Idol Speculation

From this week’s Marilyn von Savant column (not yet available online):

I believe that the voting for American Idol is inherently flawed. The performer with the fewest votes must leave the show. But we vote for the one we like best! If we voted for the performer we think is worst, the person who leaves might be someone else.
– Vic Gregorini, Belle Vernon, Pa.

[Marilyn:] That’s true, but if you kept voting “against” a performer each week, I think you’d probably still end up with the same person as the winner – except the show would seem negative instead of positive.

Marilyn was wise to include that “probably,” because it gives her some wiggle room. But she’s making an empirical claim here without any basis. As a theoretical matter, it’s simple to find patterns of voter preferences that yield significantly different results depending on the voting rule.

Suppose we have three contestants denoted A, B, and C, and 100 voters with preferences like so:

A pref B pref C (40)
C pref B pref A (35)
B pref A pref C (15)
B pref C pref A (10)

With these preferences, contestant B is the Condorcet winner; that is, B would win in a one-on-one vote against either A or C. The very notion of a “general will” of the public is terribly problematic, but if anyone has a claim to it, it’s B. (One reason the “general will” is problematic is that a Condorcet winner does not always exist; for example, see here.)

Now, what will happen under a Vote-for-the-Best (VFB) rule versus a Vote-for-the-Worst (VFW) rule? Assume people vote their true preferences, rather than voting strategically based on knowledge of others’ preferences. Under VFB, contestant B will get eliminated in the first round of voting, as she has the fewest number of voters who consider her the best. Then A will face off against C, and A will win. Under VFW, on the other hand, contestant C is eliminated in the first round, and then B beats A in the face-off. So VFW awards the prize to the Condorcet winner, while VFB does not.

Thus, for some preferences, the existing rule (VFB) is inferior, as Marilyn’s reader claims. But with different preferences, the opposite is true. Suppose we keep the 100 voters above, and add two more voting groups:

A pref C pref B (10)
C pref A pref B (30)

(We now have 140 voters.) With these preferences, the Condorcet winner is C. Under VFB, contestant B is eliminated in the first round, and then C beats A in the face-off. Under VFW, contestant C is eliminated in the first round, and then A beats B in the face-off. So VFB gives the prize to the Condorect winner, while VFW does not.

The point, of course, is that every voting system is flawed for some voter preference distributions. To decide one system is superior (as Marilyn’s reader says) or that they are “probably” equivalent (as Marilyn herself says), we have to know a lot more about the structure of people’s preferences.

(Readers familiar with voting theory may note that VFW failed to select the Condorcet winner in the latter case only because we included non-single-peaked preferences. So if we assume single-peaked preferences, VFW might look superior. But why should we assume single-peaked preferences?)