Do you buy in short or deep in cash games?

This is one of the questions I got after the Swarthmore math talk. It came from the student who found this blog before the talk and commented that he had read Chen and Ankenman's The Mathematics of Poker. This is a question I've addressed previously on this blog.

The answer is that I usually buy in relatively short but will sometimes buy in for more if the situation is right. One reason I mentioned is that, by buying in short, you leave the option of buying in for more later. By buying in deep immediately, you pigeon-hole yourself, because you are not allowed to take chips back off the table. I also explained some factors that make me think buying in short can be advantageous. The question was a little outside the scope of the talk, so I first explained the question to the audience; hopefully my readers here will know what buying in short and deep means.

In no-limit holdem, there is a small mathematical advantage to having a shorter stack. This is for two reasons:

Reason 1. Most players with big stacks will not even bother trying to play well against players with short stacks. They maximize their EV by focusing their strategy on playing well against other players with big stacks. This means they will be exposing themselves to being exploited by you if you have a short stack.

Reason 2. If you are all-in against two or more deeper-stacked opponents, one of your opponents might fold a hand that would have won, allowing you to take the pot. Or, you might get to keep a hand that you would have folded if you were not all-in.

This second point was made by Sklansky in his tournament strategy book. Although I originally thought it was a very significant factor, I have come to think it's less important than the first reason I list here. Situations relevant to Reason 1 are fairly common. An example would be if you buy in for $500 and everyone else is playing behind $2k-$5k. They will simply not mind paying you off for $500 if they think they have a chance to win $2k+ from another player. Situations relevant to Reason 2 are comparatively rare. A lot of uncommon factors all have to fall into place. You would need to get all-in with a hand that would have come in second place. The first place player would then need to fold after you are all-in. Even when this does happen, it might actually have hurt you to be all-in, because it's possible you would have won a bigger pot if you had been able to wager more. Because of this it's not entirely clear that being all-in is really mathematically advantageous at all.

During the talk I also mentioned that in my experience, players got used to me playing a short stack, and so when I did have a big stack, they tended to underestimate me. I was able to take advantage of that. (In truth, this probably argues for buying in deep more often, but I didn't bring that up during the talk.)

Against very good opponents I believe it is actually best to buy in short, mostly for Reason 1. Against very weak opponents, it is probably better to have a deep stack, because this allows you to take full advantage of your skill differential. This is especially true if your weaker opponents happen to have deeper stacks than your stronger opponents at a certain table; this is likely to be a temporary situation, but it is usually worth taking advantage of.

Against normal opponents, it's tough to say which is better, but I prefer to buy in short because it allows me the option of staying short stacked if the table becomes tougher.

Frankly, this last consideration is much more important if you are in a situation like I was in as a prop, where you do not always have the option to leave a table if the game is very tough. As a prop, I valued having the option to have a short stack against tough opponents; for most people, the better option is to just leave the game. I didn't think to mention this during the talk, but I think it's an important point.

There is actually a third reason to buy in short that I didn't mention during the talk.

Reason 3. Having a deep stack is stressful and draining. As a prop player, I valued my ability to keep the game as low-stress as possible, because I did not have the option to just leave. It is very difficult to focus intently on poker for eight straight hours; by buying in short, I allowed myself some mental breaks during the day. Most players who are not in the abnormal prop situation should probably buy-in deeper against moderately weak opponents. However, there is something to be said for pacing yourself at the poker table by keeping the stress level low.

How often do you raise?

This was the simplest question I got after my talk. However, I didn't know the answer; I simply don't keep track of my play that closely. I probably should have been more prepared for this question, because the last third of my talk was about how we can use Bayesian statistics to model our opponents' strategies, and I used raising rates as an example. In truth, a player's raising rate is (for most players) highly dependent on the situation. The same exact situation never really presents itself twice (at least if we take into account stack sizes and meta-game issues), so it is difficult to gather relevant data. We need to look at similar situations and hope that they are relevant to each other. Poker players do this intuitively: if player A raises way too often in one situation, we guess that he will raise too often generally. I think a very interesting research topic would be to develop a better model of how to generalize from one situation to another, and how to quantify the "similarity" between two different situations. The best analysis I've seen in this regard simply looks at how often players raise given a particular seat position (eg "on the button"), without regard to other relevant factors such as opponents and stack sizes.

An interesting anecdote is that there were a few players at the Bike's 20-40 limit game who thought that I intentionally raised whenever it was their turn to put in the blind bet, and they would call and reraise me more than they would against other players; in turn, I actually did start raising their blinds slightly more often (that is, with slightly weaker hands than I would normally) in order to maximize my EV against their relatively weak calling and raising ranges.

At the talk, I interpreted this question as referring to raising before the flop in no-limit holdem. Even given this rather narrow interpretation of the question, I considered it too broad to give an accurate answer (I didn't venture a guess at the time, but if I had to now, I would say 15%). Instead, I explained two of the most important factors that go into my raising rate. First, the number of players and my position at the table. If I have only one or two players I need to beat, I will raise with way more hands than if I have seven or eight players I need to beat. In the latter scenario, the chances are just too good that someone else has a hand stronger than mine. This position-based thinking is a basic poker concept that should be familiar to any self-respecting poker player. The other factor I mentioned during the talk is the skill level of my opponents. If my opponents are very weak, I will play a lot more hands. Against stronger opponents, many of those hands would probably yield negative EV if I tried raising with them. One thing I forgot to consider in my answer is the possibility of just calling before the flop. I do this so rarely that in my mind I equated "raising" with "not folding," but I do limp sometimes, especially if there are limpers in front of me and my opponents are weak and are unlikely to raise behind me.

In closing, I admitted that I should probably keep better track of how I play. If I played online I would be more likely to keep track, but in live poker it would be very distracting to record all the relevant data.

Why didn't you play online instead of live?

A commenter suggested I answer this question, which is similar to one I was asked after the talk.

It's true that playing online makes a lot of sense for a professional poker player. You can play from home, you can play way more games per hour, and you can keep track of very useful data on yourself and your opponents much more easily than in the casino. Although the competition is tougher (or so I have heard and believe), if you can beat the games online, it makes way more financial sense to play online than in a casino. If you get good enough at multitabling shorthanded games to win even 25 cents per hand, you can earn more per hour than someone playing live and winning $4 per hand. In addition, it's likely that the ratio of your win rate to your volatility will be relatively low because every $X that you win will come over many more hands, evening out the wins and losses. So, why didn't I play online?

There are three main reasons. All of them have to do with focusing on abstract issues related to my quality of life as opposed to my finances.

First of all, I simply don't enjoy playing online and playing in a casino remains fun. I just prefer being around people to sitting in my house at my computer. I like the structure of going out and avoiding the distractions of being at home. I like the culture of live poker, playing with real cards and real chips. I also like the aspects of pure poker that get obscured by online play, which is actually my next point.

The second reason is that online poker isn't real poker, but merely a good substitute. I want to be able to look at and talk to my opponents. I actually think this is a minor aspect of real poker, but it still matters and it still makes the game more fun and interesting. Perhaps most importantly, I think live poker is more "pure" because there is probably less cheating. It's way too easy to collude in online poker ("hey, I've got pocket aces, can you raise for me now?"). Being able to see your opponents not only makes it harder for them to cheat but also makes them less likely to want to cheat. Several people have admitted to me that they have colluded online. I doubt most of these people would be willing to cheat people in live poker because it just feels so much meaner to cheat a real person than an avatar. All of this is ignoring the "superuser" problem of poker site employees helping their friends cheat by revealing opponents' hands. (Online poker scandals listed here.)  We also have the issue of poker bots, which are probably beatable but certainly diminish the "purity" of the game.

Third, it simply didn't seem worth getting back into online poker after the UIGEA was passed, which was a year after I moved to Vegas. I actually did try to deposit money once or twice since then, but when I realized it would be a complicated process I gave up and never looked back. I also try to be lawful whenever possible (I'm one of the few pro poker players who makes an effort to report his earnings to the IRS accurately and I don't attend obviously illegal home games), so avoiding online poker lets me sidestep the ambiguous legal standing of online poker in America nowadays.

So, basically I had a strong personal preference for live poker, cheating seems more rampant online, and the laws make things complicated.

Would you recommend professional poker or grad school to people graduating with a math major?

I got a question similar to this one at my mathematical poker talk last week: Would you recommend professional poker or grad school to people graduating with a math major?

The answer certainly depends on the individual. Five or ten years ago, poker was much easier and if you liked the game and studied it for a few weeks you could probably beat the players in the casino. The games in the casinos (as well as online) have gotten much tougher since then, and it really takes a lot of work to develop the skill to make over $5/hour now.

Having never been to grad school, it would be difficult for me to say whether it's a good decision. Obviously, I'm planning to quit poker in order to begin grad school, so in my case I decided that grad school is probably the best option for me. However, this doesn't mean that I think I should have gone straight to grad school out of college. If I had done that, I don't think I would have known why I wanted to go to grad school, which might have made it difficult to motivate myself into being a good student.

For me, the desire to go to grad school developed over a number of years of feeling more and more socially irrelevant. Swarthmore encourages social consciousness, and I guess this might have rubbed off on me a bit. Over the years, the feeling that I could be doing something more worthwhile with my analytical abilities has been gnawing at me. As I've become more politically conscious, my desire to try to make a difference has really increased. I'm not sure that grad school is necessarily the ideal way for me to make myself more relevant to society, but I think it is a reasonable enough way to get started on that path.

That's more or less how I answered the question during the talk. Let me now expand on my answer a bit. The lifestyle of a poker player has many benefits that I could document, but the bottom line is that it can be very stressful if you don't have BOTH the right mindset and the ability to win comfortably. It also helps to be financially stable from the outset. I had the good fortune to have no debt and in fact quite a bit of savings when I started. I also happen to have an ideal temperament for poker: I don't get at all upset or angry about bad luck. This is part of what I mean by having the "right mindset." The right mindset also includes focusing on making good decisions as opposed to focusing on your financial results. Basically, having the right mindset involves adjusting your perception of money in order to subordinate its importance to the importance of good strategic play. Even if you have the ability to have right mindset, though, it's not going to work unless you are also a substantial long-run winning player. If you're just barely a +EV player, I think it's going to wear on you. A $1000 loss is going to take an average of 200 hours to make up if you only win $5 an hour. That is likely to be stressful even if you are predisposed to having the mindset to focus on strategy instead of results.

For me, poker was an ideal way to take a step back after college and a short stint at a sort of dreary job and decide how I wanted to proceed with my life. Some people may be able to do this while working for a year or two, and some people may already have it figured out by the time they graduate from college. One big change for me (and it may have something to do with getting married and having a kid) was that I decided it was worth the extra work and stress in order to have a greater impact on society. Different people may come to different conclusions on this point in different points in their lives, but now when I look at people who dedicate their lives to poker, part of me is surprised that they aren't plagued by the same sense of irrelevance that haunted me the past few years.

Question: Is it really optimal to play pure strategies in the [0,1] no-fold one-raise game?

I got this question, or something like it, while presenting the example game theory solution in my mathematical poker talk. The solution to that game is that player X will check-call with hands worse than 4/9, check-raise with hands better than 8/9, and bet the hands in between. If X bets, Y will raise with 2/3 or better and call otherwise. If X checks, Y will bet with 1/3 or better and check otherwise.

This is one of the games in The Mathematics of Poker, except, as with all the [0,1] games in that book, Chen and Ankenman define lower hands to be better, so 0 is the best hand and 1 is the worst. Also, there is a typo in one of the indifference charts for this game, but it does not affect the resultant indifference equation.

The answer I gave during the talk was correct: yes, they do really play pure strategies here. I hadn't reached the point in the talk where I explain pure and mixed strategies, so I used this opportunity to introduce those concepts. A pure strategy would be if the players always play the same way given any particular hand. This is true everywhere except for the indifference points. As an example, I pointed out that in this game, if X held the hand 1/3, he always checks and then calls. An example of a mixed strategy would be if X sometimes check-called when holding 1/3 but sometimes chose another play (such as betting) when holding this hand.

That's what I said during the talk. Let me expand on it a bit.

In Texas Holdem and other common poker games, there are only a few "indifference point" hands where mixed strategies make sense (like X with 4/9 or 8/9). The rest of the possible hands require pure strategies. This is complicated a bit by the fact that real poker games are dynamic, with hands changing value as new cards are dealt, but I think the principle still holds that most hands have one play that has greater EV associated with it than the other available plays. This is why I so often argue against randomizing your play and instead advocate just choosing which hands are your indifference points. This wouldn't have been appropriate to address during the talk, since I didn't want to discuss any real poker games such as Texas Holdem; many people in the audience weren't familiar with the rules of those games.

One thing I wish I did add during the talk was an explanation of why, for this particular [0,1] game, mixed strategies were not appropriate. It would have been very difficult to succinctly prove the point without preparation, but I could have at least elucidated the general principle that was at play: players in a Nash equilibrium will never make a play that has less EV than another available play in any situation, nor will they make a play that has the same EV if that play can then be exploited by an opponent.

With the hand of 7/9, for example, player X has the same EV whether he check-raises or bets. However, this is not his "indifference point" because if he check-raises here instead of betting like he "should" in the Nash equilibrium, he allows player Y to change his strategy and improve his (Y's) EV. This violates the definition of Nash equilibrium (in which no player can change his strategy to improve his EV). I think the more fundamental point that the professor was getting at with this question may have been this: isn't it worth mixing up your strategy with every hand in order to avoid becoming predictable to your opponent? The answer to this is no. I'm not sure why this answer meets with such resistance among poker players (necessitating my repeated refutations), but it really does seem to offend most players' basic idea of what good poker looks like. I guess they think that their opponents will be able to figure out what they are holding if they play their hand the "right" way every time. However, this problem of predictability is completely solved by simply playing other hands the same way. Player Y cannot know which of the hands in the range [0,4/9] and [8/9,1] X has checked with. [Edit.] No need to sacrifice any EV trying to throw your opponent off any further. Sacrificing EV is, by definition, always the wrong play unless you gain it back somehow. In this case you gain nothing back.

Four more points about randomizing your play...

First, I'm not saying that X should always play the hand "1/3" the same way against all opponents, only that he should play it the same way against player Y, who plays optimally. Against other players, X might maximize his EV by betting with the hand 1/3. In any given situation, X should not be randomizing his play between two or more actions; he should be choosing the single action that seems to maximize his EV based on all the information that is available to him.

Second, the very fact that X sometimes plays this hand differently based on his opponents and other situational factors serves as a sort of pseudo-randomizing of X's play. That is to say, to other players, X's strategy will look random even though it's not. X knows the reason he is playing differently, so he is not "randomizing" his play; he is always maximizing his EV for a given situation. X's opponents, unable to completely discern all the factors X has taken into account before making his play, will view his play as being random. Even if you believe you need to make your play look random to your opponents (which I think is wrong), this should already be accomplished because the situational factors are just to complex for all players to evaluate in the same way.

Third, your play will be randomized because you will make mistakes. No need to add extra "intentional mistakes" when you are playing a game that is hard enough already!

Fourth, your opponents are not likely to be paying close enough attention to take advantage of your predictability anyway, and if they try they will likely do it wrong.

*****

I'm writing this from a hotel in Baltimore; tomorrow begins the recruitment weekend for the Johns Hopkins Biostatistics department.

*****

Edit (3/1/12). This originally said "[0, 1/3] and [2/3, 1]". I fixed it to reflect X's actually check-calling and check-raising ranges of [0, 4/9] and [8/9, 1].

Questions from the Swarthmore Math Talk

I've been thinking about some of the questions I got during and after the talk on Tuesday. I'm pleased with the way I responded for the most part - despite nervousness, I was able to think clearly and give what I think were thoughtful answers to each question. I'm going to use some of those questions as fodder for future posts here on the blog. I know that at least one person other than myself thought each of these questions was worth having answered, which is one more than usual for this blog! I'll share what I can remember from what I said during the talk and add anything else I think is interesting.

Some sample questions I remember:

Is it really optimal to play pure strategies in the [0,1] no-fold one raise game?
Would I recommend professional poker or grad school to undergrad math majors?
How often do I raise?
Do I buy in short or deep in cash games?
Why don't I play tournaments?
Do players at casinos usually know who the props are?

I'll also try to answer other questions if anyone asks in the comments section on this post, especially if you were at the talk (but even if you weren't).

Poker Presentation Slides

I just finished the Mathematical Poker presentation for Swarthmore College Mathematics and Statistics department. There were at least sixty people (I'm guessing), it went well, and there were some good questions afterward. Anyway, for those who are interested, here are the slides I used for the talk.