(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 The old obligation about offering two-sided bets on all your beliefs is obsolete, use this one from now on.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?
Of course I can. I can represent my beliefs about the probability as a distribution, a meta- (or a hyper-) distribution. But I'm being told that this is "meta-uncertainty" which right-thinking Bayesians are not supposed to have.
No one is talking about inventing new fields of math
Clearly not since the normal distribution goes from negative infinity to positive infinity and the probability goes merely from 0 to 1.
That 0.5 is conditional on the distribution of r, isn't it? That makes it not a different question at all.
Notably, if I'm risk-averse, the risk of betting on Coin 1 looks different to me from the risk of betting on Coin2.
Can you elaborate? It's not clear to me.
Hm. Maybe those people are wrong??
That's right; I should have either said "approximately", or chosen a different distribution.
Yes, it is averaging over your distribution for _r_. Does it help if you think of probability as relat... (read more)