Dutch Book

A Dutch Book is a set of bets that you consider individually fair, but which collectively guarantee a loss

This usually happens when people commit probabilistic fallacies (e.g. the conjunction fallacy, believing $P(A\wedge B|E)>P(A|E)$ when this can never be the case). Another common mistake is double counting probabilities

For example, if J believes that $P(heads)=P(tails)=\frac{2}{3}$, we can propose two bets

1. Pay $2;win$3 if heads, $0 if tails
2. Pay $2;win$3 if tails, $0 if heads

Both bets make sense for J. However, if J takes both bets, then he faces a guaranteed loss of $1

Have the agent bet for propositions with credences (or FBQs) that are too high, and against propositions with credences (or FBQs) that are too low

For any given bet (set $p$ to be $1-p$ for the against case):

Player wins bet	Player loses bet
$S-pS$	$-pS$

Dutch Book Theorem

Based on the Kolmogorov probability axioms,

1. If any axiom is violated, a Dutch Book can be made.
2. If no axiom is violated, then no Dutch Book can be made.