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evand comments on Open Thread August 31 - September 6 - Less Wrong Discussion
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I'm looking for a good demonstration of Aumann's Agreement Theorem that I could actually conduct between two people competent in Bayesian probability. Presumably this would have a structure where each player performs some randomizing action, then they exchange information in some formal way in rounds, and eventually reach agreement.
A trivial example: each player flips a coin in secret, then they repeatedly exchange their probability estimates for a statement like "both coin flips came up heads". Unfortunately, for that case they both agree from round 2 onwards. Hal Finney has a version that seems to kinda work, but his reasoning at each step looks flawed. (As soon as I try to construct a method for generating the hints, I find that at each step when I update my estimate for my opponent's hint quality, I no longer get a bounded uniform distribution.)
So, what I'd like: a version that (with at least moderate probability) continues for multiple rounds before agreement is reached; where the information communicated is some sort of simple summary of a current estimate, not the information used to get there; where the math at each step is simple enough that the game can be played by humans with pencil and paper at a reasonable speed.
Alternate mechanisms (like players alternate communication instead of communicating current states simultaneously) are also fine.
Bridge, the card game. Bidding is the process of two players exchanging information about the cards they hold via the very limited communications channel (bids). The play itself is also used to transfer more information about which cards remain in the hand.
I don't know if that will work as a demonstration of the Aumann's Theorem, though, bridge gets very complicated very fast :-/
That's an excellent practical example, though it doesn't really have the explicit probability math I was hoping for.
In particular, I like that you'll see stuff like which player thinks the partnership has the better contract flips back and forth, especially around auctions involving controls, stops, or other specific invitational questions. The concept of evaluating your hand within a window ("My hand is now very weak, given that I opened") is also explicitly reasoning about what your partner infers based on what you told them.
I think the most important thing here might be that bridge requires multiple rounds because bidding is limited bandwidth, whereas giving a full-precision probability estimate is not.
If you want explicit probability math, you might be able to construct some kind of cooperative poker (for example, allow two partners to exchange one card from their hands following some very restricted negotiations). The probabilities in poker are much more straightforward and amenable to calculation.
The two-coins example might be useful as a first step, even if you then present a more difficult one.
How about some variation on Bulls and Cows?
That seems like fertile ground for exploration, but no probability / agreement variation immediately springs to mind. Did you have something specific in mind?
Have several people try to guess the same number, with everyone able to see everyone's guesses and results.
But then everyone has the exact some information, right? I'm specifically looking for something that's like Hal Finney's game, in that the different players have different information, and communicate some different set of information (some sort of knowledge about the state of the world, like their posteriors on the joint data).
Based on simple coin flip; other games:
I am sure there are more small games that have a similar "known" problem space.
What change would you make that results in multiple rounds being required?
For example, if each player flips multiple coins, and then we share probability estimates for "all coins heads" or "majority of coins heads" or expectations for number of heads, in each case the first time I share my summary, I am sharing info that exactly tells the other player what information I have (and vice versa). So we will agree exactly from the second round onwards.
example I was thinking:
each player flips 3(? 10) coins of their own. (giving them various possibilities on what they think the whole coin-space looks like) They present their 90%, 99% confidence intervals on there being more than 4 (9) heads. Round 2 repeat. (also make statements based on what they think the state of play is ++ try to get to the answer before the other person. So make statements that can be misleading maybe?)
Not sure how easy it is to tease out that information for a human. maybe a computer could solve it. but not so much a human...
"I flipped 10 coins; My 90% confidence that there are at least 7 of each heads and tails is 90%. 99% confidence is 60%."
confidence for "at least 10 heads and 6 tails" etc.
Here's how that goes. I flip 3 coins. Say I get 2 heads. My probability estimate for "there are 4+ heads total" is now 4/8 (the probability that 2 or 3 of your coins are heads). For the full set of outcomes I can have, the options are: (0H, 0/8) (1H, 1/8) (2H, 4/8) (3H, 7/8). You perform the same reasoning. Then we each share our probability estimates with the other. Say that on the first round, we each share estimates of 50%. Then we can each deduce that the other saw exactly two heads, and on the second round (and forever after) both our estimates become 100%. For all possible outcomes, my first round probability tells you exactly how many heads I flipped, and vice versa; as soon as we share probabilities once, we both know the answer and agree.
(Also, you're not using "confidence interval" in the correct manner. A confidence interval is defined over an expectation, not a posterior probability.)
I still don't see any version of this that's simpler than Finney's that actually makes use of multiple rounds, and when I fix the math on Finney's version it's decidedly not simple.
My version of making this work would be choosing to only share limited information.
i.e. estimates of 33% heads. or estimates of >10% heads and >80% tails. Where they don't sum to 100%, and will be harder to work out the "unknown space" in the middle. Limiting the prediction set to partial information. Also playing with multiple people should make it more complicated. Also an optional number of coin flips (optional to the person flipping coins and unknown to others within parameters)