**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.

This post inspired me to give the Dutch book a shot over at Pokerstars!

Oh wait. The government seized that domain name. Because poker is definitely gambling and there is definitely no skill.

Watch out Briggs! Your blog may be seized next…unless you’re willing to give the government a cut of your massive profits.

In the illegal activity known as betting on football, the point spread may be different in different cities. This allows anyone with a telephone and credit with an out-of-town bookie to create his own Dutch Book. Of course it would be illegal to do so and any illegal winnings would have to be declared to the IRS as income.

While not a Dutch Book, state lotteries have large profit margins.

Includes numbers by state.

Speed,

Note the footnote: “The amounts under “ticket sales” exclude commissions paid to vendors”.

The takeout is obviously higher than stated here. Commissions aren’t operational costs? Hmmm..

The ticket buyer likely doesn’t care how the takeout is divided.

Using the Maryland Lotto as an example, the minimum takeout is 40% while the maximum exceeds 75%. This has been so for longer than 5 years. Both make me wonder at the veracity of the table.

Briggs,

That happens for any takeout rate. The pool in a parimutuel system can go “negative”. Tracks handle (no pun) this by rounding off the odds to the nearest nickel or dime. The round off goes into a pot used for negative pool payouts.

DAV, you said, “Using the Maryland Lotto as an example, the minimum takeout is 40% while the maximum exceeds 75%.”

I don’t see those numbers in the chart. The only annual reports I could find for the Maryland Lottery (the difficulty of finding up to date numbers should be troubling) are from 2004 and 2005.

Year 2005 2004

Total Sales 1,485,732,850 1,395,408,458

Income from Operations 480,548,166 465,746,987

Income as percent 32.3% 33.4%

http://www.msla.sailorsite.net/resources/annualreport.pdf

Income is reported after taking out for prize expense, retailer commissions, gaming vendor and data processing fees and instant ticket printing and delivery.

Sorry for the format fail above. WordPress appears to strip out the multiple spaces I used to make the columns line up.

Speed,

The numbers I quoted aren’t in the chart. They come from the back of the ticket used in the MD Keno (I meant Keno though I said Lotto — my bad) and they aren’t given directly. The payout for 2-spot is $10/$1 while the probability is between 1/17 and 1/16. Works out to just a little over 40% takeout. The 10-spot has more than 75%.

Does anyone beside the state really care how that takeout is divided? Hopefully, the state doesn’t make any profit at all but ploughs its take back into services. I wonder why the table omits vendor commissions. It’s as if they don’t count toward operational costs. Weird. What else was glossed over?

Might try using the html table tag:

Month

Savings

January

$100

Did it work?

Nope got stripped. Too bad.

DAV says, “I wonder why the table omits vendor commissions.”

That information is included at the link to the annual report beginning at page 13.

A state lottery isn’t an illegal gambling operation in the same way collecting taxes isn’t stealing.

Also, it’s a sin tax on stupidity.

Adam H,

Surely you meant “extortion” and not “stealing”. Under extortion, the victim is given a choice, e.g., Trick or Treat. Stealing is outright confiscation. Another example, in the 10CC song “Dreadlock Holiday” the tourist is given a choice: give me your silver chain at a ridiculous price or I’ll steal it plus some:

Dr Briggs, your interesting article mentions a couple

of on-line references for the “Dutch Book”. Do you

recommend any good off-line references (i.e. books

and journals)?

PS I just recently learned the existence of Parrondo’s

Paradox while browsing a book. Fascinating.

Sweet! I have plenty of students who have trouble finding the buttons on their calculators but will tie themselves into knots learning how to work the angles on a sports bet, so this is a little treasure. And you just wrote me a wonderful probability lecture. Now all I need to do is beef it up with some homework problems, and I’m golden.

DAV, ah you misunderstood me – it was poor wording on my part. Taxes would be stealing if done by any other entity, and lotteries would be illegal gambling operations if done by any other entity. That was all I was trying to say.

I don’t think you can (or meant to) argue that collecting taxes is akin to extortion…

The US track uses a parimutuel system. But the English have mulitple competing bookmakers. Each bookmaker may have slightly different odds. It is possible, though rare, for each bookie to make a book which should favor the bookies if the bets are spread fairly uniformly, but favors the punter if he takes the best odds from each bookie.

DAV, Adam H,

If you refuse to pay taxes due, men with guns will show up at your door and this is more akin to extortion (or would that be coersion). But, for most of us our taxes slip out from our accounts and our paychecks with us even having a chance to put up a fight, so in reality taxation is theft.

Another blast from the past. Briggs, February is a ratings month. You should save the reruns for Christmas.

DAV, Very true. My excuse is that I am too busy today. The roulette stuff is new, at least.

This article describes what also happens naturally, to a usually tiny degree, and for only brief periods of time in a variety of markets…such as currency trading. Many firms employ math wiz’s to identify & then exploit the tiny imperfections on grand scales to risklessly eke out a tiny per-profit transaction that adds up big. Its called arbitrage.

The modern bookmakers markets are very competitive. Have a look at sites like oddschecker to verify that you can get 95%+ payouts if you spread bets between bookies.

Perhaps the best opportunity for arbitrage is when the odds change, perhaps due to injury, and not all the bookmakers react at once. This may be especially true of international markets. Of course, somebody is probably doing this already, and the bookies may have algorithms in place to make sure they are not caught out.

All is not lost, you have 2 more opportunities to make a profit – set yourself up as a site promoting bookmakers and you can get a percentage of losses each month and you can get “sign up bonuses” on your accounts. Eke out a bit more profit by playing two of your accounts play against each at different bookies and you get a percentage of losses paid back on the account that loses.

Be careful though; putting too much money on insignificant matches will get you banned as this is deemed suspicious. Also, make sure your odds scraper (software to update your database) fakes a recent browsers, ideally cookies and all, and does not hammer the bookmakers website.

You can turn a small profit this way but will be breaking the terms of service if you keep signing up new accounts.

One more thought; if the market is competitive and payouts are near 100%, it might make sense to bet on the underdog given that many people will be paying accumulators and choosing favorites without deep consideration of the odds. It should be possible to test this theory with the right data. If this does not work in at least one sport in one country or region I would be surprised and blame it on electronic arbitrage. tsk.

“Suppose the bookie has been taking too much Dutch courage (are we still allowed to say that?)”

Yes, as long as you lot cancel the “The Hague Invasion Act”.

genemachine,

Actually, accepting bets with payouts that are at or below what the probability of success dictates is rarely worth the effort regardless of how the payouts are calculated. If (Pay – cost/Pwin) is less than zero, you are effectively in the same position as one who manufactures an item for $300 then sells it for $295 hoping to make a profit.

DAV,

My point was that you can get close to the magic 100% if you compare odds between multiple bookmakers.

To win you need some other advantage, perhaps just a few percent, to beat the bookmaker; referral through your own site, sign up bonuses, an idea that the odds will shift when some sports injury becomes widely known, predicting pattern from previous games etc.

Bookmakers need to worry about how people are betting and predicting pattern from previous games, you only need to worry about the former. There just might be a gap here.

Can’t you only make a Dutch Book if you get to fiddle with your odds after all the bets are already placed? This seems clear from your first example, where at the point you’re manipulating the odds you already know how much has been bet on each possible outcome. But in real life surely no gambler would accept that – they want to know the odds before they bet, and they choose where to bet by looking for odds they disagree with.

In the extreme case, if all the gamblers bet on the same event and that event occurs, the bookmaker must lose – unless he’s allowed to change the odds to “0 to 1” behind the scenes, in which case he breaks even.

I meant the latter, not former.

George,

In a Parmutuel system, common in the US, none of the odds are fixed. The track takes in all of the bets, rakes a fixed percentage, and pays out the pot to the winners. The track puts out indications of where they expect the post time odds to be, and adjusts those indications as bets are placed. An outside auditor periodically check to see that the track isn’t skimming more for itself.

Other bookmaking opperations, are more of a contract between the client and the bookie. The bookie may offer 3:1 odds to you, and 4:1 odds to annother client.

You wrote:

Odds are a one-to-one function of probabilities. The function isprobability = 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 â€œ1 to 2â€³ = 1 / 2 = 0.5, and â€œ3 to 1â€³ = 3 / 1 = 3. Bookies often state odds like â€œ3 to 2â€³, but here to keep a common denominator of 1, this is written â€œ1.5 to 1â€³ (since 3 / 2 = 1.5), etc.I’m not sure if this could have been more confusing.

First, I think it is p=1/(1+odds).

Second, and less serious, in some of your examples you have a decimal fraction in the denominator, and I think a fraction would have made things much clearer.

Since I don’t bet, I’ve never bothered understanding odds, so I was looking to this for some enlightenment. Ended up at Wikipedia and figured out (I think, trusting Wikipedia, etc.) what is correct.

marcel,

You’re welcome to work through the formula, starting from odds = p / (1 – p) and solving for p.

Second, it was the New York Lottery themselves that provided the decimal-form odds. They did so to normalize everything to a $1 bet; this is also why the expected payouts are given per dollar.