From the postbag: Doubles

Of course, Splittter is the only one writing to me at the moment, which makes me feel a bit like Willie Thorne in the Fantasy Football League sketches. Anyway, here is his wisdom:

Your post on doubles has been bugging me since I read it basically because the accepted gambling wisdom is simply "don't do doubles", full stop, no exceptions... yet your maths looked correct.

I had a sneaking suspicion that it had to do with your bet size relative to your bankroll, and that hidden in the double is the fact that you're essentially sticking an amount larger than your actual stake on the 'second' outcome.

So, to test that theory I imagined the following:

There are two bets for which you'll get 3.00: event 1 you reckon will come in 37%, event 2 35%, both clear value bets.

He goes on to analyse the situation in excruciating detail. As I refuse to be out-mathsed by anyone, let alone Splittter, I'll do the same but more clearly - and reach a slightly different conclusion. His experiment suggests Kelly staking.

With Kelly staking, you would place a fraction k = p - (1-p)/(o-1) of your bankroll on each bet. Your expected return is p(kB(o-1)) - (1-p)(kB) = Bk(po-1)

Betting singles, your Kelly stake on the first game is 5.5% of bankroll; on the second, 2.5%. The outcomes are as follows:

Win-win: (12.95%) +16.50%
Win-lose: (24.05%) + 8.23%
Lose-win: (22.05%) - 0.78%
Lose-lose: (40.95%) - 7.86%

The weighted average of these - trust me - is 0.73%.

By contrast, if you bet the double, your Kelly stake is 2.07% of bankroll, and your outcomes are:

Win-win: (12.95%) +16.5%
Any other: (87.05%) - 2.1%

So, on average, you come out 0.34% ahead. So far, so good for the singles. However, let's examine the bets in terms of risk vs. reward:

Expected risk for two singles: 6.99%
Expected return: 0.73%
Value for singles: 10.44%

Risk for double: 2.07%
Expected return: 0.34%
Value for double: 16.42%

You might argue that we're not comparing apples for apples - that if we're betting singles, we're forced to make the second bet even if the first fails. However, if we don't make the second bet, we do even worse - as you'd expect, failing to make a value bet lowers your expected return (in this case, to 0.66%). The risk in that case is fixed at 5.5%, making the value 12.00% even.

How about the order of the bets? In fact, it doesn't make a difference to the expected return. It does make a difference to your expected risk, though, which drops to 6.26%. That makes the value 11.66% - still lower than the double. Without the second bet if the first loses, the expected return falls to 0.34%, with a risk of 2.5%, making the value 13.59%.

My correspondent challenges me to prove things in general. I scoff, mainly because I ought to do some work. I may leave that for a later post.

All of which seems to show that a double on two value bets gives better value than two singles. The singles give a higher expected value, but at the cost of an increase in risk which reduces the value below the double's.

Exotic bets

For some people, singles - betting on one selection in one event - simply aren't exciting enough, and they need to bet on several selections in several events.

The simplest multiple is the double. The proceeds of winning one bet (a stake of s at odds of d1) are reinvested as your stake in the other (at d2), so you win (s.d1.d2 - s) if both win. A treble is the same with three selections; an accumulator is the same with more.

Another common multiple is the forecast in racing. You predict the first and second place horses/dogs/etc (in order, for a straight forecast, in either order for a reverse forecast which costs twice as much for the same payout*. The winnings are calculated by a Secret Formula known only to the bookies. Exacta and trifecta bets are the same kind of thing, but with three horses.

Now comes the fun bit - exotic bets. The first of these is the patent: three singles, three doubles and a treble on the same three selections, all with the same stake. It thus costs 7 stakes. If all of the selections are at 1.83, you would need to win two to break even (3.66 from the singles, 3.34 from the doubles, minus 7 staked). You need odds of about 7.00 on each to break even with a single win, and three wins will make a profit at any stake at all.

A trixie is similar, but without the singles - so it's four bets: three doubles and a treble. This kind of bet requires at least two of your selections to win to get any kind of payout. At 2.00 each, if two of your selections win, you break even. Three winners, as before, bring you a profit at any odds.

A yankee consists of four selections and every combination of doubles (6), trebles (4) and a 4-way accumulator, making 11 bets altogether. Odds of 3.33 on each selection are enough to break even with two wins, whereas 1.56 would do the trick for three wins.

A lucky 15 adds four single bets to the yankee. At odds of 15.00 each selection, a single winner will break even. Odds of 3.00 will break even with two winners, while 1.52 will guarantee a profit if three of them win.

There are many other types, including the Super Yankee (or Canadian), involving five selections (all combinations except singles - 26 bets). A Heinz is the same thing with six selections (57 bets - hence the name), a Super Heinz the same with seven (120 bets) and a Goliath does the same with eight (247 bets). Lucky 31 and Lucky 63 are the same as Lucky 15 but with five and six selections respectively.

There are two main reasons that exotic bets are interesting. Firstly, for the average punter, there's the chance of a huge payout. If your four singles came in at 2.00, you'd win 8 stakes minus the four you placed. If you'd had a Super 15, you'd have won 80 stakes, minus the 15 you put in.

Secondly, and this is critical for us, they multiply value. This is best explained with a double, or else it gets really complicated. Let's say you have two selections you believe are worth 1.80, but the bookmaker has them available to back at 2.00. On their own, each would give a value of 11% - a nice markup. If you backed them in a double, the true odds of both winning would be 1.80 * 1.80, or 3.24. According to the bookie, the odds of the double are 4.00, giving you a value of 23%, more than double.

Unfortunately, the best odds are usually to be found at betfair, whose structure doesn't lend itself to simultaneous multiples. However, if you're prepared to do a bit of maths, you can figure out equivalent stakes for consecutive events - with the double example above, after the first selection won, you would place the entire winnings on the second selection. I'll see if I can knock up some code to determine optimal staking plans... at some point in the future.

* because it's really two straight forecast bets.