreme case, you might end up being the only voter and decide the entire election. Suddenly voting would pay.

At what point does voting tip from being not worth it to being worth it? You might not find it worthwhile to vote in a landslide, but you would get out of your house for a close election. It turns out there is no “bright line” for voting.

Here is one interesting narration of the problem:

The absence of a “bright line” dates back to the Greek philosopher Zeno, who tells the paradox in terms of creating a mound from grains of sand one at a time. It seems true that no one grain can turn a non-mound into a mound. And yet, enough grains will turn a molehill into a mountain. A vote is much like a grain of sand. It is hard to imagine how one additional vote will change anyone’s perception of the outcome. (source: Thinking Strategically).

In this light, deciding whether to vote is much like deciding whether to visit a popular bar. You only want to go when the bar is not too crowded, but you have to decide in advance, and everyone is thinking the same thing. The bar situation is known as the El Farol Bar problem and has some interesting conclusions.

Of course, game theory only tells part of the story. Many people find value in voting besides being the pivotal voter. It is a civic duty and brings a happiness that’s hard to measure.

So for the next two points, let’s assume you want to vote. The next topic to consider is how can you make your vote more powerful?

4. Voting blocs, or the power of groups

A voting bloc is a group of voters that choose the same decision. In high school elections, perhaps the cheerleaders or the debate team acted as voting blocs. In political races, certain well-organized groups can be thought of like voting blocs.

Voting blocs gain power through unity, and they can be resilient to change. A good example comes earlier this year when Israel was considering a change in its Supreme Court. At the time, rulings were made by a simple majority of 5 out of 9 judges. The law committee wanted to raise the standard to a larger number to encourage discussion and potentially limit strategic voting. Would it work?

One problem is that voting blocs could retain power. Nobel Laureate Robert Aumann spoke to the law committee and explained that a voting bloc of three judges would hold the same power, whether the majority was 5, or 6, or 7. The calculation depends on using a “power index,” which is a numerical way to estimate the power of voters or blocs of voters. The idea is technical, so see an intuitive explanation in this article. (By the way, power indices have applications in areas besides voting. Here is a surprising application in genetics research).

Registered party members can also be thought of like voting blocs, though members don’t always vote for the party’s favored candidate.

In fact, that is one of the reasons elections can be gamed.

5. Vote splitting, or how third party candidates spoil elections

Vote splitting is the phenomenon where people divide votes among similar candidates, giving dissimilar candidates a voting advantage. In U.S. races, people say vote splitting “spoils” elections because it can result in the second-most popular candidate getting elected. (If enough votes to the most-popular candidate were split or “stolen” by a third-party candidate).

The phenomenon is documented in William Poundstone’s book Gaming the Vote. The book is very enjoyable and there is a whole chapter devoted to the history of vote splitting. The examples are worth knowing and can be good cocktail party stories. The best part of the chapter is the conclusion, which is a summary of how vote splitting has spoiled elections:

In 1844, an abolitionist spoiler put a slave-owner in the White House.

In 1848, a former Democratic president sabotages the Democratic Party’s changes.

In 1884, a Prohibition Party candidate helped elect a supposedly “ally of the saloon.”

In 1912, a former Republican candidate prevented the reelection of a Republican president.

In 2000, a consumer and environmental advocate elected the favored candidate of corporate America.

Poundstone explains that spoilers have affected 5 out of 45 elections since 1828, a failure rate of over 11 percent. If you are considering voting for a third-party candidate over your party’s front-runner, and still don’t realize you could be throwing your vote away, I ask that you read those examples again.

Vote splitting allows for much strategy during campaigns. The 2004 election had some very interesting punches and counterpunches. Democrats, aware that Ralph Nader might spoil the election, sent emails urging supporters to attend Nader’s conventions to crowd out authentic supporters. In response, Republicans encouraged supporters and essentially paid people to collect signatures so he would get on the ballot.

Nader has been making political calculations as well. In the 2000 election, Nader was aware of his spoiler role. Why did he run anyway? Some quotes indicate he was confident Gore was going to win anyway, so his spoiler role was not that harmful and it would help give attention to his causes. Other quotes suggested he was content being the spoiler because allowing Bush to win over Gore would be a punishment. The administration would act so poorly that it would create a backlash and a leftward trend in America, which would be better in the long-term. That’s quite the view of looking ahead and reasoning back.

It’s eight years later and we have to wonder: was it worth it? Only time, and perhaps this election, will tell if the strategy will paid off.