Yet another case where knowing one’s fallibility leads to fairly conservative probabilistic actions. Does this simplify to “perform the action with a probability such that if all participants have the same posterior, that will be the chance that any of us take the action”?
Or does it?
It’s absolutely worth noting that this is NOT what a unilateralist would do to maximize the outcome. It’s well-known that if you assign 51% to a binary proposition, you should bet on yes, not randomize to 51% yes and 49% no. In the 51red and 49 blue balls in a jar, bet on the color of the next pick, it’s suboptimal to use any strategy except “guess red”.
I really wonder (and am not quite willing to write the model and find out, for which I feel bad, but will still make the comment) where the cutoff is in terms of correlation of private signal to truth, and number of players, where the true unilateralist does better. Where players=1 and correlation is positive, it’s clear that one should just act if p(G) > 0.5. For players=2, it’s already tricky, based on posterior and correlation—my intiution is if one is confident enough, just write the paper, and if one is below that range one should be probabilistic. And it all goes out the window if the noise in the private information is not independent among players.
The discussion option is clearly best, as you can mutually update on each other’s posteriors (you have mutual knowledge of each others’ rationality, and the same priors) to get a consensus probability, and then everyone will act or not, based on whether the probability is greater than or less than 50%. In the absence of discussion, there’s ALREADY a huge loss in expectation with the need to be probabilistic, even if you all have the same information (but don’t know that, so you can’t expect automatic unanimity). I don’t see a way to do better, but it’s definitely unfortunate.
Yet another case where knowing one’s fallibility leads to fairly conservative probabilistic actions. Does this simplify to “perform the action with a probability such that if all participants have the same posterior, that will be the chance that any of us take the action”?
Or does it?
It’s absolutely worth noting that this is NOT what a unilateralist would do to maximize the outcome. It’s well-known that if you assign 51% to a binary proposition, you should bet on yes, not randomize to 51% yes and 49% no. In the 51red and 49 blue balls in a jar, bet on the color of the next pick, it’s suboptimal to use any strategy except “guess red”.
I really wonder (and am not quite willing to write the model and find out, for which I feel bad, but will still make the comment) where the cutoff is in terms of correlation of private signal to truth, and number of players, where the true unilateralist does better. Where players=1 and correlation is positive, it’s clear that one should just act if p(G) > 0.5. For players=2, it’s already tricky, based on posterior and correlation—my intiution is if one is confident enough, just write the paper, and if one is below that range one should be probabilistic. And it all goes out the window if the noise in the private information is not independent among players.
The discussion option is clearly best, as you can mutually update on each other’s posteriors (you have mutual knowledge of each others’ rationality, and the same priors) to get a consensus probability, and then everyone will act or not, based on whether the probability is greater than or less than 50%. In the absence of discussion, there’s ALREADY a huge loss in expectation with the need to be probabilistic, even if you all have the same information (but don’t know that, so you can’t expect automatic unanimity). I don’t see a way to do better, but it’s definitely unfortunate.