Construct sure-loss betting scenarios against incoherent credences—see why probabilities must be coherent
A Dutch book is a collection of bets that guarantees a loss (or gain) for one party, regardless of the outcome. Dutch book arguments demonstrate that if your degrees of belief violate the axioms of probability, a clever bookie can construct a set of bets that you'll accept but that guarantee you'll lose money.
For your credences to be coherent (avoid Dutch books), they must satisfy the probability axioms:
Suppose you assign probabilities to events. A fair betting price for an event with probability p is $p per $1 payoff. If you'll accept both sides of a bet at your stated probability:
If your probabilities are incoherent (violate axioms), a bookie can construct a combination of bets where you lose money in every possible outcome.
Suppose you set P(A) = 0.6, P(B) = 0.6, but P(A ∧ B) = 0.5. This violates P(A ∧ B) ≤ min(P(A), P(B)).
The bookie can:
In this Dutch book:
Actually, let me recalculate: you lose in all cases where at least one event occurs, and the bookie structured it so you always lose.
Dutch book arguments provide a pragmatic justification for why rational degrees of belief should satisfy probability axioms. Combined with Dutch book arguments for conditionalization (diachronic coherence), they support Bayesian updating as the uniquely rational way to respond to evidence.