Errors of Enchantment

The election is nearly upon us and now seems an appropriate time to remember why democracy is so messed up.
Imagine three voters with three different opinions about the Iraq war. Our voters are Abel, Bobby and Carl. Abel is a dove. Her first preference is to have a small number of troops in Iraq. If she can’t have her first preference, Abel wants to keep troop levels where they are now. Her last preference would be an increase in troop levels. Bobby doesn’t like change. He would like to see the number of troops remain as they are. If he can’t have this, his second choice is to increase the number of troops to many. His third preference is to decrease troop levels to few. Finally, we have Carl. Carl’s first choice is to have many troops in Iraq. But if he can’t have this, Carl would like to see few troops there. Comparing the current conflict to Viet Nam, Carl believes that the worst choice we can make is to leave troops at their current, intermediary level. For him, it is best to have either many troops or just a few. Summarizing, our voters have these preferences:
________Abel____________Bobby_____________Carl
1st_____Few_____________Current____________Many
2nd____Current__________Many_______________Few
3rd_____Many____________Few_______________Current
Let us assume that Abel, Bobby and Carl all live in a direct democracy governed by the simple majority rule. What will be the outcome of the election? Let us say that this November, the choice presented to the voters is between many troops and the current level (the option few troops is not on the ballot). Abel and Bobby will vote for the current level, winning the election over Carl’s vote for many troops. Let us then suppose that in December, there is another election, this one asking voters to choose between the current level and a lower level. Abel and Carl will then vote for few troops, beating Bobby’s vote for the current level. Finally, in January, voters are asked whether they prefer few troops to many troops. Now, Bobby and Carl will vote for many troops while Abel will vote for few.
So, without changing underlying preferences, voters will first vote to keep troops at the current level. They will then vote to decrease the level. Then they will then vote to put troops at the highest level. From there, they might vote to put troops back at the initial level. Voter theorists term this problem “cycling.” If we saw it in a person rather than an electorate, many of us would conclude that he or she were crazy! Someone preferring milk to juice and juice to water, ought to prefer milk to water! Notice, moreover, that each of the individuals in our little democracy has completely rational preferences. It is only when we try to aggregate preferences that we run into trouble. I should note that democracy doesn’t always produce this odd result. But it is quite possible under reasonable assumptions. Vote rules other than simple majority can make the cycling less likely, but none can rid us of it.
Perhaps we can avoid the problem by letting voters vote for all three options simultaneously? To see how this might work, assume that 30 percent of the public has Abel’s preferences, another 30 percent has Bobby’s and the remaining 40 percent have Carl’s. Now, if given all three choices, 40 percent of the public would vote for many troops, 30 percent would vote for current levels and another 30 percent would vote for few troops. The electorate would choose many troops. But, referring again to the chart, we see that though the election selected many troops, 60 percent of the public would have preferred current levels to more! (Gore supporters angry with their Nader-voting friends no doubt understand this frustration.)
As far as we know, this problem with democracy was first discovered by a French nobleman named the Marquis de Condorcet in 1785. Though cycling (also called Condorcet’s paradox) received some attention at the time, people seemed to forget about it. The mathematician, C.L. Dodgson (better known as Lewis Carol to readers of Alice in Wonderland) rediscovered the problem in the late 1800s. But again the idea was lost. The economists Duncan Black and Kenneth Arrow independently rediscovered the possibility of cycling in the middle of the 20th century. Though volumes have been written about the problem in academic forums, it has been almost entirely ignored in everyday discussions about democracy.
For me, cycling takes its place alongside many other problems with democracy such as tyranny of the majority and tyranny of the interest group. Given the fairly obvious undesirability of other forms of government, I suppose we are left to deal with it.
We can take solace in the fact that our Constitution severely curtails the power of everyone’s vote. We cannot, for instance, vote to deprive someone of life, liberty or property without due process. Nor can we vote to establish a religion or infringe upon the right to bear arms. There was a time when courts held that we could only vote to have government do those things which were specifically enumerated in the Constitution. So let us celebrate the Constitution which limits our power to vote and lament the passing of an age when the Constitution limited much more.