Singles or Doubles?

Some background

Back in the long ago, bookmakers were surly beasts who wouldn't accept anything less than a treble on football bets (or so I'm told!).

In this day and age, with easy access to the Internet for many punters, the increased competition has forced the end of this practice. Punters have the option of checking a wide range of bookies to find the best odds, leading in many cases to the virtual elimination of the bookmakers margin, or 'overround' .

"Bookmakers were surly beasts who wouldn't accept anything less than a treble on football bets."

That freedom to choose prices from a range of bookmakers, means that even those who bet on a large number of games may only have a single bet with a particular bookmaker at any one time.

"Many betting sites would have strongly advocated betting in single bets, rather than multiples. But is this correct?"

Meanwhile, many betting sites, this one included, would have strongly advocated betting in single bets, rather than multiples - and certainly staying away from large accumulator bets.

But is this correct? We assume that the bettor is only placing his/her money when they believe they have an 'edge' - that is the return being offered is higher than their estimation of the risk. This is the 'value principle' that underlies all successful gambling.

However we know that the edge is multiplicative rather than additive - that is, if I have an edge of 20% on two bets (highly optimistic, but indulge me!), then the edge on a double is 44% rather than 40%. So I actually achieve a higher edge on multiple bets. Given that the edge is the average profit I can expect to make on any given bet, we appear to have an answer, yes?

Other things being equal, yes we do - but other things are seldom equal. We also need to account for the fact that I am now staking at longer odds, so will expect less winners. To maintain my risk at reasonable levels I need to reduce my stake size when betting longer odds. And after all, a double bet can be viewed as a single (where I bet that both events happen) with longer odds.

It would seem therefore that there may be some optimum in the middle where the balance of higher risk (for the doubles) is compensated for by the higher expectation . This article sets out to explore this middle ground.

The approach

"We'll need to make certain simplifying assumptions, which hopefully will allow us the insight to generalise to more realistic scenarios."

If we are going to look at a problem like this we'll need to make certain simplifying assumptions, which hopefully will allow us the insight to generalise to more realistic scenarios.

What I have done here is to examine a wide range of possibilities in several distinct groups. I assumed that the bettor began with a bank of 100 units and had a season of 300 'events', all of which were at the same odds, and all of which had the same edge.

An event was considered as two parallel 'matches'. For each 'event' there were three possibilities - both 'matches' won, both lost, or one won and the other lost. I considered on one hand a bettor who bet doubles (using the recommended Kelly stake for this combination of edge and odds), and on the other hand a singles bettor who bet half the recommended Kelly stake (for the single) on each of the two singles.

This constituted a single simulation run, at the end of which I recorded whether the bettors bank (in each of the cases of singles and doubles bettors) had dipped below 80 units, 60 units, 40 units, 20 units or 0 units (i.e. bankrupt). Each simulation run was repeated 1000 times. Finally the above process was repeated for every combination of odds and edge, using the following odds:

• 1.4, 1.6, 1.8, 2.0, 2.5, 3.0, 5.0, 10.0

and the following edges

• 1%, 2%, 3%,.........19%, 20%

In this article I'll look at the risks to your bank when staking singles with various odds and edges. In a subsequent article, I'll consider how extending your range to doubles impacts on your risk and profitability. The staking in both cases will be 'fixed-Kelly' - that is calculated based on a 100 unit (i.e. starting) bank, not on the current bank as is the case with 'pure' Kelly staking.

The results

The graph below shows the risk of dropping below 80 units bank during the course of a 'season' when betting with varying edges at odds of 1.4.

e.g. with an edge of 5% there is a 40% risk of the bank dipping below 80 units at least once during the season. We must be careful about drawing too many conclusions from this one graph, and from a limited simulation run (we would prefer 50,000 or 100,000 simulations rather than just 1,000 - to smooth out the 'noise' in the graph). However, we can note certain things, which we can then keep an eye out for in the other results.

"It appears that the risk rises fairly dramatically with edge at small values of edge"

It appears that the risk rises fairly dramatically with edge at small values of edge (i.e. from 1% to 3%), reaching a peak at about 5% and then falling off again. To understand why this is happening, we can look at the way in which Kelly stakes are calculated. The typical approximation (which we use here - a future article may look at the implications of this approximation) is that the stake size is given by the formula (edge)/(odds-1).

Our odds are fixed here (for this graph), so that the above reduces to edge/0.4 = (edge)(2.5). This means that our stake size is directly proportional to our edge, so therefore rises very fast at small values of edge - i.e. it doubles as we go from 1% edge to 2% edge, increaseses by a further 50% as we go to 3% edge etc.

Balanced against that is that as our edge increases we see more winners at any given level of odds. At odds of 1.4, with a 1% edge we expect to win 72%, at 4% edge it is 74%, at 10% edge it is 79% and so on. These two opposing effects give rise to the maximum (relative) risk in the region of 5% edge. We can see similar effects when we plot the equivalent graph for odds of 2.0.

although this time the noise makes it a little more difficult to see the pattern. A look at the equivalent for 10.0

shows a similar trend, albeit we don't appear to reach a peak of maximum risk. The peak is moving in these cases because of the dependence of the stake size on the odds, i.e. stake=edge/(odds-1). Another point worth noting is that the maximum level of risk doesn't vary that much accross the different odds - showing the critical importance of adjusting your stake size for the odds involved.

"A more psychologically important threshold might be when over half of the initial bank is gone."

Of course most of us with any betting experience wouldn't be too perturbed if our betting bank dipped to this level - after all we started with 100 units and we still have 80, with time recover. A more psychologically important threshold might be when over half of the initial bank is gone. I didn't examine specifically for 50% of bank in this study but the following graph is for the risk of dipping below 40% of bank.

Again we see the risk peaking for the short-odds betting around an edge of 5% - albeit the risk is low enough at about 7.5% - i.e. over 92% of the time we bet we won't see our bank dip to these levels. For the odds of 2.0 the peak is at about 9% edge, while for the longer-odds, in this case 5.0, the peak appears to be at very large edge. Similar results can be seen for the risk of total bank loss below.

The results are more 'noisy' due to small amount of data - a result of only having 1000 simulations. We can extrapolate however from the totality of the data to say that similar conclusions to earlier may be drawn. The final graph we will look at in this article will compare risk to potential return directly. We can calculate the expected bank growth as follows:

• Total Wagered=Stake size*Number of bets
• Expected Profit=Total Wagered*%Edge

It is possible therefore to achieve the same expected profit level through a range of difference combinations of edge and odds. The following graph shows the risk of losing varying amounts of the bankroll versus expected profit.

"It is possible therefore to achieve the same expected profit level through a range of difference combinations of edge and odds."

The most important conclusion that can be drawn from this (and the other) graph(s) is that the risk rises very rapidly with expected profit up to the point of bank doubling/trebling, before either levelling out or even falling.

The corrollorary of this is that the risk is very sensitive to small changes in stake size in this region, allowing risk to be substantially reduced for a small (relatively speaking) change in expected profit. Conversely, getting greedy here really ramps up your risk!

In the follow-on to this article I'll examine what happends when we use doubles rather than singles as our bet of choice.