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 1ifJohnSmithwinstomorrow′selection,and0 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. 1wageredattheseoddswillproduceeitheralossof1 (if Smith loses) or a win of 7(ifSmithwins).Ifthe1 is placed in pledge as a condition of the bet, then the 1willalsobereturnedtothebettor,shouldthebettorwinthebet.DutchbookApersonwhohassetpricesonanarrayofwagers,insuchawaythatheorshewillmakeanetgainregardlessoftheoutcome,issaidtohavemadeaDutchbook.WhenonehasaDutchbook,one′sopponentalwaysloses.ApersonwhosetspricesinawaythatgiveshisorheropponentaDutchbookisnotbehavingrationally.SothefollowingDutchbookargumentsshowthatrationalagentsmustholdsubjectiveprobabilitiesthatfollowthecommonlawsofprobability.Therulesdonotforbidasetpricehigherthan1, 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.
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In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which the gamblers bet. It is associated with probabilities implied by the odds not being coherent. In economics, the term usually refers to a sequence of trades that would leave one party strictly worse off and another strictly better off.