Tuesday, August 26, 2014

A "Poker Showdown" on Andrew Gelman's blog and the problems with applying "Risk of Ruin" to poker

The two main influences on my poker presentation from 2012, primarily Chen and Ankenman's Mathematics of Poker, and secondarily my friend Rick Schoenberg's Introduction to Probability with Texas Hold'em Examples (in which I am quoted deriding Sklanksy's Fundamental Theorem of Poker as being neither fundamental nor a theorem), have been pitted against each other based on the latter's criticism of the former's misapplication of a mathematical identity involving with regards to the Risk of Ruin, featured in a post on the widely read statistics blog by Andrew Gelman.

Being a professional poker player with unusually close knowledge of these two texts, I thought I would surely have something worthwhile to bring to the discussion. However, while I was catching up with the dispute, it seems that it has been resolved in part, with both sides agreeing that while the error that Rick identified was indeed a problem, the implications for the Chen and Ankenman's conclusions were probably not as dire as Rick first thought.

As for me, my interest in the subject is actually rather mild. The truth is that while Risk of Ruin is a fun mathematical concept to investigate, it has surprisingly limited appeal to me as a poker player.

I usually like mathematical models because although they only examine idealized situations, the insights can be deep and are generally rock solid (within the model's assumptions). As long as we keep the simplifications solidly in mind, quite a lot can be learned from these models. In other words, I can decide how to apply my tidbits of knowledge without having to worry at all about the veracity of these tidbits. For example, from reading The Mathematics of Poker, I now know that blocking bets can be advisable and that in No Limit games, it is okay to bet different amounts with different hands even against the most perceptive players, as long as you have a corresponding bluffing hand with which you will bet that same amount. By integrating this appropriately into the rest of my poker strategy, I profit from this knowledge regularly.

There are some relevant poker insights to be had from looking at the RoR literature; notably, it is a really terrible idea to splurge after scoring a big win. But you should probably have been able to realize this, anyway. The simplifications that quashes RoR's poker relevance is that it assumes you will keep playing the same game even after you have lost most of your money.

Let's say you want to ensure you have under a 1% risk of going broke. Like me, you decide to be responsible and do some calculations and decide that with your bankroll and a conservative expected win rate and volatility, you could play 20-40 limit poker indefinitely, making $25/hour with only a .9% risk of ruin. You could also have played 100-200 limit, making $70/hour indefinitely, but the huge increase in volatility would have carried with it a 15% risk of going broke, according the the RoR analysis.

If you forgo the 100-200 game using this logic, I believe you are making a big mistake. The reason is that in real life it is very easy to stop playing a certain game or limit if you start losing. What the you should do is play the 100-200 limit game, but stop if you lose, say, 10% of your bankroll. If this happens, then you should move down in stakes. More often, you will win, and after a while you can actually move up in stakes. I believe your expected win rate would be much higher with this strategy, and if you are scrupulous about moving down in stakes when called for, your RoR could still be well under 1%. After all, if you find yourself back at the 20-40 game you would have been playing under the naive RoR strategy and you lose 10% of your bankroll, you will be moving down in stakes, which will significantly reduce your RoR. Under the naive strategy of playing 20-40 limit into oblivion, win or lose, you will keep playing at this level even after your bankroll is at 90%, 50%, 10% and 1% of its original total, which in practice no responsible poker player would be willing to do.

Moreover, if you want to employ a more savvy game selection strategy, occasionally you should also play "above your head" for a few hours (in a game where your mathematical RoR might be as high as 20% if you played indefinitely) to get a sense of what the competition is like. There are certain players who play high stakes games who are so bad that their presence in a game make it at least twice as profitable. If you never try these other games out, you might never be able to identify these fish. While there is something to be said for having a "regular" game with which you are very comfortable and can become highly expert at, I think some people underestimate the value in trying different games out. For most of my career, inspired by my RoR calculations, I made the mistake of sticking with one game that I knew would never ruin me. I played well below my abilities and I probably forwent quite a bit of potential winnings (not to mention experience playing against better players) by over-applying the mathematical implications of the risk of ruin analysis.

Like Rick, I seem to have misplaced my copy of Mathematics of Poker, so I can't say whether these issues are addressed in the book. I bring up these topics just because I wanted to explain my disaffection with applying RoR to poker and why I don't plan to become more involved in the discussion on Gelman's blog.

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