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Many skilled gamblers who can calculate expected value (EV) often ask me how to control volatility, how much capital to prepare, how much to bet, and how long it will take to see returns.
Asking these questions indicates a lack of skill! Knowing EV is not enough; managing capital is essential, as is understanding the reasons behind strategies.
In fact, the capital management of skilled players is not about economics, statistics, or management, but mathematics!
Firstly, I ask a simple question: Is there a betting strategy that allows you to never go bankrupt, even if you keep losing?
Please take a minute to think before looking at the answer.
The answer is to use proportional betting, which will never lead to bankruptcy!
For example, if I bet 10% of my total capital each time, if I lose, I bet 10% of the remaining 90%, and if I win, I bet 10% of the remaining capital. This way, no matter how many times I lose, I always keep 90% of my previous bet, ensuring that I always have money left.
Now the question is, what percentage of the total capital should be bet for optimal results?
Let's start with a zero-sum game.
Playing two hands, losing one and winning one, if I bet 10%, whether I lose first (remaining 90%, then bet 10% of 90% = 9%), or win first (remaining 110%, then bet 10% of 110% = 11%), I always have 99% left. The formula is (1 + 10%) * (1 - 10%) = 99% loss of capital!
Playing 200 hands, losing 100 and winning 100, if I bet 10%, the result is (1 + 10%) ^ 100 * (1 - 10%) ^ 100 = 36.6% loss of capital! The more hands played, the more is lost.
This experiment shows that using a betting method that never leads to bankruptcy, even though you never exhaust your losses, you will still end up losing more as you play more.
Next, everyone knows that we play a +EV game, where the winnings outweigh the losses. In a +1% game, for example, winning 101 out of 200 hands, if we use the strategy that never leads to bankruptcy, will we make money?
Playing 200 hands, losing 99 and winning 101, if I bet 10%, the result is (1 + 10%) ^ 101 * (1 - 10%) ^ 99 = 44.7373% better than the zero-sum game but still not profitable.
Now, if we bet 5%, (1 + 5%) ^ 101 * (1 - 5%) ^ 99 = 86.051%, much better than betting 10%.
And if we bet 1%? (1 + 1%) ^ 101 * (1 - 1%) ^ 99 = 101.005%, finally a way to never go bankrupt and make a profit.
But is this really the best strategy? To save time, I've created a table showing the betting proportions and their corresponding results:
| Bet Percentage | Capital Result |
|-----------------|----------------|
| 0.1% | 1.001902 |
| 0.2% | 1.003606 |
| 0.3% | 1.005114 |
| ... | ... |
| 2.0% | 0.999997 |
| ... | ... |
| 2.5% | 0.987569 |
| ... | ... |
| 3.0% | 0.970424 |
From this table, we can draw some conclusions:
1. For a +1% EV game (winning 101 out of 200 hands), betting between 0% and 2% of the total capital will yield profits.
2. Betting 1% of the total capital yields the highest profit for a +1% EV game. Betting smaller or larger amounts yields less profit.
3. Each bet should be precisely calculated, and betting should strictly follow the proportion. Any deviation from the optimal bet size will reduce profits.
Kelly, in fact, only deduced this principle and found that when the EV is +1%, you should bet 1% of your capital; when the EV is +2%, you should bet 2%... However, Kelly's calculations are based on a one-to-one condition, and in reality, the optimal betting proportion may vary depending on the odds of different gambling games. By creating a simple Excel sheet, you can find the optimal betting proportion for your game.
I hope this helps novice advantage players who don't understand betting strategies. |
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Post time 11-3-2024 20:38:16