# Dutch book

Short description: Gambling term

In gambling, economics, and the philosophy of probability, a Dutch book or lock is a set of odds and bets that ensures a guaranteed profit (regardless of the outcome being bet on).[1] It is generally used as a thought experiment to motivate Von Neumann–Morgenstern axioms or the axioms of probability by showing they are equivalent to philosophical coherence or Pareto efficiency.

In economics, the term usually refers to a sequence of trades that would leave one party strictly worse off (effectively "burning money"). Typical assumptions in rational choice theory (namely, the Von Neumann–Morgenstern axioms) rule out Dutch books. These assumptions are weakened by behavioral economics.

In philosophy the Dutch book argument is used to explore degrees of certainty in beliefs,[2] showing that rational agents must be Bayesian (i.e. assign probabilities to events).

There is no agreement on the etymology of the term.[3]

## The arguments

The standard Dutch book argument shows that rational agents must have subjective probabilities for random events, and that these probabilities must satisfy the standard axioms of probability. In other words, any rational person must be willing to assign a (quantitative) subjective probability to different events.

Note that the argument does not imply agents are willing to engage in gambling in the traditional sense. The word "bet" as used here refers to any kind of decision under uncertainty. For example, buying an unfamiliar good at a supermarket is a kind of "bet" (the buyer "bets" that the product is good), as is getting into a car ("betting" that the driver will not be involved in an accident).

### Establishing willingness to bet

The Dutch book argument can be reversed by considering the perspective of the bookmaker. In this case, the Dutch book arguments show that any rational agent must be willing to accept some kinds of risks, i.e. to make uncertain bets, or else they will sometimes refuse "free gifts" or "Czech books"--a series of bets that leaves them better-off with 100% certainty.

### Unitarity

In one example, a bookmaker has offered the following odds and attracted one bet on each horse whose relative sizes make the result irrelevant. The implied probabilities, i.e. probability of each horse winning, add up to a number greater than 1, violating the axiom of unitarity:

Horse number Offered odds Implied
probability
Bet price Bookmaker pays
if horse wins
1 Even $\displaystyle{ \frac{1}{1+1} = 0.5 }$ $100$100 stake + $100 2 3 to 1 against $\displaystyle{ \frac{1}{3+1} = 0.25 }$$50 $50 stake +$150
3 4 to 1 against $\displaystyle{ \frac{1}{4+1} = 0.2 }$ $40$40 stake + $160 4 9 to 1 against $\displaystyle{ \frac{1}{9+1} = 0.1 }$$20 $20 stake +$180
Total: 1.05 Total: $210 Always:$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 always reap 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 axioms

Other forms of Dutch books can be used to establish the other axioms of probability, sometimes involving more complex bets like forecasting the order in which horses will finish. 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 all the axioms of probability.

## Economics

In economics, the classic example of a situation in which a consumer X can be Dutch-booked is if they have 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. This method is a money pump, where Y exploits X using 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.