Arrow’s Theorem

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Arrow’s theorem (named for Kenneth Arrow) is the kind of statement that will almost certainly surprise you the first time you hear it. For someone (like me) without enough experience in game theory to have developed the right intuitions, it can be really hard to understand the theory in context, so you’ll hear lots of people overreacting (“Arrow’s Theorem says that democracy is impossible”) or dismissing it as trivial (“no one thought voting was fair in the first place”). With that in mind, here’s a quick summary, adapted from the Wikipedia page and my class notes.

Imagine you’ve just founded a new democracy, and are writing the section of the constitution relating to elections. You’ve heard about some unappealing aspects of other prominent democracies, such as the candidate with a plurality of the popular vote losing the election because of something called the “Electoral College”, and so you want to design a better system without these quirks. You therefore make a list of reasonable guidelines for elections in your country, and then attempt to construct a method of voting that satisfies all of them. Here’s a version of the list that Kenneth Arrow made:

1. The voting mechanism must be complete – meaning that every possible voter preference can be expressed on the ballot. In this example, it’s roughly equivalent to saying that voters must be allowed to rank candidates in whatever order they want.

2. The voting mechanism must be transitive – so if candidates X, Y, and Z are running, and if the people prefer X over Y and Y over Z, then the voting mechanism should also favor X over Z.

3. The voting mechanism must be independent of irrelevant alternatives – that is, if candidate X is preferred over Y, then adding candidate Z to the ballot should not change the fact that X is preferred over Y (although Z could be preferred over both X and Y).

4. There must be no dictator – the voting mechanism must account for everyone’s preferences, not just one person’s.

I think, and I hope you’ll agree, that all four of these guidelines seem pretty desirable in general. Unfortunately Arrow proved that these four guidelines are mutually exclusive – no ranked voting mechanism can satisfy all four. America fails automatically because of the aforementioned electoral college fiasco, but even if we abolished that and went with the candidate who received the most votes, it would still fail to satisfy all four guidelines. Majority rule fails the second guideline in general because when people only get one vote it’s possible to choose carefully candidates who can split coalitions and steal that vote. For example, imagine if the general election had been held in late March, when the Obama-Clinton rancor was near its peak: it’s easy to believe that a majority at that time would have preferred Obama over Clinton, and Clinton over McCain, but that Obama might still have lost to McCain (see here if you don’t believe me). It’s possible to have a transitive system if you allow people to vote for multiple candidates, but this inevitably fails one of the other three guidelines. In short, Arrow’s Theorem shows that it’s impossible (except for very simple situations like only two candidates or only three voters) to accurately and repeatably measure the “will of the people”. Thus (at least to a strict scientist), there’s actually no such thing as a “will of the people”.

There are many directions to go from here. I see this theorem mostly as a warning not to oversimplify: masses of people are really complicated, and attempts (historical or modern) to jam them into simple groups are foolishly misguided. Oh, and I guess it’s not overreacting to conclude that a perfectly representative democracy is impossible, but I think most of us have guessed that by now anyways.

Two final points. First, a quick plug for my favorite voting mechanism, range voting, which actually does manage to satisfy all four guidelines listed above without violating Arrow’s Theorem, by allowing people to express their preferences with more nuance than simply ranking them. And second, it’s fun to consider Arrow’s Theorem in light of research claiming that each person contains many different personalities (with different preferences) continuously vying for control. If there exists no such thing as a “will of the people” among your constituent selves, then what exactly are you?

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4 Responses

  1. […] – an unambiguous expression of the “will of the people”. (Well, unambiguous except for my own misgivings about that […]

  2. well, what can you say about the four conditions stated by arrow? Are these conditions collectively exhaustive… is there not any other condition? It dosen’t matters though because the voting systems are not satisfying even the stated conditions but, maybe there is a flaw in the 4 conditions…!!!

  3. I really enjoyed reading your blogpost, keep up posting such interesting posts.

  4. Dear Michael,

    I read your notes on Arrow’s Theorem and I do not seem to grasp the theory and why the conclusion is that democracy can not exist and impossible to achieve. I have a quite difficult assignment on this subject that is due this Thursday so I was searching the net in hope of finding the light at the end of the tunner. I am still in the dark and no matter how much I read, I do not move forward. Could you maybe explain a little bit this concept of Arrow and how it relates democracy? I would so much appreciate your response. Thanks a lot in advance for any feedback on this topic.

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