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 $P(A∧B∣E)>P(A∣E)$ when this can never be the case). Another common mistake is *double counting* probabilities

For example, if J believes that $P(heads)=P(tails)=32 $, we can propose two bets

- Pay $2;win$3 if heads, $0 if tails
- 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,

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