(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 The old obligation about offering two-sided bets on all your beliefs is obsolete, use this one from now on.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?
That would be missing the point. The vNM theorem says that if you have preferences over "lotteries" (probability distributions over outcomes; like, 20% chance of winning $5 and 80% chance of winning $10) that satisfy the axioms, then your decisionmaking can be represented as maximizing expected utility for some utility function over outcomes. The concept of "risk aversion" is about how you react to uncertainty (how you decide between lotteries) and is embodied in the utility function; it doesn't apply to outcomes known with certainty. (How risk-averse are you about winning $5?)
See "The Allais Paradox" for how this was covered in the vaunted Sequences.
Obviously you're allowed to have different beliefs about Coin 1 and Coin 2, which could be expressed in many ways. But your different beliefs about the coins don't need to show up in your probability for a single coinflip. The reason for mentioning sequences of flips, is because that's when your beliefs about Coin 1 vs. Coin 2 would start making different predictions.
Would it? My interest is in constructing a framework which provides useful, insightful, and reasonably accurate models for actual human decision-making. The vNM theorem is quite useless in this respect -- I don't know what my (or other people's) utility function is, I cannot calculate or even estimate it, a great deal of important choices can be expressed as a set of lotteries only in very awkward ways, etc. And this is even besides the fact that empirical human preferences tend to not be coherent and they change with tim... (read more)