Posted 11:28 AM by Robert

A Note On The Mathematics Of Conventions

The political process is often compared to a "marketplace of ideas." Within a particular political party, the purpose of the presidential primary season is in large measure to force the competitors to eliminate each other from competition. If that process is not completed by the primaries, then the convention must finish the job and establish a "monopoly" by choosing a single nominee.

What happens in a convention in which nobody commands a majority of delegates? Well, if one candidate commands a near majority, the brokering of the convention will likely be a deal cut by that that candidate and another player who can put the leader over the top. In other words, the convention will bear a stong resemblance to a kind of oligopoly.

If no contender has a near-majority, things get much more complex very fast. How fast? Well, continuing the economic analogy, one might say that as the "market" within the convention gets less concentrated, the convention functions more more like "perfect competition" - which is exactly the chaotic situation that the primaries and convention are supposed to eliminate.

How fast does a convention degenerate into perfectly competitive chaos? Probably mush faster than the degree by which the candidates fail to obtain a majority of the delegates. In the economic arena, a standard measure of market concentration is the Herfindahl index:

[T]he Herfindahl index is a measure of the size of firms in relationship to the industry and an indicator of the amount of competition among them. It is defined as the sum of the squares of the market shares of each individual firm. ... The major benefit of the Herfindahl index in relationship to such measures as the concentration ratio is that it gives more weight to larger firms. Take, for instance, two cases in which the six largest firms produce 90 percent of the output:

Case 1: All six firms produce 15 percent, and
Case 2: One firm produces 80 percent while the five others produce 2 percent each.

We will assume that the remaining 10% of output is divided among 10 equally sized producers.
The six-firm concentration ratio would equal 90 percent for both case 1 and case 2, but in the first case competition would be fierce where the second case approaches monopoly. The Herfindahl index for these two situations makes the lack of competition in the second case strikingly clear:

Case 1: Herfindahl index of 1360
Case 2: Herfindahl index of 6430

This behavior rests in the fact that the market shares are squared prior to being summed, giving additional weight to firms with larger size.

If the economic analogy holds, then a political convention in which, say, five contenders each hold about 20% of the delegates will pretty well approximate chaos. In other words, such a situation would mean that the entire primary season had decided essentially nothing.