Perhaps the most famous proponent of the Kelly Criterion is Edward Thorp. He founded the M.I.T. Blackjack Club, published various papers on gambling and investing, and became both a professor of mathematics and a billionaire investor. The Kelly Criterion played a key role in most of these; he dubbed it “Fortune’s Formula”.

Thorp has authored various articles about the Kelly Criterion over the years, e.g. The Kelly Criterion and the Stock Market. These typically list six properties related to the Kelly formula which I will now attempt to paraphrase:

- If your expected compound rate of growth is positive, your wealth will approach \(\infty\) over time
- If your expected compound rate of growth is negative, your wealth will approach zero over time
- If your expected compound rate of growth is zero, your wealth will approach both \(\infty\) and zero (i.e. make arbitrarily wide swings) over time
- The ratio between the performance of the Kelly strategy and that of any other strategy will approach \(\infty\) over time
- The expected time to reach any fixed target wealth is shorter for Kelly than for any other strategy
- To maximize your expected rate of growth over many rounds, you can simply maximize the expected logarithm of your wealth each round, even if the exact probabilities and payoffs change from round to round

At least a couple of these results were first established by Thorp himself in the 60s.

To reiterate the context: We assume you have some “edge” in gambling or investing, and you are going to make a large sequence of bets/investments using that edge, compounding your results over time. These properties — and the Kelly formula itself — are about your strategy for sizing each bet. (If you have no edge, you should not be making bets in the first place.)

Properties (1), (2), and (3) say you do not have to use the Kelly formula to do (very) well; smaller or even somewhat larger bets will work fine. But be careful not to make your bets *too* large or you are very surely going to do (very) poorly.

Property (4) is essentially the one I have mentioned already: As the number of bets goes up, Kelly is increasingly likely to outperform any other strategy, and that outperformance is likely to grow toward \(\infty\) over time.

Property (5) says Kelly bets are the fastest expected way to reach a betting/investment target.

Property (6) says it is valid to apply the Kelly Criterion to situations more complex than (e.g.) my little toy example with the dice.

Other billionaire investors known or strongly suspected of using Kelly methods include Warren Buffett, George Soros, and James Simons. That is some impressive company.

…

On the flip side, the most prominent critic of the Kelly Criterion was probably Paul Samuelson. A Nobel prizewinner in Economics, he wrote about the Kelly formula several times, the most amusing surely being his NSF-funded academic paper consisting entirely of one-syllable words. He was presumably trying to make it accessible to the less gifted; Prof. Samuelson was apparently a bit of an a**hole. My kind of guy.

Now, nobody likes to laugh at economists more than I do. And stodgy academics telling colorful billionaires how to invest certainly seems a ripe opportunity.

But this is really not fair. Those same academics would also tell a lottery winner he should never have bought a ticket, and *they would be right*. The details, and not the outcomes or personalities, are what matter.

We will ponder some of those details in the next installment.