Coherence principle4/30/2023 ![]() 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. Price ( Red Sox ) + Price ( Yankees ) ≠ Price ( Red Sox or Yankees ) One may set the prices in such a way that 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. These lose-lose situations parallel the fact that a probability can neither exceed 1 (certainty) nor be less than 0 (no chance of winning). 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. 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. So the following Dutch book arguments show that rational agents must hold subjective probabilities that follow the common laws of probability. A person who sets prices in a way that gives his or her opponent a Dutch book is not behaving rationally. When one has a Dutch book, one's opponent always loses. 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.Ī 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. $1 wagered at these odds will produce either a loss of $1 (if Smith loses) or a win of $7 (if Smith wins). This arbitrary valuation - the "operational subjective probability" - determines the payoff to a successful wager. 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. The price one sets is the "operational subjective probability" that one assigns to the proposition on which one is betting. ![]() In other words: Player A sets the odds, but Player B decides which side of the bet to take. 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. One must set the price of a promise to pay $1 if John Smith wins tomorrow's election, and $0 otherwise. Operational subjective probabilities as wagering odds 3 Conditional wagers and conditional probabilities. ![]()
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