Focusing your choices, guided by the Kelly algorithm.

Have you ever had a situation where you have a problem or opportunity, and you have multiple options,, but you’re not sure which one(s) to pick? I certainly have! One approach is to try all of the options, but by then the problem (or again the opportunity) is likely gone, and you’re left flat-footed. It reminds me of the old proverb, “If you chase after two rabbits, you’ll lose both.” 

Another approach is to use your experience to choose. But it’s helpful to use it in a certain rigorous way. The method is called “the Kelly Criterion method”. It’s one of several useful “algorithms for innovation”. Here’s the Wikipedia link. The steps below might take 30-60 minutes to do, but they might save you days of effort and many $1000s in expense … AND give you a better outcome! Here are the steps (example below):

1. Make a table with top-row headings for “Option”, “Loss ratio (a)”, “Win ratio (b)”, and “Probability”. The table should then have a row for each option you have.

2. For each option, make an estimate of the loss ratio (a), which is the amount you expect to lose if your effort fails (e.g., in dollars, freed up time in days or hours), divided by your investment of dollars or time. In many cases a = 1 (but not with the stock market).

3. For each option, make an estimate of the win ratio (b), which is the profit you expect to receive back if your effort succeeds (e.g., in dollars, freed up time in days or hours), divided by your investment of dollars or time.

4. For each option, make an estimate of the probability (p) of success. Thus the probability of losing is then (1 - p). Estimating a, b, and especially p might seem hard, and so at first pass, just estimate (guestimate?) based on your experience. Don’t allow your brain to say, “I have no idea.” Usually our experience does give us some idea for a, b, and p.

5. This step is where the magic happens. Kelly derived a rigorous and simple formula to tell you what fraction (f) of your resources (money, time) should be bet on each option. He found that f = p/a  -  (1-p)/b. If f is negative, that bet is a loser on average, and should be avoided.

Let’s give an example. Let’s say that you have 3 investment opportunities at your business:

Option 1. Invest $5,000. If it fails, you lose it all, but if it succeeds, you stand to win an additional $15,000 (i.e., Net Present Value = $20k). You estimate that the probability of success = 0.70. a = 1, b = 15/5 = 3, p = 0.70.

Option 2. Invest $13,000. If it fails, you have the opportunity to get half back. If it succeeds, you win an additional $3,000 (i.e., NPV = $16k). You estimate that the probability of success is 0.60. a = 0.5,  b = 3/13 = 0.23, p = 0.60.

Option 3. Invest $2,000. If it fails, you lose it all. If it succeeds, you win an additional $14,000. It’s a big win but it’s a long shot, since your probability of success is 0.20. a = 1, b  = 14/2 = 7, p = 0.20.

How would you invest? It can seem confusing at first glance. Kelly can help us here.

Option a b p = probability f = p/a -  (1 - p) / b

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Option 1 1 3 0.70 0.60

Option 2 0.5 0.23 0.60 -0.54  (negative)

Option 3 1 7 0.20 0.086

There are several conclusions here:

- Since Option 2 has a negative f, we don’t bet there. On average, it’s a losing gamble. Based on looking at the other numbers, that might not be a big surprise. Kelly just tells you to do what you think -- avoid #2.

- How to  split your investment between #1 and #3? Put 60% into #1, and 8.6% into #3. That might not have been so obvious. Kelly often tells us to bet a small amount on long shots.

- What about the remaining 31.4%? Kelly tells you to hang onto it in CASH! That might seem counterintuitive. Aren’t I leaving money on the table if I don’t invest? No! Holding onto the 31.4% is your OPTIMAL path to future growth! That’s true not even counting the fact that the 31.4% might be used for some better and unexpected opportunity that comes up in the middle of the year!

Below are three questions that might arise. If you have questions, or if you want a PDF of my paper on this, just connect.

1. How to estimate these outcomes and probabilities? This can take some skill (and maybe even art!), but there are methods for doing this. I’ll discuss the Delphi method in a future post.

2. What if the sum of the f’s is greater than 1? Then you must decide whether to leverage your position on margin (i.e., borrow money), or whether to set the absolute limit at f = 1. If f must equal 1 or less, that’s a bit more complicated.

3. What if the probability of success increases with investment? For instance, what if investing $2k into #3 increases p? That’s another important variation. Again, if you have questions, or if you want a PDF of my paper on this, just contact me.

The Kelly method can be applied to business as given above, and can also be applied to your health, your family, your community, or really anywhere there are various options in the face of uncertainty with estimated outcomes and probabilities.