# Dutch book

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This article is about gambling. For books in the Dutch language, see Dutch literature.

In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, regardless of the outcome of the gamble. It is associated with probabilities implied by the odds not being coherent.

In economics a Dutch book usually refers to a sequence of trades that would leave one party strictly worse off and another strictly better off. Typical assumptions in consumer choice theory rule out the possibility that anyone can be Dutch-booked.

## Gambling

In one example, a bookmaker has offered the following odds and attracted one bet on each horse, making the result irrelevant. The implied probabilities, i.e. probability of each horse winning, add up to a number greater than 1.

Horse numberOffered oddsImplied
probability
Bet PriceBookie Pays
if Horse Wins
1Even$\frac{1}{1+1} = 0.5$$100$100 stake + $100 23 to 1 against$\frac{1}{3+1} = 0.25$$50$50 stake +$150
34 to 1 against$\frac{1}{4+1} = 0.2$$40$40 stake + $160 49 to 1 against$\frac{1}{9+1} = 0.1$$20$20 stake +$180
Total: 1.05Total: $210Always:$200

Whichever horse wins in this example, the bookmaker will pay out $200 (including returning the winning stake) - but the punter has bet$210, hence making a loss of $10 on the race. However, if Horse 4 was withdrawn and the bookmaker does not adjust the other odds, the implied probabilities would add up to 0.95. In such a case, a gambler could guarantee himself a profit of$10 by betting $100,$50 and $40 on the remaining three horses, respectively, and not having to stake$20 on the withdrawn horse, which now cannot win.

Other forms of Dutch books can exist when incoherent odds are offered on exotic bets such as forecasting the order in which horses will finish. With competitive fixed-odds gambling being offered electronically, gamblers can sometimes create a Dutch book by selecting the best odds from different bookmakers, in effect by undertaking an arbitrage operation. The bookmakers should react by adjusting the offered odds in the light of demand, so as to remove the potential profit.

In Bayesian probability, Frank P. Ramsey and Bruno de Finetti required personal degrees of belief to be coherent so that a Dutch book could not be made against them, whichever way bets were made. Necessary and sufficient conditions for this are that their degrees of belief satisfy the axioms of probability.

## Economics

In economics, the classic example of a situation in which a consumer X can be Dutch-booked is if he or she has intransitive preferences. Suppose that for this consumer, A is preferred to B, B is preferred to C, and C is preferred to A. Then suppose that someone else in the population, Y, has one of these goods. Without loss of generality, suppose Y has good A. Then Y can first sell A to X for B+ε; then sell B to X for C+ε; then sell C to X for A+ε, where ε is some small amount of the numeraire. After this sequence of trades, X has given 3·ε to Y for nothing in return. Y will have turned X into a "money pump", exploiting an arbitrage opportunity by taking advantage of X's intransitive preferences.

Economists usually argue that people with preferences like X's will have all their wealth taken from them in the market. If this is the case, we won't observe preferences with intransitivities or other features that allow people to be Dutch-booked. However, if people are somewhat sophisticated about their intransitivities and/or if competition by arbitrageurs drives epsilon to zero, non-"standard" preferences may still be observable.

## References

• de Finetti B., Machi A., Smith A. (1993). Theory of Probability: A Critical Introductory Treatment. New York: Wiley. ISBN 0-521-41850-X.
• Maher P. (1992). Betting on Theories. New York: Cambridge University Press. ISBN 0-471-58882-2.
• de Finetti, B. (1931). "Sul significato soggettivo della probabilità". Fundamenta Mathematicae. 17: 298– 329.