Recently one of my potential clients pointed out a website that claims to debunk the Kelly criterion. For those that don’t know the Kelly criterion is a money management strategy that works out how much you should bet on a sports outcome based on the probability of the team winning, the odds that the team are at and your bank size.

It is not the first time I’ve noticed the article on http://professionalgambler.com/debunking.html as the article has been up there for quite a while. Several years ago another of my friends said it was “embarrassing for the bloke” that the article was still there. But considering that I highly use the criterion in my betting, I hate to have potential clients find articles that criticise the method that I use. Well, they have every right to criticise me, but when they get it completely wrong, that’s where I come in.

So this is my article, debunking the debunking of the Kelly criterion

PG, for short, suggest that we get a couple of decks of cards and do a test. That’s so old school. I’ll do the same thing with a statistical simulation. Using exactly his test, but I might do it say, 50,000 times instead. Computer programming is handy you know.

And guess what? The results actually favour PG’s argument. When betting with the Kelly criterion the average bank size after 100 bets was $49,320.70, up quite a bit from the original $10,000. When working out how much each bet was using the average size bet of the Kelly criterion, the average finishing bank size over 50,000 simulations was $53,763.75. In fact 65.7% of the results favoured PG’s flat betting strategy.

So perhaps PG is correct, we should all be flat betting no matter what? Not so. You see PG decided in his article to compare the average size of all the Kelly bets and bet the same amount flat betting. That sounds all very good, but in reality you have no idea what your average bet size is several years into the future.

To show why this method doesn’t work, I will do another simulation where instead of just betting 100 bets (which one could easily do in a day or two), but with 1000 bets.

The results are incredible and obviously not realistic, but they are there to prove a point. The average finishing bank using the Kelly criterion was just over $100 Billion, whilst the average flat betting strategy was just over $1,000 Billion, or 10 times the size.

Seems like the flat betting wins again yes? No. Because the average size bet for flat betting was $288 Million, which considering that you only have a $10,000 bank is pretty impressive.

The comparison is assuming that even with a $10,000 bank, that you are going to bet $288 Million each bet, and if you lose your original $10,000 (which you could do on the very first bet), you just keep going betting $288 Million at a time.

Quite clearly, the comparison is not legitimate.

You see PG doesn’t seem to understand what exactly the Kelly Criterion is. The Kelly criterion suggests that you bet more when the probability is greater, you bet more when your advantage over the bookmaker is greater, and you bet more when you bank size is greater.

He incorrectly labels this a progressive betting technique when it is far from it. Lets look at each criteria in the Kelly betting system.

You bet more when the probability is greater. If something was a 99% chance to win, you surely would bet more than if something was a 1% chance to win. Id hate to be risking 5% of my bankroll on a 200/1 shot. PG probably has never thought of this in this case, because like a lot of Americans, he only bets on the line where something is around a 50-60% chance to win each time.

You bet more when your advantage over the bookie is greater. Surely if you rated a team a 90% chance to win and they were paying $2.00, wouldn’t you bet more than if you rated a team a 50% chance to win and they were paying 2.01?

It seems that the only thing that PG doesn’t like about the Kelly criterion is the fact that as your bank grows larger (or smaller) then you bet more (or less). Incorrectly calling this a progressive betting system, the reason why one should adjust their bet size based on their bank size is given by a few examples.

Remember when you were young child and you would save all week to go down to the corner shop and buy a $1 bag of mixed sweets? Would you do the same now? Save all week to buy some sweets? Of course not, now that you are more wealthy, you spend more. You could save all week to buy a TV or a dining room table that would cost you $1,000, and if you did, here you have spent a percentage of what you are worth.

When I first started betting, I put $100 into a bookmaker and bet $10 a time. As I realised my edge was significant, and my bank kept increasing, I increased by bet size. Imagine me now betting $10 a bet. What would be the point? Some might say that of course you change your bet amount at the end of the season or after each month, but isn’t that just adjusting your bet size to your bank? Why not do it after every bet?

The truth is, to accurately compare the Kelly Criterion to flat betting one has to keep constant the percentage of the bank that one bets. With PG’s card game example, the average bet size using the Kelly criteria based on odds of 1.95 is 14.47%. So lets do the simulation example, comparing how the Kelly Criterion went compared to flat betting with each bet at $1,447 (14.47% * $10,000).

Whilst the average Kelly bet bank $49,320.70 over the 50,000 simulations, the average bet size when flat betting was only $29,623.16. That’s a $39,000 profit compared to $19,000 profit – Less than half the profit. In fact Kelly betting had a greater profit 32% of the time.

But what happens if we compare with say, 1000 bets like previously? Kelly betting finished, as previously, with an average bank size of just over $100 Billion, whilst flat betting finished with an average bank size of $205,966.82. That’s 500,000 times smaller. Ouch!

In fact flat betting only had a higher bank than Kelly betting 0.6% of the time.

PG finally digs himself a hole by stating that it is “futile” to try to estimate the probability of a team winning because there are so many “countless variables involved” and hence “you can never know what your winning expectation might be”

Quite clearly, PG is from the old school of professional gambling. One who hasn’t woken up to the power of mathematics. I wish J. R. Miller all the best of luck of course, but in the mean time, consider his article on debunking the Kelly System, debunked.

Would it be possible to give an example of the difference Kelly made with, say, ATP betting in 2011.

e.g. What would the profit have been if you flat bet $X on every bet suggested by the model. And how this would compare to betting at the suggested kelly stakes.

yep good idea andy, however looking at one sport will provide different results for flat betting and kelly betting. In doing atp tennis 2011, better results were given for flat betting than kelly. This is because the last few weeks of tennis 2011 lost, which effects the kelly betting more than flat betting, as the bank size is a lot larger at this time.

I said above that about 2/3rds of the time kelly will be better than flat betting, tennis 2011 was one of the times it wasn’t, however the theory is there. Kelly betting is prefered.

Thanks a lot.

I do find there are a fair few issues with Kelly. Not so much for tennis, but bookmaker’s limits in stuff like Euro basketball cause problems so the theoretical returns in the bet history there are probably a bit misleading. I’d also hazard a guess the low limit matches are in the leagues where there’s the most value. It also literally takes one small bet with pinnacle to move the prices in those leagues.

Kelly is best when you do know true probability, the maths tell us this …

but as a sports better, you don’t need to know the true probability to win, you just need to know that you will have a edge on that bet over time, if PG doesn’t believe he knows the true probability, then flat betting is the best way

When I started out almost 10 years this guy was touting Kelly as the sure way to make money.

It used to be “don’t worry about the short, in the long term using the Kelly principle you will always come out ahead”. It was words to that effect.

I don’t even believe he or his son even use it. I really do think they just play underdogs.

Stay away from this guy. A lot of punters have read the same books this guy has. No point paying for the picks.

andy you are right, you are limited in sports by how much you can get on, which in a sence, limits the big bets, which have the biggest overlays (or the bets on the favourites).

LRBC & savagedude – I think PG is really from the old school, that doesn’t believe that you can work out probabilities of sporting events that you dont even watch. I did however feel I needed to make this article, as I have had people being skeptical of my website because I use Kelly and PG apparently discredits it. Not so.

I think Kelly betting is way too aggressive. The problem is people’s lives are limited and the Kelly is not. I have run simulations with given expectations and really it is a great way to understand how your risk per bet corresponds to your risks and goals. Many people would rather have a 90% probability of doubling bankroll every year and only a 5% probability of greater than 20% decline. Some would be okay with the alternative of a higher average expectation skewed towards the top 30% each year with a 10% probability of a 70% drawdown of something egregious but in the “long run” a better geometric rate of return.

The Kelly assumes an infinite time horizon, not being effected by volatility, complete tolerance of drawdowns, normally distributed fixed probabilities with complete certainty in your edge (this is the most important because if you are wrong enough about your edge and play the aggressive side, you can actually drawdown to a magnitude you cannot recover from in your lifetime).

Furthermore, a graphical representation is available if you search which is plotting the results for given risk percentage and it shows that you can still get 75% of the results with half the risk with a half kelly strategy, and I think like 45% or something with 25% of the volatility risking 1/4 Kelly. In some situations, even 1/4 Kelly is too much. Additionally, a Kelly strategy effects strategy in some situations. Doubling down soft hands for example in blackjack may be optimal from an expected value perspective, but not once you factor in that it delivers greater bankroll volatility.

Finally, if you could bet simultaneously on 80 coin flips that pay 3:2 with 1% of capital per bet or 1 coinflip with 80% of capital betting simultaneously on 80 coinflips over 1 flip period would NOT disobey the Kelly and would actually make a lot more money over time than 1 bet… When you look at multiple opportunities with zero correlation, it actually is best to weight according to your edge and diversify across as many bets as possible…

In the stock market it gets more complicated because there exists correlation and much more uncertainty in your edge which changes considerably along with market environment.

nice reply mike,

“complete certainty in your edge (this is the most important because if you are wrong enough about your edge and play the aggressive side, you can actually drawdown to a magnitude you cannot recover from in your lifetime).”

This is why noone bets fully kelly and most people bet anywhere between 1/4 kelly to 1/20th kelly

I was wondering if there is a statistics based method for determining what kelly fraction to use for different models. Obviously, it varies depending on personal risk profile. But it also depends on confidence in the model.

For example, assuming that I chose a kelly fraction of 1/5th for the AFL line model, based on my own risk tolerance, is there a statistics based method to determine how to weight bets on, say, the NFL totals, or some other model that would maintain the same relative risk?

It is relatively easy to subjectively compare the models, based on their history. Can they be objectively compared?

Hi Zac,

yes there is a statistical way to determine what kelly fractional is best for each model. Basically you need to work out what %ROI the model has returned, and what the average overlay is for every bet suggested. If the average overlay for every bet suggested is 25% (as an example), and the %ROI that the model produces is 5%, then the optimal Kelly to be used is 25/5 = 5 or 1/5th Kelly. This can change with different overlays. A models larger overlays may result in more profit than smaller ones etc. But yes, this is an article I probably should look into and write, but that is the idea. Of course, this assumes that the past results will continue in the future.

“Assuming that the past results will continue in the future”…. Kind of fundamental to the whole modelling/forecasting concept, isn’t it?

If you do write that article, it’d be great if you could include a bit on how large the sample size needs to be to have confidence in it…. But that is probably simple enough statistics that a bit of Googling should answer my question.