Arrow’s Impossibility Theorem and The Voting Paradox

Growing up, I couldn’t stand spicy food. My brother loved it. This was a constant source of conflict. If dinner was too spicy, I complained. If it was not spicy enough, my brother complained.

This is why separate portions of spicy and non-spicy were typically made. But that was impractical for some dishes. Hence, one of us was often complaining that the other person was being favored. We tried to get “votes” from other family members to support our taste preference. It did not feel good to lose out.

Such situations are far from unusual. People are always complaining that they are the victim of an unfair majority, especially in situations when votes are collected. I think about situations like hiring decisions, the presidential election, and the baseball hall of fame. When you can’t get people to support your case, life can seem unfair and dictatorial.

But we should not be surprised. Situations where votes are collected are full of conflicting preferences. By their very nature, it’s likely someone will feel shafted.

How often does this happen? Choice theory says it happens very frequently. There’s a famous theorem about making optimal social decisions. It basically says that for many decisions (3 or more choices, 2 or more people), it is impossible to aggregate individual preferences in a meaningful way.

Here’s how one graduate level-text summarizes the theorem: “Either we must give up the hope that societal preferences could be rational [in the economics sense] … or we must accept dictatorship.”

Before I get into a specific example, I’ll digress about one of the great results from choice theory.

Arrow’s Impossibility Theorem

Kenneth Arrow is among the economists I admire the most. He won the Nobel Memorial Prize in Economics at the age of 51, becoming the youngest ever recipient for the award. He made contributions to general equilibrium analysis, the economics of information, growth theory, and social choice theory.

I’m not the only one to admire him. My friend’s dad, an economist in Korea, admired Arrow so much that he was compelled to name his son Kenneth in honor. It’s amusing because my friend Kenneth is sharp at economics and math, just like the economist he was named after. I’m not aware of any other people being named after economists, and that further reinforces how highly I think of Arrow’s research.

I discuss Arrow in this article because he proved what he called a “general impossibility theorem.” The theorem is so important that it is now named “Arrow’s impossibility theorem” in his honor. This work is part of why he was awarded the National Medal in Science in 2004, the nation’s most prestigious award for scientists. To give you a sense of Arrow’s contributions, I’ll point out that this theorem was just the beginning. I mean it literally was just the beginning of his research, as it was the topic of his Ph.D. thesis. Talk about starting at the top.

The articulation and proof of the theorem are technical so I won’t get into details here. I think it’s only fun to read for hard-core theoretical economists. If you wish to go through the nitty-gritty details, you can read three separate proofs in this document (pdf).

The executive summary is that whenever there are at least 2 people and at least 3 options, it’s impossible to aggregate individual preferences without violating some desired conditions, like Pareto efficiency. You either have to accept that society will not act rationally like an individual would, or you have to accept that society’s preferences will exactly mimic one person’s preferences. In a sense, that makes the individual a dictator.

You can get a sense of Arrow’s theorem from a small-scale example, which I discuss next. The example also illustrates how voting problems were known almost two hundred years before Arrow. It’s a philosophical idea that dates all the way back to the age of the Enlightenment.

The Voting Paradox (Condorcet’s Paradox)

In 1785, Marquis de Condorcet wrote one of his most important works, Essay on the Application of Analysis to the Probability of Majority Decisions. It was one of the first applications of probability to the social sciences, and it includes a stunning example of a possible problem with elections.

Imagine that there are three candidates running for office. I’ll call them A, B, and C.

In this concocted example, let’s just suppose there are only three voters (or there are three political constituencies, all voting in party line).

Each voter has a preference over the three candidates. I’ll write the preferences in shorthand, with the most favored candidate first, and the least favored last.

Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B

(For example, voter 1 prefers A the most, then B, and then C.)

Which candidate best represents the majority preference?

Let’s consider putting A in power. The result would be great news for voter 1, but both voters 2 and 3 would be unhappy as their candidates lost.

Voter 3 counters that the election is unfair. Voter 3 argues that C is a much better politician than A. This is clearly a biased opinion because voter 3 wants C to win it all. But voter 3 also discovers that voter 2 is unhappy. And that’s the secret to the paradox.

Voters 2 and 3 meet and realize that they both prefer C over A. Take a look.

Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B

Voter 3 has C as a first preference, and voter 2 has C as a second preference over A. This means both voters 2 and 3 would prefer C over A. So voters 2 and 3 decide that C would be a better person than A. They gang up and overrule voter 1, placing C in power.

The peace only lasts a few minutes because soon voter 2 has regrets about recommending C. Sure, C is better than A, but only by a little bit. In fact, voter 2 can only live with B being in power, his first preference. Is there a way to make that possible? Voter 2 decides to talk with voter 1, and soon learns something he likes. It turns out voter 1 also prefers B over C.

Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B

So voters 1 and 2 now combine forces to decide that B would be a better person than C. Using the power of majority, they gang up and place B in power. I’m sure you’re getting the picture by now. B is in power for only a very short time when voters 1 and 3 join forces because they both prefer A over B.

Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B

And once again, the case is made to put A in power.

Using majority preference, it turns out that A loses to C, C loses to B, B loses to A, and A loses to C again, and so on ad infinitum. This is an example of how society’s preferences are much more complicated than individual’s. Using the language of Arrow’s Impossibility Theorem, the voting paradox would be an example of “irrational” societal preferences.

In this situation, no candidate can claim to be favored by the majority. It’s impossible.

Now incorporate some real world practicalities: that people change their mind and are susceptible to miscalculations in snap decisions. That’s the workings of a democracy. It’s no wonder that people are whining about some thing.

I only kid about applying this paradox to a real situation, like the 2008 U.S. presidential election. There are all sorts of others issues to keep in mind. Don’t take me too seriously or literally. But for the sake of the rest of us, please don’t take losing too personally. If it’s not you, it would be someone else.

Further Reading

Looks like Arrow’s impossibility theorem is on other people’s mind as well. Today, I discovered a couple related articles with more U.S. presidential election specific details. I think the both authors understate the assumption of “independence of irrelevant alternatives”–since it’s not how humans actually behave–but I’ll leave that discussion for another week.

Real Clear Politics On McCain’s Voting Coalition
An argument being proffered by Romney supporters is that McCain’s victories in the early states have been due to the conservative vote being split among many candidates. By this thinking, McCain would have lost South Carolina and maybe Florida if conservatives had coalesced around a single anti-McCain candidate. Michael Medved commented on this theory yesterday. I would like to toss in my two cents, as this sort of matter is up my alley.

At first blush, this theory might seem compelling. However, if we take a closer look at the exit polls in New Hampshire, South Carolina, and Florida, we can see that this is actually a problematic assertion.

First of all, we have to clear away some of the theoretical underbrush. What we are talking about here is the idea that the electoral results to date display social irrationality. Is this possible? Absolutely.

Salon.com: Your presidential candidate: Hot or not?
eb. 12, 2008 | Elections aren’t fair. This much you know, and if you’ve observed American politics at all, you’ve likely even drawn up a bill of particulars: Start with the Electoral College, which offers the loser of the presidential popular vote a chance to win. Add the absurdly undemocratic primary process, in which Iowa is accorded a greater role than California in selecting our leaders. Then, of course, there are the hackable voting machines, the wildly inconsistent state-by-state voting rules, the scandalous campaign finance loopholes, and … Every gear in the system grinds against the popular will, subverting the democratic mojo that we tell the rest of the world bleeds from our veins. George W. Bush got the White House because five judges and Ralph Nader put him there; if administered truth serum, would anyone, even Bush himself, call such a system fair?

Yet these are only the small problems. Even if you were to abolish the Electoral College, ban touch-screen voting, and institute public financing for campaigns, American elections — and not just ours — would be murky affairs still. That’s because trouble with voting is not entirely human. It’s also mathematical.