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 if his opponent is prudent.

"Dutch books"

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.

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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

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:mbox{Price}(mbox{Red} mbox{Sox})+mbox{Price}(mbox{Yankees}) eqmbox{Price}(mbox{Red} mbox{Sox} mbox{or} mbox{Yankees}).

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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 probability is additive (see probability axioms).

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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.

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