Gambling
In one example, a bookmaker 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) - but the punter has bet $210, hence making a loss of $10 on the race.
However, if Horse 4 was withdrawn and the bookmaker does not adjust the other odds, the implied probabilities would add up to 0.95.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of $10 by betting $100, $50 and $40 on the remaining three horses, respectively, and not having to stake $20 on the withdrawn horse, which now cannot win.
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.
Read more about this topic: Dutch Book
Famous quotes containing the word gambling:
“Someone once asked me why women don’t gamble as much as men do, and I gave the common-sensical reply that we don’t have as much money. That was a true but incomplete answer. In fact, women’s total instinct for gambling is satisfied by marriage.”
—Gloria Steinem (b. 1934)
“At the gambling table, there are no fathers and sons.”
—Chinese proverb.
“As Jerome expanded, its chances for the title, “the toughest little town in the West,” increased and when it was incorporated in 1899 the citizens were able to support the claim by pointing to the number of thick stone shutters on the fronts of all saloons, gambling halls, and other places of business for protection against gunfire.”
—Administration in the State of Ariz, U.S. public relief program (1935-1943)