Dutch Book

Gambling and prediction markets are known to produce a wide range of probabilities that may not always align with reality. In gambling, this can lead to the bookmaker benefiting at the expense of the gambler, regardless of the outcome. However, in prediction markets, these inconsistencies can offer an opportunity for arbitrage traders to generate risk-free profits. This is known as a Dutch book, and it arises when inconsistent probabilities assigned to different outcomes across markets are exploited.

A Dutch book can emerge in several ways. For instance, the market may still be in its early stages or a major trade may shift the odds significantly, leading to a mispricing. When this occurs, arbitrage traders can buy and sell in multiple markets to lock in a profit, thereby eliminating the mispricing. While this process may be complex due to regulatory barriers and the need to lock up funds in multiple markets, Dutch book opportunities are often short-lived, as the market tends to correct itself quickly once mispricing is detected.

Let’s explore two examples of Dutch books:

Rain or No Rain

Consider the question, “Will it rain tomorrow?” and assume that the following odds are being offered in two different markets:

Individually, the odds in both markets are too expensive and cannot be Dutch booked. However, there is a Dutch book here if we consider them together. The probabilities of no rain and rain add up to less than 1. This violates the axiom of probability.

Let’s trade them with the following step:

• Buy $33 worth of “Rain” in the second market
• Buy $42 worth of “No Rain” in the first market

With a total cost of $75, this set of bets is guaranteed to yield $100 regardless of the outcome, earning a 33% risk-free return.

Incoherent Preference

Consider the following market:

These probabilities are incoherent because the probability of “Rain” should always be equal to or greater than the probability of “Rain and Snow”. While we don’t need to know the true probabilities of the weather events to identify this inconsistency, it is enough to recognize that the inequality is in the wrong direction.

Knowing that arbitraging would correct the mispricing, we can guess that the risk free trade is pushing things at that direction with proportion to the probabilities:

1. Buy 100 “Yes” shares for “Rain”, for $6
2. Buy 100 “No” shares for “Rain and Snow”, for $90

The possible outcomes are:

With this Dutch Book, we will always be able to pocket $4 when the bet settles, and even have a chance of netting $104.

In conclusion, Dutch books in prediction markets are a result of inconsistent probabilities assigned to different outcomes, often across markets. This inconsistency presents a window of opportunity for arbitrage traders who can take advantage of the situation to make risk-free profits by buying and selling in multiple markets. Despite the complexities involved, such as the need to lock up funds in multiple markets and regulatory hurdles, Dutch book opportunities are straight-forward to spot using the axioms of probability. As prediction markets continue to grow and mature, these opportunities may become scarcer, but for now, the astute trader can profit from them and help steer market towards coherency.