Estimated Edge Error: A simulation of the Kelly conundrum

There have been countless posts in the Think Tank regarding the virtues of Kelly staking versus flat betting. Most naysayers of the Kelly Criterion are convinced that inaccurate edge estimation creates a dangerous environment for Kelly staking. Because of this, they insist that flat betting is a more suitable alternative for sports betting staking. To address these concerns, along with a personal aversion to the constant rehashing of this topic, I've provided an Excel simulator that allows you to see the differences yourself.



The simulator randomly chooses a line between the maximum line and minimum line settings (max = highest implied win probability). It then applies a randomly chosen edge between the maximum and minimum edge percentages. This is simply a representation of your estimated edge for the wager. Your estimated win percentage is then derived from the line you would be betting and your estimated edge for the wager. An error in your estimated edge is introduced (-10% > edge error > 10%). This is also randomly chosen between the maximum and minimum edge estimation error settings. The error is then subtracted from your edge and an actual win percentage is derived. The simulator then determines whether the play is a win or loss based on the actual win percentage.

Some other settings:

Plays: Self-explanatory I hope. (The number of plays that the simulator runs.)

Kelly Multiplier: The fraction of Kelly that you choose to simulate. This can range from anywhere between 0 and 8.

Flat Betting Percentage: The percent of your bankroll that is flat bet. This is the amount bet TO WIN.

Minimum Bet: The minimum bet that can be placed. If the wager that Kelly or flat betting attempts to place is smaller than this, the bet will be set to zero. Keep this in mind when setting your edges, Kelly multiplier, and starting bankroll. Small edges may not be bet with Kelly where they might with flat betting in certain scenarios.

The simulator also lists each individual flat/Kelly wager and the respective running bankrolls.

What you'll likely find is that when there is an equal chance of your estimated edge being either higher or lower than the actual edge, it tends to even out over time and Kelly remains far superior to flat betting. Only when flat betting approaches the average Kelly wager, does it come close to competing.

When the sample underperforms against the REAL win percentage, Kelly does moderately more poorly than flat betting. However, this is made up and then some when the win percentage regresses to the mean.

And finally, when you think you have an edge on the majority of your wagers but you don't, full Kelly does moderately poorer than flat betting. It's certainly NOT as treacherous as many would have you think. However, half Kelly and other variants show similar results to flat betting. But let's face it, if you THINK you have an edge but really don't, no staking system is going to make you money over the long haul.


In any case, you can come to your own conclusions with this simulator. And any time that the question of flat betting versus the Kelly Criterion is posed, hopefully people will be pointed to this simulator rather than rehashing the topic ad-nauseum once again.

good job sir

it seems to me that kelly increases variance, makes the highs higher and the lows lower

Great stuff. The big take away for me is that it is extremely easy to confuse variance (randomness) with actual edge, which becomes apparent when running this on a large number of plays.

Monkey, I adjusted the code to run faster on large number of plays so 100,000 plays are completed in < 1 sec. If you care, I can attach it in another post but I won't do so unless you approve.

Thanks Monkey.

I did something similar myself a few years back.

Originally Posted by SportsMushroom

good job sir

it seems to me that kelly increases variance, makes the highs higher and the lows lower

That is always going to be the case as long as the average wager size is bigger. If you were to increase the flat betting percentage to the 2.5-3.0% range you would see similar peaks/troughs.

Originally Posted by thom321

Great stuff. The big take away for me is that it is extremely easy to confuse variance (randomness) with actual edge, which becomes apparent when running this on a large number of plays.

Monkey, I adjusted the code to run faster on large number of plays so 100,000 plays are completed in < 1 sec. If you care, I can attach it in another post but I won't do so unless you approve.

That would be great, I was running these big sims a few times.

Might be interesting to automate multiple runs instead, to explore variance in bankroll growth over a more realistic number of plays (eg run 100 sims of 1k plays, rather than 1 x 100k).

Very smart guy. I would probably blow my brains out after that big drop in the middle though.

Originally Posted by thom321

Great stuff. The big take away for me is that it is extremely easy to confuse variance (randomness) with actual edge, which becomes apparent when running this on a large number of plays.

Monkey, I adjusted the code to run faster on large number of plays so 100,000 plays are completed in < 1 sec. If you care, I can attach it in another post but I won't do so unless you approve.

Sure. Go ahead.

I was going to optimize it but I got lazy.

Originally Posted by HeeeHAWWWW

That would be great, I was running these big sims a few times.

Might be interesting to automate multiple runs instead, to explore variance in bankroll growth over a more realistic number of plays (eg run 100 sims of 1k plays, rather than 1 x 100k).

I figured people could use the base code and add a loop for multiple sims or track it in another spreadsheet to analyze separately. It certainly wouldn't be hard to do and perhaps when I get some time, I'll add a charted feature like that myself.

In the version I attached, I store the data in arrays and write the arrays to the worksheet when the simulation is done, rather than write the data to the worksheet as the simulation is running. I also disabled screen updating and automatic calculation when the simulation is running.

A simulation of 100,000 plays now take less than a second on my computer.

I also disabled the message box and instead write the message to a cell in the sheet. Everything else is the way Monkey did it.

What if you give naysayers the ultimate credit by assuming there might be a correlation between your model predicting a huge edge and the possibility that there's some information you're unaware of?

i.e. if your model predicts a 20% edge when normally it's 1-2%, your reaction as a modeler is usually "I must have ****** up", not "grandma I need to borrow 100k to bet the Pirates"

Okay. Here's my final version.

It includes multiple iteration simulations as requested. It also includes the optimized code that Thom provided.

Originally Posted by mathdotcom

What if you give naysayers the ultimate credit by assuming there might be a correlation between your model predicting a huge edge and the possibility that there's some information you're unaware of?

i.e. if your model predicts a 20% edge when normally it's 1-2%, your reaction as a modeler is usually "I must have ****** up", not "grandma I need to borrow 100k to bet the Pirates"

Garbage in, garbage out. Like I said in my original post, if you're miscalculating your edge to the point of negative expectation (or grossly overbetting) with regularity, you won't be successful with whatever staking strategy that you use. If your model is puking on you then you shouldn't be betting it regardless. If you were flat betting would you place a wager on that game? You probably shouldn't be.

But that also assumes that you're a modeler. You don't need to model games in order to use Kelly.

Appreciated guys, very impressive. I was going to give it a crack tomorrow, but usually my attempts at programming/scripting result in things so unoptimized you need a supercomputer just to run them.

Very interesting..so if one is going to use half kelly, is that about the same as flat Betting? It seems like it anyways..