A few days ago on Armstrong & Getty I mentioned the Arrow Impossibility Theorem and how a majority vote does not, and typically can not, actually reveal what most people want. I'm afraid I wasn't very good at explaining it briefly on the air. Here's the explanation I like, by Prof. Donald Boudreaux, who taught it to me. The basic idea is this. Let's say there are three people who want to decide what to have for lunch. The possible choices are apples, bananas, and cantaloupe. Tom wants apples. If he can't have apples, he wants bananas. If he can't have bananas, he'll pick cantaloupe. Richard wants bananas first, then canteloupe, then apples. Bob would prefer canteloupe first, and if he can't have that, he'd rather have apples, and if he can't have that, he'd prefer bananas:
Tom: 1. apples, 2. bananas, 3. canteloupe
Richard: 1. bananas, 2. canteloupe, 3. apples.
Bob: 1. canteloupe, 2. apples, 3. bananas.
Now, what does the majority prefer? Well, nothing. There's a 3-way tie. But notice: both Tom and Richard prefer bananas to cantaloupe. Both Richard and Bob prefer cantaloupe to apples. Both Tom and Bob prefer apples to bananas.
Time to vote! They can only pick one, so they'll have a "primary" election first. In the primary, it's apples versus bananas. Tom votes apples. Richard votes bananas. Bob votes apples. So apples wins and moves on to the "general" election, between apples and cantaloupe. Tom votes apples. Richard votes cantaloupe. Bob votes cantaloupe. So...Cantaloupe wins! Yay! That was what the majority wanted, right? But no. Only Bob wanted cantaloupe. The only thing the majority agreed upon was that it did not want cantaloupe.
Now let's switch it up, just for fun, and this time in the "primary" let's do apples versus cantaloupe. Bob votes cantaloupe. Richard votes cantaloupe. Tom votes bananas. Cantaloupe wins! And in the general election, between canteloupe and bananas, Bob votes cantaloupe; Richard votes bananas; Tom votes bananas. Bananas wins! Yay! that was what the majority wanted, right? But no. Only Richard wanted bananas. The majority wanted something else.
But notice something else. We've done two elections now, and they came out differently. But between the two elections, nobody's preferences changed. The only thing that changed was the order in which we ran the "primary" and "general" elections.
This is called the Condorcet ("con-dor-say") Paradox. It's the basis of the Arrow Impossibility Theorem, which explains that there is no way to eliminate this influence between how an election is run and the outcome of that election. So long as there are more than two choices, what matters is what order the "primary" versus "general" elections operate.
This is only one reason why a pure democratic system does not do what its supporters claim: i.e., demonstrate the "true will of the people." (Another is the difference between stated and revealed preferences. There's also the problem of rational ignorance.)
Update: Back in the 70s, some economics professors, knowing of this phenomenon, used it to rig some elections in their favor. Thanks to Jonathan Wood for the link.
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