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Dutch Book

Last updated Oct 19, 2022 Edit Source

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

  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.