This single line of mathematics is a way of finding the optimal investment size to grow your bankroll as fast as possible (or portfolio in the investment based scenarios the Criterion was originally developed for).
Because betting too much at once increases the risk of losing your whole bankroll the formula finds the ideal middle ground between the extremes of maximising profit every roll and minimising the risk of your bankroll being wiped out.
So what is it?
Its usefulness is most easily explained via a game of coin-flipping. Assume you have a particular talent for coin-flipping and can get a 55% rate of heads against any other coin-flipper.
The standard bet for all coin flippers is evens. Bet one dollar against your opponents dollar and the winner takes all.
You’ll make an average profit of 10 cents for every $1 you wager so your first thought might be to maximise the number of dollars. More profit means faster growth, right?
If you can bet any amount one any roll then to maximise your average profit per roll you should bet all fifty bucks for an expected value (EV) of $5.00 on your first flip.
Obviously in any one flip, you would either double your money or lose it all. So it is worth noting that the ‘expected value’ is a mathematical abstraction based on the odds and payouts (basically an average payout if you played an arbitrarily large number of games this way) as a result it is different from your actual profit on a given hand which will have to be either +$50 or -$50 given the 1-to-1 payout.
Then if you win you bet it all again for an EV of $10.00 on your second flip. Than $20 when you go double-or-nothing for the third time and so on.
On the other hand, your risk of ruin (RoR) is also maximised. It takes just one bad flip to knock your entire bankroll out (because you are wagering it all each time) and you have to factor in that 45% chance every time you flip.
So it’s about also minimising your risk of ruin
On the other extreme you can bet the smallest possible amount on your first bet, so $0.01, and maintain that proportion of your roll investment each time i.e. betting 1/5000th of your bankroll on each bet.
Then your risk of ruin is tiny you will have to lose 5,000 bets in a row. (Actually, 5,000 bets in a row is a minimum. If your losing streak starts after you’ve grow your bankroll then the 1/5000 bet will shrink along with your roll until you hit the minimum bet of $0.01).
Although your roll will only grow an average of 5% of 1/5000th of your roll each flip, it will take an astonishing run of bad luck to deplete your roll entirely.
The Kelly Bet
If those two strategies represent the daddy bear and baby bear strategies. The Kelly Criterion is the momma bear strategy – by pushing the bet up until the RoR begins to counter the advantages of the larger payouts, we find the Kelly Bet.
Let’s call it a happy medium.
The Kelly bet maximises the overall rate of bankroll growth in the long run. To calculate it you just take the odds of you winning a given bet (p), the odds of you losing a given bet (q), and the payout when you win (b).
These three factors are fed into an equation and the answer is the fraction of your bankroll you should bet (f*).
The equation looks like this:
f* = (bp-q)/b
So for your coin flipping bankroll we can work the Kelly Bet out as follows:
b = 1.0 (ie. for every $1 dollar you bet you get $1 dollar paid out)
p = 0.55 (your 55% win rate expressed as a decimal)
q = 0.45 (you 45% lose rate expressed as a decimal
Which we can put into the equation above as:
f* = (1 * 0.55 – 0.45)/1
f* = 0.1/1
f* = 0.1 OR 10%
So in the example above the best way to grow your bankroll over all is to bet 10% of your bankroll on each flip of the coin.
Some complications (sorry…)
The above formula describes the simple situation where there are two outcomes (win or lose). And you lose 100% of your stake everytime you lose.
With multiple payouts or partial bet losses the maths does get more complicated. However, one way to get a good estimate for those more complicated situations is to look up the variance for the game you play and divide that by the advantage.
Variance is the square root of the standard deviation of possible outcomes of a bet – which sounds pretty complicated until you find out you don’t need to calculate it in most cases; you can normally look up for casino and other types gambling games.
Once you know the variance you get a good approximation of the Kelly Bet using the formula:
f* = Advantage/Variance
When you note that advantage is equivalent to b*p-q you can immediately see the relationship between the formulae.
So that’s an Intro to the Kelly Criteria. For the more mathematically minded among you there is plenty more to explore in this area, but for the average gambler that should be all you need to manage your bankroll rationally.
Clearly a good starting place would be for games offering simple win or lose bets, like roulette, blackjack and baccarat.
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