Dutch Book: Or, How to Gamble

This post first appeared sometime before 2012, but I lost the original date due to the hacking.

If you are a gambler, you'll be delighted to learn that there is a way to create a system of wagers such that you are guaranteed to win money no matter what happens. The system of sure profit is called a Dutch Book. Unfortunately, it only works if the bookie with whom you lay the bet has set his odds incoherently.

That doesn't happen often; but what does is that the bookie (or stock broker, or insurance agent, etc.) usually and purposely makes Dutch Book against his gamblers (or investors, or customers, etc.). That is, the bookie is usually certain of making a profit no matter what happens. No surprise there, right?

There are few on-line sources which explain a Dutch Book. The Stanford Encyclopedia of Philosophy and Wikipedia entries are fine, but short on details and examples. This is because, as you will soon see, it is a subject difficult to understand.

Bets

A bookie sets odds for events, and then gamblers place bets based on those odds. If the event the gambler selected occurs, the bookie must pay the gambler the original amount of the bet plus the bet times the odds; this amount is called the payout. For all the other events (which do not occur) the bookie keeps the bet and there is no payout. The classic example is a horse race. One horse must, and only one horse can, win. A win by one preselected horse is the event of interest to the gambler. The bookie must however set odds on all horses.

If the odds for an event are "even", i.e. 1 to 1, and a gambler places a bet of (say) 10, then if the event occurs the bookie must pay the gambler the original 10 plus the bet times the odds, i.e. 10 + 10 x 1 = 20. If the event does not occur the bookie pockets the 10. If the odds are 3 to 1 with the same bet, the bookie pays 10 x 3 = 30 for the bet plus the original 10 for a payout of 40. Again, if the event does not occur, the bookie keeps the 10.

The bookie doesn't just set odds for one event, of course. He always, at least tacitly, sets them for at least two, with the second always being the "event" that the first event doesn't happen. Make sense? For example, if the event is that the Detroit Tigers win tomorrow's game, the second "event" is that the Tigers lose. The odds for the event and non-event, or for all events as in a horse race, should always be such that the probabilities add to 1. If they do not, then they are not coherent and a Dutch book can be made. The explanation follows.

Odds

Odds are a one-to-one function of probabilities. The function is

   probability = odds / ( 1 + odds).

For even odds, probability = 1 / (1 + 1) = 0.5; for 3 to 1 odds, probability = 1 / (1 + 3) = 0.25; for 1 to 2 odds, probability = 1/(1 + 0.5) = 0.67. The odds are, of course, a fraction, which is why "1 to 2" = 1 / 2 = 0.5, and "3 to 1" = 3 / 1 = 3. Bookies often state odds like "3 to 2", but here to keep a common denominator of 1, this is written "1.5 to 1" (since 3 / 2 = 1.5), etc.

Dutch Book for the Gambler

Suppose the bookie has been taking too much Dutch courage (are we still allowed to say that?) before setting his odds and comes up with the following system of odds: the odds for the event are even, i.e. 1 to 1, and the odds for the non-event are 3 to 1. The probability implied by these odds sums to 0.75 (as shown in the table). Something has gone wrong. It is now possible for a gambler to make Dutch book against the bookie.

Book		Odds		Probability		Bet		Payout
Event		Even		1 / (1 + 1) = 0.50		20		20 + 20 = 40
Non-event		3 to 1		1 / (1 + 3) = 0.25		10		30 + 10 = 40
				0.5+ 0.25 = 0.75		30		40

 

One way is that the gambler makes a bet of 20 on the event, which is at even odds. If the event occurs, the gambler takes 40 (the original 20 plus 20 more). The gambler, or a confederate, also makes a bet of 10 on the non-event. If this non-event occurs, the gambler also takes 40 (the original 10 plus 30 more). The gambler paid 20 + 10 = 30 to play, but no matter happens what he wins 40, which is a sure profit of 10 regardless whether the event occurs or not. Dutch book!

The example is contrived, but it is easy to show that the gambler can always find a way to take money from a bookie if the bookie miss-estimates the odds such that the probabilities implied by the odds sum to a number less than 1. That is, whenever you see a set of odds for a set of events which sum to a probability less than 1, you can be certain of making a profit. Of course, this works for more than two events in a set as well (such as horse races, stocks, and so forth).

Dutch Book for the Bookie

Now turn everything around and look at it from the bookie's perspective. Is there a way the bookie can find a Dutch book against the gamblers? Yes, and they routinely do it. Dutch book is the very means Las Vegas, racetracks, and brokerage houses make their dough. The parimutuel system of betting is a perfect example of the house creating a Dutch book.

The exact details of how bets are placed are not interesting to us here. Suffice to say that bets are made by gamblers who each estimate their own odds, perhaps using their guts or a pre-event estimate of odds given by the bookie. Suppose the end results are in this table.

Book		Bet		Probability		Odds		Payout
Event		30		30/100 = 0.3		2.33 to 1		30 + 70 = 100
Non-event		70		70/100 = 0.7		1 to 2.33		70 + 30 = 100
		100		0.3+ 0.7 = 1				100

 

The pool of money bet is 100, which implies that the pre-event probabilities as judged by the gamblers are 30 / 100 = 0.3 and 70 / 100 = 0.7. This gives the odds 2.33 to 1 and 1 to 2.33, with the payouts being 100 no matter if the event occurs or not. Since the bookie only took in 100 and has to pay out 100 no matter what, he breaks even—and of course makes no money. To fix this, the bookie or broker or racetrack takes a cut off the top. He makes it so he pays out no more than the total amount bet minus some percentage.

Suppose that percentage is 10%. Then there is only 100 * (1 – 0.1) = 90 left for payouts. The table is then adjusted:

Book		Bet		Probability		Odds		Payout
Event		30		30/90 = 0.333		2 to 1		30 + 60 = 90
Non-event		70		70/90 = 0.777		1 to 3.5		70 + 20 = 90
		100 – 10 = 90		0.333+ 0.777 = 1.111				90

 

The implied probabilities have both shifted higher and now sum to 1.111, which is greater than 1, thus the system of bets is incoherent. The odds have also shifted, though they do not immediately appear strange, which is a disadvantage of working with odds for the uninitiated. The payout is always 90 no matter if the event occurs or not, and since the amount bet is 100, the bookie makes a profit of 10 no matter what happens. The bookie has made a Dutch book against the gamblers. Not necessarily against any individual gambler, you understand—some gamblers will still make money—but as a whole the gamblers are taking less than they should.

The bookie can put himself into deep kimchee if he cuts too much, however. Suppose he wants to skim 40% off the top, leaving only 60 in the payout pool. If the non-event occurs, the gambler who bet 70 is going to want not just his 70 back but will demand some kind of profit, however minimal. But with only 60 in the payout pool, he cannot even get his original 70 back. So the bookie must be content with taking less than 40%. Actually, for this system of bets, he must take less than 30% (the maximum skimmable amount is a simple function of the sizes of the bets for the various events and which event occurred).

The amount skimmed is called the "juice", "vig", "transaction costs", among other things. Las Vegas sports bookies usually set the Dutch book so that the odds sum to a probability of about 1.05, which means they skim about 5% from the pool of bets.

Any sum of probabilities greater than 1 also guarantees a Dutch Book for the bookies, just as any sum of probabilities less than 1 guarantees a Dutch Book for the gamblers. The only "fair" bet is where the sum of probabilities equals 1.

It can happen that no gambler picks the event that occurred. Some racetracks fix this situation by refunding all tickets. Brokers manage it by not returning your calls.

Casinos

Update A roulette wheel is a Dutch Book for the house. For American roulette, there is a 1 in 38 chance, i.e. a probability of 2.63% of hitting any number on the wheel. The odds, however, are 34 or 35 to 1, depending on the casino (35 to 1 is the most common). That implies a probability of 2.86% or 2.78% for the single number/slot.

Here is the tricky part: since the odds are 34 or 35 to 1 for the single number, they are that way for each number. That means the implied probability of hitting any number is 38 * 0.0286 = 1.086 or 38 * 0.0278 = 1.056. Since both of these are greater than 1, you can see that the house has guaranteed itself a Dutch Book in its favor.

Almost. The Dutch Book applies only if all numbers on the wheel are bet by gamblers on each roll. If only one gambler bets on one number, then a Dutch Book isn't technically present, but the odds are still against the gambler. Technically, the Dutch Book calculation is for large numbers of spins of the wheel. That is, it is one of the calculations the casino uses to figure their take.

Philosophy

Dutch Books figure prominently in the foundations of subjective Bayesian philosophy. There, axioms and theorems say that degrees of belief/probabilities should be coherent in the same way that bets are. Objective Bayesian foundations don't start with Dutch Books, but say they arise from even more basic axioms. However, these are subjects for another day.

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