CardSharp

Part 1, Part 2, Part 3

It’s time to talk personal playing philosophy. Over the last month or so, I’ve written a ton about strategy rhos and game theory. I’ve talked about several different rhos:

• The big strategy rho (part 1, part 2)
• bluffing & calling on the end (part 1, part 2)
• defending against a draw with a made hand (part 1, part 2)

If you read these articles carefully, you’ll notice that I’ve adopted three different and indeed incompatible approaches to poker strategy in the three series. In the changing gears article, I advocated trying to in essence “time” the rho and figure out which strategy your opponent was selecting, and then in turn select the best counter. I’m going to call this approach of deducing your opponent’s strategy and exploiting it the “exploitative approach”. In the bluffing and calling on the end article, I used a strict game theory approach of making your opponent’s decision not matter. I’ll call that the “game theoretic approach”. The defending against the draw series ultimately advocated a combination of a bigger preflop bet and keeping the pot size small so the difficult decision was avoided entirely. I’ll call this process of using the the structure of the game to avoid trouble the structural approach. Each one of these approaches could indeed be called a meta-strategy, or a strategy for selecting strategies.

My thesis for this article is simply that these three meta-strategies are not equally good. I believe the exploitative approach is the best when feasible, followed by the structural approach, and that the game theoretic approach is the choice of last resort. My position on this subject is not shared by everyone. Many noted poker scholars who’s play and writings I respect disagree with me. These include Bill Chen and Jerrod Ankenman (authors of The Mathematics of Poker) and Dan Harrington who advocates game theoretic approaches in his recent Harrington on Cash Games series, so what I’m saying here is controversial. However, I hope to show by simple examples how the various meta-strategies relate, and that the order of preference I stated is correct.

The Relationship Between Game Theoretic and Exploitative Meta Strategies

Let’s recall a previous fact about game theoretic optimum strategies: they make opponent decisions not matter as long as those decisions are chosen from the head of the rho. Or, to re-phrase the same thing, if you adopt a game theoretic strategy, the only mistake your opponent can make is to choose a strategy outside the head of the rho. Since poker is more or less a “fair” game, if your opponents aren’t making mistakes then you aren’t making money. The question then becomes how often your opponents select strategies outside the head of the rho. I claim it’s not as often as you might like, and I think an example will illustrate this. Consider the bluffing & calling on the end problem. The first player’s two available strategies (bluff & don’t bluff) are both in the head of the rho. The second player to act has three possible strategies – call, fold, and raise. Two of those (call and fold) are in the head of the rho. Raising however is in the tail of the rho – it’s strictly inferior to calling since it will fold out the bluffs (calling would get the same result) and be called by the strong made hands (calling would get a better result). So, what does this have to do with game theoretic vs. exploitative meta-strategies? Quite a bit, actually. From the perspective of the would-be bluffer, if he adopts the game theoretic strategy, the only way he can profit from the remaining play of the hand is if his opponent raises. Otherwise, all he can hope for is to break even in a fundamental-theorem sense. Notice something in this case, however – the play which lies outside the head of the rho is just plain dumb, and obviously so at that. It’s a play that no player who analyzed the situation to any depth would ever make. I believe it is a fact that a large part of what separates beginning from intermediate players is the process of eliminating these tail-of-the-rho plays from their game. What that means is that the game theoretic strategy can only extract additional expected profit from an opponent who’s very unskilled. Against intermediate opponents who won’t make that play, game theory has nothing to offer except the prosepct of breaking even.

There’s a more insidious problem, however – exploitative meta-strategies work even better against the same bad opponents that game theoretic strategies require to work at all. These opponents are the least tricky in terms of mixing up their strategies, and are thus the most easily predicted and exploited. When facing a pure calling station, I say it make no sense whatsoever to occasionally bluff him just because game theory says it’s “optimal” to do so. In terms of real money in your pocket, you’re better off never bluffing him. In other words, the opponents who select tail-of-the-rho strategies are easy to beat with an exploitative strategy.

There is another class of opponents against whom some might claim game theory offers an advantage. These are opponents so skilled at exploitative meta-strategy that if you both adopt it, they will win. Game theory does offer you an option to break even against such opponents. However, I claim that such an option is rarely useful. Most of the time when faced with opponents with superior skills, quitting is the best option. There’s no money to be made playing against such opponents, and I do not believe breaking even against them is a worthwhile goal as I see little value in break-even gambling for its own sake. There’s an even more fundamental issues at stake here: the reason one plays poker in the first place. Personally, I play to use my superior skills against weaker players and take their money. In order to acquire those superior skills at exploitative play, I had to practice at it. If I had always selected game theoretic strategies when the opportunity arose, I would not have gotten that practice, and as a result would not be as skilled against strong players as I now am.

The Relationship Between Game Theoretic and Structural Meta Strategies

This relationship is a bit more murky. If you recall the defending against a draw series, there were basically two routes we could have taken. The first was to select a rate to fold our made hand when the draw hit such that the draw was break-even. The second option was to manipulate the pot size such that the draw was break-even or worse even though we never folded. These approaches were not totally distinct. Game theory was used to inform the structural approach by providing the math to figure out what size pot control should hold the pot to. Similarly, game theory served as a backup to the stuctural play – in the event that pot control failed, folding (possibly randomly) some percentage of the time was the alternative.

The major thing to note here is that the structural strategy was far more balanced than the game theoretic strategy would have been. While both could successfully defend against a draw, the structural strategy was better when dealing with weaker made hands preflop and bluffs. This tendency towards ballance is a nice property of structural strategies.

A note to mathematically advanced readers: yes, I know that the lack of balance in the game the game theoretic strategy was due to the way I computed it. Had I done the computation using ranges of hands and ranges of bets across multiple streets, the game theoretic strategy would have been well balanced as well. However, such realistic game theoretic computations are computationally infeasible, so they’re useful only in theory, not in practice. What I’m really claiming here is that what you can practically achieve via the structural method in terms of balance is superior to what you can practically achieve with game theory.

The Relationship Between Structural and Exploitative Meta Strategies

This is the easiest of the three relationships to understand, and the defending against a draw example will illustrate it. The need for a structural remedy in the form of pot control came from the fact that we couldn’t easily distinguish whether or not our opponent might be representing a set on a bluff. If that problem were removed and we had an effective means of determining if our opponent was bluffing this time, everything would change. If he was bluffing, there would be no need for pot control as we would be more than happy to let him put all his money in the pot. And likewise no pot control would be needed if we knew he was not bluffing – once he repsresented a hand that beat aces, we could fold with total confidence.

In other words, we chose our structural meta-strategy because we lacked an effective explotative strategy. Had one been available, it would clearly be superior to the structural strategy.

Conclusion

Based on the relationships discussed above, I believe that my thesis is unavoidable – the exploitative meta-strategy is superior to the structural, which is in turn superior to the game theoretic. This is not a meta-rho but rather a strict ordering since the exploitative meta-strategy is also superior to the game theoretic. This belief in large part determines my play. I will use game theory as a way to construct structural strategies, or to inform the rough rates at which I will employ exploitative strategies, but I do not randomize at the table and do not believe skilled players should have any need to do so.