Feb 21, 2023
Feb 21, 2023

Developing the Root-Unit staking method

Understanding Maximum Expected Growth

Once a bettor decides to place a bet on a market, what’s the best way to decide how much to stake on it? There are several methods that are commonly used, but which one is optimal?

Or is there another way? Read on to find out.

In his recent book ‘Monte Carlo or Bust’, Joseph Buchdahl introduced a new staking method that he called “unit-z”.

The goal of this method is to adjust your stake size between bets of different odds such that the statistical z-score is the same for each one. I won’t embarrass myself by trying to explain to you what that means, so please read Joseph’s explanation if you haven’t already.

The great thing about this new method is that when he tested its effectiveness against a huge dataset of profitable soccer bets, it estimated the Expected Value (EV) attainable at any given odds much more accurately than the more common unit-loss (i.e., flat staking) and unit-win (i.e., bet “to win” one unit) methods.

Even the creative unit-impact method (introduced by Andrés Barge-Gil and Alfredo García-Hiernaux of the tipster platform Pyckio and published in the Journal of Sports Economics) proved inaccurate at longer odds. The not-so-great aspect of his method is that the formula he had to derive to calculate your Maximum Probable EV (or Expected Edge) based on the z-score is quite difficult.

To save you from too many complicated formulas, I won’t repeat it here. Instead, I’ll recreate his graph of the results of EV available vs. decimal odds:

Since the unit-z staking curve very closely matches the real-life data, let’s use Joseph’s method as a reference to represent actual results.

Why do you think the curve takes this shape, and is there any significance to it? What if, like with so many of the other questions about sports betting that I’ve investigated, the answer is that it depends on the Expected Growth (EG) available to the bettors?

Since each bettor has a different bankroll size, we would need to use the Maximum Expected Growth (MEG) to compare the EV available at different odds.

If we compute the MEG for one of the unit-z data points, and then plot a Line of Constant Optimal EG (LOCO EG) across the same range of odds as in the first graph, then it looks like this:

It’s an exact match to unit-z. In fact, the relationship between the EV at any given odds and the EV at even money becomes incredibly simple if we use fractional odds to calculate and plot it instead of decimal ones, like so:

The black trend line, Power (LOCO EG), overlaying our LOCO EG curve in the graph above is a perfect fit too. Since the y-axis is the EV or edge in this graph, and the x-axis is the fractional odds “b”, the formula for the edge “e” on our curve is:

e = 1.46% * √b

Why 1.46%? Because that’s exactly the Expected Edge for even Money Lines in the dataset that Buchdahl used to set up the variables for his unit-z formula. If all you need to do is compute the same MEG at a given odds as you’d get at even money, it’s not hard to derive the formula for your Expected Edge. First, we must use a handy approximation for the MEG:

MEG = edge2/(2*odds)

This “edge squared over twice odds” amount is typically very small, because it calculates the median change to your total bankroll from just one bet. Yet, it’s very powerful because, by setting it equal across all possible odds, we can derive the same exact formula that fits our unit-z curve:

MEG = edge2/(2*odds) = edge02/(2*1)

e2/2b = e02/2

e2 = e02b

e = e0√b

e = 1.46% * √b

Where:

e = edge or EV at the given odds

e0 = edge or EV at even money

So, what does this all actually mean? Well, over a dataset of 55,000 points, the best available line in the market produced the same amount of MEG for bettors when compared to Pinnacle’s margin-free line regardless of the actual odds that were offered.

Favourites, longshots, and pick ‘ems all had the same bankroll growth potential if you could get your bet down on the best line before it was gone, and if you staked optimally.

Is the treasure trove of European soccer data that Joseph used representative of all sports?

That’s the key question, because if so, it would mean that we could easily set our expectations for how much value can likely be found for different values of true odds based on our expected ROI at even odds. Then we can stake appropriately so that we take optimal advantage of our edge but don’t overbet.

“Root-unit” staking

For us to stake the optimal amount based on the LOCO EG, we just need to remember the old mantra from the Kelly Criterion method that your stake size should be “edge over odds.”

When we write the simple Kelly equation that way, and substitute our estimate of the edge at even money, we get another very simple equation:

f = edge/odds

f = e/b

f = e0√b/b

f = e0/√b

And, since the optimal fraction to bet at even money, f0, is the same as e0, then we end up with this formula:

f = f0/√b

In other words, to stake an amount that will give you the same EG at the current odds as you would get in a similar market at even money, just bet your usual even-money unit size divided by the square root of the odds.

That’s why I call this new technique the “root-unit” staking method.

Say that you are a one-unit bettor for NBA spreads and Totals, but spotted a good Money Line bet on the Pistons to beat the Sixers in tomorrow night’s game at +400 (or 5.0 in odds)?

Then you’d want to bet one unit divided by √4 in that case. Which means you bet half a unit. What if your model says the value is on the Sixers at -400 (1.250 decimal) instead?

Then you’d bet one unit divided by √0.25, or two units. Since you don’t have to get these values exactly right (and you may look far too sharp by betting amounts like \$103.58 even if you did), you can use the following table as a guide:

 American Odds Decimal Odds Net Fractional Odds Units Staked -1000 1.100 1/10 3.00 -600 1.167 1/6 2.50 -400 1.250 1/4 2.00 -200 1.500 1/2 1.40 -110 1.909 10/11 1.05* +100 2.000 1 1.00 +200 3.000 2 0.70 +400 5.000 4 0.50 +600 7.000 6 0.40 +1000 11.000 10 0.30

*If you still prefer to stake 1.1 units to win 1 for accounting purposes, you can.

Make a copy and tape it up next to your monitor if you want to.

If you have equal confidence in your ability to sniff out an edge at different odds, this method is how you can quickly stake in a Kelly-like way adjusted for the different levels of risk you’re accepting.

Remember that you can use root-unit staking for any fraction of Kelly that you feel comfortable with. It will simply allow you maintain that same balance of risk vs. reward for any odds you bet on.

Read more insightful and educational articles from Dan Abrams at Pinnacle Betting Resources. Sign up today and bet on your favourite sports with Pinnacles exceptional odds.

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