Coherence (philosophical gambling strategy)

In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent precisely if it does not expose the wagerer to certain loss regardless of the outcomes of events on which he is wagering, provided his opponent chooses judiciously.

Operational subjective probabilities as wagering odds
You must set the price of a promise to pay $1 if John Smith wins tomorrow's election, and $0 otherwise. You know that your opponent will be able to choose either to buy such a promise from you at the price you have set, or require you to buy such a promise from him/her, still at the same price. In other words: you set the odds, but your opponent decides which side of the bet will be yours. The price you set is the "operational subjective probability" that you assign to the proposition on which you are betting.

"Dutch books"
A person who has set prices on an array of wagers in such a way that he or she will suffer a net loss regardless of which outcome eventuates is said to have made a Dutch book.

A very trivial Dutch book
The rules do not forbid you to set a price higher than $1, but if you do, your prudent opponent may sell you that high-priced ticket, and then your opponent comes out ahead regardless of the outcome of the event on which you bet. Neither are you forbidden to set a negative price, but then your opponent may make you pay him to accept a promise from you to pay him later if a certain contingency eventuates. Either way, you lose. These lose-lose situations parallel the fact that a probability can neither exceed 1 nor be less than 0.

A somewhat less trivial and more instructive Dutch book
Now suppose you set 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. You may set the prices in such a way that


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

But if you set the price of the third ticket too low, your prudent opponent will buy that ticket and sell you the other two tickets. By considering the three possible outcomes (Red Sox, Yankees, some other team), you will see that regardless of which of the three outcomes eventuates, you lose. An analogous fate awaits you if you set the price of the third ticket too high relative to 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. You must set the prices of three promises:


 * to pay $1 if the Red Sox win tomorrow's game; the purchaser of this promise loses his 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. You may set the prices in such a way that


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

(where the second price above is that of the bet that includes the refund in case of cancellation). Your prudent opponent writes three linear inequalities in three variables. The variables are the amounts he will invest in each of the three promises; the value of one of these is negative if he will make you buy that promise and positive if he will buy it from you. 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 and only if the determinant of the matrix is not zero. That determinant is:


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

Thus your prudent opponent can make you a sure loser unless you set your prices in a way that parallels the simplest conventional characterization of conditional probability.

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 (but not necessarily countable additivity).