They've both got pros and cons but can you ever really definitively say one is better than another?
Doctor of Danger says "yes".
And on the first day, DoD said.......
"We are seeking some elusive optimum where the risk of what we lose is balanced, to our own comfort level, against the potential winnings."
At the infancy of our gambling careers, we've all considered the perennial betting question - how much should I bet?
We will know, of course, that the more we bet, the bigger our potential win is. Yipee! Big win-good!
Quickly following this epiphany is the realisiation that the more we stake, the more we can potentially lose. Doh! Big loss-bad!
You still with me brother? Of course you are! This is simple stuff and we've all been there. What we may not have considered, till now, is that we are taking the first tentative steps towards forming a staking plan - i.e. deciding how much to place on any bet. We are seeking some elusive optimum where the risk of what we lose is balanced, to our own comfort level, against the potential winnings.
That comfort level is a personal thing - how risk-averse are you? - but the fundamental principles involved will remain constant for all bettors. What this article will do is to compare, using some computer simulations, two common (and reputable!) staking plans, namely fixed profits (where the bettor wages a varying amount, calculated to return a fixed profit in the event of a win) and fixed stakes (where the bettor wages a fixed amount, returning a variable amount in the event of a win).
And on the second day, DoD said..... .
"To get a fair comparison it is important that the same amount is risked in each staking plan"
Now that we know what we want to do, it is important to set out the basis on which the comparison was carried out. What I did was to consider a scenario where someone had an edge of 5% (i.e. on average they would expect to return 5% profit an any given wager), and made a sequence of 100 wagers. In each case the bettor would start with a notional bank of 100 units.
With the fixed stakes scheme, they would be risking one unit on each wager. With the fixed profits scheme they would be risking varying amounts - depending on the odds available for that wager. To get a fair comparison it is important that the same amount is risked in each staking plan. To do this, the individual stake sizes on the fixed profits scheme were all scaled by a common factor so that the total stake for the 100 wagers was equal to 100 units, i.e. the same as for the fixed stakes scheme.
And on the third day, DoD, after a short nap, got down to specifics......
For each staking plan I used the same sequence of wagers, and ran 1000 simulations. For the wagers, I chose to use an example of laying horses, but exactly the same prinicples would apply when backing and/or when considering another sport.
In the interests of realism I used real-life data taken from Adrian Massey's site, covering over 45,000 horses. For the purposes of this simulation, for each 'run' of 100 bets I chose 100 'horses' (or more correctly odds) randomly from these 45,000, but rejecting any whose odds were higher than 9/1. Each run consisted of using the two staking plans in parallel and noting the following results:
- % Profit after 100 bets
- Maximum and minium bank during the 'run'
- The average change in bank per bet, i.e. on average how much (in absolute terms) did the bank go up or down on average after each bet
- % time during each run that the level profits system bank was ahead of the level stakes system bank
On the fourth day, DoD, produced some results.....
The distrubtion is also 'narrower' - the fixed profits system tended to be more reliable in producing the theoretical yield.
What the graph above shows is how often a specific yield occurred for each staking system. What you can see fairly clearly is that the fixed profits system tended to produce a profit more often, and a better profit when it did. The distrubtion is also 'narrower' meaning that the yield is more predictable - i.e. while both systems produced something like a 5% yield (the theoretical average yield) most often, the fixed profits system tended to be more reliable in producing this yield. The next graph, shown below, is analogous to the above one, but this time uses cumulative frequency (i.e. how often this yield or worse was produced).
What can be seen from the above graph is that for any given level of yield it is more likely to be achieved by fixed profits staking. The small anomoly visible at the top right of the graph is an artefact of the small (statiscially speaking!) number of simulations.
On the fifth day, DoD had a phone call from people wondering about how the bank behaved during the simulations.....
The above graphs show the results at the end of the simulation, but the behaviour of the bank during the 'season' is also of crucial importance. Wild fluctuations can be scary, and in a real-life scenario may cause us to cry halt, even if profit might potentially be just around the corner. There are several ways to measure this - the most simple being to keep track of the maximum and minimum levels of the bank during each simulation run. These results are shown below.
What we see here is interesting. Higher levels of maximum bank occur more often with the fixed stakes system - notice the way the blue curve is above the pink one from about 112 onwards, but that the lower max bank levels are less likely. The cumulative frequency version of the same data tells us pretty much the same thing
although what we do see is that we have greater likelihood of any given level of maximum bank with the fixed profits system.
On the sixth day, the pessimists wondered what was happening at the other other end of the scale....
With minimum bank we get much the same picture
This time we see, more clearly from the cumulative frequency graph, that the fixed stakes system is more likey to produce a lower minimum bank.
On the seventh day, Doc was demanding his statuory day of rest but while the pessimists were somewhat appeased, there were doubts - in between the extremes, how did the the two systems compare?...
Well, this is indeed an interesting question. To answer it isn't straightforward, so what I did was to measure two quantities
- for each run of 100 simulations, I kept track of the percentage of times (during the 100 bets) that the level profits system bank was ahead of its' level profits counterpart.
- how the bank varied after each bet with each system. I did this by recording the absolute (i.e. neglecting positive or negative) changes in bank after each bet, and then averaging for the run of 100 bets.
With regard to the first quantity, the graph below shows the results
On the evidence of the above, there is really only one conclusion that can be drawn - fixed profits staking is where it is at.
This graph takes a little explanation to make clear! The x-axis shows the a percent of times that the fixed profits was ahead of the fixed stakes system. The pink line is for reference. A 'good' system will be above the line before 50% and below it afterwards. We can see that our fixed profits system
(relative to the fixed stakes system) conforms to this rubric.
On the eighth day, Doc finally took a well-earned rest, but drew some conclusions before kicking back in his recliner...
On the evidence of the above, there is really only one conclusion that can be drawn - fixed profits staking is where it is at. One final piece of evidence is in regard of the second of the two bullet points listed above - i.e. how the bank changed after each bet. The results were:
- Fixed Profits - The bank changed on average by 0.298 units after each bet
- Fixed Stakes - The bank changed on average by 0.423 units after each bet
So, there you have it folks - under pretty much any criteria you mention the fixed profits system performs better. Now you just need to be able to pick some winners to take advantage!
