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:

4 majors: 0.9% (TradeSports: 8.5%)
3 majors: 8.2% (TradeSports: 12.7%)
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.