Guest post by Colin Beveridge
The experts will always tell you that single bets are better for your bank balance than doubles or trebles or more complicated bets. What they don’t usually tell you is why that should be the case.
Let’s take the simplest case we can think of: two races, one after the other, with the favourite available at odds of evens (2.00) for each of them. Let’s also say that each favourite has a true winning probability of 60% – i.e., the true odds are 4/6. The bookie also offers you a double at odds of 3/1 (4.00).
[A quick aside: I’ll be talking about the PERCENTAGE EDGE of a bet from here on: we work it out as (DECIMAL ODDS x PROB – 100), so a 4/1 shot with a 25% chance of winning would have an edge of ( 5.00 x 25 – 100 ) = 25%. ]
OK, this is good: we have three different value bets on offer here. A single on either horse has an edge of 20% (2 x 60% – 100%); the double will come in 60% x 60% = 36% of the time, so the edge on the double is a whopping 44% (4 x 36% – 100%). So, at first glance, it looks like a better bet than the other two combined.
Obviously, there’s more to it than that. Or else it’d be a really short article.
Even when you use flat stakes, the punter who backs the doubles seems to come out ahead over the long term, comparing return to stakes (in fact, Mr Doubley-Pants makes 44% profit, and Mr Sensible-Shoes only 20% – exactly the edges we worked out before). However, it’s not fair to count Mr Sen
The trouble is, there’s a horrible bit of money-management going on in the second bet. Find the nearest mirror you can look into and say out loud: “I’m going to double my stake on the second horse if the first one wins, and walk away if it doesn’t”. Now check out your reflection looking at you oddly. ‘What are you thinking?’, it’s thinking.
How about, if instead of blindly putting our entire winnings on the second horse if it wins, we stake what we’d _expect_ to put on instead. In the double, we back 60% of the time with 2 units, and 40% of the time with nothing – on average, that’s 1.2 units we expect to place. Let’s back the first horse with one unit, and the second with 1.2 units, regardless of the first result. Here’s what happens:
Prob Res E P/L
Both win: 36% +2.2 +.792
First wins: 24% -0.2 -.048
Second wins: 24% +0.2 +.048
Neither wins: 16% -2.2 -.352
=====
Expected: +.440
Brilliant! Just the same as before. There’s a little bit of sleight of hand (we can now lose more on the pair of races) but over the long term, this set of bets has a much lower VARIANCE than the double (2.536 vs. 3.88). What that means is (just like with the whisky) doubles lead to more instability than singles. The double is more of a gamble: you can get big upswings, but these are exactly countered by big downswings… and that means more chance of blowing up and losing all your money.
We’re not done, though! There’s an even better way to split your stakes to get the same return. It certainly doesn’t make sense to bet different amounts on two identical propositions. Let’s back with equal stakes of 1.1 each time:
Prob Res E P/L
Both win: 36% +2.2 +.792
First wins: 24% -0.0 .000
Second wins: 24% +0.0 .000
Neither wins: 16% -2.2 -.352
=====
Expected: +.440
This has a slightly lower variance (2.5168) – and it’s a simple, logical pair of bets.
So what has all this shown? Only that backing doubles may not be a complete mug’s game when you have value, but it’s certainly a drunkard’s. With a little work, you can design simple singles bets with the same outcome but much lower variance.
Colin is a successful bettor and math tutor. Having worked for NASA/NSF he decided to come back to the UK in 2008.
