Tuesday, October 26, 2010

Arrow's Impossibility Theorem

One of the more interesting and not-well-understood things that economists know is Arrow's Impossibility Theorem.  An econ blog has recently made the claim that this needs to be popularized more.  Since this blog exists partly to share economics insights with my friends and family, and since I don't have anything else to write about today, I'll discuss this.

The theorem basically says that any mechanism for aggregating preferences into a choice will either be irrational, dictatorial, or potentially useless. It is impossible to design any voting system that satisfies a small set of requirements that most people would want a voting system to satisfy.

The definition of 'irrational' used here gets technical; there are several criteria involved.  I am not going to go through the proof here, nor will I explain things in too much detail*, but I will give an example.  All other examples I have seen are public choice example, analyzing groups of people trying to make a collective decision.  I will shake things up a bit:

Suppose that you are deciding who to marry.  The choices are Alex, Elliot, and Pat.  Also suppose that your instincts, emotions, and rationality have different rankings:

Your instincts, those animal desires that care about physical attractiveness, tell you that Pat is better than Elliot and both are better than Alex.
Your emotions tell you, based on feelings of trust and bonding, that Alex is better than Pat and both are better than Elliot.
Your rationality, which measures competence and character, tells you that Elliot is better than Alex and both are better than Pat.

So, to summarize:
Instinct says P>E>A
Emotion says A>P>E
Reason says E>A>P

It is impossible to choose a winner by looking at all of them at once, because each part of you has a different favorite.  A vote would be deadlocked.  Even if you did something more complicated, like assigning points for second place, they are still tied.

Now look at what happens if you choose two of them and compare them to each other:

If you compare Alex to Elliot, then both Instinct and Reason say Elliot is better, so Elliot wins.
If you compare Elliot to Pat, then both Instinct and Emotion say that Pat is better, so Pat wins.
If you compare Pat to Alex, then both Emotion and Reason say that Alex is better, so Alex wins.

Your preferences are, to use a technical term, non-transitive.  Elliot is better than Alex, who is better than Pat, who is better than Elliot.  You could chase this chain of reasoning in circles forever and get nowhere.

Most discussions of Arrow's Impossibility Theorem get to this point and start talking about how we just have to accept irregularities in voting systems and sports tournaments.  They also talk about how it is foolish to say 'what society wants' because society often has no clear preferences over multiple options.

I will use it to make a different, and more practical point.  The only way to be a sane and decisive human being is to simply anoint a dictator from among your various mental processes.  It is impossible to satisfy them all.  You have to choose one, or end up dithering endlessly, or act chaotically.

I have chosen rationality, and I suggest you do the same.  Just ignore the rest of your mental chatter, except in those cases where rationality has no preference.

*Go here and here if you are interested in more details.

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