More fun with voting

Kevin Miller continues

to maintain that it's incoherent to suggest that it's worth trying to influence the Dem presidential nomination, but only if Lieberman is available.

Let's see what we can do with this:

Assuming that the purpose of government is to serve the common good, let's first pretend we can estimate the service a candidate would offer the common good if he were elected president. Call S_L the service of President Lieberman, S_B that of President Bush, and S_F that of the Democratic field (i.e., the product of a Liebermanless nomination process). We'll assume the field is worse than Lieberman, who is worse than Bush, so S_F < S_L < S_B.

Now, given a Lieberman-Bush race, there's a certain probability Lieberman will win; call it P_L. The probability Bush will win is 1-P_L. Similarly, in a {Democratic field}-Bush race, the probabilities of winning are P_F and 1-P_F, respectively.

The expected service to the common good of the winner of a Lieberman-Bush race is

E_L = P_LS_L + (1-P_L)S_B

For a {Democratic field}-Bush race,

E_F = P_FS_F + (1-P_F)S_B

We expect the common good to be better served by a Lieberman-Bush race when E_L > E_F, which is to say when

P_L < P_FR

where

R = (S_B - S_F) / (S_B - S_L)

We know R > 1, because we've already assumed S_F < S_L.

What all this says, in short, is that if Lieberman is better for the country than the rest of the Democratic field but worse than Bush, then a Lieberman-Bush election is expected to be better for the country even if Lieberman has a better chance of winning than the rest of the Democratic field.

Suppose Bush would be ten times better than Lieberman, and any other Democratic president would be worthless. Then S_B = 10S_L, S_F = 0, and R = 1.11111. So Bush can be 11% more likely to lose to Lieberman than to any other Democratic candidate and we would still expect a Lieberman-Bush race to be better for the country. (That's an 11% better percentage, not an 11% higher percentage; for example, if the field had a 50% chance of winning, Lieberman could have a 55.5% chance (11% of 50% is 5.5%), not a 61%.)

You may not think Lieberman is better for the country than the rest of the Democratic field, but I don't think it's "incoherent" for those who do to recommend influencing the Democratic nomination process if and only if he's available.

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