In the previous blog we looked at the reduction in vigorish that arose from adding a single bookmaker to the portfolio of options when wagering on the AFL. There we found significant reductions in average vigorish from this simple inclusion.

I wondered aloud in that blog about how the addition of bookmakers might further reduce average vigorish were we to choose the best price from across the portfolio.

Since I posted that blog, a number of people have generously provided interesting data sources and links that I might be able to exploit to expand that analysis purely in the context of the AFL. 

Today though, I'm going to use some data that I've found at a football data site, which provides bookmaker data for, among other things, EPL games going back for over a decade. Specifically, I'm going to use data for about 1,900 EPL games spanning seven European bookmakers across the period 2012/13 to 2016/17.

Those seven bookmakers are:

• Bet365
• Bet & Win
• Interwetten
• Ladbrokes
• William Hill
• VC
• Pinnacle Sports

Note that, since we're talking football, three prices are offered, one each for the possibility of a home win, draw, or away win. As such, total overround is calculated as 1/Home Price + 1/Away Price + 1/Draw Price - 1.

Vigorish is still calculated as Overround/(1+Overround).

What I'll look at is how the average vigorish varies as we choose portfolios of between 1 and 7 of those bookmakers, the chart for which appears below.

Each dot in the chart refers to the average vigorish from a different portfolio of bookmakers - 7 of them where the portfolio contains just one bookmaker, 21 where it contains two or five bookmakers, 35 where it contains three or four bookmakers, and so on.

The portfolio labelled at the bottom of each "column" is the optimal portfolio for the given number of bookmakers, the average vigorish for which is shown in the table at right.

What we see is a fairly significant reduction as we move from 1 to 4 bookmakers, but far less meaningful reductions as we add a 5th, 6th and 7th.

Now, of course, we have no solid evidence that what we see in European EPL bookmaker markets will generalise to other sports in other continents. Nonetheless, the notion that there might be a practical optimum to the number of bookmakers that we might have in our portfolio, seems reasonable.

Adding bookmakers to a portfolio is not costless because it requires us to forego interest on any monies that we might keep with that new bookmaker, and also adds to the "transaction costs" that we personally endure in pfaffing around with yet one more bookmaker.

One aspect of the optimal EPL portfolio that is especially interesting is the composition of it as we add bookmakers.

Were we able to choose just a single bookmaker, Pinnacle would seen to be the best choice from amongst those whose data we have, since its stand-alone average vigorish is smallest across the period we're investigating. Allowed a second bookmaker, we'd add VC, which makes sense given that its stand-alone average vigorish is second only to Pinnacle's.

But, our optimal choice of a three bookmaker portfolio would see Interwetten added to the group ahead of William Hill, Bet365 and Ladbrokes, despite the fact that all three of them, alone, have lower average vigorish figures. Simply put, adding Interwetten creates a better portfolio for us than does adding William Hill, Bet365, or Ladbrokes. I am, strongly, reminded of Modern Portfolio Theory and the approach it prescribes for creating optimal stock portfolios. The best stock to add isn't necessarily the one with the highest stand-alone expected return but, instead, the one that adds most to the portfolio's expected return.

There's a lot more data to be investigated on the Football Data site, and I intend to do that over coming months, but for now I think it's telling that this initial analysis has confirmed that a portfolio of about 4 bookmakers, empirically chosen, provides a portfolio with near-minimum vigorish.