If you have played Blackjack before, you may have heard about the story of the MIT Blackjack Team to “beat the dealer” by counting cards. The strategy used by MIT Blackjack Team originated from the book “Beat the dealer” by Dr. Edward Thorp. The strategy generally estimate the probability of having “large cards” and “small cards” next round from the remaining cards in deck by counting cards already shown. This set of rules helps the Blackjack Team won millions from casinos. Today, I would like to put our focus on part of the strategy to decide our betting size.
Recalling what we have learned in Markov Chain Lessons, the Gambler’s Ruin tells us that if a gambler enters a negative expected value game, eventually this gambler would go bankruptcy.
So we must choose a game with positive expected value if we don’t want to lose all our money. In Blackjack, card-counting skills enables gamblers to decide when to hit and when not to by estimating the probability of winning. But we have only solved half of the question. Now suppose we have entered a gamble with positive expected value, how much should we bet each time? It would be irrational to bet all our wealth one time. So, the next question is: how much we should bet each time so as to maximize our profit?
Let’s do some very simple Maths.
Denote f as the fraction of our current wealth we bet, 
![E[\frac{1}{N}\log{\prod_{i=1}^N{(1+fX_i)}}]=\frac{1}{N}\sum_{i=1}^N{E[\log{(1+fX_i)}]}](https://s0.wp.com/latex.php?latex=E%5B%5Cfrac%7B1%7D%7BN%7D%5Clog%7B%5Cprod_%7Bi%3D1%7D%5EN%7B%281%2BfX_i%29%7D%7D%5D%3D%5Cfrac%7B1%7D%7BN%7D%5Csum_%7Bi%3D1%7D%5EN%7BE%5B%5Clog%7B%281%2BfX_i%29%7D%5D%7D&bg=ffffff&fg=444444&s=0 "E[\frac{1}{N}\log{\prod_{i=1}^N{(1+fX_i)}}]=\frac{1}{N}\sum_{i=1}^N{E[\log{(1+fX_i)}]}")
If we assume each bet is independent, then we would maximize
![E[\log{(1+fX_i)}]](https://s0.wp.com/latex.php?latex=E%5B%5Clog%7B%281%2BfX_i%29%7D%5D&bg=ffffff&fg=444444&s=0 "E[\log{(1+fX_i)}]")
By Taylor Expansion, we have
![E[\log{(1+fX_i)}]=E[fX_i-\frac{1}{2}f^{2}X_{i}^{2}+...]](https://s0.wp.com/latex.php?latex=E%5B%5Clog%7B%281%2BfX_i%29%7D%5D%3DE%5BfX_i-%5Cfrac%7B1%7D%7B2%7Df%5E%7B2%7DX_%7Bi%7D%5E%7B2%7D%2B...%5D&bg=ffffff&fg=444444&s=0 "E[\log{(1+fX_i)}]=E[fX_i-\frac{1}{2}f^{2}X_{i}^{2}+...]")
Approximately, we have
By differentiation, we got the Kelly’s Criterion:
With this f*, the expected growth rate in each bet would be
As we can see from the diagram above, if we bet twice Kelly, we would have an expected growth rate of zero, and further increasing our bet would be a suicide in long run, even if we have the advantage in the gamble. The expected long-term growth rate will also decrease when the gambler bets aggressively, yet with higher risk than Kelly. This is why some gamblers would use a “Half Kelly”, half betting size of Kelly’s Criterion, to protect them from being over-aggressive. The size is halved while the expected growth rate is 75% of Kelly’s, which makes “Half Kelly” a welcoming choice.
Now, let’s look at the expected long-term growth rate = 
Have you seen something similar before?
In our Corporate Finance Course on Mean-Variance Optimization, we aim at minimizing
for vector w as portfolio we have, vector 
It is equivalent to Kelly’s criterion with
We may in fact consider f* as the proportion of asset we invest in market portfolio, while the remaining (1-f*) to be invested in risk-free rate. We would dynamically rebalance f* by the change of expected return and variance of market portfolio.
Hence, Kelly’s criterion is one of the candidates for multi-period portfolio optimization.
For more details, you may refer to http://www.bjmath.com/bjmath/thorp/paper.htm
Reference: Paul Wilmott on Quantitative Finance
