Reasons to Ignore the Kelly Criterion – Analytics.Bet

In “sophisticated” betting circles, the Kelly Criterion is accepted as the single correct way to determine your optimal bet sizing.In plain English, the Kelly criterion says: Bet bigger when your edge is bigger and your chance of winning is higher, but bet less when your edge is smaller and your chance of winning is lower.  At face value, this makes perfect sense and it leads to a precise formula that balances the two competing factors to achieve the best long-run results:

(**) Kelly = Edge / Odds

I won’t discuss the mathematical details of where this formula comes from, as have been (too) many articles written on the topic.  For a detailed discussion of the Kelly Criterion see [PlusEV article on The Real Kelly Criterion].  For our purposes, all we need to know is that the math is correct.  The problem is how this math gets applied in actual betting.

The theory tells us that if we have an edge and we bet according to (**), then our bankroll will grow like this:

Multiply your initial bankroll by about 7.5x after only 500 bets. Who wouldn’t want those results?

And yet no sharp gambler would ever recommend betting the full Kelly fraction.  Maybe they’re not so sharp after all!

Or maybe the Kelly Criterion is optimized for a different problem than what bettors actually care about?

The Kelly Criterion is the optimal bet size to achieve fastest possible bankroll growth for a given edge and payoff odds.  The Kelly Criterion doesn’t care about volatility, doesn’t consider the possibility that you’ve miscalculated your edge, doesn’t entertain the possibility that you might not get paid if you win, doesn’t account for minimum and maximum bet sizes, …

So, if you’re an emotionless, omniscient bettor dealing with bookmakers always take your action and can be 100% trusted to pay when you win, then by all means apply the Kelly Criterion.

Everyone else can keep reading.

There are three primary reasons to be careful with the Kelly Criterion (especially in sports betting contexts).

1. It is volatile.
2. Your edge is probably smaller than you think it is.
3. The situations where you think we have the biggest edge are most likely to be those in which you are the most wrong.

The first two of these are obvious, but the last one is less understood and is a reason you might want to ignore (or seriously revise) the Kelly criterion when betting actual money.

Kelly Criterion is Volatile

Growing our bankroll is great, but even better is growing our bankroll and being able to sleep at night.

The plot above only shows where you’ll end up by betting Kelly, without saying anything about how you end up there.  A typical experience for the Kelly bettor looks something like this:

With a 10% edge betting at +120 odds (2.20 decimal), we expect to multiply our bankroll by 7.5x after 500 bets.  But in the course of realizing this “expectation”, we very easily could find ourselves up more than 10x after 400 rounds of betting, and then up only 3x by the 450th round.  In other words, there’s a big difference between theoretical expectation and actual realization

The untold cost of attaining the theoretical expectation is having to live through the actual realization, which means a lot of volatility, sleepless nights (and possibly subscription to Prozac).

The easiest way to cut volatility is to bet less.  Instead of betting the full amount in (**), suppose we bet 25% of it.  We’ll make less money in the long-run, but we’ll sleep better at night:

This fractional Kelly approach has been applied by savvy bettors to reduce volatility. But its not just for nits who can’t handle the swings.  Even if you have the stones to handle the ups and downs of Kelly betting, you still shouldn’t try it.  Not to burst your bubble but …

Your edge is smaller than you think it is

Even if you’re sharp, there’s always information you haven’t accounted for.  Usually, the information you haven’t accounted for explains why you think a bet is favorable and the market doesn’t.  So, if you think your edge is 10%, it’s probably more like 5%, or even 2-3%.

What this means is that, even if we don’t mind the volatility of Kelly, we should still consider fractional Kelly in order to guard against overestimating our edge.

In our running example, we assume a 10% edge at +120 odds, so that the optimal bet size is 8.33% of our bankroll.  But since we have a positive edge, wouldn’t we make more money (on average) by betting more than 8.33%?

The interesting thing about the Kelly criterion that betting less can sometimes increase how much money we make, even when we have an edge.  This is related to the idea of overbetting, which happens whenever we bet an amount that is greater than the Kelly fraction in (**).

As the graphic below shows, if we bet 12.5% of our bankroll (or 1.5x-Kelly), then we expect our bankroll to grow slower than when we be 8.33%.  And if we bet 16.66% (or twice Kelly), we shouldn’t expect our bankroll to grow at all!

But why does this lead to fractional Kelly?

Read carefully. The Kelly criterion (**) says to bet in proportion to what our edge is, not what we think our edge is.  So if we think our edge is 10%, meaning our edge is more like 2-3%, then we should bet as if our edge is 2-3%.  This translates to betting around 20-25% of the Kelly fraction.

For illustration, suppose we adopt a half-Kelly approach, so that instead of betting 8.33% we bet 4.17% of our bankroll.  Also suppose that we have overestimated our edge, so that it’s actually 5% instead of our estimated 10%.  Now consider the difference between what we think we would be getting by betting full Kelly (orange line) to what we would actually get (red line) to what we get by betting half-Kelly (blue line).

If we bet full Kelly, we would expect a lot of growth but get none.  But by being conservative (betting half-Kelly), we actually are betting the exact correct Kelly fraction given our actual edge.

So, in addition to diminishing volatility, fractional Kelly also helps us avoid overbetting (betting a higher amount than we should given our edge and bankroll).  If we bet too much, even when we have a positive edge, we’ll stunt the growth of our bankroll. 

By now you’re probably convinced that betting 10-50% fractional Kelly is useful practical advice, both to reduce volatility and avoid overbetting.  But you could have learned this from just about this just about any other article on the Kelly criterion.  What you probably haven’t read is that you might be better off forgetting about the Kelly criterion altogether and simply flat betting—picking a unit size (say, $100) and betting the same amount every time.

The bigger you think your edge, the more likely you made a big mistake

It’s common to talk about edge as if it’s a fixed physical constant of the universe.  We interpret having an edge of 2% to mean that every bet we place expects a profit of 2%.  But in reality, having a 2% edge in sports betting really means a 2% average edge.  

Sports betting isn’t like craps or roulette, where every bet is placed under the same conditions with the same probabilities and edges.  In sports betting, every future outcome is unique, which means our information about that potential outcome is unique (in its own way), and therefore subject to variation due to the natural fluctuations of information flow, human psychology (public opinion reflected in the market odds), and human psychology (our own opinions).

We might win 2% over the long run, but in practice this 2% is made up of some bets that have a 10% edge, others a 1%, others a -3% edge, and it all averages out to an ROI of 2%.

Of course, if we know exactly what our edge is, then we would bet more when we have the 10% edge, less with the 1% edge, and not bet at all with the -3% edge.  But, as we’ve already observed, we don’t know our edge.  We’re just guessing.

We discussed this, and how we can address it, when we talked about fractional Kelly above.  But we didn’t talk about how our actual edge and our perceived edge may not be related by the simple relationship assumed by fractional Kelly betting.  For example, half-Kelly betting assumes that our actual edge is half of whatever we think it is.  So if we think our edge is 1%, then we bet as if it were 0.5%; if we think our edge is 10%, we bet as if it were 5%; and if we think our edge is 100%, we bet as if it were 50%.

Sense a problem?

First, why should we think that the gambling gods would be so kind to allow for a nice proportional relationship between our perceived and actual edges, regardless of their size?  It can’t be this easy!

Second, and more to the point, aren’t the situations in which we think our edge is biggest most likely to be the ones where we either made a mistake or are missing a crucial piece of information that the markets (and everyone else) are already accounting for?

Consider the following very simple situation.  Suppose our strategy uses input from a statistical model which computes probabilities by processing data from an API feed.  Under ordinary conditions, this approach yields a 2% return on average.  However, suppose that we have (unknowingly) ignored a number of rare edge cases in our data processing.  These cases occur pretty rarely, so we didn’t notice them at first, but when one of these edge cases occurs it leads to distortions in the data that causes our model to project a 20% edge.

(The kinds of data errors to imagine are: an error in the starting lineup, a mismatch between a player’s name in the API feed versus our historical dataset, a misreported injury announcement.  These can be pretty rare — the spelling can be wrong for just 1 or 2 player’s names in the league — yet extremely costly when they occur.)

For simplicity, suppose all of our bets are at even odds, and that the Kelly fraction determined by our model is either 2% of our bankroll (when there is no data error) or 20% (when there is an error).

Here’s the thing.  The reason our model thinks the edge is 20% isn’t because of ordinary randomness in the process that generated the historical data.  It’s due to a data processing error of our own doing.  Meaning that our mistake is likely to perceive an edge on a random scenario for which no edge exists.  In these situations, we are most likely assessing a 20% edge on a random outcome, meaning that we can expect to lose the vig over the long run.

For the sake of illustration, let’s assume that your actual edge in these situations is -10%.  Also, let’s assume that when we assess the edge at 2%, it is actually 2%.  (The exact values of vig, edge, etc. don’t matter for this example.) 

Under these assumptions, we have an expected profit of 0.04% of our bankroll each time our model finds a 2% edge.  And we have an expected loss of -2.0% of our bankroll each time our model finds a 20% edge (20% Kelly fraction times our actual edge of -10%).

Therefore, even if our data issue occurs only 1 time out of 50, our 2% “edge” amounts to a 0% ROI.  If it happens more often, then our profitable model is now a loser. 

Even more troubling is that fractional Kelly is helpless in this scenario.  Suppose we were betting half-Kelly instead of full Kelly.  Then we cut both our expected gains and losses in half, so our 2% bets become 1% bets with an expected profit of 0.02% of our bankroll, and our 20% bets become 10% bets with an expected loss of -1.0% of our bankroll.  Still, 1 mistake in 50 is enough to wipe out our edge entirely.

If, on the other hand, we ignore Kelly and always bet the same amount no matter what edge is (say $100), then we profit $2 (on average) when our model correctly identifies a 2% edge, and we lose $10 (on average) when it incorrectly assesses a 20% edge.  Now our data error isn’t so fatal because we are capping our exposure to it: if we have a data issue 1 time out of 50, then we still earn an ROI of about 1.8%, which is a little bit less than 2.0% but a whole lot better than 0%.

The observation raises a bit of a paradox.  On the one hand, we want to bet the most when our edge is highest.  The only way we can possibly do this is if we trust our evaluation of how large our edge is.  But we also know that the times when we think our edge is highest are likely to be those where we are most likely to be missing something or making a mistake.

Of course, the best way to address this issue would be to fix all of our data errors, remove all bugs from our code, and make sure we have all of the information that everyone else has.  A perfect plan, in theory. 

In practice, mistakes are inevitable regardless of our approach (top-down, bottom-up, statistical, technical, or anything else).  We can only do our best to eliminate them, mitigate them, and then protect against the ones that survive our attempts at elimination and mitigation.  Forgetting about the Kelly criterion, or severely curtailing our use of it, may just be the best way to do this.