It's NP-hard. Here's a reduction from the complement problem of 3SAT: let's say you have n clauses of the form (p and not-q and r), i.e., conjunctions of 3 positive or negated atoms. Offer bets on each clause that cost 1 and pay n+1. The whole book is Dutch iff the disjunction of all the clauses is a propositional tautology.

I've written some speculations about what this might mean. The tentative title is "Against the possibility of a formal account of rationality":

http://cs.stanford.edu/people/slingamn/philosophy/against_rationality/against_rationality.pdf

I really like the Less Wrong community's exposition of Bayesianism so I'd be delighted to have feedback!