# Bookmakers love accumulators

You probably know about accumulator, or so-called ‘acca’, bets. Rather than betting individually on several different matches, in an accumulator any winnings from a first bet are used as the stake in a second bet.  If either bet loses, you lose, but if both bets win, there’s the potential to make more money than is available from single bets due to the accumulation of the prices. This process can be applied multiple times, with the winnings from several bets carried over as the stake to a subsequent bet, and the total winnings if all bets come in can be substantial. On the downside, it just takes one bet to lose and you win nothing.

Bookmakers love accumulators, and often apply special offers – as you can see in the profile picture above – to encourage gamblers to make such bets. Let’s see why that’s the case.

Consider a tennis match between two equally-matched players. Since the players are equally-matched, it’s reasonable to assume that each has a probability 0.5 of winning. So if a bookmaker was offering fair odds on the winner of this match, he should offer a price of 2 on either player, meaning that if I place a bet of 1 unit I will receive 2 units (including the return of my stake) if I win. This makes the bet fair, in the sense that my expected winnings – the amount I would win on average if the game were repeated  many times – is zero. This is because

$(1/2 \times 2) + (1/2 \times 0) -1 = 0$

That’s the sum of the probabilities multiplied by the prices, take away the stake.

The bet is fair in the sense that, if the match were repeated many times, both the gambler and the bookmaker would expect neither to win nor lose. But bookmakers aren’t in the business of being fair; they’re out to make money and will set lower prices to ensure that they have long-run winnings. So instead of offering a price of 2 on either player, they might offer a price of 1.9. In this case, assuming gamblers split their stakes evenly across two players, bookmakers will expect to win the following proportion of the total stake

$1-1/2\times(1/2 \times 1.9) - 1/2\times (1/2 \times 1.9)=0.05$

In other words, bookmakers have a locked-in 5% expected profit. Of course, they might not get 5%. Suppose most of the money is placed on player A, who happens to win. Then, the bookmaker is likely to lose money. But this is unlikely: if the players are evenly matched, the money placed by different gamblers will probably be evenly spread between the two players. And if it’s not, then the bookmakers can adjust their prices to try to encourage more bets on the less-favoured side.

Now, in an accumulator bet, the prices are multiplied. It’s equivalent to taking all of your winnings from a first bet and placing them on a second bet. Then those winnings are placed on the outcome of a third bet, and so on. So if there are two tennis matches, A versus B and C versus D, each of which is evenly-matched, the fair and actual prices on the accumulator outcomes are as follows:

Accumulator Bet A-C A-D B-C B-D
Fair Price 4 4 4 4
Actual Price 3.61 3.61  3.61 3.61

The value 3.61 comes from taking the prices of the individual bets, 1.9 in each case, and multiplying them together. It follows that the expected profit for the bookmaker is

$1-4\times 1/4\times(1/4 \times 3.61) = 0.0975$.

So, the bookmaker profit is now expected to be almost 10%. In other words, with a single accumulator, bookmakers almost double their expected profits. With further accumulators, the profits increase further and further. With 3 bets it’s over 14%; with 4 bets it’s around 18.5%. Because of this considerable increase in expected profits with accumulator bets, bookmakers can be ‘generous’ in their offers, as the headline graphic to this post suggests. In actual fact, the offers they are making are peanuts compared to the additional profits they make through gamblers making accumulator bets.

However… all of this assumes that the bookmaker sets prices accurately. What happens if the gambler is more accurate in identifying the fair price for a bet than the bookmaker? Suppose, for example, a gambler reckons correctly that the probabilities for players A and C to win are 0.55 rather than 0.5. A single stake bet spread across the 2 matches would then generate an expected profit of

$0.55\times(1/2 \times 1.9) + 0.55\times (1/2 \times 1.9) -1 = 0.045$

On the other hand, the expected profit from an accumulator bet on A-C is

$(0.55\times1.9) \times (0.55\times1.9) -1 = 0.092$

In other words, just as the bookmaker increases his expected profit through accumulator bets when he has an advantage per single bet, so does the gambler. So, bookmakers do indeed love accumulators, but not against smart gamblers.

In the next post we’ll find out how not knowing the difference between accumulator and standard bets cost one famous gambler a small fortune.

Actually, the situation is not quite as favourable for smart gamblers as the above calculation suggests. Suppose that the true probabilities for a win for A and C are 0.7 and 0.4, which still averages at 0.55. This situation would arise, for example, if the gambler was using a model which performed better than he realised for some matches, but worse than he realised for others.

The expected winnings from single bets remain at 0.045. But now, the expected winnings from an accumulator bet are just:

$(0.7\times1.9) \times (0.4\times1.9) -1 = 0.011,$

which is considerably lower. Moreover, with different numbers, the expected winnings from the accumulator bet could be negative, even though the expected winnings from separate bets is positive. (This would happen, for example, if the win probabilities for A and C were 0.8 and 0.3 respectively.)

So unless the smart gambler is genuinely smart on every bet, an accumulator bet may no longer be in his favour.

## 7 thoughts on “Bookmakers love accumulators”

1. Richard Greene says:

Key question – would Smartodds recommend accumulator betting (based on the models) to its clients, were that to be available?!

1. Stuart Coles says:

that is a good question, and i’d be interested to know what other people think. i guess arguments against are: i) as i showed in the post, you really need your numbers to be superior in every bet to be sure of improving value; ii) it would involve moving to a regime with a lower frequency of wins, but with bets at a bigger overall price. this might improve expected winnings – if model were genuinely positive value on all bets – but it would also increase variance: i.e. bigger swings in returns; iii) the logistics of trading multiple events in a single bet – ensuring model reliability, market value etc etc – are likely to be considerable, even if such betting options are actually available. this is really outside my knowledge-base though, so i’d really appreciate other points of view.

2. Ian Rutherford says:

Richard my gut instinct is no Smartodds would not recommend accumulator betting from a profitability point of view, unfortunately I’m wrong nearly as often as I’m right. Best to get the maths guys to put all the information in a computer and see what it comes up with.

3. Ian Rutherford says:

Having re read what Stu wrote, given it 5 minutes thought, I’ve changed my mind, accumulators better re profit.

1. Stuart Coles says:

Thanks for comments Ian. Yeah, subject to the comment I made that each bet has to have positive value to be sure that the accumulator increases overall value, I think the maths supports the use of such bets. But the caveats remain, including what might be the biggest issue of having to simultaneously monitor several markets to determine whether and when to place a bet. There are likely to be others reading this post who know about such practicalities far better than I do though, and hopefully one of them can comment.

1. Sam.Johnston@smartodds.co.uk says:

Yes definitely agree that as long each selection is +ev then an accumulator would definitely be a good idea. Especially as, because of the muggish perception of them, they’d be a good way to mask action

But speaking from a placement side of things, they are not really worth the effort. Stake sizes through our client’s normal outlets would be extremely small and any route offering decent volume would likely not last.

There was a brief period last season where they were able to place a couple of doubles (none of them won!) but it didn’t last long

2. Stuart Coles says:

Thanks Sam… that clears things up nicely.