Dutch book
Encyclopedia
In gambling
a Dutch book or lock is a set of odds
and bet
s which guarantees a profit, regardless of the outcome of the gamble. It is associated with probabilities
implied by the odds not being coherent
.
In economics
a Dutch book usually refers to a sequence of trade
s 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.
has offered the following odds and attracted one bet on each horse, making the result irrelevant. The implied probabilities, i.e. probability of each horse winning, add up to a number greater than 1.
Whichever horse wins in this example, the bookmaker will pay out $200 (including returning the winning stake) - hence making a profit of $10 on the race.
If Horse 4 was withdrawn and the bookmaker does not adjust the other odds, the implied probabilities would add up to 0.95. In such a case, a gambler could lock in (guarantee himself) 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.
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.
, 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 turned X into a "money pump", exploiting an arbitrage opportunity by taking advantage of X's intransitive preferences.
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.
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...
a Dutch book or lock is a set of odds
Odds
The odds in favor of an event or a proposition are expressed as the ratio of a pair of integers, which is the ratio of the probability that an event will happen to the probability that it will not happen...
and bet
Bet
Bet or BET may refer to:* A wager in gambling* Basic Economics Test * Bet , the second letter in many Semitic alphabets, including Aramaic, Hebrew, Phoenician and Syriac* Brunauer-Emmett-Teller isotherm. See BET_theory...
s which guarantees a profit, regardless of the outcome of the gamble. It is associated with probabilities
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
implied by the odds not being coherent
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...
.
In economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
a Dutch book usually refers to a sequence of trade
Trade
Trade is the transfer of ownership of goods and services from one person or entity to another. Trade is sometimes loosely called commerce or financial transaction or barter. A network that allows trade is called a market. The original form of trade was barter, the direct exchange of goods and...
s 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.
Gambling
In one example, a bookmakerBookmaker
A bookmaker, or bookie, is an organization or a person that takes bets on sporting and other events at agreed upon odds.- Range of events :...
has offered the following odds and attracted one bet on each horse, making the result irrelevant. The implied probabilities, i.e. probability of each horse winning, add up to a number greater than 1.
| Horse number | Offered odds | Implied probability |
Bet Price | Bookie Pays if Horse Wins |
|---|---|---|---|---|
| 1 | Even |  |
$100 | $100 stake + $100 |
| 2 | 3 to 1 against |  |
$50 | $50 stake + $150 |
| 3 | 4 to 1 against |  |
$40 | $40 stake + $160 |
| 4 | 9 to 1 against |  |
$20 | $20 stake + $180 |
| Total: 1.05 | Total: $210 | Always: $200 |
Whichever horse wins in this example, the bookmaker will pay out $200 (including returning the winning stake) - hence making a profit of $10 on the race.
If Horse 4 was withdrawn and the bookmaker does not adjust the other odds, the implied probabilities would add up to 0.95. In such a case, a gambler could lock in (guarantee himself) 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
Trifecta
In horse racing terminology, a trifecta is a parimutuel bet in which the bettor must predict which horses will finish first, second, and third in exact order. The word comes from the related betting term, "perfecta". A trifecta is known as a tiercé in France and Hong Kong and as a tris in Italy.It...
. With competitive fixed-odds gambling
Fixed-odds gambling
Fixed-odds betting is a form of wagering against odds offered by a bookmaker, an individual, or on a bet exchange.-Calculating fixed odds:It is customary with fixed-odds gambling to know the odds at the time of the placement of the wager , although this category also includes wagers whose price is...
being offered electronically, gamblers can sometimes create a Dutch book by selecting the best odds from different bookmakers, in effect by undertaking an arbitrage
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
operation. The bookmakers should react by adjusting the offered odds in the light of demand, so as to remove the potential profit.
In Bayesian probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
, Frank P. Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
and Bruno de Finetti
Bruno de Finetti
Bruno de Finetti was an Italian probabilist, statistician and actuary, noted for the "operational subjective" conception of probability...
required personal degrees of belief to be coherent
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...
so that a Dutch book could not be made against them, whichever way bets were made. Necessary and sufficient conditions
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...
for this are that their degrees of belief satisfy the axioms of probability.
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 generalityWithout loss of generality
Without loss of generality is a frequently used expression in mathematics...
, 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
Numéraire
Numéraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
. After this sequence of trades, X has given 3·ε to Y for nothing in return. Y will have turned X into a "money pump", exploiting an arbitrage opportunity by taking advantage of X's intransitive preferences.
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.
External links
- http://plato.stanford.edu/entries/epistemology-bayesian/supplement2.html, Dutch Book Arguments in the Stanford Encyclopedia of Philosophy
- http://info.phys.unm.edu/~caves/reports/dutchbook.pdf Probabilities as Betting Odds, report by C. Caves
- http://www.stat.berkeley.edu/~census/dutchdef.pdf Notes on the Dutch Book Argument, by D. A. Freedman
See also
- Coherence (philosophical gambling strategy)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...
- DutchingDutchingIn gambling, Dutching is sharing the risk of losing across a number or runners by backing more than one selection in a race or event. The process calculates the correct stake to place on each selection so that the return is the same if any of them wins...
- Arbitrage bettingArbitrage bettingBetting arbitrage, miraclebets, surebets, sports arbitraging is a particular case of arbitrage arising on betting markets due to either bookmakers' different opinions on event outcomes or plain errors. By placing one bet per each outcome with different betting companies, the bettor can make a profit...
- Mathematics of bookmakingMathematics of bookmakingIn betting parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger and gives the English language the term bookmaker for the person laying the bets and thus...