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
So if your edge is 10% on a bet at even-odds, Kelly says to bet 10% of your bankroll; but for the same 10% edge on a bet at 10-1 odds, Kelly says to bet 1% of your bankroll. The smaller fraction in the latter case offsets the additional risk presented by betting on a 10-1 outcome versus a 1-1 outcome.
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 The Real Kelly Criterion by PlusEVAnalytics. For our purposes, all we need to know is that the math is correct. But as bettors who care about winning (real) money, we need the math to work in practice, not in theory.
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 optimized 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 we’ve miscalculated our edge, doesn’t entertain the possibility that we might not get paid when we win, doesn’t account for minimum and maximum bet sizes, …
So, if you’re an emotionless, omniscient bettor dealing with bookmakers who always take your action and can be 100% trusted to pay when you win, then by all means apply the Kelly Criterion.
For everyone else, please can keep reading.
There are three primary reasons to be careful with the Kelly Criterion (especially in sports betting contexts).
1t is volatile.
Your edge is probably smaller than you think it is.
The situations where you think you have the biggest edge are most likely to be those in which you are the most wrong.
Taken together, these three reasons might be enough to make you want to ignore (or seriously revise) the Kelly criterion when betting actual money.
Kelly Criterion is Volatile
Growing your bankroll is great, but even better is growing your bankroll and being able to sleep at night. And even better is growing your bankroll and knowing that your bankroll is growing because you’re good (not because you’re lucky).
The plot above only shows where you should 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. That’s the difference between theoretical expectation and actual realization.
The untold cost of attaining Kelly’s theoretical expectation is having to live through the actual realization, which means a lot of volatility, sleepless nights (and possibly a subscription to Prozac).
The easiest way to cut volatility is to bet less. Instead of betting the full amount suggested by (**), suppose we bet 25% of that amount. 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 it’s not just for nits who can’t handle the swings. Even if you think you can handle the ups and downs of Kelly betting, you still shouldn’t try it.
As bettors, we know that winning and losing is part of the deal, and we need to train ourselves to handle ups and downs. We may be tempted to dismiss the volatility shown above as something only a “lesser bettor” would concern themselves with. We only care about what makes the most money, and living through the volatility is the price we pay to achieve that goal.
All well and good — if you know that your violent down swings are just a result of ordinary Kelly volatility. In the simulation shown above, we know the down swings are only caused by random chance. If we let the simulation run longer, the line will recover, and we’ll live happily ever after in math fantasyland.
But in real life, we’ll never know for sure that the downturns are just a result of ordinary statistical variation. When we have a losing streak, it might be because of temporary bad luck, or it could be because something has changed either in the betting markets, or in our betting strategy, or in our data, or somewhere else that has eliminated our edge. Thoughts will (and should) run through our heads:
Did we introduce a bug in our code?
Is our data corrupted?
Has the betting public gotten stronger?
Are our models stale?
Does our strategy no longer work?
These are questions we are always asking ourselves because all of these things can (and sometimes do) happen. But by subjecting ourselves to an approach that is extremely volatile in the best case scenario — as is the case with Kelly betting — we make it impossible to quickly diagnose when any of the above (very serious) issues pops up.
This last discussion about all the bad things that can go wrong without us knowing about it leads us to our next reason to reconsider the Kelly criterion. Not to burst your bubble but …
Your edge is smaller than you think it is
No matter how sharp you are, there’s always information you haven’t accounted for. Often, this mismatch of information explains you think a bet is favorable in the first place. If you knew what others in the market knew, then you probably wouldn’t think the bet was as favorable as you do. So, if you think your edge is 10%, it’s probably more like 5%, or even 2-3%.
This means that even if we don’t mind the volatility of Kelly, we should still consider fractional Kelly as a way to guard against overestimating our edge.
The interesting thing about the Kelly criterion that betting less can sometimes increase how much money we make. This relates to overbetting, which happens when we bet more than the Kelly fraction in (**). If we bet too much, even when we have a positive edge, we’ll stunt the growth of our bankroll.
In our running example, we assume a 10% edge at +120 odds, so that the Kelly 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%?
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 our edge is 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. And since we are very likely to be overestimating our edge, fractional Kelly may actually bring our bet size closer to the optimal amount.
By now I hope you’re convinced that less is more when it comes to the Kelly criterion: 10%-50% fractional Kelly is useful practical advice, both to reduce volatility and to avoid overbetting. But you could have learned this from just about this just about any other article about the Kelly criterion. What you probably haven’t read elsewhere 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 situations you think you have the biggest edge are those in which you’re the most wrong.
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 and human psychology.
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 knew 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 really 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%.
But why should we expect 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!
More to the point, aren’t the situations in which we think our edge is biggest also most likely to be the ones where our opinion is most wrong?
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 our projected edge is 2%) or 20% (when our model senses a 20% edge).
Here’s the thing. The reason the model thinks our edge is 20% in some cases isn’t because of ordinary randomness in the process being modeled. It’s more likely because our model is wrong in a systematic way, meaning that we might be assessing a 20% edge on a random outcome. In such cases, we can expect to lose the vig (or worse) over the long run.
For the sake of illustration, let’s assume that our 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 (2% Kelly fraction times our edge of 2%) 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 a mistake happens only 1 time out of 50, our 2% “edge” reduces 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 our 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 mistakes aren’t so fatal because we’ve capped our exposure to them: if we’re wrong 1 time out of 50, then we can still earn an ROI of about 1.8%, which is a little bit less than 2.0% but a whole lot better than 0%.
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.