In gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, no matter what the outcome of the gamble. It is associated with probabilities implied by the odds not being coherent. Gambling (or betting) is any behavior involving risking money or property (making a wager or placing a stake) on the outcome of a game, contest, or other event in which the outcome of that activity depends partially or totally upon chance or upon ones ability to do something. ... In probability theory and statistics the odds in favor of an event or a proposition are the quantity p / (1 − p), where p is the probability of the event or proposition. ... Black Entertainment Television (BET for short) is the first television network geared towards African-Americans. ... The word probability derives from the Latin probare (to prove, or to test). ... 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. ...

In economics a Dutch book usually refers to a sequence of trades that would leave one party strictly worse off and another strictly better off. Typical assumptions in consumer choice theory rule out the possibility that anyone can be Dutch-booked. U.S. Economic Calendar Economics at the Open Directory Project Economics textbooks on Wikibooks The Economists Economics A-Z Institutions and organizations Bureau of Labor Statistics - from the American Labor Department Center for Economic and Policy Research (USA) National Bureau of Economic Research (USA) - Economics material from the organization...

Gambling

In one example, a bookmaker has offered odds and attracted bets that make the result irrelevant; in this case the implied probabilities will add up to a number greater than 1. A bookmaker, or a bookie, is an organisation or a person that takes bets and may pay winnings depending upon results and, depending on the nature of the bet, the odds. ...

In this case, whichever horse wins, the bookmaker will pay out 200 (including returning the winning stake) and so make a profit of 10.

If for some reason Horse 4 was withdrawn and the bookmaker was foolish enough not to adjust the other odds, the implied probabilities would add up to 0.95 and a gambler could lock in a profit of 10, by betting 100, 50 and 40 on the remaining three horses respectively.

Other forms of Dutch books can exist when incoherent odds are offered on exotic bets such as forecasting the order in which horses will finish. With competitive fixed-odds gambling being offered electronically, gamblers can sometimes create a Dutch book by selecting the best odds from different bookmakers, in effect by undertaking an arbitrage operation. The bookmakers should react by adjusting the offered odds in the light of demand, so as to remove the potential profit. Fixed-odds gambling is a form of gambling against odds offered by a bookmaker, an individual, or on a bet exchange. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two or more markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ...

In Bayesian probability, Frank P. Ramsey and Bruno de Finetti required personal degrees of belief to be coherent so that a Dutch book could not be made against them, whichever way bets were made. Necessary and sufficient conditions for this are that their degrees of belief satisfy the axioms of probability. Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. ... Frank Plumpton Ramsey (February 22, 1903 – January 19, 1930) was a British mathematician, philosopher and economist. ... Bruno de Finetti (Innsbruck, June 13, 1906 - Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the operational subjective conception of probability. ... 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. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... The probability of some event (denoted ) is defined with respect to a universe or sample space of all possible elementary events in such a way that must satisfy the Kolmogorov axioms. ...

Economics

In economics the classic example of a situation in which a consumer X can be Dutch-booked is if he or she has intransitive preferences. Suppose that for this consumer, A is preferred to B, B is preferred to C, and C is preferred to A. Then suppose that someone else in the population, Y, has one of these goods. Without loss of generality, suppose Y has good A. Then Y can first sell A to X for B + ε; then sell B to X for C + ε; then sell C to X for A + ε, where ε is some small amount of the numeraire. After this sequence of trades, X has given 3·ε to Y for nothing in return. Y will have exploited an arbitrage opportunity by taking advantage of X's intransitive preferences. In grammar, an intransitive verb is an action verb that takes no object. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ... Without loss of generality or simply WLOG is a frequently used expression in mathematics. ...

Economists usually argue that people with preferences like X's will have all their wealth taken from them in the market. If this is the case, we won't observe preferences with intransitivities or other features that allow people to be Dutch-booked. However, if people are somewhat sophisticated about their intransitivities and/or if competition by arbitrageurs drives epsilon to zero, non-"standard" preferences may still be observable.

See also

• Coherence (philosophical gambling strategy)
• History of the term Dutch Book