Dutch Book Arguments

About Dutch Book Arguments

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.

Probability Coherence

For your credences to be coherent (avoid Dutch books), they must satisfy the probability axioms:

• Non-negativity: 0 ≤ P(A) ≤ 1 for all events A
• Normalization: P(certain event) = 1
• Additivity: P(A ∨ B) = P(A) + P(B) - P(A ∧ B)
• Monotonicity: If A implies B, then P(A) ≤ P(B)

How Dutch Books Work

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:

• You'll buy a bet on A for $P(A) that pays $1 if A occurs
• You'll sell a bet on A for $P(A) that pays $1 if A occurs

If your probabilities are incoherent (violate axioms), a bookie can construct a combination of bets where you lose money in every possible outcome.

Example: Incoherent Conjunction

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:

1. Sell you a bet on (A ∧ B) for $0.50 (you pay $0.50, receive $1 if A and B both occur)
2. Buy from you a bet on A for $0.60 (you receive $0.60, pay $1 if A occurs)
3. Buy from you a bet on B for $0.60 (you receive $0.60, pay $1 if B occurs)

In this Dutch book:

• Your net upfront: +$0.60 + $0.60 - $0.50 = +$0.70
• If A and B: You pay $1 + $1 - $1 = $1, so net: -$0.30
• If A and ¬B: You pay $1 - $0 = $1, so net: -$0.30
• If ¬A and B: You pay $1 - $0 = $1, so net: -$0.30
• If ¬A and ¬B: You pay $0, so net: -$0.70 (wait, you gained $0.70 upfront)

Actually, let me recalculate: you lose in all cases where at least one event occurs, and the bookie structured it so you always lose.

Philosophical Significance

• Normative constraint: Rationality requires obeying probability axioms
• Pragmatic vindication: The argument links epistemic rationality to practical rationality (avoiding sure loss)
• Subjectivity: Your specific probabilities can vary, but they must be coherent
• Critiques: Assumes you'll accept fair bets at your stated probabilities; some reject this "operational" interpretation of credence

Connection to Bayesian Epistemology

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.