Though a free election seems to be the fairest and wisest system for making policy decisions and electing officials, statisticians sometimes fear that a paradox political scientists call "vote cycling" can result in a win for the minority. There's a better way to hold an election.
When I was a little child statistician, my parents would occasionally allow me to make choices about personal thingswhat to wear, what to eat, which story book to read at bedtime, and so on. I noticed that sometimes the choice was open-ended: "Your choice, Bruce: when would you like to go to bed?" And sometimes the choice was presented as a set of alternatives for me to choose between: "Your choice, Bruce: would you like to go to bed now or in five minutes?"
Of course, the second choice isn't much of a choice, really. When I had to choose between various alternatives, my true opinion wasn't reflected as accurately as when I could choose anything I wanted.
Democracy works like that as well. When it is time to vote for President, or Mayor, or Dogcatcher, we usually must choose between several alternatives. We might not be happy with any of the options, but we vote anyway (at least statisticians do).
Did you ever leave the voting booth, though, and feel that somehow your real feelings weren't represented very well by those choices? Political scientists know that feeling. They have analyzed the sometimes unsatisfying outcomes of votes between alternatives and discovered that such a process can result in outcomes in which no one is happy (except the winner, of course).
There are a variety of ways that elections can be structured. Imagine that an electorate (such as the residents of a city, members of a club, or faculty at a university) is asked to vote on a policy and there are three choices. Imagine, also, that there are three groups of supporters that each favor one of the three options over the others. The election could ask people to vote for their favorite policy. Under that system, the policy favored by the largest group is likely to win the most votes. This seems fair, and this is the system we most commonly see.
Another system that makes good sense, too, at least on the surface, would be to present each pair of options against each other and have a kind of runoff election, in which A is compared to B, B is compared to C, and C is compared to A. The biggest vote getter in this sort of system should result in an equally fair decision. It turns out, though, that this type of system, called vote cycling, is difficult to use fairly because the order in which you present the options can determine the outcome of the election!
How It Works
Here's an example of how vote cycling can work. Imagine that your scout troop has to decide what color to paint the inside of the troop clubhouse (or wherever scouts meet these days). As a group, you will be voting for Red, White, or Blue. Different political "groups" have formed among your colleagues who favor different color choices.
There are the Apples who prefer red, the Elephants who favor white, and the Jayhawks who like blue best. The groups also differ on which color they like second best and which color they like least. Table 6-22 shows the three groups and their political agendas.
To determine the will of the scouts, you could hold a two-stage election. Stage one presents two alternatives. The winner of that stage then "competes" with the third alternative to pick a winner. The two stages and results could look like this:
So, blue paint must be the will of the people! This is a paradoxical outcome, though, because only one group, representing 40 percent of scouts, liked blue best. An equal number liked white best, and another 20 percent hated blue. The order of decision making affected the outcome. Let's do it again in a different order:
We have a different outcome than before, just because of the order of matchups. This is fun; let's do it one more time. Maybe we can arrange for red to win this time:
Three potential orders of matchups result in three completely different policy decisions.
Getting Off the Votercycle
If we think of voting systems as measurement systems, this matchup method of making decisions has low validity. There is information that could be gleaned from the voters that is being lost here. However, there are a couple solutions that come to mind to solve the problem of vote cycling.
If the designers of the voting system are interested in the rank-order preferences of voters, voters could be asked to rank-order all candidates. The lowest mean rank wins. This is a fairer method that uses all the information available, but it can lead to choices that no one is really thrilled about.
Another solution is to make all candidates available for a single vote, with the majority winning. This is the most common system, but it does have the disadvantage of sometimes choosing candidates that have no majority support.
For elections in which there are many candidates (in some mayoral or governor elections, for example), there is often a runoff election in which the larger number of candidates is whittled down to a smaller number. This doesn't have the weakness of vote cycling, because all alternatives are considered at the same time. It also eliminates the weakness of the single-trip-to-the polls approach because it increases the likelihood of a winning candidate with majority support.