Part 3
So last time I said I would begin to look at KPIs and investment metrics. I did some cursory reading and realised it is very difficult to quantify with my approach. The main reason is down to how I define my exits.
Investment financials and analysis are a bit confusing at first - especially if you’re not familiar with them. They tend to revolve around a basemark value. For example, if I were assessing the success of a fund, I would compare that fund to the base stockmarket (FTSE 100 for example), and then use that to determine how well it does. There are metrics to determine the percentage returns, as well as metrics for the volatility and risk. Because there’s a lot more than I’d expected, I’m going to avoid getting into all of that right now.
Keeping it even more simple, lets just play with the three variables I discussed in my previous post: the exit multiple, the starting value and the minimum bet. I’m going to keep the number of trials consistent. I just want to see how my probability of success changes as I vary the different parameters.
I use the same code as earlier, as you can see some of my outputs below:
| Starting balance | Exit balance | Minimum bet | Returns (%) | Success rate |
|---|---|---|---|---|
| 100 | 110 | 0.50 | 10% | 89.6% |
| 100 | 125 | 0.50 | 25% | 77.3% |
| 100 | 150 | 0.50 | 50% | 68.4% |
| 100 | 200 | 0.50 | 100% | 51.5% |
| 100 | 1000 | 0.50 | 900% | 9.4% |
| 100 | 110 | 1 | 10% | 89.3% |
| 100 | 125 | 1 | 25% | 75.4% |
This all seems to make sense intuitively. As exit value increases, our risk of success decreases. The longer we stay at the table, the more chance we have of going broke. What’s most interesting is the fact that we have a 51.5% chance of doubling our money. In pure roulette, we could double all of our money by betting on a colour (Red/ Black) and then winning. This has odds of ~48%, and this acts as a pretty good control. We can immediately see that the Martingale method is (very slightly) better than pure roulette when it comes to doubling our money.
There’s also some sort of relationship between minimum bet and success, and again this seems to make sense. The minimum bet dictates how many times we can lose before we go broke. The lower the minimum bet, the longer we can play for. However this relationship isn’t linear - it actually scales logarithmically.
Let’s run through it - if we start with £50, a minimum bet of £1, and keep losing then our balance looks like this:
| Attempt | Starting Balance | Bet |
|---|---|---|
| 1 | £50 | £1 |
| 2 | £49 | £2 |
| 3 | £47 | £4 |
| 4 | £43 | £8 |
| 5 | £35 | £16 |
| 6 | £19 | £32 |
| - | £0 | £0 |
So we can lose 5 bets in a row before we run out of money. However, adding an extra pound or two to our starting balance doesn’t actually help. For us to last another bet, we would need to start with £64 rather than £50. Its linked to powers of 2, and this is because we’re doubling our bet each time. So we can find how many bets we last from the below formula:
bets = log_2(starting amount/minimum bet)
We need to round the answer down to the nearest integer, and then we’ve got it. So we understand the relationship between minimum bet and how many times we can fail.
However the minimum bet also determines the rate at which we grow. Each successful round ends with us winning the minimum bet (and no more). So a lower minimum bet means we can have more attempts, but we need to win a lot more games to exit. Let’s look into this in more detail. Each successful round of roulette wins my minimum bet. I define a round as a series of rolls resulting in a win. This means*:*
- loss, loss, win
- loss, win
- loss, loss, loss, win
- win
are all rounds. If we start with £50, exit multiplier of 1.1x and minimum bet of £1, we need to win 5 total rounds to exit. If our minimum bet is £0.50 however, we need to win 10 total rounds. It seems like this doesn’t matter: a lower minimum bet is better. Lets test this further just to be certain. In the below table, I look at the success rate if we start with £50 and aim to make £5 (exit multiplier = 1.1x). I vary the minimum bets, and the percentages can be seen below:
| Minimum bet | Success rate |
|---|---|
| 0.10 | 92.3% |
| 0.25 | 90.6% |
| 0.50 | 89.5% |
| 0.75 | 93.6% |
| 1 | 88.9% |
| 2 | 89.3% |
| 4 | 86.3% |
At first these results really surprised me - especially the fact that our success rate peaked at £0.75. I’m really not sure why that is. I even increased the number of trials to 10,000 rather than 1,000 just to be certain. My suspicion is that £0.75 is the ideal in terms of number of rounds needed and number of possible losses. As our minimum bet decreases, we can afford to lose more times in a row (remember the log_2() formula), but we have to win many more games. The opposite occurs with a high minimum bet - with a minimum bet of £4 we can only lose 3 bets in a row, but we only need two rounds to exit.
I’m going to end this post here. I’m still shocked at the results of the above table. I’m going to re-review my simulation as well just to see if there’s an issue there. For my next post I’m going to try to understand the ideal minimum bet - is there a way of mathematically working out the ideal minimum bet to increase our success?