Centrism begets extremism, in a precise game-theoretic sense

(Excavated from recent unpublished drafts.)

This is not rocket science.

Suppose a couple* is getting divorced and in joint possession of an indivisible good, such as a house. They have agreed that the wife should keep the house, but pay the husband for 50% of the value of the house. Now they get into an argument about the value of the house: the wife will try to claim that the house is worth less, and the husband will claim that the house is worth more. Who will help them resolve this dilemma?

They first take the claim to Carol the Centrist, whose arbitration strategy is "split the difference" because she believes that "whenever two parties disagree, the truth is always somewhere in the middle". Carol will take the average of the wife's estimate and the husband's estimate. Immediately, the wife will claim that the house has value negative infinity, and the husband will claim that the house has value positive infinity. Since the average of negative infinity and positive infinity is undefined in the general case, no agreement will be reached and Carol's arbitration strategy is fucked. Meanwhile, Carol's silly faith in centrism has driven the husband and wife even further apart than they were to begin with.

The husband and wife realize that Carol is a fool, and they take their dispute to Dave the Decider. Dave understands game theory, and therefore he says: "Each of you write your claimed value down on a piece of paper. I will get an independent assessment of the house's value, and just pick whichever of your claims is closest to the assessment." The husband and wife immediately realize that it is in their interest to bid as close to the true value as possible. Given access to the same information, they will simply bid exactly the same price and Dave will not even have to hire his own assessor. He collects a fat arbitrator's fee and goes home. Meanwhile the husband and wife's set of disagreements has been reduced by one.

This is an extremely simple illustration — in fact, literally a textbook illustration — which I learned from my undergrad class with S. J. Brams (previously mentioned). I say all this by way of commenting on M. Yglesias's and H. Hertzberg's posts about a month ago on the wages of "centrism".

What's amusing is that centrists imagine themselves as essentially a moderating influence, when they are exactly the opposite.

This also puts the lie to the claim, which I sometimes hear journalists making, that if you're getting criticism from both the left and the right, you must be doing something right. Journalism that presents "both sides" without taking sides will, in the long term, lead to an objectively less well-informed public, as it creates bad incentives for the actors. As astonishing as it seems, it would be far preferable to attempt to determine the objective truth, and then print verbatim whichever press release is closer to that truth (so long as you announce this strategy beforehand).

Of course, it's not necessary to go to that extreme. One could instead devise a type of journalism which simply says, "A says X, which is close to the truth. B says Y, which is more of a lie compared to X. Here are the details: ..."

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*Given the rest of this post, can you guess the name of the wife and husband?