GetBackJack: The Kelly Criterion

Kelly Criterion

One formula that helps avoid Gambler’s Ruin (losing it all), while betting the optimum amount is the Kelly Criterion, first described in 1956 by J.L. Kelly. It is used by some traders and professional gamblers, and I have started considering it when deciding the right amount to bet on any single lawsuit.

This formula comes complete with a mathematical proof and tells a trader, gambler, or investor how much one should optimally bet, based on the probability of winning the bet and the payoff amount if the bet wins. The formula assumes no ties.

Shown below is the general formula:

Here is an example where you have a chance to bet $1, and collect $2 if you win ($ 1 original bet plus the $1 profit [the variable labeled “b” in the formula]). I call this a “2X” bet. In this example you have a 60% probability of doubling your money; the Kelly Criterion says that you should bet 20% of your bankroll (or your entire net worth, if your net worth were all in the form of cash).

This formula demonstrates quite logically that:

1) for a 2X bet you should not bet anything unless the probability of winning is above 50%. In fact, if the probability is below 50%, you should try to find a way to take the other side of the bet or trade (go short).

2) At a 51% probability of winning, Kelly suggests that you should bet 2% of your bankroll.

3) At an 80% probability of winning, Kelly suggests that you should bet 60% of your bankroll.

Kelly’s original paper made the point that the criterion is only valid when a series of bets are made. Even though the formula says that one should bet 98% of your bankroll when you have a 99% chance of winning a 2X bet, that still leaves you with a 1% chance of going broke—too high for me, at this stage in my life.

Although I am comfortable taking risks and making substantial bets, the Kelly Formula feels more aggressive than my inner voice approves. Obviously, for a lawsuit investment I have to adjust the formula based on the probabilistic estimate of when the bet will pay off, and then adjust it based on my assumed discount rate (the amount that reflects the discount for getting paid later.) And a minor detail is that I never know the exact odds. But even if I did know the exact odds, I can’t conceive of putting 60% of my net worth on the line, even if I had a marvelous 80% probability of winning. So maybe I need to take my own advice and get a little specialized psychotherapy—so I can de-wussify and get on board with making bigger bets.

All-In

Many successful entrepreneurs, traders, and investors take measured and intelligent risks, trying to avoid at almost any cost the all-or-nothing bet. This is after all, the ultimate risk of Gambler’s Ruin. The younger you are the less reckless it is violate this aspect of the rule because you have on average a smaller bankroll and more time to recover if you lose it all. If you are poor, you are likely to have no alternative to the all-in bet. This is why I encourage young people to get started on their entrepreneurial ventures sooner rather than later. I love it when a 14-year-old puts every last dollar he owns into a lawn mower to start his landscaping business—as long as he keeps enough capital to fill up the gas tank.

And the Survey Says…

I wanted to know how my friends and family would answer questions related to the Kelly Criterion. I asked them to complete the following survey:

I have three questions for you about how much of your bankroll you would wager in three different circumstances.

I start with the premise that you are playing an absolutely fair, honest and random game. For example it is the picking a chip out of a hat. Some of the chips say “win” and the rest say “you lose”. If you win, you double the amount of your bet; if you lose then your bankroll shrinks by the size of your bet.

You know in advance exactly what the chances are of winning. In every case, you are 30 years old (please try to adjust your thinking to how you would have played or will play this game at this age.) You have a bankroll of $100,000 which is all the assets you have in this world and your bankroll is all you have to get you have to work with. You have no home, no IRA, no retirement, no job. The random contest you are playing is always in your favor but sometimes it is more in your favor than other times. The three conditions we will consider are 60%, 75% and 90%. In other words you have either a 60%, 75%, or a 90% chance of doubling the money you bet. Another way of looking at it is that you have a 40%, 25%, or a 10% chance of losing the amount of your bet.

For each of these percentages, what percentage of your bankroll would you bet? (the choices are between 0% and 100% in 10% increments.) Let’s go.

Question 1) At a 60% probability of doubling your bet, what percentage of your bankroll would you bet?

Question 2) At a 75% probability of doubling your bet, what percentage of your bankroll would you bet?

Question 3) At a 90% probability of doubling your bet, what percentage of your bankroll would you bet?

Discussion of the Survey Results

· I asked everyone to imagine they were 30 years old, and had a $100,000 cash bankroll that comprised their entire net worth. I wanted to minimize the bias that as people get older they generally shy away from bigger risks. I was intentionally silent about if the participant had children or “big responsibilities”.

· Sometimes what people say they will do differs from what they will actually do, when the rubber meets the road. If I had about $5-10 million dollars I was willing to invest in a real live experiment, I am sure I could have found 50 30 year olds to take the test with real money. It was not that important to me.

· For each of these questions, the Kelly Criterion calculates the optimum answer (based on one bet in a long series of bets). If you know exactly what the probability of winning and the exact amount of the payoff, the Kelly Criterion tells you the optimum amount to bet. Theoretically, if you always bet this amount, over many opportunities, on average after many bets, you have the greatest chance of having the biggest ending bankroll.

· If you bet less than the Kelly Criterion amount, you are being “risk averse”, to your detriment. If you are betting more than this amount, you are taking greater risks that optimum, also to your detriment, on average in the long run. If you bet zero, when you have a 90% chance of doubling your bet, you are “ultra risk averse”. If you bet 100% of your stack, you are going “all-in”.

· I was surprised that the survey results didn’t show a more risk averse average response from my friends and family—maybe I hang around some “weird” folks.

· The survey results show the “Wisdom of the Crowds”. Although the responses varied greatly, the average results for each question are reasonably close to the Kelly Criterion—although, some participants were very risk averse and others quite willing to lay it all on the line. I declare the aggregate results rational, although I might be saying this because my answers were close to the average results.

· The results showed that on average, the more the bet approached “all-in” or 100% of one’s bankroll, the more conservative the response, relative to the Kelly Criterion. I consider this reasonable. The question, as framed, never promised that the participant was going to get to make the decision 1,000 days in a row.

· The biggest diversity of answers came on the third question. We had three respondents that said they would not bet a penny (even though they had a 90%) chance of doubling their money, and four that said they would bet their entire bankroll. Slightly different risk tolerances.

Here is a summary of the results:

Probability of Win	60%	70%	90%	Total
Kelly Criterion Percentage	20%	50%	80%
Survey Average Amount Bet	25.8%	39.26%	59.26%
# of Risk Averse	28	49	56	133
# @ Kelly Percentage	15	14	9	38
# Greater than Kelly Percentage	38	18	16	72