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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, even if his opponent makes the most judicious choices.

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 him/her, 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 wins) or a win of $8 (if Smith should lose). 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.

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, his or her opponent always loses. A person who sets prices in a way that gives their opponent a Dutch book is not behaving rationally. So the following Dutch book arguments show that rational agents must hold subjective probabilities that follow the common laws of probability.

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 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).

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


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