The Kelly Criterion: Where does the logarithm come from?

This is the third post in a four-part series exploring the Kelly criterion:

The neat thing about the derivations in the last two posts is that they give a motivation for "optimizing the logarithm of wealth". The logarithm is not put in by decree, but is a mathematical technicality that arises from the repeated betting process! Kelly mentions this in his original paper:

At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies.

This argument is very general. Let's say we model the wealth VnV_n after nn rounds of betting based on the initial wealth V0V_0 in terms of a function f(R,l)=Vn+1/Vnf(R, l) = V_{n+1} / V_n as

Vn=V0f(R,l)n(1)V_n = V_0 f(R, l)^n \tag 1

where RR is a random variable describing the possible outcomes of the game, and ll is the fraction of available money to invest in each round. Then we can derive the following formula for the best ll value:

lopt=argmaxljf(rj,l)pjn=argmaxljpjlogf(rj,l)=argmaxlE[logVn+1Vn](2)\begin{aligned} l_{opt} & = \operatorname*{argmax}_l \prod_j f(r_j, l)^{p_j n} \\ & = \operatorname*{argmax}_l \sum_j p_j \log f(r_j, l) \\ & = \operatorname*{argmax}_l E \left[ \log \frac{V_{n+1}}{V_n} \right] \tag 2 \end{aligned}

where rjr_j are the possible investment outcomes and pjp_j are the associated probabilities. The crucial change from random variable RR to outcomes and probabilities rjr_j and pjp_j is justified by the law of large numbers. Based on the exponential nature of the formula, switching to a logarithmic view feels very natural.

Consequently, neither the details of the game — represented by the random variable RR — nor the exact form of the per-round return ff matter. Any iterative scheme with reinvestment of profits should be representable in the form of equation (1), leading to the logarithm in solution (2). Beyond the origin of the logarithm, this analysis also shows the universality of the Kelly derivation.

Unfortunately, this argument is mostly skipped in online discussions. Often the logarithm is not justified at all, or it is treated as "genius from the 1950s says: use log\log". Sometimes the result is also linked to utility theory, which posits that having twice the money is not twice as useful. While utility theory may be true, reasonable people can disagree on their utility function — exactly how useful more or less money is to them. However, Kelly's result is not grounded in utility, and the log\log does not represent logarithmic utility of money. Consequently, even people who disagree on their utility function should agree that the Kelly criterion is the fastest way to gain wealth.

I hope this post shed some light on the reasoning behind the Kelly decision scheme. If you're interested in interactive plots that really helped me understand this material, you can find them in this Jupyter Notebook.

In the next post, we'll take a closer look at the relation of the Kelly criterion to expected values.