This is another interview-style question that seemed like the type I might get after my Mathematical Poker talk back on February 14. Since I haven't been playing poker recently, these questions have provided a good excuse to keep this blog alive for the past several weeks. I hope to address some other poker issues in the coming weeks, but this will probably be the last in this question-answer format unless a commenter has another question worth answering.
The one thing I think would most help my poker income would be to put more effort into game selection. I'm not sure if this qualifies as a "weakness": most players neglect game selection as much as I do, so I'm not actually particularly weak. Doing game selection has some drawbacks, especially in live poker: it is conspicuous to scout out various tables before playing, it cuts into the time you could be playing, and it annoys the floorpeople if you keep changing tables (especially at the Commerce, in my experience). Still, the effect of having a good game can overwhelm any effect of marginal improvements in my own strategy when it comes to my win rate. I state this without data; perhaps I'll find the motivation to do a bit of analysis to back up (or falsify) this idea. Drawing from my memory, however, it really does seem that some tables at a casino can be clearly seen to be better than others, I am pretty confident that selecting a good game is far more important to my winnings than any small fluctuations in my own play or adding a few extra minutes of playing to my session. In a similar vein, I have rarely tried to make stabs at higher limits, despite my belief that this can be a useful endeavor, and I've had several strong players tell me I should be playing higher stakes. I guess I just don't consider myself much of a "gambler" and I would like to keep it that way.
Historically, my approach to game selection has been as follows: spend 20 seconds scanning the room for known fish or for a table seeming to have an unusually good time. (Occasionally, it helps to play with players who are good but who are particularly skilled at making other players play badly. Rarely, it can also help to notice recognize a player I know focuses on game selection, and I can be confident that any game he is in will be a good one.) Then, when I'm in a game, I am rather aggressive about selecting against staying there. That is to say, I will ask for a table change within 5 minutes if the game is not great. (This can annoy the floorpeople and gives your opponents some information about the type of player you are.) To improve my table selection would require that I spend an extra couple of minutes watching a few hands at each potential game until I find one that is quite juicy, and there are usually one or two of these if you look hard enough. If I don't find a juicy game, I should still be able to select one that is at least pretty good.
At the table, my biggest weaknesses are probably concentration and exploiting strong players, which are related. Against weak players, I am perfectly capable of focusing and probing their weaknesses, but I tend to just model strong players as "perfect" and try to play optimally against them in a game-theory sense. This creates a sort of feedback loop of being distracted from the game if I'm up against only solid players, because I don't feel the need to concentrate on them and this creates boredom. In truth, I do think even good players are exploitable, and this becomes especially interesting once they start trying to exploit me; I tend to be a step behind the competition in such situations (not a big deal, since I think my standard game is unusually difficult to exploit in the first place). I think the problem is that finding opportunities to exploit these weaknesses seldom arise, and it takes an awful lot of patience for the effort to pay off. Perhaps other players are fooling themselves into thinking that it's more worthwhile than it truly is, but the bottom line is that most other players seem to have deeper wells of patience and concentration to draw from than I do when it comes to studying their opponents.
Showing posts with label math talk. Show all posts
Showing posts with label math talk. Show all posts
Monday, April 02, 2012
Sunday, March 25, 2012
What are some common weaknesses of your opponents?
This is another question I was NOT asked at my poker talk last month, but that I figured would make for an interesting blog post.
This question is basically the same as the following: what do you do as a poker player that sets you apart from most other players and enables you to win? It's not strictly necessary to be able to answer this question in order to be a +EV player. For example, it could be that your income comes entirely from the 5% of players who are truly terrible, and that against most players you have no real strategic advantage. In practice, though, a winning poker player will try to identify and exploit weaknesses in all his opponents.
Below I list several common weaknesses of players. You will notice a common theme of players being conflicted between the "correct" goal of maximizing EV and some other goal, usually related to psychological satisfaction. I believe one of my primary advantages over other players is that I am not an emotional player. To the extent that my emotions are involved in my play, they are usually tied to my ability to make +EV plays, since it gives me satisfaction to think I am playing well.
The most common weakness of players is to play too many hands. Any player who has heard even a little poker advice knows that they should be playing very few hands. To a new player in a nine-player NLHE game, it is shocking how often they should be folding, and it often takes a while for them to really believe get used to it. Among weaker players, I think another problem is that they have trouble reconciling conflicting motives: they want to make money but they also want to have fun. Folding is not fun, so they play lots of hands even if they know they shouldn't.
Another common weakness I see is that players overestimate their skill level. This often manifests itself in the first weakness I mentioned: playing too many hands. Players think they can make more hands win than they really can. In good players, overconfidence can also result in "Fancy Play Syndrome," which means deviating too much from standard play, usually in an attempt to make a "hero call" or a big bluff. These plays are psychologically extremely rewarding when they work, which suggests that good players, too, succumb to conflicting motives.
A more specific weakness is that players are reluctant to call with hands that probably losing but are +EV because of good pot odds. A classic example is if you hold QQ and you figure your opponents has raised with AA, KK, or AK. You are behind your opponent's range here but you will likely want to call if you are getting good pot odds. (This is one reason why I am a big fan of semi-bluffing with AK before the flop.) Again, I think this can be traced back to a problem of conflicting motives. Good players often care deeply about their reputation, and since it can be embarrassing to call with a hand that is behind, they are reluctant to do so.
Finally, I think good players spend too little effort considering their own hand ranges and too much trying to "outthink" their opponents by thinking one "level" higher. It's much more efficient to just make sure you have balanced your play in a game-theoretical sense. For example, before you check on the river with a weak hand consider your hand range given the play so far. If your hand range contains some hands that you would like to value bet with, it should also contain some hands that you will bluff with. In fact, for those who saw my talk at Swarthmore or who have read The Mathematics of Poker by Chen and Ankenman, you know that there is a specific ratio of the number of hands you should be betting to the number you should be bluffing. Unless you are trying to exploit a suspected weakness of your opponent (for example, you think he calls too much on the river), you should balance your range in agreement with an optimal strategy. (As an aside, I think that this approach is one reason why people's assessment of my play tends to be polarized, with many people thinking my strategy is carefully measured, and other thinking I'm maniacally aggressive because I find opportunities to bluff in situations that most players wouldn't bother. These assessments seem contradictory, but both are essentially correct.) I think this weakness usually stems from the fact that most players simply haven't learned to think about poker in this way. However, for some players it may be that this is yet another example of conflicting motives like we saw in the first few examples. That is, it is certainly fun to try to outsmart your opponents (rather than just balancing your own range), and so it seems likely that players will put a bit more effort into that aspect of the game than is really called for.
This question is basically the same as the following: what do you do as a poker player that sets you apart from most other players and enables you to win? It's not strictly necessary to be able to answer this question in order to be a +EV player. For example, it could be that your income comes entirely from the 5% of players who are truly terrible, and that against most players you have no real strategic advantage. In practice, though, a winning poker player will try to identify and exploit weaknesses in all his opponents.
Below I list several common weaknesses of players. You will notice a common theme of players being conflicted between the "correct" goal of maximizing EV and some other goal, usually related to psychological satisfaction. I believe one of my primary advantages over other players is that I am not an emotional player. To the extent that my emotions are involved in my play, they are usually tied to my ability to make +EV plays, since it gives me satisfaction to think I am playing well.
The most common weakness of players is to play too many hands. Any player who has heard even a little poker advice knows that they should be playing very few hands. To a new player in a nine-player NLHE game, it is shocking how often they should be folding, and it often takes a while for them to really believe get used to it. Among weaker players, I think another problem is that they have trouble reconciling conflicting motives: they want to make money but they also want to have fun. Folding is not fun, so they play lots of hands even if they know they shouldn't.
Another common weakness I see is that players overestimate their skill level. This often manifests itself in the first weakness I mentioned: playing too many hands. Players think they can make more hands win than they really can. In good players, overconfidence can also result in "Fancy Play Syndrome," which means deviating too much from standard play, usually in an attempt to make a "hero call" or a big bluff. These plays are psychologically extremely rewarding when they work, which suggests that good players, too, succumb to conflicting motives.
A more specific weakness is that players are reluctant to call with hands that probably losing but are +EV because of good pot odds. A classic example is if you hold QQ and you figure your opponents has raised with AA, KK, or AK. You are behind your opponent's range here but you will likely want to call if you are getting good pot odds. (This is one reason why I am a big fan of semi-bluffing with AK before the flop.) Again, I think this can be traced back to a problem of conflicting motives. Good players often care deeply about their reputation, and since it can be embarrassing to call with a hand that is behind, they are reluctant to do so.
Finally, I think good players spend too little effort considering their own hand ranges and too much trying to "outthink" their opponents by thinking one "level" higher. It's much more efficient to just make sure you have balanced your play in a game-theoretical sense. For example, before you check on the river with a weak hand consider your hand range given the play so far. If your hand range contains some hands that you would like to value bet with, it should also contain some hands that you will bluff with. In fact, for those who saw my talk at Swarthmore or who have read The Mathematics of Poker by Chen and Ankenman, you know that there is a specific ratio of the number of hands you should be betting to the number you should be bluffing. Unless you are trying to exploit a suspected weakness of your opponent (for example, you think he calls too much on the river), you should balance your range in agreement with an optimal strategy. (As an aside, I think that this approach is one reason why people's assessment of my play tends to be polarized, with many people thinking my strategy is carefully measured, and other thinking I'm maniacally aggressive because I find opportunities to bluff in situations that most players wouldn't bother. These assessments seem contradictory, but both are essentially correct.) I think this weakness usually stems from the fact that most players simply haven't learned to think about poker in this way. However, for some players it may be that this is yet another example of conflicting motives like we saw in the first few examples. That is, it is certainly fun to try to outsmart your opponents (rather than just balancing your own range), and so it seems likely that players will put a bit more effort into that aspect of the game than is really called for.
Wednesday, March 14, 2012
What is one piece of advice you would give to a beginning player?
I've been using this blog to answer questions that I got during the Mathematical Poker talk I gave last month. In the coming week, I'll answer two or three other questions that I did NOT get. They are sort of standard interview-type questions that I thought I might have to answer during the talk, but they didn't come up. I feel like answering them anyway. The first one I'll address is how what advice I would give to beginning players. I am assuming they will be playing no limit holdem.
If I really had to give only one piece of advice, it would be to play as low stakes as possible and fold almost everything as soon as possible. Most new players play way too many hands, most of which they will have no idea what to do with. Even good players would lose if they played that many hands, and new players will probably lose way more with them.
However, the answer really depends on what you hope to get out of poker. The above advice is good if your goal is to play and lose as little money as possible. However, many players will have other goals. For example, if you have played very little poker and your goal is to:
1. Make money. My advice is: don't play poker. You'll probably lose because your opponents are probably better than you.
2. Get better at poker. First, read a lot. If I can give more than one piece of advice: think about your own hand ranges with each action you take. (If your opponent knew your strategy, would he be able to figure out what you have?) Third, keep track of your results and, if you play online, keep track of your statistics.
Fourth, play lots of hands and raise a lot. (This puts you in the action and gives you lots of experience and you will learn how your opponents react to you. The downside is you will probably lose a lot of money in the meantime.) Fifth, play high stakes. (Again, you'll lose a lot, but this is the fastest way to learn how to win at these stakes.)
3. Get better without losing so much. Same as the previous answer, but skip the fifth point above and moderate the fourth point. I would still advise playing hands that yield slightly -EV because this is a good way to gain experience.
4. Have fun but don't lose too much. Think hard to try to figure out what your opponents have and try to exploit them. If this isn't fun for you, you probably do not like poker. Don't try to trick your opponents too often, because this will lose you money. Play very low stakes, fold a lot, and raise a lot. Folding a lot is not fun, but it is absolutely essential to avoid losing a lot. Make your raises bigger than the size of the pot. If this means putting in more than one quarter of your stack or your opponent's stack, just go all-in. This makes it very hard for better players to take advantage of you.
*****
I didn't get into UPenn Wharton statistics or UMD economics. Those, along with JHU biostat, were probably my only chances at an academic career. I'm still waiting to hear back from three other PhD programs that are less competitive. If it doesn't work out, I always have poker to fall back on! Perhaps I could become a professional blogger. (Or I could try to get a late application in somewhere, like GW econ or UMD Law.)
*****
I signed up for a free Game Theory course offered by Stanford.
If I really had to give only one piece of advice, it would be to play as low stakes as possible and fold almost everything as soon as possible. Most new players play way too many hands, most of which they will have no idea what to do with. Even good players would lose if they played that many hands, and new players will probably lose way more with them.
However, the answer really depends on what you hope to get out of poker. The above advice is good if your goal is to play and lose as little money as possible. However, many players will have other goals. For example, if you have played very little poker and your goal is to:
1. Make money. My advice is: don't play poker. You'll probably lose because your opponents are probably better than you.
2. Get better at poker. First, read a lot. If I can give more than one piece of advice: think about your own hand ranges with each action you take. (If your opponent knew your strategy, would he be able to figure out what you have?) Third, keep track of your results and, if you play online, keep track of your statistics.
Fourth, play lots of hands and raise a lot. (This puts you in the action and gives you lots of experience and you will learn how your opponents react to you. The downside is you will probably lose a lot of money in the meantime.) Fifth, play high stakes. (Again, you'll lose a lot, but this is the fastest way to learn how to win at these stakes.)
3. Get better without losing so much. Same as the previous answer, but skip the fifth point above and moderate the fourth point. I would still advise playing hands that yield slightly -EV because this is a good way to gain experience.
4. Have fun but don't lose too much. Think hard to try to figure out what your opponents have and try to exploit them. If this isn't fun for you, you probably do not like poker. Don't try to trick your opponents too often, because this will lose you money. Play very low stakes, fold a lot, and raise a lot. Folding a lot is not fun, but it is absolutely essential to avoid losing a lot. Make your raises bigger than the size of the pot. If this means putting in more than one quarter of your stack or your opponent's stack, just go all-in. This makes it very hard for better players to take advantage of you.
*****
I didn't get into UPenn Wharton statistics or UMD economics. Those, along with JHU biostat, were probably my only chances at an academic career. I'm still waiting to hear back from three other PhD programs that are less competitive. If it doesn't work out, I always have poker to fall back on! Perhaps I could become a professional blogger. (Or I could try to get a late application in somewhere, like GW econ or UMD Law.)
*****
I signed up for a free Game Theory course offered by Stanford.
Thursday, March 08, 2012
Has mathematical analysis changed the game a lot in the past few years?
This is similar to one of the last questions I got after my Mathematical Poker talk last month. Unless a commenter asks another question, I think this will be the last one I answer here on the blog.
The game has indeed changed a lot in the past several years, but it's hard to tell how much of that change is really attributable to mathematical analysis. Certainly, the games have gotten tougher in during the years I played, but I think the main factor may just be natural selection. I would guess that most of the players int the games I was playing in Los Angeles do not know the game theory results I presented in the talk. Ten years ago, I think there really was a Moneymaker-aided poker "craze", which was the inspiration for this blog's name. That brought in a lot of new players, and so games were easy to beat. The ensuing years weeded out the bad players, while the good players were more likely to stick around. This process is pretty hard to disentangle from the fact that the good players were also getting better with practice and study, so in order to guess at the influence of analytics, I need to rely upon conversations I've had with players. My impression from such conversations is that new players tend to have a good sense of the analytical fundamentals of the game, much better than almost anybody had ten years ago. Meanwhile, the older players tend to be mostly those who had a good intuitive feel for the game and were able to survive the natural selection process. Most of the best players today (especially the young players) seem to have a very strong analytical foundation, and that is where I think analytics have caused some of the biggest and most visible changes in poker. In light of that, the continued success of Doyle Brunson is all the more impressive to me.
Another area analytics seems to have made a big difference is in tournaments. I haven't been playing tournaments much, so I can't speak from experience, but tournaments seem more ripe for mathematical analysis. Also, my poker discussions with tournament players gave me the impression that they are more keen on the technical aspects of the game.
Monday, March 05, 2012
Do players at the casinos usually know who the props are?
This was another of the questions from my Swarthmore math talk, which I'm revisiting and answering on the blog.
To varying degrees, all of the props at the casinos in Los Angeles try to blend in with the customers. Some will outright lie to customers about whether they are props, while others might make it obvious without even being asked. In general, none of the props wear any identification, and the casinos operate under the assumption that it's better if customers don't know that there are any props at all. I think we were technically supposed to tell people the truth if we were asked, and I know that the floorpeople needed to inform customers of who the props were if asked. Beyond that, there was a lot of secrecy. Shortly after I started at the Bike, we had a meeting in which one of the supervisors told us that if someone asks us if someone else is a prop, we should just say we don't know. "It's not really lying because for all you know, he quit that morning and isn't a prop anymore." If you are concerned about who the props are, just go ask a floorman.
In practice, the regular players at a casino do know who the props are, but new players usually do not. In fact, the new players usually don't even know that there are any props at all. I didn't know when I started going to casinos.
To varying degrees, all of the props at the casinos in Los Angeles try to blend in with the customers. Some will outright lie to customers about whether they are props, while others might make it obvious without even being asked. In general, none of the props wear any identification, and the casinos operate under the assumption that it's better if customers don't know that there are any props at all. I think we were technically supposed to tell people the truth if we were asked, and I know that the floorpeople needed to inform customers of who the props were if asked. Beyond that, there was a lot of secrecy. Shortly after I started at the Bike, we had a meeting in which one of the supervisors told us that if someone asks us if someone else is a prop, we should just say we don't know. "It's not really lying because for all you know, he quit that morning and isn't a prop anymore." If you are concerned about who the props are, just go ask a floorman.
In practice, the regular players at a casino do know who the props are, but new players usually do not. In fact, the new players usually don't even know that there are any props at all. I didn't know when I started going to casinos.
Friday, March 02, 2012
Why don't I play tournaments?
This is another question I got after my Mathematical Poker talk: Why don't you play tournaments?
I think tournaments are actually full of opportunities to apply mathematical principles. Any situation where your opponents' decisions are relatively unimportant is likely to be solvable mathematically, because you don't need to get bogged down in much game theory. For example, there are situations where you should go all in with any hand, and it is very useful to know when you are in such a situation. I used to play tournaments quite a bit back in Las Vegas. I even had some success. However, I stopped playing nearly as much once I got to Los Angeles.
The first reason I gave during the talk was that tournaments and cash games have slightly different skill sets. I was doing both for a while, but once I was hired by the casino to play cash games for 40 hours a week, I rarely felt like playing more in my free time. After a year or two of that, my tournament skills became very rusty. For example I do not know when exactly I should be pushing all-in with specific hands. Meanwhile, new frontiers seem to have been explored in tournament poker research. The top players seem to have integrated a knowledge of the implications of ICM into their strategies. It seems like it would take quite a bit of work to get back up to speed and make myself a solid winning tournament player again.
The second reason I gave during the talk is that there seems to be a culture of cheating in tournaments, and I think cheating works better in tournaments. Specifically, collusion is more effective in tournaments because knocking another player out helps all the other players left. I actually think I overstated this point, because it is rather hard to ensure you end up at a table with one of your cronies.
In restrospect, I think the main reason I stopped playing so many tournaments was that the fees just seemed too high (I forgot to mention this during the talk). It was harder to find big tournaments in LA, and the smaller tournaments would take as much as 30% fee plus 1-5% for the staff... and then you are supposed to tip, as well. It's tough to overcome that.
Some other factors I neglected to mention:
Tournaments just don't fit into a schedule very well. You might play for five minutes or twelve hours. I don't like playing longer than eight hours at a time. If it seemed like the tournament might take longer than ten hours to win, I wouldn't join it.
I don't like the deal-making at the end of tournaments. This is actually a bigger problem than it seems. When everyone else at a final table wants to make a deal and I am the last holdout, I become a target to get knocked out. This lowers my EV (actually increases the EV of tournament chips, but also increases my volatility, which lowers my payout EV) and is socially uncomfortable, too.
The bigger tournaments, especially the main event of the WSOP probably yield the best win rates. So, these are the most likely to be worthwhile. However, if I get to the money I will be playing for tens of thousands or even millions of dollars. At these levels, my utility curve will really start to bend, meaning my overall expected utility may be less than what $EV would suggest. Although I generally assume that maximizing the EV of my winnings is the "right" way to play poker, in truth this depends on large sample theory and the idea that we are playing for the long run. This breaks down in situations where you might get fewer than ten really big payouts over a career as a tournament player. Furthermore, I just have a constitutional preference for not having to think about money while I'm playing. This is something I could probably overcome if I wanted to.
Very similar to the previous point: Tournament winnings probably have a much higher volatility than cash games. I haven't verified this, but it is conventional wisdom, and it seems very plausible. I think it's true.
Playing online tournaments actually eliminates many of these problems. If you can play a few single table tournaments online at once, you maximize the value of mathematical play, you multiply win rate without increasing volatility, and you probably won't be asked to make deals at the end. This might actually be the ideal way for a mathematician to win at poker. The only problem is figuring out how to get money onto a poker site in America...
*****
Despite being invited to the JHU Biostatistics PhD recruitment weekend, I was not accepted. I'm still waiting on five other programs.
I think tournaments are actually full of opportunities to apply mathematical principles. Any situation where your opponents' decisions are relatively unimportant is likely to be solvable mathematically, because you don't need to get bogged down in much game theory. For example, there are situations where you should go all in with any hand, and it is very useful to know when you are in such a situation. I used to play tournaments quite a bit back in Las Vegas. I even had some success. However, I stopped playing nearly as much once I got to Los Angeles.
The first reason I gave during the talk was that tournaments and cash games have slightly different skill sets. I was doing both for a while, but once I was hired by the casino to play cash games for 40 hours a week, I rarely felt like playing more in my free time. After a year or two of that, my tournament skills became very rusty. For example I do not know when exactly I should be pushing all-in with specific hands. Meanwhile, new frontiers seem to have been explored in tournament poker research. The top players seem to have integrated a knowledge of the implications of ICM into their strategies. It seems like it would take quite a bit of work to get back up to speed and make myself a solid winning tournament player again.
The second reason I gave during the talk is that there seems to be a culture of cheating in tournaments, and I think cheating works better in tournaments. Specifically, collusion is more effective in tournaments because knocking another player out helps all the other players left. I actually think I overstated this point, because it is rather hard to ensure you end up at a table with one of your cronies.
In restrospect, I think the main reason I stopped playing so many tournaments was that the fees just seemed too high (I forgot to mention this during the talk). It was harder to find big tournaments in LA, and the smaller tournaments would take as much as 30% fee plus 1-5% for the staff... and then you are supposed to tip, as well. It's tough to overcome that.
Some other factors I neglected to mention:
Tournaments just don't fit into a schedule very well. You might play for five minutes or twelve hours. I don't like playing longer than eight hours at a time. If it seemed like the tournament might take longer than ten hours to win, I wouldn't join it.
I don't like the deal-making at the end of tournaments. This is actually a bigger problem than it seems. When everyone else at a final table wants to make a deal and I am the last holdout, I become a target to get knocked out. This lowers my EV (actually increases the EV of tournament chips, but also increases my volatility, which lowers my payout EV) and is socially uncomfortable, too.
The bigger tournaments, especially the main event of the WSOP probably yield the best win rates. So, these are the most likely to be worthwhile. However, if I get to the money I will be playing for tens of thousands or even millions of dollars. At these levels, my utility curve will really start to bend, meaning my overall expected utility may be less than what $EV would suggest. Although I generally assume that maximizing the EV of my winnings is the "right" way to play poker, in truth this depends on large sample theory and the idea that we are playing for the long run. This breaks down in situations where you might get fewer than ten really big payouts over a career as a tournament player. Furthermore, I just have a constitutional preference for not having to think about money while I'm playing. This is something I could probably overcome if I wanted to.
Very similar to the previous point: Tournament winnings probably have a much higher volatility than cash games. I haven't verified this, but it is conventional wisdom, and it seems very plausible. I think it's true.
Playing online tournaments actually eliminates many of these problems. If you can play a few single table tournaments online at once, you maximize the value of mathematical play, you multiply win rate without increasing volatility, and you probably won't be asked to make deals at the end. This might actually be the ideal way for a mathematician to win at poker. The only problem is figuring out how to get money onto a poker site in America...
*****
Despite being invited to the JHU Biostatistics PhD recruitment weekend, I was not accepted. I'm still waiting on five other programs.
Sunday, February 26, 2012
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.
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.
Thursday, February 23, 2012
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.
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.
Tuesday, February 21, 2012
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.
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.
Monday, February 20, 2012
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.
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.
Thursday, February 16, 2012
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].
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).
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).
Tuesday, February 14, 2012
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.
Tuesday, January 24, 2012
Poker Talk at Swarthmore
I've been invited by the Swarthmore College math department (from which I graduated) to give a talk about how mathematics is used in poker. The talk will be February 14th and should be added to this page in a few weeks. This colloquium series is open to the public, so feel free to show up if you like. Of course, this is my first attempt at giving such a talk, and in fact it is the first talk of any sort I've given since college, so it might not be very polished. That said, I'm working hard to make it interesting and engaging. I will be drawing inspriation mostly from The Mathematics of Poker by Chen and Ankenman and also Introduction to Probability with Texas Hold'em Examples by Frederic Paik Schoenberg.
*EDIT 5/3/12* Here is the ad for the talk.
I'm planning to argue for focusing on EV for strategic purposes and then do some analysis of abstract simple forms of poker like the [0,1] and AKQ games analyzed in The Mathematics of Poker. The audience will be mathematicians who are not necessarily familiar with poker, so I will discuss pot odds and equity using a [0,1] game example. I will introduce game theory and solve a simple no-fold [0,1] game and a full-street AKQ game that will yield the optimal bluffing to value betting ratio. I will then shift away from game theory and look at an example of using Bayes' rule to model an opponent's strategy. At the end I plan to list several other aspects of poker that can be interesting to analyze mathematically, but I won't do any of this analysis for the talk (unless I am underestimating my timing, which is unlikely... more likely, I'll have to cut out something like the no-fold [0,1] analysis). These other topics include bankroll strategy, game selection, tournament theory, online data mining, and the metagame. Also, I will allude to the sort of structured hand analysis and concept analysis that I have done here on the blog in past years. All this is supposed to fit into 45 minutes, so I will be doing a lot of rehearsing and tweaking in the next few weeks.
*****
I am quoted in the Intro to Probability book by Schoenberg (a friend and commenter on this blog) criticizing Sklanksy's "Fundamental Theorem of Poker." Beyond that, the book's examples are much more interesting to me as a mathematically-inclined serious poker player than I expected. I would recommend the book to anyone who fits that description and also to statistics departments looking to attract more students to their introductory classes. (I can't imagine going through life without a basic understanding of statistical concepts - not the doing of statistics so much as the interpreting of statistics. "Statisteracy" is a term I recently learned and I think for the health of our society it needs to be an educational focus on par with numeracy... although a catchier word than "statisteracy" would be preferable.)
*****
I said earlier that I planned to apply to some statistics Master's programs. Instead, I applied to the following PhD programs:
Johns Hopkins Biostatistics
UPenn Wharton Statistics
George Washington Statistics
UMBC Statistics
U Maryland-College Park Applied Math & Statistics and Scientific Computation
U Maryland-College Park Economics
*****
I have been playing much less poker than I expected. This is largely due to my applications taking longer than expected, but also, the games are tougher than I expected and I've mostly been losing. Last year was my first losing year, but I played less all year than I used to in a normal month. I haven't played at all this year, although I have been invited to a low-stakes home game next month.
*EDIT 5/3/12* Here is the ad for the talk.
I'm planning to argue for focusing on EV for strategic purposes and then do some analysis of abstract simple forms of poker like the [0,1] and AKQ games analyzed in The Mathematics of Poker. The audience will be mathematicians who are not necessarily familiar with poker, so I will discuss pot odds and equity using a [0,1] game example. I will introduce game theory and solve a simple no-fold [0,1] game and a full-street AKQ game that will yield the optimal bluffing to value betting ratio. I will then shift away from game theory and look at an example of using Bayes' rule to model an opponent's strategy. At the end I plan to list several other aspects of poker that can be interesting to analyze mathematically, but I won't do any of this analysis for the talk (unless I am underestimating my timing, which is unlikely... more likely, I'll have to cut out something like the no-fold [0,1] analysis). These other topics include bankroll strategy, game selection, tournament theory, online data mining, and the metagame. Also, I will allude to the sort of structured hand analysis and concept analysis that I have done here on the blog in past years. All this is supposed to fit into 45 minutes, so I will be doing a lot of rehearsing and tweaking in the next few weeks.
*****
I am quoted in the Intro to Probability book by Schoenberg (a friend and commenter on this blog) criticizing Sklanksy's "Fundamental Theorem of Poker." Beyond that, the book's examples are much more interesting to me as a mathematically-inclined serious poker player than I expected. I would recommend the book to anyone who fits that description and also to statistics departments looking to attract more students to their introductory classes. (I can't imagine going through life without a basic understanding of statistical concepts - not the doing of statistics so much as the interpreting of statistics. "Statisteracy" is a term I recently learned and I think for the health of our society it needs to be an educational focus on par with numeracy... although a catchier word than "statisteracy" would be preferable.)
*****
I said earlier that I planned to apply to some statistics Master's programs. Instead, I applied to the following PhD programs:
Johns Hopkins Biostatistics
UPenn Wharton Statistics
George Washington Statistics
UMBC Statistics
U Maryland-College Park Applied Math & Statistics and Scientific Computation
U Maryland-College Park Economics
*****
I have been playing much less poker than I expected. This is largely due to my applications taking longer than expected, but also, the games are tougher than I expected and I've mostly been losing. Last year was my first losing year, but I played less all year than I used to in a normal month. I haven't played at all this year, although I have been invited to a low-stakes home game next month.
Tuesday, June 01, 2010
UCLA Poker Panel and Another Poker Riddle
Last week, as I mentioned, I went to see Chris Ferguson and Bill Chen at UCLA. The event was entertaining, as the panelists recounted anecdotes about Chris and joked about how he well-known he is. I had been hoping it would be more informative, perhaps with some online poker-data analysis or even a mathematical proof or two. However, nothing even approached that level, and it seemed like nobody there had ever even tried looking at online data. There was some discussion of the state of poker game theory and whether poker computer programs would ever be able to beat the best human players. Ferguson and Chen agreed that heads-up limit hold'em might be solved within ten years, and that computer programs might already be as good as people (only in heads-up limit hold'em), but it would be very hard to judge who is "best" unless many hundreds of thousands of hands were played. Chris's dad added that poker in general is unsolvable, since with more than two players, you can expect more than one equilibrium point (a well-known, general result of game theory). The event seemed to be at least partly for promotional purposes, as it was put online and Bill Chen made frequent book references, while the backdrop was initially a Full Tilt advertisement.
At the reception afterward, I didn't see Chris Ferguson, but I spoke a bit with Chris's dad and a bit more with Bill Chen. I asked Bill a little about his book, and jokingly asked when the sequel would be coming out. He said that they might be writing an example book where they look at 100 hands and analyze them with game theory. Sounds interesting. In order to slip away from my group of 3-4 people, Chen left us with the following riddle: if you hold 9333 in Omaha, what are 3 ways you can make the nuts? One way is to have 999 on the board (without a bigger quads or straight flush possible, of course). The other two are harder to see... as a group it took three of us a few minutes to figure them out, but none of us are Omaha players.
Here is the video of the event. I don't recommend viewing it, but I think it's about time for this blog to become a multimedia experience. I didn't watch most of it, but it seems to begin in the middle of Professor Tom Ferguson's tribute to his son.
At the reception afterward, I didn't see Chris Ferguson, but I spoke a bit with Chris's dad and a bit more with Bill Chen. I asked Bill a little about his book, and jokingly asked when the sequel would be coming out. He said that they might be writing an example book where they look at 100 hands and analyze them with game theory. Sounds interesting. In order to slip away from my group of 3-4 people, Chen left us with the following riddle: if you hold 9333 in Omaha, what are 3 ways you can make the nuts? One way is to have 999 on the board (without a bigger quads or straight flush possible, of course). The other two are harder to see... as a group it took three of us a few minutes to figure them out, but none of us are Omaha players.
Here is the video of the event. I don't recommend viewing it, but I think it's about time for this blog to become a multimedia experience. I didn't watch most of it, but it seems to begin in the middle of Professor Tom Ferguson's tribute to his son.
Tuesday, May 18, 2010
Chris Ferguson at UCLA
A friend at UCLA just registered me for an event at UCLA with Chris Ferguson and several other people discussing "Math, Computer Science and Poker." Among the other panelists are Chris's father Tom Ferguson (one of Brigid's professors) and Bill Chen, author of The Mathematics of Poker.
I'm tempted to go back and look at my notes from Mathematics of Poker to try to come up with some intelligent questions. It might be more interesting to try to get opinions on more general poker theory, though. I don't really know if I'll get to ask any questions, but it should be pretty interesting.
I'm tempted to go back and look at my notes from Mathematics of Poker to try to come up with some intelligent questions. It might be more interesting to try to get opinions on more general poker theory, though. I don't really know if I'll get to ask any questions, but it should be pretty interesting.
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