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 they are wagering, even if their 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 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. Dutch book 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. 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.