Summary: Growth is sustainable only if you are employing a strategy that manages risk relative to your bankroll. Therefore, Probabilistic cost-benefit analysis must be framed in terms of a bankroll or else the statements are meaningless. The Kelly criterion can be used to understand how to balance EV vs Variance.
I am a gambler by trade and I feel that the current progress studies discourse is missing a full understanding of bankroll management and how that applies to decision making under uncertainty. I simply want to use some mathematical models to help build mental models of sustainable growth so that we can agree what properties are important.
First, let’s look at the Gambler's Ruin problem. Bankroll management is the process to optimally respond once we have identified a positive expectancy bet for how much we should wager. There are numerous resources to learn about bankroll management but a good first thought is learning the Kelly Criterion, aka Kelly. Others have written books about thinking in betting strategies and others specifically wrote great explainers at length on Kelly. However, allow me to indulge myself because I still google top rationalist writers and the phrase “Kelly Criterion” and find zero relevant hits1. In later posts I will discuss particulars but today we will simply look at the Kelly criterion.
Let’s start on Kelly’s Wikipedia page, the math and repeated reformulations are not very clear but let me use them to build our mental models.
Kelly’s can be framed as "What percentage of your bankroll can you risk losing on this uncertain outcome?" Note that we need a notion of a bankroll to answer this statement.
Takeaway 1: Probabilistic cost-benefit analysis must be framed in terms of a bankroll or else the statements are meaningless.
The example given on the wiki page is fantastic and I will repeat it here because it perfectly demonstrates the devastating effect of ignoring bankroll management even while employing bets that are positive expectancy:
In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. The behavior of the test subjects was far from optimal:
Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment.
Using the Kelly criterion and based on the odds in the experiment (ignoring the cap of $250 and the finite duration of the test), the right approach would be to bet 20% of one's bankroll on each toss of the coin. If losing, the size of the next bet gets cut; if winning, the stake increases. If the bettors had followed this rule (assuming that bets have infinite granularity and there are up to 300 coin tosses per game and that a player who reaches the cap would stop betting after that), an average of 94% of them would have reached the cap, and the average payout would have been $237.36.
In this particular game, because of the cap, a strategy of betting only 12% of the pot on each toss would have even better results (a 95% probability of reaching the cap and an average payout of $242.03).
There are 2 sets of people making clear blunders. One blunder is *much* worse. Please try to stop and guess at what I am referring …
Now let me elaborate, 66% of people are betting on the 40% outcome at some point which is non-sensical (It loses you money in expectancy) but occasional bets are not catastrophic since a conservative betting strategy will still be positive expectancy with a higher frequency of bets on the 60% outcome. Yet, 28% went bust, in a game with a strategy where 95% of the time you win $242! That is catastrophically stupid. These people likely bet on the right (60%) outcome with wrong bet size 2.
I really can't emphasize enough how bad it could be for society if our leaders behave like members of that 28% who lose everything. If we are playing an infinitely repeated game and you are wagering in excess of Kelly then you are nearly guaranteeing ruin. Discussing details is beyond the scope of this post, but we can all agree total ruin is awful.
It is not enough to calculate the positive EV bet at every step. You must wager an appropriate size of your bankroll to have a positive expectancy strategy. In other words,
Takeaway 2: Growth is sustainable only if you are employing a strategy that manages risk relative to your bankroll.
The keyword is focusing on the strategy rather than the individual decisions. So much of our discussions are about whether a bet is positive rather than understanding the full strategy employed. To understand if a strategy is sustainable and positive we need to know the bet size as a percentage of the bankroll and the risk, aka variance assuming a known distribution, of the bet. You can see it is right there in the formula for optimal stock investments:
Optimal_leverage = EV / σ^2
Takeaway 3: Given two bets with equivalent EV, the bet with lower variance is better.
Given the real world is much messier than brownian motion with respect to uncertainty, the ability to receive costless leverage, and ability to make infinite uncorrelated bets, I think a good rule of thumb is to simply scale our EV by the standard deviation. In finance this is the pervasive Sharpe Ratio.
So now we have a rough set of variables that must be discussed to compare strategies. First, we must know what our effective bankroll is. Then we can create a range of reasonable estimates for the expected value of each uncertain decision (less a certain alternative “risk-free rate”). Also, we need to calculate an estimate of variance.
That is all I wanted to get across in this post. In further posts I will inspect the different formulations and how they define risk, attempt to use these formulations to build different mental models, and apply those mental models to different problems.
Later edit: I found Gwern mentioning (I am blocked from the tweet) using Kelly for Pascal’s mugging. My thoughts of using Kelly for that and Newcomb’s paradox lead me to want to write this.
Notice how the ratio that wagered 100% of their bankroll on one toss was very similar to the ratio that went bankrupt.