Kelly Criterion Simulation, Optimum Bet Size for Bankroll Growth
and what it means for blackjack card counters
The following spreadsheet simulates the results of a coin-flip gambling game using a biased coin (51% heads, 49% tails). You might think that betting heads for 1,000 flips would come out ahead for sure, with bigger wins for bigger bets. The actual results might surprise you.
The color-coded line graphs show the results of four betting strategies:
------ Safe (1/4 Kelly optimum)
------ Optimum (= Kelly optimum)
------ Risky (twice Kelly optimum)
------ Crazy (four times Kelly optimum)
To run a new simulation of 1,000 coin flips, click the Refresh icon
( ) beneath the spreadsheet, or reload the page.
The Kelly Criterion
According to the Kelly Criterion, when you have an advantage in a gambling game, the optimum bet size for bankroll growth is the fraction of your bankroll equal to your long-term advantage. In the simple coin-flip game described above, a biased coin has a 51% chance of landing heads and a 49% chance of landing tails.
Betting heads has an advantage of 2% -- if you play 100 times, win 51 times, and lose 49 times, your net win of 2 bets is 2% of 100. Therefore, for optimum bankroll growth, you should bet 2% of your bankroll on heads for each flip, adjusting the bet size as your bankroll grows and shrinks.
The Simulation Graph
The spreadsheet above simulates 1,000 random flips of the biased coin and keeps track of your bankroll as you bet on heads for every flip. The green line represents the Kelly optimum bet size. The blue line (1/4 Kelly) is safer but doesn't grow the bankroll as quickly. The gray (2X Kelly) and red (4X Kelly) lines go up faster in winning streaks but lose more heavily in losing streaks, endangering your bankroll.
You start with a bankroll of $10,000. The first column on the left side shows the statistics for the run, including the net number of bets won in the 1,000-flip sequence (cell A15) and the ending bankroll for each of the four betting strategies (cells A19-A22).
What does Kelly predict?
A Kelly analysis predicts the long-term results for the following bet sizes:
Less than the Kelly optimum: Your bankroll will still grow in the long run, but at a slower rate.
Exactly the Kelly optimum: Your bankroll will grow at the fastest possible rate.
More than the Kelly optimum but less than 2X Kelly: Your bankroll will still grow in the long run, but at a slower rate due to losing streaks knocking down your bankroll.
Exactly 2X Kelly: Your bankroll will fluctuate wildly but end up where you started due to losing streaks knocking down your bankroll (zero growth).
More than 2X Kelly: Your bankroll will fluctuate wildly but end up close to zero due to losing streaks killing your bankroll (negative growth).
These results are summarized by graphing the bankroll growth rate versus bet size, which is the Kelly function G=POWER(1+B,0.51)*POWER(1-B,0.49)
Are 1,000 flips the long run?
No, 1,000 flips is not the long run, so the simulation does not always show the results predicted by Kelly, nor do you see win/loss statistics that you might expect.
The average net win for 1,000 flat bets is 20 bets (510 wins, 490 losses). However, when you run the simulation several times, you'll see wide variation, including many runs in which losses exceed wins.
The longer the simulation (or gambling in real life), the more certain it is that you will get the predicted results. Even 1,000 bets is not enough.
What does this mean for blackjack card counters?
The Kelly simulation demonstrates the folly of over-betting. You will eventually lose your entire bankroll if you bet too much, in spite of having an advantage over the casino. (Note to casino managers: This is a reason not to bar amateur card counters!)
Furthermore, the graph demonstrates the extreme difficulty and exceptional patience required to eke out a long-term gain from card counting. In typical good 6-deck games, a 2% player advantage requires a true count of +5 or better (using the Hi-Lo count). You might get this advantage for
one or two dozen hands per day.
Even after playing 1,000 such hands over several weeks, there's no guarantee that you'll come out ahead for just those hands with a 2% advantage, even while betting the Kelly optimum. Meanwhile, you'll also be playing many tens of thousands of hands with less advantage, or no advantage, or a slight disadvantage. Card counting is a very tough business!
For more information
The biggest mistake blackjack card counters make
(recorded live demo of Kelly simulation)
Kelly Criterion: Bankroll Size for Blackjack Card Counting
(explains the Kelly criterion using simple math)
Blackjack Running Count, True Count, and Player Advantage Simulator
(DIY card counting advantage simulator)
) beneath the spreadsheet, or reload the page.
