Sunday, 27 October 2013

Why The Draw Is (Almost) Never Favourite

While we await the conclusion of the weekend's matches and the official prices before the latest, eagerly awaited FTL table is published, here is a post from the Betfair Forum and my contribution to it. The original poster (the astronomy minded Swift-Tuttle) posed this question:

Does anyone have a mathematical proof (or could refer me to one) of why a football match is unlikely to end in a draw?
To put it another way, of the 3 results possible the draw is never favourite. Why not?
Please don't mention those matches in Italy where the Mafia are involved and the draw is a firm favourite. I'm discounting those.
My thoughts on this topic were:

The original poster asked if there was any mathematical proof why the draw can never be the favourite outcome in a match, excluding special cases where a draw will suit both sides such as are seen in late season Italian league matches or in the final game of round-robin group matches.

While it is not a mathematical proof in the true sense of the word, and unlikely to win me a Fields Medal (I’m too old to win it anyway), the answer lies in the fact that any credible football pricing model derives the match odds from goal expectancies, and the only combination of Home Team goal expectancy v Away Team goal expectancy which will result in the draw having the highest probability is when goal expectancy is very low.

If your model uses a simplistic method along the lines of “in the last 20 matches between a top team and mid-table team, 10 matches have been drawn” then you might put the draw probability at 50%, but such a model is fundamentally flawed and no serious punter would ever use such a model.

The probability of a draw is clearly highest when the expectancy of goals is low. Of the under 1.5 goal outcomes, one-third (the 0-0) results in a draw. Add another goal and one-third of the six possible outcomes results in a draw, but add one more goal, and the draw probability drops precipitously. The draw outcomes from the Under 3.5 goal markets total two from ten.

So unless the goal expectancy in a match is two or less, the draw will never be favourite, but the closer to zero the goal expectancy is, the more probable the draw is. The relative lack of interest in the draw as a betting outcome does, in my opinion, make the draw a value bet when the market’s expectation of goals is higher than it should be.

Results from my model show that in 1,823 such matches over four seasons in the top five European Leagues, 565 matches have resulted in draws, an implied draw price of 3.23 which, with the possible exception of the low-scoring Ligue 1, is usually an easily beatable price.


Jamie said...

Well that's not really a provable statement, but if we let ϵ>0...

Lets go with the alternative statement of when is a football match likely to end in a draw, which then leads to either low goal expectancy (for 0-0 scorelines) or it suits both teams as you suggest.

Thus, a draw doesn't usually suit both teams.

Is this because a win is worth 3 points to the winner and a draw is worth 1 point each? - ie. game theory.
Or because this is a sport we're talking about?

Maybe a model based on how much a draw would suit both teams would give a higher draw probability in some cases.
An in-play model might work with teams who play a specific style - ie. more likely to sit back and settle for a draw with x mins to go.

Nick said...

As Jamie suggests you would think the points incentive of a win over a draw would effect the draw price. But the year they changed from 2pts to 3pts in England (1981/82), there was 121 draws, actually 3 draws more than the previous season's 118.

In a way it proves Cassini's "mathematical proof" on goal expectancies. If back then there were people running their BBC Micros or Acorn Electrons to get goal expectancies to derive match odds, their models wouldn't have been any less accurate, at least for draws.

Anonymous said...

A sample of only 2 seasons?

Simon said...

Hi, first comment on your site

I use something along the lines of

Analyse home team versus similar teams as the away team and away teams against similar team to the home team. Both historical data and then a look at the recent 5 and 10 games for both sides.

If the combined draw probability was 50%, with a relative even split, I would back the draw. Especially as most EPL games the draw is 3.4 (roughly 30%).

Can you explain in a bit more detail why you think the above is flawed?