The Kelly Criterion in 3D

The Kelly Criterion is a gambling strategy which maximizes the logarithm of your expected wealth. The Kelly Criterion tells you what fraction of your bankroll to wager. It is a function of the net fractional odds[1] received and the probability of a win .

Some properties are intuitively easy to understand.

  • The Kelly wager is positive iff the expected value is positive. The Kelly wager is zero otherwise.

  • The Kelly wager is 1 for all . (Ignore .)

  • The Kelly wager is 0 for all . (Ignore .)

What surprised me is that if you fix and restrain to the region of positive then is a linear function of . This was not intuitive to me.

I expected asymptotic behavior with the greatest in a neighborhood of . In other words, I expected the fractional wager to increase slowly at first and then increase faster as approached 1. Actually, is linear.

Kelly wagers tend to be more aggressive then human intuitions. I knew this and I still underestimated the Kelly wager. I didn’t mess this up in a high stakes situation where fear throws off my calculations. I didn’t even mess this up in a real-world situation where uncertainty complicates things. I underestimated the Kelly wager on a purely conceptual level.

I have written before about the utility of my fear heuristic. My fear heuristic might be helping to compensate for my Kelly miscalibration.


I’m good at tolerating risk when it comes the small number of gigantic risky bets guiding my professional career. I’m also good at tolerating risk in the domain of painlessly small bets. (Not that there is much risk to tolerate in this latter case.) Judging by this post’s analysis, I am awful at calibrating my risk tolerance for wagers between 0.05% and 1% of my net worth. Specifically, I am insufficiently risk tolerant.

What makes this worse is that the region of 0.05% to 1% of my net worth is full of long tails. The wagers I’m skipping could easily repay themselves a thousandfold. If I take a wager like this every day for 2 years and just a single one of them repays itself a thousandfold then I win bigtime.

I need to gamble more.

Optional Practice

I used these problems is to help develop my intuitive grasp of the Kelly criterion.

Q1: If and then what is the corresponding ?


The above number is way higher than what my intuition tells me is appropriate.

Q2: If and then what is the corresponding ?


The above number is higher than the answer to Q1. This result was, again, unintuitive to me. I expected it to be smaller because is smaller in absolute terms. But I didn’t pay sufficient attention to . The average return is your initial investment.

Q3: If and then what is the corresponding ?


Q4: If and then what is the corresponding ? (Not that means you get back your original wager plus double you wager for a total of your wager.)


Q5: If and then what is the corresponding ? (In practice, opportunities like this are so rare you will usually not get to wager a full Kelly.)


I was a little surprised; I had expected a higher result. The logarithmic value function is doing the work of keeping Kelly down.

Q6: If and then what is the corresponding ?


  1. The “net fractional odds” indicate how much you win in the case of a win. If you wager and lose then you lose . If you wager and win then you get your back plus an additional . ↩︎