# 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 $f^*$ of your bankroll to wager. It is a function of the net fractional odds[^1] received $b>0$ and the probability of a win $p\in(0,1)$.

[^1]: The "net fractional odds" $b$ indicate how much you win in the case of a win. If you wager $x$ and lose then you lose $x$. If you wager $x$ and win then you get your $x$ back plus an additional $xb$.

$$
f^*=\frac{p(b+1)-1}b
$$

![](https://s3-us-west-2.amazonaws.com/www.lsusr.com/lesswrong/kelly/kelly-3d.png)

Some properties are intuitively easy to understand.
- The Kelly wager is positive iff the expected value $bp-(1-p)$ is positive. The Kelly wager is zero otherwise.
- The Kelly wager is 1 for all $p=1$. (Ignore $b=0$.)
- The Kelly wager is 0 for all $b=0$. (Ignore $p=1$.)

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

![](https://s3-us-west-2.amazonaws.com/www.lsusr.com/lesswrong/kelly/kelly-2d.png)

I expected asymptotic behavior with the greatest $\frac{df^*}{dp}$ in a neighborhood of $p=1$. In other words, I expected the fractional wager to increase slowly at first and then increase faster as $p$ approached 1. Actually, $p$ 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](https://www.lesswrong.com/posts/qHpazCw3ryvBojGSa/my-fear-heuristic). My fear heuristic might be helping to compensate for my Kelly miscalibration.

# Recalibrating

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 $p=0.01$ and $b=1000$ then what is the corresponding $f^*$?

:::spoiler
0.9%
:::

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

**Q2:** If $p=0.1$ and $b=20$ then what is the corresponding $f^*$?

:::spoiler
5.5%
:::

The above number is *higher* than the answer to Q1. This result was, again, unintuitive to me. I expected it to be smaller because $b$ is smaller in absolute terms. But I didn't pay sufficient attention to $bp=2$. The average return is $2\times$ your initial investment.

**Q3:** If $p=0.51$ and $b=1$ then what is the corresponding $f^*$?

:::spoiler
2%
:::

**Q4:** If $p=0.51$ and $b=2$ then what is the corresponding $f^*$? (Not that $b=1$ means you get back your original wager *plus* double you wager for a total of $3\times$ your wager.)

:::spoiler
26.5%
:::

**Q5:** If $p=0.65$ and $b=100$ then what is the corresponding $f^*$? (In practice, opportunities like this are so rare you will usually not get to wager a full Kelly.)

:::spoiler
65%
:::

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 $p=0.05$ and $b=100$ then what is the corresponding $f^*$?

:::spoiler
4%
:::