The Aumann Game

Aumann’s agreement theorem says that Bayesian agents cannot “agree to disagree” — their subjective probabilities must be identical if they are common knowledge. This is true regardless of differences in private knowledge. When agents take turns stating their estimates, updating each time based on the information contained in the other’s estimate, private knowledge will “leak out” and the probabilities will converge to an equilibrium.

This theorem makes some big assumptions. One is common knowledge of honesty. Another is common priors. Another is common knowledge of Bayesianity. However, Robin Hanson has shown that uncommon priors require origin disputes, and has discussed agents who are “Bayesian wannabes” but not Bayesians.

It may be interesting to see how this process plays out with real humans in a simplified test bed. Below are 25 statements.

To play, for each statement, you have to say your honest subjective probability that it’s true. Make sure to take into account the estimates of previous commenters. You are strongly encouraged to post estimates multiple times, showing how the estimates of others have caused yours to change. We will then see whether, as the theorem suggests, everyone’s estimates converge to the same equilibrium over time, and whether that equilibrium is any good.

Scott Garrabrant (seemingly) independently invented a very similar game, but designed for in-person play rather than playing online via comment section. It has printable instructions, including a convenient scoring table for an adjusted Bayes score which (a) is a proper scoring rule, and (b) is relatively fair (ie, minimizes the importance of whether you happen to get correct or incorrect answers).

Created by Vladimir_Nesov at 4y