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
For any given bet (set $p$ to be $1-p$ for the against case):
|Player wins bet||Player loses bet|
# 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.