Kelly Betting System

Example

In the betting game, instead of all in, you probably only consider using a proportional betting each time. Now, let’s imagine a 1-1 odds game with winning probability $p=2/3$, and suppose at each time, you put a fixed portion $\pi$ of the fortune at each stage. Then after $n$ stages, say you have won $Z_n$ games, then you fortune will become,

$$ X_n=(1+\pi)^{Z_n}(1-\pi)^{n-Z_n} X_0, $$

Consider the log of average geometric growth rate of the fortune, which is $(1 / n) \log \left((1+\pi)^{Z_n}(1-\pi)^{n-Z_n}\right)$, since the fortune evolves $n$ times and we have $\frac{X_n}{X_0}$ as the total growth. The limit of it is the rate of convergence, that is when $n$ becomes large, $Z_n\approx \frac{2}{3}n$, then the rate of convergence converges to $(2 / 3) \log (1+\pi)+(1 / 3) \log (1-\pi)$.

We now maximize it by choose the $\pi=1/3$.

The Kelly Betting System and Log Utility

Similarly, let’s just assume a general $p$, the rate of convergence (follow the same analysis as above) is now,

$$ p \log (1+\pi)+(1-p) \log (1-\pi) $$

And the best betting portion becomes,

$$ \pi(p)= \begin{cases}0 & \text { if } p \leq 1 / 2 \\ 2 p-1 & \text { if } p>1 / 2\end{cases} $$

Notice that we take log of the geometric growth to derive, which equally mean that we are maximizing the expectation of log fortune at each stage.

$$ \begin{aligned} &\mathrm{E}\left(\log \left(X_k\right) \mid X_{k-1}\right)\\ &=p \log \left(X_{k-1}+b_k\right)+(1-p) \log \left(X_{k-1}-b_k\right) \\ &=p \log \left(X_{k-1}\left(1+\pi_k\right)\right)+(1-p) \log \left(X_{k-1}\left(1-\pi_k\right)\right) \\ &=\log \left(X_{k-1}\right)+\left[p \log \left(1+\pi_k\right)+(1-p) \log \left(1-\pi_k\right)\right] \end{aligned} $$

Suppose we are making the decision to transfer fortune from $X_{k-1}$ to $X_k$, take the expectation of the log fortune, and separate the known part $X_{k-1}$, we get the same optimization term as initially. This means that Kelly betting system uses a logarithm of fortune as the utility function.

Now you see how to make decisions at stage $k$ according to the probability $p_k$, and you may wonder if the decision is still optimal if you have estimates for the future $p_i$ (look ahead and adjust). It can be proved the following

The betting system that maximizes E(log(Xn)|Z0) is the Kelly betting system with bk = π(pk)Xk−1, where pk = P(Yk = 1|Zk−1).

So you do not need to look ahead and only need to react by the current information.

Extension

In a more general case, where we have $p$ winning and $q$ losing with $b$ as the wining odds $a$ as the losing odds (for every dollar you bet, you get $b$ additional dollars back, so your fortune grows at $\pi b$). The expected geometric growth rate is therefore,

$$ r=(1+\pi b)^p \cdot(1-\pi a)^q $$

Similarly, we want to maximize the log of the growth, which yields

$$ \log (r)=p \log (1+\pi b)+q \log (1-\pi a) $$

Maximize this log rate, we have $\pi=\frac{p}{a}-\frac{q}{b}$.

Summary

Kelly betting system decides the optimal portion of the fortune to wager in a betting game, so that you maximize the expectation of log wealth at each stage. This holds with a assumption of playing repetitive games and should converge in the long term.