Dutch book theorems

In decision theory, economics, and probability theory, the Dutch book arguments or coherence theorems 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. A Dutch book is a set of bets that ensures a guaranteed loss, i.e. the gambler will lose money no matter what happens. A set of beliefs is called coherent if it cannot result in a Dutch book.

The Dutch book arguments is used to explore degrees of certainty in beliefs, and demonstrate that rational agents must be Bayesian; 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.

In economics, is used to model behavior by ruling out situations where agents "burn money" for no real reward; models based on these assumptions are called rational choice models. These assumptions are weakened in behavioral models of decision-making.

The thought experiment was first proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, and was more thoroughly explored by Leonard Savage, who developed them into a full model of rational choice.

Operational subjective probabilities as wagering odds
One must set the price of a promise to pay $1 if John Smith wins tomorrow's election, and $0 otherwise. One knows that one's opponent will be able to choose either to buy such a promise from one at the price one has set, or require one to buy such a promise from them, still at the same price. In other words: Player A sets the odds, but Player B decides which side of the bet to take. The price one sets is the "operational subjective probability" that one assigns to the proposition on which one is betting.

If one decides that John Smith is 12.5% likely to win—an arbitrary valuation—one might then set an odds of 7:1 against. This arbitrary valuation — the "operational subjective probability" — determines the payoff to a successful wager. $1 wagered at these odds will produce either a loss of $1 (if Smith loses) or a win of $7 (if Smith wins). If the $1 is placed in pledge as a condition of the bet, then the $1 will also be returned to the bettor, should the bettor win the bet.

The arguments
The standard Dutch book argument concludes 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 leaving 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: 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.

Dutch books
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. When one has a Dutch book, one's opponent always loses. A person who sets prices in a way that gives his or her opponent a Dutch book is not behaving rationally.

A very trivial Dutch book
The rules do not forbid a set price higher than $1, but a prudent opponent may sell one a high-priced ticket, such that the opponent comes out ahead regardless of the outcome of the event on which the bet is made. 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. In either case, the price-setter loses. These lose-lose situations parallel the fact that a probability can neither exceed 1 (certainty) nor be less than 0 (no chance of winning).

A more instructive Dutch book
Now suppose one sets the price of a promise to pay $1 if the Boston Red Sox win next year's World Series, and also the price of a promise to pay $1 if the New York Yankees win, and finally the price of a promise to pay $1 if either the Red Sox or the Yankees win. One may set the prices in such a way that


 * $$ \text{Price}(\text{Red Sox})+\text{Price}(\text{Yankees})\neq\text{Price}(\text{Red Sox or Yankees}) \, $$

But if one sets the price of the third ticket lower than the sum of the first two tickets, a prudent opponent will buy that ticket and sell the other two tickets to the price-setter. By considering the three possible outcomes (Red Sox, Yankees, some other team), one will note that regardless of which of the three outcomes eventuates, one will lose. An analogous fate awaits if one set the price of the third ticket higher than the sum of the other two prices. This parallels the fact that probabilities of mutually exclusive events are additive (see probability axioms).

Conditional wagers and conditional probabilities
Now imagine a more complicated scenario. One must set the prices of three promises:


 * to pay $1 if the Red Sox win tomorrow's game: the purchaser of this promise loses their bet if the Red Sox do not win regardless of whether their failure is due to their loss of a completed game or cancellation of the game, and
 * to pay $1 if the Red Sox win, and to refund the price of the promise if the game is cancelled, and
 * to pay $1 if the game is completed, regardless of who wins.

Three outcomes are possible: The game is cancelled; the game is played and the Red Sox lose; the game is played and the Red Sox win. One may set the prices in such a way that


 * $$\text{Price}(\text{complete game})\times\text{Price}(\text{Red Sox win}\mid\text{complete game}) \neq \text{Price}(\text{Red Sox win and complete game})$$

(where the second price above is that of the bet that includes the refund in case of cancellation). (Note: The prices here are the dimensionless numbers obtained by dividing by $1, which is the payout in all three cases.) A prudent opponent writes three linear inequalities in three variables. The variables are the amounts they will invest in each of the three promises; the value of one of these is negative if they will make the price-setter buy that promise and positive if they will buy it. Each inequality corresponds to one of the three possible outcomes. Each inequality states that your opponent's net gain is more than zero. A solution exists if the determinant of the matrix is not zero. That determinant is:


 * $$\text{Price}(\text{complete game})\times\text{Price}(\text{Red Sox win}\mid\text{complete game})-\text{Price}(\text{Red Sox win and complete game}).$$

Thus a prudent opponent can make the price setter a sure loser unless one sets one's prices in a way that parallels the simplest conventional characterization of conditional probability.

Another example
In the 2015 running of the Kentucky Derby, the favorite ("American Pharaoh") was set ante-post at 5:2, the second favorite at 3:1, and the third favorite at 8:1. All other horses had odds against of 12:1 or higher. With these odds, a wager of $10 on each of all 18 starters would result in a net loss if either the favorite or the second favorite were to win.

However, if one assumes that no horse quoted 12:1 or higher will win, and one bets $10 on each of the top three, one is guaranteed at least a small win. 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). A win by the second favorite would produce a payoff of $30 plus the original $10 wager, for a net $10 increase. A win by the third favorite gives $80 plus the original $10, for a net increase of $60.

This sort of strategy, so far as it concerns just the top three, forms a Dutch Book. However, if one considers all eighteen contenders, then no Dutch Book exists for this race.

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.

Coherence
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.