Imagine you go to a carnival for fun. You buy a fried Twinkie, see the world’s largest living pig, and perhaps ride the whirl-and-puke. You then head over to the midway with the intent of playing some silly games for entertainment. This carnival, however, is different – along with usual staples like the ‘milk bottle ring toss’ and themed shooting galleries, there’s a coin flipping game. You are given a penny, and you get to flip this penny 1000 times. The number of heads you get is recorded, and whoever gets the most heads is awarded a monetary prize. There is no monetary cost to play the game, but you do have to sit there and flip the coin 1000 times. A question arises: How large would the prize have to be in order to convince you to play the game?
Let’s say you can flip the coin once every second. That’s a pretty generous estimate, but I’m a generous guy so we’ll go with it. At that rate, it takes around 17 minutes of sitting there flipping a coin to play the game. Flipping a coin for 17 minutes would get pretty boring. Is it worth it, though, for a shot at, say, $10,000? The odds of you winning are 1 divided by the number of people who play the game. This is where it’s important to mention that the fair is really, really big – let’s say around 200 million people big. Suppose only half of those people decide to play the game. Your expected value, then, for this game is $10,000 / 100,000,000 which is $0.00001. Even if the prize were an astronomical $100,000,000, you’d still only expect to earn one dollar, for 17 minutes of boredom flipping that coin. A rational person would conclude that it makes no sense to play the game. The thing is, most people are irrational and a large number will still play the game even after you explain the math to them. If you doubt me, look at the number of people who buy lottery tickets. My game has a positive expected value – lottery tickets have a negative expected value and people still buy them. Unless you really enjoy flipping coins, it just doesn’t make sense to play the coin flipping game.
What does this have to do with voting? We can easily look at voting systems as large, multi-player games. The payout if you win (your vote chooses the guy who gets elected) is very high. However, the probability that your vote will be the deciding vote is vanishingly small. There is no monetary cost to voting, but it does take time. Going to the polling place and voting takes probably 15 minutes, and you could spend hours each week educating yourself on the issues. Unless you really enjoy doing these things, it just doesn’t make any sense to spend your time attempting to influence an event, when the probability of your actually having an effect is so small that it might as well be zero.
A common rebuttal to this argument is ‘what if everybody thought this way?’ If everybody was rational and decided voting didn’t make sense, it would suddenly become rational to vote, because one vote would decide the election. The same logic works for the coin flipping game, however – if everybody realized how stupid it was to play the game, nobody would play it, and then any one guy who decided to play would win. The fact that a decision makes sense in an alternate reality where everybody behaves perfectly rationally doesn’t in any way mean it makes sense in our reality. The ‘what if everybody thought this way argument’ is like arguing ‘well, if we lived on the moon it would make sense to play basketball with 30 foot high hoops. ‘ Yes, but we don’t live on the moon, so it doesn’t make any sense.
Do you owe it to society to educate yourself and vote? Even people who are very educated about world affairs disagree with each other on the best course of action. The Pew Center for People and the Press does a typology survey every few years, breaking the electorate down into different groups based upon common values and shared attributes. In the 2004 survey, the two groups that followed world affairs most closely were the Enterprisers, who were Bush’s staunchest supporters, followed by the Liberals, who aligned themselves with Kerry. More education most likely isn’t going to change your mind – it’s just going to give you more reasons to believe that you’re right and that your political opponents are idiots.
What about convincing other people to vote? By taking a stand and convincing others around you to vote your way, surely you’re making a difference, right? Unfortunately, the answer is still ‘no’ unless you’re very persuasive. A statewide election decided by 10,000 votes is still considered very close. Unless you can convince 10,000 of your friends to change your mind and vote your way, you really aren’t making a difference.
What does all of this mean? It means that our political decisions are made by people who are irrational – they don’t understand the math governing the logic of making decisions in massively multi-player games. This means that pure democracies, where everything is decided by a majority vote, are probably the worst form of democratic government possible. How can we do better? The electoral college is a start, because it increases the probability that you will be able to influence an election. Increased federalism is another huge step in the right direction – instead of choosing a president who is essentially a king in a giant winner-take-all election, choosing a president who has a limited set of powers prescribed by a constitution interpreted strictly literally lessens the blow of having a bunch of irrational people make decisions.
A final improvement could be gleaned from implementing a recursive system of representatives. Suppose every 1000 people chose a single representative among them. Each 1000 of those representatives would choose a meta-representative, who represents 1,000,000 people. We could have each 100 of those meta-representatives choose a single meta-meta-representative, and so on. Due to the exponential nature of the math in this system, the government would only need 3 or 4 layers. Each of the different layers would be charged with different levels of power, with most of the power at the bottom levels of the system, where it makes the most sense for individual voters to participate. When only 1000 people are voting, your vote matters much more and suddenly it becomes much more rational to vote and to try to convince others to vote your way. Will this ever happen? Probably not. Still, a guy can dream, can’t he?