## Friday, February 01, 2008

### An inconsistency in Tiger Woods betting markets?

There are many PGA golf tournaments throughout the year. A specific four of those tournaments are called "majors."

What are the chances that Tiger Woods will win all four majors?

You can start by assuming that Tiger has the same chance of winning any of the four. This probably isn't true, but it's a good start. A friend of mine notes that at his online bookmaker, the current odds (expressed as probabilities) of Tiger winning are:

42% Masters
44% US Open
37% Open Championship
37% PGA Championship

Taking an average of about 40%, we can calculate that the chance of Tiger winning all four events is 0.4 to the fourth power, which is 2.56%. Without showing my work, here are the binomial odds of Tiger winning various numbers of majors:

4 majors: 2.6%
3 majors: 15.4%
2 majors: 34.6%
1 majors: 34.6%
0 majors: 13.0%

However: at
TradeSports.com (choose "Golf" from the left menu, then "Tiger Woods Props"), the current odds are very different. Here they are, taking the halfway point between the bid and ask. (My friend's bookie's odds for these bets are almost the same.)

4 majors: 8.5% (-not the theoretical 2.6%)
3 majors: 12.7% (not the theoretical 15.4%)
2 majors: 24.5% (not the theoretical 34.6%)
1 majors: 35.0% (yes the theoretical 34.6%)
0 majors: 22.5% (not the theoretical 13.0%)

There appears to be a mismatch between these odds and the individual tournament odds. It could just be that TradeSports bettors are assuming a different win probability than 40%.

What would it take to give Tiger an 8.5% chance of winning all four majors, as Tradesports estimates? He'd have to have a 54% chance of winning each major (.54 to the fourth power is about .085). But if he DID have a 54% chance, the other odds should be different. Here's the full table assuming 54%:

4 majors: 8.5%
3 majors: 29%
2 majors: 37%
1 majors: 21%
0 majors: 4%

These are very, very different from the posted odds. If you really thought Tiger had a 54% chance of winning each major, you should also believe he has only a 4% chance of losing them all. You should sell the 22.5% contract for \$2.25, knowing it's only worth 40 cents, and make lots of money.

But what if you don't believe the "4 majors" odds of 8.5%? What if you believe that the "0 majors" odds of 22.5% is the correct number?

In that case, you'd have to assume Tiger had a 31% chance of winning each tournament and a 69% chance of losing, since .69 to the fourth power equals 22.5%. Based on that assumption, the true odds are:

2 majors: 27 % (TradeSports: 24.5%)
1 majors: 41 % (TradeSports: 35.0%)
0 majors: 31 % (TradeSports: 22.5%)

In this case, you'd believe that Tradesports is hugely overestimating the "4-majors" odds. You'd be happy to sell a "4 majors" contract at 8.5%, knowing the true odds are really only about 1%. You'd be making, on average, a 700% markup!

No matter what probability you assume for a Tiger win, the odds just don't seem to make sense. It looks like there are some serious opportunities here to make money. If you believe Tiger is a 54% winner, lay odds on "0 majors." If you believe Tiger is a 31% winner, lay odds on "4 winners". And so on. Whatever you believe Tiger's correct odds are, there's an advantageous bet for you.

And not just a *slight* advantage – a HUGE one! If you go with the 54% estimate, and lay odds on "0 majors," you're getting a 500% markup. If you go with the 31% estimate, and lay odds on "4 majors," you're getting that 700% markup.

Indeed, there is no single-tournament probability that causes the posted odds to make enough sense that there isn't a huge opportunity one way or another.

That confused me. Aren't markets supposed to be efficient? How can these probabilities be so far out of line?

Then it occurred to me: in the arguments above, which bet was advantageous depended on Tiger's true odds of winning. But what if that itself is unknown? Not just unknown, but REALLY unknown – random, in the sense that no amount of study can figure it out. Maybe Tiger has off-years and on-years, and there is absolutely no way of knowing which it's going to be. In fact, Tiger's skill level might not yet be set – it might depend on his practice level, or his health, or his personal life. Suppose it would be randomly decided just before the Masters?

More specifically, let's suppose that Tiger's single-tournament probability might be 15%, or 20%, or 25%, or so on, up to a maximum of 70%. And suppose each of those 12 probabilities is equally likely. If we do the math based on that assumption, we come up with:

4 majors: 08.0% (market odds 8.5%)
3 majors: 19.6% (market odds 12.7%)
2 majors: 26.2% (market odds 24.5%)
1 majors: 26.9% (market odds 35.0%)
0 majors: 19.2% (market odds 22.5%)

On a more practical level, the theoretical odds now come reasonably close to the market odds. Not perfect, but much better than any of the situations based on a single, fixed probability. Indeed, maybe there's a model that might come even closer to market odds: maybe, for instance, if Tiger has a 50% chance of being way off, with a 10% probability, but a 50% chance of being on, with a 70% probability, we get the TradeSports odds exactly. I haven't tried that, or any of the infinity of other possibilities. The closest I came, though trial and error, is the chart above. But there might be a way to solve this mathematically, instead of by hacking away like I did.

(Another interesting thing about this method is that the assumptions have Tiger's expected single-tournament odds at 42.5% (the sliding scale from 15 - 70% averages 42.5%). The chance of winning four straight tournaments at 42.5% is about 3.3%. At first glance, that might lead you to believe that the odds of "4 majors" should be 3.3%. But, because the probability of a sweep isn't linear on the probability of a single tournament, it doesn't work that way. The actual chance is 8%, not 3.3%. This could be the explanation of why my friend's bookie has each tournament at 40%, but the parlay of all four tournaments at much more than 40% to the fourth power.)

In any case, I originally thought there was a seriously advantageous betting opportunity. There's less than I thought. However: in everything I tried, the "3 majors" number was significantly higher than the "4 majors" number. It's only 50% higher at TradeSports. I think money can be made by selling the "4 majors" contract at 8.5%, and buying the "3 majors" contract at 12.7%. It's not a sure thing, but I think the expectation has got to be positive.

Here's a challenge: can you come up with some plausible set of assumptions under which the "3 majors" probability is only 50% higher than the "4 majors" probability? My gut says you can't – that this is one of those examples of "
favorite-longshot bias," and one that you can easily take advantage of right now.

But I'm not certain of this, and markets are a lot smarter than I am. There might be something I'm missing. Don’t blame me if Tiger hits the grand slam and you lose your shirt.

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At Friday, February 01, 2008 6:13:00 AM,  j holz said...

Some betting markets are far more efficient than others. Propositions like "Will Tiger Woods win the Grand Slam" or "Will the Patriots go undefeated?" are mainly bet by unsophisticated bettors. How many people are willing to estimate the odds of the Pats winning every game and multiply them together? It's much easier to say "I think they can rewrite history" or "It's never been done and it won't happen now," and back that opinion with some cash.

On SportsCenter and NFL pregame shows this past season, they constantly asked the hosts if they thought the Patriots could go 16-0. The hosts were never interested in discussing probabilities; they gave a yes or no answer and stuck with it. That's the way most people think about it.

At Friday, February 01, 2008 10:55:00 AM,  tangotiger said...

Phil, I agree with the last half of your post. His true winning percentage is not fixed, but some number that follows a normalish distribution around some true mean.

So, the consecutive 4 slams is a result of the chances that he is really better than his expected mean, and the average of that number to the 4th, and the chances that he's worse than his mean, must be higher than the mean to the 4th.

It's interesting in trying to find a model that explains the odds. Certainly, if you can't create a model that explains the odds, you have an arbitrage opportunity here.

And even if you can create a model that explains the odds, the underlying assumptions could be rather optimistic or pessimistic, again offering an opportunity to beat the line.

Fascinating.

At Friday, February 01, 2008 11:07:00 AM,  Fredrik Gyllensten said...

I make my living from golf betting, and I can tell you that Tiger has a much bigger chance for winning all the 4 majors then what the bookmakers thinks..

I seriously believe that it is at least a 15 % chance for Tiger to win a Grand Slam this year. Seriously , he is propably going to win 75-100 % of his tournaments this year, and the chances are bigger for majors.. And if he win the 3 first, he got a lot bigger chance of winning the PGA then any of the other majors; he is just so good mentaly.

You should not be surprised if Tiger wins every single tournament he plays in '08.

At Friday, February 01, 2008 11:09:00 AM,  Phil Birnbaum said...

Tango: agree completely. In this case,

f(E(x))

is not equal to

E(f(x))

because, as you point out, the odds of winning four straight are proportional to x^4, not to x.

Agreed, also, that the distribution is probably normal around some mean, but we don't know what that mean is. It's probably reasonable to start with 0.4, since that's what the individual tournament odds say. I just didn't want to do the programming for the normal distribution.

Anyway, for "4 majors" to be 2/3 as likely as "3 majors," the probability of winning would have to be very high. At 70%, there's a .24 chance of 4/4, and a .41 chance of 3/4.

You have to go to 80% to get equal chances of 4/4 and 3/4 (about 40% each). And there's no way Tiger should ever be a 1:4 favorite to win a major.

So I think the "short 4/4 and go long 3/4" strategy would work.

At Friday, February 01, 2008 11:12:00 AM,  Phil Birnbaum said...

Frederik: agreed, if you think Woods will win 75% of his tournaments, then there's a 32% chance he'll win the grand slam. I disagree with you that he'll win 75% of his tournaments, though.

If you do, you're probably better off betting the individual tournaments, where you can get a 75% chance for a 40% bet. The advantage is less, but you get lots and lots of bets for the odds to kick in, and you're more likely to make money.

At Friday, February 01, 2008 11:39:00 AM,  Brian Burke said...

I generally agree. But aren't you assuming independence. Normally, that's a good assumption, but if Tiger wins the first 3 majors of the year then we might be able to make some suppositions:

-Tiger is healthy
-His swing mechanics are doing well
-His primary threats are not as competitive this year
-Or his primary threats are injured

Knowing Tiger has already won 3 majors, the outcome of the 4th is not independent of the previous tournaments. (It's not direct dependence, but codependence on other underlying factors.)

So maybe Tiger really would have a higher probability of winning the fourth major. Instead of .4^4, Tiger's chance of winning all 4 majors .4 x .4 x .5 x. 6 or something like that.

That said, I prefer j holz's explanation.

At Friday, February 01, 2008 11:57:00 AM,  Phil Birnbaum said...

Yup, I'm assuming independence. But if Tiger's swing is working well, the odds on *all four* majors should be 0.6 shouldn't they? Same if his competition isn't healthy, and so on.

I agree with your .4/.4/.5/.6 example, but those would be the *before-the-fact* estimates of the odds. After Tiger wins three straight, we would realize that his chances really were .6/.6/.6 for all three. Which would make them all still independent.

Put another way: non-independence would mean that when Tiger wins, that actually changes the odds that he'll win the next one. What is happening here is that when Tiger wins, all that's changing is *our perception of the odds* that he'll win the next one. Our estimates change, but the events are still independent.

If someone wants to say that when Tiger wins, he demoralizes his opponents to the point that his odds actually do increase, then independence wouldn't be satisfied.

I think that's what you mean by "it's not direct dependence, but codependence on other underlying factors." But I think that still qualifies as independence.

At Friday, February 01, 2008 8:27:00 PM,  tangotiger said...

I'll ditto Phil's last post. He was thinking what I was, but said it so much better.

Fred: no offense, but either you are a yapper or a bettor. If you truly believe that the oddsmakers are wrong, then bet the mortgage on it.

At Friday, February 01, 2008 9:39:00 PM,  Brian Burke said...

Ok. I see.

At Monday, February 04, 2008 2:13:00 PM,  Nate Hebel said...

Could an injury to Tiger be considered a non-independent event? For example, Tiger cruises through the first 2 majors by a wide margin. But then he breaks his wrist in the 3rd, that "loss" in major #3 would directly impact his performance in the 4th major. Thus in this case the odds of winning 3 majors could be justified as lower than independence would suggest...

I'm not sure if I buy my own argument or not, but it could be a factor.

Regardless, I think the "buy 3, sell 4" strategy should work. And you could even hedge yourself along the way to eliminate risk. That is, sell a volume of 4 greater than the purchase of 3 to cover the price ratio. (That is, be revenue-neutral if tiger wins 0-2). Then your overriding interest would be in Tiger losing a major. You could then buy Tiger to win the Masters to hedge that interest. If he wins, you use that \$ to re-hedge in the next major. If he loses, you are in good shape as the "win 4" is out of play - now you need to hedge in the opposite direction (sell Tiger to win the next major)... other than the bid-ask spread, it should work.

At Tuesday, February 05, 2008 6:15:00 PM,  Brian Burke said...

One other idea is that the courses are unique. Certain courses favor certain types of golfers and perhaps that is being factored into the odds for Tiger--not just for him but for his primary competitors as well.

At Friday, February 08, 2008 2:56:00 PM,  Anonymous said...

tangotiger; I make my livivng from sports betting.

But I have a lot of learn when it comes to calulating the chances for long-term bets like this.

How do I calculate the chanses for him winning 4 straight? If this is the % for the individual tournaments;
Tiger to win The Masters: 80 %
Tiger to win the US Open: 80 %
Tiger to win The Open: 67 %
Tiger to win US PGA: 70 %
I know this is really high percentages, but I seriusly believe that it is close to correct..

Remmember;
- Tiger have won 7 of his last 8 tournaments..
- He won 4 of his last 8 tournaments by 7 or 8 shots.
- He is more fit then ever, hes swing is better then ever, he puts better then ever; And he still says that he need to improve, he says that he has holes in his game that he need to fix.

Btw; I did put a good slump of money on Tiger when the odds was better, 41 in odds for Grand Slam. I'll win 5 figure if he make a Grand Slam, plus 4 figure if he beats Jack's record of 18 majors by the end of 2009 - I got 26 in odds for that on ladbrokes, but the removed the bet just hours after I have out tips for it..

BTW; my ROI over the last 36 weeks (which is the time I have been giving out tips) is at about 170 %.

At Friday, February 08, 2008 2:57:00 PM,  Fredrik said...

The post above was from me, 'Fredrik'.

Feel free to answer me by e-mail to fredarn@fredarn.com

At Wednesday, February 13, 2008 10:21:00 AM,  parinella said...

I'm a little late here, but I just saw it yesterday and had a thought today.

This strikes me as a projection problem. PECOTA (or SNEAD, maybe, for golf) comes up with a range of estimates for his performance, which is a combination of that player's true skill level and luck. My initial thought on the model was something like your "50% chance at having a true skill level of 10% (in any major), 50% at being at 70%". Then, after the first major, we can come up with a much better estimate of where his true talent is in 2008.

But then I thought about how insane it would be (or at least tango and mgl would mock a person for doing so) to modify a player's projections for the rest of the year based on how he performed in April.

But is it? What if we split up players into "high variability" and "low variability" based on PECOTA (or other system that had confidence intervals), noted how they did in April, then compared whether April had any value in predicting their rest-of-year performance, and whether the high variability group did indeed play closer to their April performance than the low variability group.