In decision theory, economics, and probability theory, the Dutch book arguments are a set of results showing that agents must satisfy the axioms of rational choice to avoid a kind of self-contradiction called a Dutch book.
The Dutch book arguments are used to explore degrees of certainty in beliefs, and demonstrate that rational agents must be Bayesian;[2] in other words, rationality requires assigning probabilities to events that behave according to the axioms of probability, and having preferences that can be modeled using the von Neumann–Morgenstern axioms.
The thought experiment was first proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability,[citation needed] and was more thoroughly explored by Leonard Savage, who developed them into a full model of rational choice.
This arbitrary valuation — the "operational subjective probability" — determines the payoff to a successful wager.
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 leaving them better-off with 100% certainty.
[citation needed] In one example, a bookmaker has offered the following odds and attracted one bet on each horse whose relative sizes make the result irrelevant.
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.
A person who has set prices on an array of wagers, in such a way that he or she will make a net gain regardless of the outcome, is said to have made a Dutch book.
A person who sets prices in a way that gives his or her opponent a Dutch book is not behaving rationally.
The rules also do not forbid a negative price, but an opponent may extract a paid promise from the bettor to pay him or her later should a certain contingency arise.
The favorite (who did win) would result in a payout of $25, plus the returned $10 wager, giving an ending balance of $35 (a $5 net increase).
After running out of money, Jane leaves the market, and her preferences and actions cease to be economically relevant.
Experiments in behavioral economics show that subjects can violate the requirement for transitive preferences when comparing bets.
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.
It can be shown that the set of prices is coherent when they satisfy the probability axioms and related results such as the inclusion–exclusion principle.