Sunday, September 28, 2008

"The lesser of two evils": an indifference analysis

It's time to get graphical again.

In economics, we talk about indifference curves. An indifference curve represents a set of tradeoffs between two goods to which a person is indifferent. To derive an indifference curve, we require a few assumptions. The first is transitivity--a person cannot be infinitely milked by being sold A, B, C, and A in succession. This would imply that preferences must be consistent and two indifference curves on the same graph can't intersect. Another is the existence of utility--a person derives utility by consuming the goods in question (but only to a point, as you'll see). The last is that more consumption of any of the goods, and no less of another, yields more utility (again, only to a point, and there's at least one major exception to this rule). There are some others, but they're unimportant for this analysis.

When we add it all together, we get something that looks like this, with X and Y both being goods of some kind:



In the above graph, I2 is preferred to I1 because every point on the line has more of every good than in I1. The arrow represents the direction of utility--as you move northeast on the graph, the person in question gains more utility. If we were to zoom out quite a bit, it would look like this:



Indifference curves are ultimately circular. After a certain point, a good becomes a bad (for various reasons)--something which gives negative utility if there is any more of it. So, to the southeast of the Bliss Point, X is a bad. To the northwest of the Bliss Point, Y is a bad. To the northeast of the Bliss Point, both X and Y are bads.

Now imagine that X and Y are not goods but policies. A person will still have policy preferences that resemble an indifference curve. Perhaps Y is "funding to artists" and X is "welfare." There is not unlimited money in the economy, so posing the two as a tradeoff makes sense.

Now imagine two people with the following indifference curves:



The line between the curves is the tangent to both circles. The line connecting the two Bliss Points is, necessarily, at a 90 degree angle to the tangent line. Furthermore, any point along the tangent line that is closer to the intersection between the two lines than another point on the same line is preferred by both voters.

However, as we've seen with our cyclical voting model, three voters with intransitive choices (ie different bliss points) can't choose a "preferred" candidate. The indifference curve analysis allows us to extrapolate the agenda-setter problem outward to degrees. Instead of choosing among three candidates, an agenda setter can pose a vote between any two points on an indifference graph. In the following graph, A, B, and C represent three voter Bliss Points, while Z0-Z4 represent policy proposals:



Since I'm stealing this example, I might as well quote Alex Tabarrok's analysis:

Suppose that the status quo is point Z0 and Z1 is brought to vote. Voters A and C prefer Z1 to Z0 and so Z1 will beat Z0 by majority rule. . . Can we find a point which beats Z1? Yes, note that line Z2Z1 is perpendicular to line BC and along this perpendicular Z2 is closer to BC than Z1. By our rule it follows that [B and C vote for Z2 over Z1]. Similarly, [A and B vote for Z3 over Z2 and A and C vote for Z4 over Z3]. . . If we don't limit the number of votes, majority rule is incapable of choosing a "best" policy, voting will cycle over an infinite number of issues without ever reaching a stopping point. Suppose, however, that only four votes are taken so the final policy is Z4. But everyone prefers Z0 to Z4! Majority rule can lead a group of people to choose a policy which everyone agrees is worse than another possible choice!

This is no different than the problem explained in the last entry--agenda setting of pairwise voting is enormously more influential than voting itself.

In the U.S. system, third parties are inviable strategically, not procedurally. But adopting a "lesser of two evils" strategy rather than a "vote your preferences" strategy yields results very similar to agenda-set, pairwise voting. In other words, it can lead to Pareto inefficient--demonstrably suboptimal--results.

1 comment:

Anonymous said...

circle indifferent curve is cool as the way you explain it is
thanks you've been great help.