# Perplexed comments on Dutch Books and Decision Theory: An Introduction to a Long Conversation

• Ok, I un­der­stood that, but I still don’t see what it has to do with Dutch books.

• P(X) < P(X and Y) gives you a dutch book.

OH I SEE. Reve­la­tion.

You can get pwn’d if the per­son offer­ing you the bets knows more than you. The only defense is to, when you’re un­cer­tain, have bets such that you will not take X or you will not take -X (EDIT: Be­cause you up­date on the in­for­ma­tion that they’re offer­ing you a bet. I for­got that. Thanks JG.). This can be con­cep­tu­al­ized as say­ing “I don’t know”

So yeah. If you sus­pect that some­one may know more math than you, don’t take their bets about math.

Now, it’s pos­si­ble to have some­one pre-com­mit to not know­ing stuff about the world. But they can’t pre-com­mit to not know­ing stuff about math, or they can’t as eas­ily.

• You can get pwn’d if the per­son offer­ing you the bets knows more than you. The only defense is to, when you’re un­cer­tain, have bets such that you will not take X or you will not take -X. This can be con­cep­tu­al­ized as say­ing “I don’t know”

Another defense is up­dat­ing on the in­for­ma­tion that this per­son who knows more than you is offer­ing this bet be­fore you de­cide if you will ac­cept it.

• OH I SEE. Reve­la­tion.

That’s not the Dutch book I was talk­ing about.

Let’s say you as­sign “X” prob­a­bil­ity of 50%, so you gladly take 60% bet against “X”.

But you as­sign “X and Y” prob­a­bil­ity 90%, so you as gladly take 80% bet for “X and Y”.

You just paid \$1.20 for com­bi­na­tions of bets that can give you re­turns of at most \$1 (or \$0 if X turns out to be true but Y turns out to be false).

This is ex­actly a Dutch Book.

• Given that they are pre­sented at the same time (such as X is a con­jec­ture, Y is a proof of the con­jec­ture), yes, ac­cept­ing these bets is be­ing Dutch Booked. But upon see­ing “X and Y” you would up­date “X” to some­thing like 95%.

Given that they are pre­sented in or­der (What bet do you take against X? Now that’s locked in, here is a proof Y. What bet do you take for “X and Y”?) this is a mal­ady that all rea­son­ers with­out com­plete in­for­ma­tion suffer from. I am not sure if that counts as a Dutch Book.

• Given that they are pre­sented in or­der [...] I am not sure if that counts as a Dutch Book.

It is triv­ial to re­for­mu­late this prob­lem to X and X’ be­ing log­i­cally equiv­a­lent, but not im­me­di­ately no­tice­able as such, and a per­son be­ing asked about X’ and (X and Y) or some­thing like that.

• Yes, but that sounds like “If you don’t take the time to check your log­i­cal equiv­alen­cies, you will take Dutch Books”. This is that same mal­ady: it’s called be­ing wrong. That is not a case of Bayesi­anism be­ing open to Dutch Books: it is a case of wrong peo­ple be­ing open to Dutch Books.

• “If you don’t take the time to check your log­i­cal equiv­alen­cies, you will take Dutch Books”

You’re very wrong here.

By Goedel’s In­com­plete­ness The­o­rem, there is no way to “take the time to check your log­i­cal equiv­alen­cies”. There are always things that are log­i­cally equiv­a­lent that your par­tic­u­lar method of prov­ing, no mat­ter how so­phis­ti­cated, will not find, in any amount of time.

This is some­what spe­cific to Bayesi­anism, as Bayesi­anism in­sists that you always give a definite nu­mer­i­cal an­swer.

Not be­ing able to re­fuse an­swer­ing (by Bayesi­anism) + no guaran­tee of self-con­sis­tency (by In­com­plete­ness) ⇒ Dutch booking

• I ad­mit defeat. When I am pre­sented with enough un­re­fus­able bets that in­com­plete­ness pre­vents me from re­al­is­ing are ac­tu­ally Dutch Books such that my util­ity falls con­sis­tently be­low some other method, I will swap to that method.