A probability question

[jack]

I've attempted to formalise mathemetically the most interesting question in http://cartesiandaemon.livejournal.com/179726.html. It should have a solution now, I'm not sure if a trivial one or an unfindable one.

Consider Pa and Po in [0,1]. Have a series of trials (independent random variables taking values 0 or 1) X1,X2,..., succeeding with probabilities x1,x2,..., and an event H that the trials are in the subset of possible results S∈{0,1}^N.

(We should add the constraint that being in S is determined by some finite initial segment of the sequence, or is so almost certainly, but can do so later if necessary.)

Each xi must be either (a) equal to Po, or (b) a function of Pa (with values in [0,1]).

Question: Can we choose x1,x2... and S thusly such that for any Pa and Po:

P(H)=Po if Pa=Po
P(H)<Po if Pa>Po

?

Comments

[gjm11.livejournal.com]

Has something gone wrong with the formatting? The condition in the final question seems garbled; or perhaps it's just my brain that's garbled. I can make some plausible guesses about what you had in mind, and view-source kinda-sorta helps, but I'm still not confident that I've reconstructed the question you intended...

Returning to the original question, here's an observation: Suppose the person whose job it is to choose the new rules is Really Stupid. Then there's nothing you can do that will *enable* him, never mind *incentivize* him, to make the new rules do the right thing with the winning probability.

I think there's a fundamental difference between this and the traditional sharing problem. With that one, you have two one-sided problems to solve simultaneously: make sure A thinks he has at least half the cake, and make sure B thinks he has at least half the cake. Similarly with the first couple of rule-choosing problems you mention: we just need to make sure that A and B each think they do at least as well as if White's winning probability is really (say) e/pi. But in so far as I've understood the final problem, it looks like two *two-sided* problems: each player is supposed to end up believing that the final probabilities are just right. Or something.

Having explained why I think the problem might not be solvable, here's a solution. First A makes up some new rules. Then B chooses between (1) playing the new game as black and the old one as white, or (2) playing the old game as white and the new one as black. If the balance between white and black is different, then B can exploit that to do better in the two-game match than he would have with two instances of the old game. If you find this unsatisfactory, then clarifying what's unsatisfactory about it may possibly make it clearer just what's being asked for...