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.
I’ve divided the statements into a few categories. For the “statistics” category I used NationMaster and StateMaster. For “history” I used Wikipedia. For “future”, please answer all questions conditional on no disruptive technologies like molecular nanotechnology and artificial general intelligence being invented. This makes the questions rather vague, so I’m not really happy with this category. For “counterfactual”, please answer conditional on the many worlds interpretation of quantum mechanics being true; even if it isn’t, it’s still a well-defined model, so the question is meaningful either way. For “internet”, I always included quote marks.
The answers in the “statistics”, “history”, and “internet” categories are easy to look up, but that would defeat the point. So no peeking allowed. Looking up any relevant information is peeking.
Discussion of the statements other than through stating probabilities is also against the spirit of the game. Feel free to ask for clarifications, though.
To reward honest estimates, in the end I may score people on the answers, using the rule where your number of points is the logarithm of the probability you assigned to the right answer.
(update: this was tried again less messily and with more suitable questions here, here, and here)
1. Oregon has more inhabitants than Slovakia.
2. Ghana has a greater GDP (PPP) than Luxembourg.
3. In 1900, Denmark had a greater GDP per capita than Spain.
4. Ohio emits more CO2 than Poland.
5. Afghanistan has more land area than Alaska.
6. Croatia has a greater GDP per capita than Mexico.
7. George Orwell was born before 1900.
8. Vladimir Putin was born before 1955.
9. The tenth emperor of Rome wore a beard.
10. More than 5000 Americans died in the attack on Pearl Harbor.
11. If the USA has a president in 2067, it will be a woman.
12. A 1000-qubit quantum computer will exist in 2020.
13. A nuclear (fusion or fission) weapon will be used in an attack before 2010.
14. Switzerland will join NATO before 2100.
15. Proof of life on Mars (past or present, not originating on Earth) will be found before 2050.
16. In a randomly selected parallel Everett world splitting from ours on 1 Jan 1940, Hitler invades England before 1950.
17. IARSPEWSFOO 1 Jan 1940, Hitler invades the USA before 1950.
18. IARSPEWSFOO 1 Jan 1, a technological singularity happens before 1500.
19. IARSPEWSFOO 1 Jan 1, nuclear war kills at least ten million people in any five year period before 2000.
20. IARSPEWSFOO 1 Jan 1900, nuclear war kills at least ten million people in any five year period before 2000.
21. “brain” gets more google results than “heart”.
22. “Ray Kurzweil” gets more google results than “Sonic the Hedgehog”.
23. “John Paul II” gets more google results than “Ron Paul”.
24. “Iraq” gets more google results than “Italy”.
25. “death” gets more google results than “purple”.