Thanks for posting this. I've always been skeptical of the idea that you should offer two sided bets, but I never broke it down in detail. Honestly, that is such an obvious counter-example in retrospect.
That said, "must either accept the bet or update their beliefs so the bet becomes unprofitable" does not work. The offering agent has an incentive to only ever offer bets that benefit them since only one side of the bet is available for betting.
I'm not certain (without much more consideration), but it seems that Oscar_Cunningham's solution of always taking one half of a two sided bet sounds more plausible.
Partial analysis:
Suppose David is willing to stake 100:1 odds against Trump winning the presidency (before the election). Assume that David is considered to be a perfectly rational agent who can utilise their available information to calculate odds optimally or at least as well as Cameron, so this suggests David has some quite significant information.
Now, Cameron might have his own information that he suspects that David does not and Cameron know that David has no way of knowing that he has this information. Taking this info into account, and the fact that...
(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?