Bet or update: fixing the will-to-wager assumption

(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 5050 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?