Every so often on your trawls through the Byzantine corridors of the World Wide Web, you happen upon a site that makes you say “Yeah! I must bookmark this”
Every so often on your trawls through the Byzantine corridors of the World Wide Web, you happen upon a site that makes you say “Yeah! I must bookmark this”. In the context of gambling this is typically one where someone gives really good tips etc, and occasionally (albeit more rarely) one which makes you profitably examine the way you bet.
One such site for me, and I suspect many who have visited it, is J.R. Millers site:
www.professionalgambler.com
which is packed full of useful betting strategies and information, albeit centred much more around American sports that what we usually bet on this side of the pond. Nonetheless the lessons are in general directly transferable in most instances, and the site can be heartily endorsed.
But….but….there was one little article there nagging away at me, which I’d never had time to consider properly till now:
“ HOW TO TEST THE KELLY CRITERION
Here's how to compare betting systems
http://www.professionalgambler.com/debunking.html
Miller states: Miller states: "The Kelly criterion as applied to sports betting would be better called the Kamikaze criterion."
I’ll cut right to the chase. None of the variations work; - at least, not against sports betting. I’m going to explain to you right here, right now, once and for all why the Kelly criterion as applied to sports betting would be better called the Kamikaze criterion. You can prove it for yourself, and here’s how:
Here’s what you’ll need, along with at least a half-hour of time:
1. A hand calculator
2. Two decks of ordinary playing cards
3. Lined paper
4. Pen or pencil
5. A ‘Thank You’ note to send me after you complete this exercise and realize how much money I've saved you.....
He goes on to describe an experiment involving two decks of cards where you bet on whether the next card will be higher than a 6 (Aces count as a ‘push’, and don’t affect the outcome). As each card is drawn it is set aside (i.e. not replaced in the double deck), until 50 cards have been drawn. After 50 cards have been dealt, the (double) deck is shuffled and the process repeated, effectively allowing 100 betting opportunities (although obviously one doesn’t bet if the deck is ‘cold’, i.e. the odds are not in our favour).
The exact odds can be calculated at any point of winning on the next ‘draw’, allowing a comparison to be made between Kelly staking (where the stake is varied according to the perceived (in this case explicitly calculable) ‘edge’ on each bet, and level staking where each bet is the same size, regardless of edge.
Miller goes on to summarise the results of his tests as follows:
The only way to fairly compare the Kelly system …… to flat betting is to use a flat bet the same size as the average size of all the Kelly bets. ……..Time to check the profits from flat betting against the record of the Kelly criterion, and ta-daa! There’s your proof. Using the average size of your Kelly bets as your flat bet, the Kelly loses, and it loses every time.
To someone who has advocated Kelly staking as optimal for many years, this seems anathema
Now, to someone who has advocated Kelly staking as optimal for many years, this seems anathema. However, being wrong is often a blessing in these areas as it means you have learned something new! My curiosity aroused, I decided to have a look at this using some computer brute force and running a monte carlo simulation of the tests outlined by Miller.
This consisted of writing some code in MS-Excel to simulate the dealing of a (double) deck of cards, and then repeating the experiment N times, where N is sufficiently large to give us a statistically significant comparison of the results.
I used Millers basic template of shuffling after 50 cards have been drawn, and then re-dealing once – for 100 possible bets. The results were recorded for both level stakes and Kelly staking and then the experiment repeated N times. I chose N as 20,000 in this instance, and a starting bank of £10,000 was used. Win odds of 1.91 were used. Betting was stopped on any particular run where the level stakes bank was exhausted. The nature of Kelly Betting is that bank is never actually exhausted, although practically it may dwindle to such a small amount as to be in all intents and purposes gone. However, in those situations, the likelihood of recovering to any significant bank level over only 100 bets is negligible.
This is not a trivial question, and involves a certain amount of trial and error Taking Millers point (about fair comparison of systems) on board I needed to ensure that the amount staked on the level stakes system was the same on average as the amount staked on the Kelly system. This is not a trivial question, and involves a certain amount of trial and error. What I did was to try various sizes of fixed stake until they matched the amount staked according to Kelly stakes. After 20,000 runs the average total amount using Kelly stakes was £190,392 (over 100 bets) and after several iterations to the average Level stake, I fixed it at £2,450, which gave me agreement between the total amount risked on each system to less than 2%.
This size of level stake (24.5% of starting bank) is massive, reflecting the advantage present in this contrived experiment. Nonetheless I persisted, at least initially (see later in this article) for the course of this experiment. The results are shown below:
The horizontal axis shows the ‘yield’ (i.e. the total profit/loss divided by the total amount staked), while the y axis shows the percentage of the 20,000 runs that the ‘game’ achieved at most a given level of profit. Thus, for example, the Kelly stakes system failed to make a profit (i.e. the 0% on the x-axis) 16.8% of the time, while the level stakes system failed to make a profit 27.3% of the time. In terms of comparison between the systems, the key feature to note is that the pink line (representing level stakes) is above the blue line at all profit levels – i.e. level staking is more likely to achieve less than any given profit level than Kelly staking. The average Kelly profit over all runs was 10.1%, while with level staking it was 1.2% - a direct result of the highly aggressive staking.
Most sensible bettors will use a ‘% Kelly’ system, where they wager a set % of the recommended Kelly stake.A potential flaw in the above analysis of course is that in some case the Kelly bank will on occasions be so small as to be effectively zero, so that betting would stop in reality. This is an artefact of the aggressive nature of Kelly staking which seeks to maximise bank growth rate without regard to risk. Most sensible bettors will use a ‘percentage Kelly’ system, where they wager a set percentage of the recommended Kelly stake.
Repeating the above experiment, but with a scaled version of Kelly (7.5%) chosen to make the average stakes similar to what a lot of American sports pro-bettors would use (~2% of betting bank at level stakes), but still maintaining equality in terms of average amount staked, the results are very interesting. The ‘strange’ behaviour at negative yield (due to the level stakes bank going broke) more or less disappears, but the overall trend is still conclusive.
The average Kelly profit over all runs was 14.7%, while with level staking it was 11.7%. Once again, the key feature to note is that the pink line (representing level stakes) is above the blue line at all profit levels – i.e. level staking is more likely to achieve less than any given profit level than Kelly staking. The average Kelly profit over all runs was 14.7%, while with level staking it was 11.7%.
In the conclusion of his article, Miller states:
But y’know what? I don’t expect a big upsurge in my mail due to Thank You notes. Hardly any believer in progressive betting systems will do the test. People want very hard to believe what they wish to be true, and they simply choose to disbelieve what they wish to be untrue.……… Users of progressive betting schemes want very much to believe they can earn more than they deserve. They refuse to be confused by facts.
So where does Miller get it wrong?
So where does Miller get it wrong? For an answer to that, I think you'd have to ask him for some more details on his experiments, but I’d like to conclude my article by pointing to the above, and asking any ‘Miller advocates’ (on this particular point – as noted above, I agree with him on 99% of everything else!) to examine the above and then consider who is right!
BBC
