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# Dutch Book Theorem

## What Is the Dutch Book Theorem?

The Dutch Book Theorem is a type of probability theory that postulates that profit opportunities will arise when inconsistent probabilities that violate the Bayesian approximation are assumed in a given context.

### Key Takeaways

• The Dutch Book Theorem is a probability theory which states that profit opportunities will arise when inconsistent probabilities are assumed in a given context.
• The assumed probabilities are a direct result of human error in calculating the probability of an event occurring.
• The Dutch Book Theorem is often associated with gambling and enables professional bettors to avoid losses.

## Understanding the Dutch Book Theorem

The assumed probabilities can be rooted in behavioral finance, and are a direct result of human error in calculating the probability of an event occurring. In other words, the theory states that when an inaccurate assumption is made about the likelihood that an event will occur, a profit opportunity will arise for an intermediary.

For example, assume there is one insurance company and 100 people in a given house insurance market. If the insurance company predicts that the probability that a homeowner will need insurance is 5%, but all homeowners predict that the probability of needing insurance is 10%, then the insurance company can charge more for home insurance. This is because the insurance company knows people will pay more for insurance than what will be needed. The profit comes from the difference between premiums charged for insurance and the costs the insurance company incurs through settling insurance claims.

## Gambling Usage of the Dutch Book Theorem

The Dutch Book Theorem is often associated with gambling, especially betting on horse racing, and the first use of the word was in a scholarly journal, The Journal of Symbolic Logic. Author R. Sherman Lehman wrote that if a bettor is not careful in setting up their bets, an opponent can win money from them no matter what happens.

Professional bettors, especially bookmakers, know to avoid its occurrence at all costs. They refer to this losing book as a "Dutch book." In summary, the Dutch Book Theorem concerns the conditions under which a set of bets guarantees a net loss to one side, or a Dutch Book.

As an example let's say that a bookie takes in a pool of \$100 from people wagering on a horse race and the odds are that the payouts will be \$100, regardless if a certain horse wins or not. The bookie took in \$100 and will pay out \$100, so he breaks even. To remedy this, the bookie, broker or racetrack, often takes a percentage off the top from the pool and thus will pay out the total amount minus some percentage.

For example, Las Vegas sports bookies usually set the Dutch book so that the odds are equal to a probability of 1.05; i.e., they skim 5% from the pool of bets and thus establish a Dutch book. If a bookie sets the skim too high, they might be caught short if the bettors win big.

### Article Sources

Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
1. R. Sherman Lehman. "On Confirmation and Rational Betting." The Journal of Symbolic Logic, Volume 20, No. 3, 1955, Pages 251-262.
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