# 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 $P(A \land B | E) > P(A | E)$ 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

- 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.