(Warning: completely obvious reasoning that I'm only posting because I haven't seen it spelled out anywhere.)
Some people say, expanding on an idea of de Finetti, that Bayesian rational agents should offer two-sided bets based on their beliefs. For example, if you think a coin is fair, you should be willing to offer anyone a 50/50 bet on heads (or tails) for a penny. Jack called it the "will-to-wager assumption" here and I don't know a better name.
In its simplest form the assumption is false, even for perfectly rational agents in a perfectly simple world. For example, I can give you my favorite fair coin so you can flip it and take a peek at the result. Then, even though I still believe the coin is fair, I'd be a fool to offer both sides of the wager to you, because you'd just take whichever side benefits you (since you've seen the result and I haven't). That objection is not just academic: using your sincere beliefs to bet money against better informed people is a bad idea in real world markets as well.
Then the question arises, how can we fix the assumption so it still says something sensible about rationality? I think the right fix should go something like this. If you flip a coin and peek at the result, then offer me a bet at 90:10 odds that the coin came up heads, I must either accept the bet or update toward believing that the coin indeed came up heads, with at least these odds. I don't get to keep my 50:50 beliefs about the coin and refuse the bet at the same time. More generally, a Bayesian rational agent offered a bet (by another agent who might have more information) must either accept the bet or update their beliefs so the bet becomes unprofitable. The old obligation about offering two-sided bets on all your beliefs is obsolete, use this one from now on. It should also come in handy in living room Bayesian scuffles, throwing some money on the table and saying "bet or update!" has a nice ring to it.
What do you think?
Bayesian rationality is also about decision making, not just beliefs. Usually people take it to mean expected utility maximization. Just assume my post said that instead.
My betting behavior w.r.t. the next coinflip is indeed the same for the two coins. My probability distributions over longer sequences of coinflips are different between the two coins. For example, P(10th flip is heads | first 9 are heads) is 1/2 for the first coin and close to 1 for the second coin. You can describe it as uncertainty over a hidden parameter, but you can make the same decisions without it, using only probabilities over sequences. The kind of meta-uncertainty you seem to want, that gets you out of uncomfortable bets, doesn't exist for Bayesians.
You are just rearranging the problem without solving it. Can my utility function include risk aversion? If it can, we're back to the square one: a risk-averse Bayesian rational agent.
And that's even besides the observation that being Bayesian and being committed to expected utility maximization are orthogonal things.
I have no need for something that can get me out of uncomfortable bets since I'm perfectly f... (read more)