The Kelly Criterion Revisited

How much should we stake when we have the edge?

In a Nutshell

One of the most important questions in betting is also one of the most obvious. Suppose you think you have found a good bet. How much should you stake on it? Getting this wrong can matter just as much as getting the selection wrong.

This is where we meet the Kelly Criterion.

TAKING ADVANTAGE OF THE ODDS

The Kelly Criterion takes its name from John L. Kelly Jr., an engineer at Bell Labs who developed the idea in the 1950s. The basic insight is simple enough. If we have an edge at the odds, the amount we bet should depend on how large that advantage is. We also need to be careful because the downside of a loss or string of losses can be devastating.

So there are two questions we need to address.

First, do we have an edge at the odds? Second, if we do how much should we risk?

Let’s take an example.

Suppose we are betting on a coin toss at even money – £10 to win a net £10 if we call right, we lose £10 if we call wrong. That’s a fair bet if it’s a fair coin. But what if we know with certainty that the next toss will come up heads. In this case, the strict Kelly criterion advises a maximum bet. Of course, absolute certainty is a rare commodity. There might be something wrong with our information, or something we haven’t thought of. But as a thought experiment it makes the point.

Take another example. We believe the coin has a 60% chance of landing heads and a 40% chance of landing tails. We’re still being offered even money. Now we have an edge, but we can still lose – indeed, we can expect to lose four times in every ten.

How much should we now bet? The Kelly answer is 20% of our available capital.

HOW DOES IT WORK?

For those who want the formula, it can be written as:

F = Pw − (Pl/W)

where:

F is the fraction of our capital to bet;

Pw is the probability of winning;

Pl is the probability of losing; and

W is the amount won for each unit risked, excluding the return of the stake.

Take our coin example. We think heads has a 60% chance and tails a 40% chance. The bet is even money, so W is 1.

The calculation becomes:

F = 0.60 − 0.40 = 0.20

So Kelly tells us to stake 20% of our capital.

Now suppose instead that we are offered odds at which we win £2 for every £1 we risk, and we believe our chance of winning is 50%.

In that case:

F = 0.50 − (0.50/2) = 0.25

So the Kelly stake is 25% of our capital.

The important point is not really the arithmetic, but the principle behind it – the bigger our edge, the more as a proportion of our available capital we should be prepared to bet. And if there is no advantage at all, Kelly tells us not to bet.

That last point is easily overlooked.

WHY KELLY?

Kelly is designed to maximise the long-run growth rate of our capital when we have opportunities to bet at favourable odds. If our capital grows, our stakes grow with it. If our capital falls, our stakes fall as well. So, instead of chasing losses when things go badly, we do precisely the opposite. The idea is to take proper advantage of any edge we have without exposing ourselves to unnecessary and potentially catastrophic risk.

BUT THERE’S A CATCH

There is, however, a fairly obvious problem. The formula assumes that we know the true probability of winning and therefore our edge. But usually we don’t. This matters, because overestimating our edge leads directly to over-staking.

FRACTIONAL KELLY

This brings us to a commonly used compromise: fractional Kelly. Instead of staking the full amount recommended by the formula, we stake only a fraction of it.

If a traditional Kelly strategy advises a stake of 20% of our betting capital, a half-Kelly strategy would recommend 10%, a quarter-Kelly would see us stake 5%.

It also gives us some peace of mind – in a world of rollercoaster volatility we prefer the slightly quieter ride.

SO HOW MUCH SHOULD WE BET?

Kelly gives us an elegant answer to a difficult question. If we know our edge and the available odds, there is a mathematically defined stake that maximises the expected long-run rate of growth of our capital.

The more uncertain we are about our supposed advantage, however, or the more defensive we are against rollercoaster volatility, the stronger becomes the case for reducing the stake.

That is why fractional Kelly has such obvious appeal. There is also a broader lesson here. Having an edge is not enough. We have to survive long enough to exploit it.

Kelly changes the way we look at optimal staking. Applying it in the real world is a more nuanced task.