Closer Analysis of Folding into the Money

Last post, I mentioned some lessons learned from my experience of folding nearly every hand toward the end of a tournament in order to back my way into the money. I think this strategy deserves a little closer attention.

First of all, I should point out that this strategy is not even worth considering unless the payout structure is deep. If it is a winner-take-all tournament, for example, it is a complete waste of time.

There are a couple of aspects of the strategy that can be formalized a little further. The important question is what place you can expect to get in the tournament if, at some point, you stop playing all your hands and simply fold everything. This breaks down to two simpler questions: 

1. How many more rounds will I survive by folding?

2. How many of my opponents can I expect to be knocked out in that many rounds? Or, how many more rounds do I need to last until I make the money?

If not for the blinds increasing, the answer to Question 1 would simply be your M-ratio, which I am in the habit of keeping track of, anyway. So, if your M is 9, you would last exactly 9 more rounds. Of course, the blinds keep going up, which complicates things. Now you have two more questions:

1a. How many hands are played per round, on average? For the tournament I have been playing, the answer is about 1, which simplifies things. This is confounded by the fact that with 1-4 tables left, the games are often shorthanded, but you can use "effective M" to account for that. (Using standard M might work fine, too; it will overestimate both how long you will last and how long it will take for others to get knocked out, so the net result should not be too far off the mark.)

1b. How fast do the blinds go up each round? Off the top of my head my guess is that it goes up on average 50% per round, but this also varies across tournaments.

The formula for how many rounds you can last with a given M when the blinds are increasing by 1.5 times each round involves (I think) a logarithm, and so is not conducive to calculation at the table. Instead, let's just look at a chart:

#of rounds     M needed
1                    1
2                    2.5
3                    4.75
4                    8.125
5                    13.1875
6                    20.78125
7                    32.171875

8                    49.2578125
9                    74.88671875
10                  113.330078125


That should be enough to get a sense of how far a given M will get you. Now we need to address Question 2: How many people can we expect to get knocked out in that many rounds?

In order to answer this question, it would be helpful to know the exact distribution of M's of all the players left in the tournament, but we can content ourselves with calculating their average M. Often, the average stack size is listed on the tournament TV's, making this an easy calculation. Three other complicating factors could be relevant:

A: Late in the tournament, the stack sizes are relatively more widely distributed than earlier (after all, the variance in stack size begins at 0 and can only go up from there!). With more variability in stack sizes, there will be more players with short stacks about to go out.

B: As the bubble approaches, more players will tighten up in order to survive to the money.

C: The number of players at each table can vary quite a bit once there are fewer than 45 players left.

Factor A can probably be safely ignored, but B and C are going to be a problem. Rather than try to calculate bounds and approximate the right answer, I think the best approach here is to simply guess. I will try to update this guess with some hard data from real tournaments in the coming weeks. What would be really nice would be to get a whole probability distribution so we could say something like "with my current M, I have a 95% chance of coming in at least 7th if I fold every hand from now on," but that is probably beyond what I'm willing to do here.

In any case, first we want to start with a guess. What is a good guess at the percentage of the field that will be eliminated in X rounds for a given average M when approaching the bubble?

Let's look at a few scenarios along with my wild guesses.

X    MAVE     guess at %age knocked out
1     1           70
1     3           40

1     5           28
1     10         15

2     1           90
2     3           80

2     5           60
2     10         35
2     15         22
3     3           90

3     5           75
3     10         60
3     15         40

5     15         90

5     20         77
5     30         55








Ok, I'll leave it at that for now. If this still seems worthwhile in a couple of weeks, I'll try to update with some real data and maybe make some charts and graphs.

*****

By the way, I forgot to mention that after I was knocked out and took the $350 bubble money, the rest of the table seemed to agree to chop the rest of the winnings for about $2300, each. The casino does not officially condone that, so the players were going to have to "play out" the remaining hands until there was a winner and then trust each other to hand over the money. Plus, one player agreed to "come in first," which involves having taxes applied. I left before this happened, but I wonder how it all worked out.

Bubble Cash and Fold-Strategy Calibration

In my previous two posts, I discussed my second-place tournament finish, the deals that were offered during that tournament, and (to some extent) the strategic adjustments I intended to make. Due to illness and scheduling conflicts, I was only able to play two tournaments in January, and I bubbled both of them. Fortunately, I still netted $50 because I won three $25 bounties in a $125 tournament and a $350 "bubble offering" in a $250 tournament.

The $250 tournament has a much more severe bubble. (With 76 entries, seventh place gets $980 compared to only $1750 for third place. Thus, it's hardly worth risking going out 9th or 10th unless you have a great shot of at least 2nd place, which pays over $3k.) This, combined with the somewhat reckless strategy of the other players, makes it worth playing very tight once you have enough chips to cruise into 7th place. That's exactly what I tried to do this past Saturday. However, I initiated my "outlast through super-tight play" strategy a bit too early, and came in 8th place instead of 7th. Fortunately, I also knew that it is common practice for players at the final table to create a small kitty to be given to whomever goes out on the bubble. This creates a sort of safety net for the "outlast"strategy, and I was able to come away with $50 from each of the seven players who cashed.

I initiated my super-tight strategy sort of accidentally. There were under 30 players left and I had 50k or 60k in chips. I really wish I had taken note of the specifics so that I would have a better idea of how to modify my strategy in the future, but the blinds were already up to 1500-3000 and the ante was something like 300. With about eight players at my table, I was second after the blind and limped with AJs after another limper. I had already decided to tighten up a bit; otherwise, I would probably have raised with this hand. The player to my left raised to 17k and everyone else folded. I took a long time considering pushing this hand all-in. My opponent's raise could have been a squeeze play, so I think my fold equity would have been good, and AJs fairs decently against his calling range if I push. In tournament chips, I think my EV is positive here if I push all-in, but in real money, I decided it wasn't worth risking what was very likely going to be at least a $980 payday. I folded and decided to play only the very strongest hands from then on.

I got only one more playable hand until the final table. It was A8o on the small blind that I pushed in and won the (substantial) blinds and antes, when there were about 16 players left. I folded literally every one of my other hands until my very last hand of the tournament (55 under the gun when I had 8.5K and would have had to pay a 1K ante next hand plus a 6K big blind). I was prepared to play my strongest hands until the final table, but no strong hands came. At the final table I was prepared to fold KK pre-flop, but I probably would have pushed with AA. The best hand I got at the final table was 88, and I folded it.

Although I didn't intend to adopt an "always fold" strategy when I was at 50K and about 26 players left, the weakness of my cards made that the strategy that I nearly de facto employed. If not for that one hand where I won about 9k with A8o, I might not have even gotten the $350 with that strategy. This provides a useful lower bound to refer to when employing an "outlast through super tight play" strategy. I can now say with confidence that, in this tournament with 76 entries, 50K is not enough to safely get me into the money when the blinds are at 1500-3000 and there are about 28 players left.

I can simplify this a little further. Since the 76 entries (some of them were re-entries, but that is not really relevant) each started with 15,000 in chips, that makes 1,140,000 total chips, and my 50K represented less than 1/20 of the chips in play.

This analysis is far from complete, but I find in instructive to examine extreme scenarios when they come up. This does not thoroughly point to what strategy would be optimal, of course; my sense is that it would be best to play somewhat tight in the scenario I described above. However, I have learned an approximate boundary for when an always-fold strategy goes from +EV to -EV, and I can incorporate that into my decision-making and intuition.

Tournament deal analysis

Although it's not officially condoned by the casino, deal-making is pervasive in tournaments, and when I decided to start playing tournaments I knew that I would need to improve my understanding of this aspect of the game. As I've mentioned before, I think optimal deal-making is best viewed as an extension of sound poker strategy, even though I was originally reluctant to consider making deals. For this post, I'll quickly discuss the three deals that were under consideration in the tournament last week when I came in second place.

The Bubble Offer: The first deal offered came when we got down to eight players. The seventh place player would win nearly $1000, but eighth place gets nothing, so this is a rather severe bubble. It was suggested that we should all give $30 to whomever was knocked out next, essentially giving eighth place have a payout of $210.

A severe bubble like this creates a strategic advantage for the players with the biggest stacks. If the smaller stacks are playing optimally and mostly folding and waiting until one more player gets knocked out, the chip leaders can win all the blinds and antes hand after hand by simply putting everyone else's tournament lives at risk. As I said in my last post, the other players were playing far too recklessly, and so this effect was greatly diminished. However, the larger stacks still have some advantage to the extent that the other players realize they should be playing extra tight.

When the deal was offered, I was at or near the chip lead, and I refused the $30 bubble idea. My explanation to the other players was that, as the chip leader, I was clearly not going to be the next one knocked out, so it would make no sense for me to donate $30. I knew this made me look cheap and unsporting, but I didn't want to explain to my opponents the more subtle reason for my refusal: I didn't want to give up my strategic advantage against everyone as the table's chip leader. If the bubble gets $210 instead of $0, it's psychologically much easier to deal with, and so my opponents would not be as desperate to hang on to get into the money. In other words, I would not be able to steal the blinds and antes as easily. The other players, muttering about how cheap I was, agreed to make the deal without me. This didn't help my situation much (the bubble player would still be getting $180), so I objected that this was collusion against me. The floorman agreed. The players pointed out that there was nothing I could do to stop them. Indeed, I knocked out the next player before long and everyone gave the guy some money (I think one guy gave him $60 to cover my part). This is exactly the sort of awkward situation that made me dislike tournaments in the first place, as I mentioned in an earlier post about why I didn't play tournaments.

According to the other players, this $30 bubble deal is commonplace. I think I will continue to refuse it if I'm near the chip lead, but accept it otherwise. I wonder how long the other players will tolerate that!

Another option is to try to influence tournament policy to make something like this deal "official." If the eighth place player had been officially lined up to receive $400 of something, I could avoid the unpleasantness of needing to discuss under-the-table deals with poker players. Considering the strategy of trying to make more final tables but with smaller chip stacks that I discussed in my previous post, this extra payout place would also be advantageous to my EV if it were officially adopted. I think if I discussed the situation I experienced where the players all colluded against me (as the floorman agreed), the tournament administrators would consider making this change.

The proportional chop offer: The first deal of the traditional "chop" variety (where the rest of the money is split and the tournament is over) that was suggested was a proportional chop. This means that each player would get a proportion of the pot equal to the proportion of the remaining chips. I was well aware that this clearly favored the bigger stacks. In fact, it's easy to imagine the smallest stack(s) getting paid less than last place money and the biggest stack(s) getting paid more than first place money. It was not clear to me how exactly this would play out, but I was not about to be the lone holdout on this deal. At this point I had about 400K of the 1M chips in play, and there was about $14,000 left in the pot. This deal would have paid me about $5600, not far from the $6300 or so that was awarded to first place. I wasn't sure what my EV would be if I kept playing, but it seemed like it must be less than that. I simply started counting my chips and hoping everyone would go through with it. After about two minutes, one of the short stacks decided to back out, so the deal was off. (The defector happened to be one of the two people most upset with me for refusing to take the earlier Bubble Deal.) We continued playing.

I used an online calculator (which I don't trust because I found a logical inconsistency that suggested an EV of greater than first place money for someone holding 90% of the chips) that suggested my actual EV was $4340 at this point, assuming all players are equally skilled. If this were accurate, I guess my EV was closer to $4600 because I was almost certainly better than the average player at the table.

As I said before, the casino does not officially condone deals, although they do look the other way. This means that in order to end the tournament and make a deal, the players would have all had to go all-in blindly and have the dealer deal out hands until just one person was left. The casino would then hand out the official winnings to everyone based on where we were each knocked out, and it would be up to us to dole out the agreed-upon amounts. This requires a level of trust in other players that I'm not completely comfortable with. I do think it's pretty unlikely someone would have the guts to back out on a deal like this, but something similar did happen at the Bicycle Casino once (although not in a tournament*). In this case, whoever came in first would also have to fill out a tax form because it is IRS policy that anyone who wins over $5k needs to do that. I pay my taxes anyway, but this is still another unwanted complication. I'm also not entirely sure who would be doing the calculations. Together, these factors diminish the value of the chop somewhat - perhaps as much as 20% should be discounted from the EV of a chop to account for these concerns, but 10% seems more reasonable. Still, 80% of $5600 is still $4480, which is probably greater than the EV I would have by playing the tournament out. (I'm assuming the poker calculator was overestimating my EV as the chip leader. Perhaps my EV was actually around $4000)

Making the deal also has the added benefit of greatly lowering the volatility of our winnings. If we discount by 10% instead of 20%, it comes to $5040, so I do think the deal would have been well worth it considering all the above factors. As I will discuss below, however, there are still other factors yet to be considered. All told, I think the deal was worth it for me, but the fact that someone backed out was not the disaster for me that it seemed like at the time.

The First Place offer: After we had been playing with three players for about ten minutes, I had around 60% of the chips, and my younger opponent suggested a chop wherein I would take first place money (around $6300), and he and my older opponent would share second and third place (around $2400 each). I could hardly believe this offer - I couldn't lose! - but the older player did not seem at all interested, barely acknowledging that an offer had been made. We continued playing.

In retrospect, I think this older player, who later informed me he had trouble hearing, probably did not even realize a deal had been offered. The lesson I learned here is to be a little more assertive and persistent in pursuing lucrative deals. If this third player had still demurred after the offer was explained to him, as I expect he would have, I should have offered him $100 to take the deal, and been willing to go up to $500. I probably lost quite a bit of value with my passivity.

*****

I mentioned that there were some factors worth considering that I skipped over in the above analysis. As a poker player, I have many objectives beyond just trying to make optimal strategic decisions by maximizing EV. When it comes to making deals to split the winnings at the end of tournaments, one objective in particular competes with my primary goal of maximizing profit: I would also really like to play the tournament out to the end because that would allow me to improve through experience. I learned quite a bit about my competitors at the Charles Town casino by playing out the three player and two player scenarios in the hour or two after The First Place Offer was passed over. In order to effectively balance these two considerations (immediate EV versus gaining valuable experience that will help my EV in the future), it would be helpful to put a monetary value on the otherwise qualitative value of this experience. I will explore the question of how to do this in a future post. I expect it will involve discounting the value of my future income, which is a standard practice in economic theory.

* One Bicycle Casino promotion was to give away about twelve keys to a car over the course of a day, eleven of which were fake. At the end of the day, all the keys would be tried and one of them would work and that person would win the car. The twelve people agreed that whoever won the car would give the others some amount of money, maybe $1000 each. The lady who won the car backed out.

Comeback Update: First Tournament Cash

In my sixth tournament since deciding in October to return to poker on a part-time basis, I finally made my first cash (aside from $225 in bounties from a couple of bounty tournaments).

Yesterday, I came in second in a $250 re-entry tournament at Hollywood Casino in Charles Town. There were seventy entries. I made only a single entry (I doubled up early) and was paid about $3150, for a net of about $2900. First place paid about twice that, and I very nearly won. We played for quite a while with three players and then also heads-up. At one point I had about 80% of the total chips. At two points, we nearly made deals that would have been extremely advantageous to me, but both times we had one holdout. The winner was a large older guy with a pretty thick accent originating, I guess, from West Virginia. The tournament lasted from noon to 8:15; after the tournament was over, the winner made a late entry into the 7:00 tournament, and was just sitting down as I was heading for the parking garage!

This was my third time playing this tournament. It starts with blinds of 25-50 and each player gets 15,000 in chips, which is a good amount (the more chips, the more chance for "poker skill" to come into play - although I think my clearest advantage over my competition is due to their poor short-stack strategy). Like most tournaments, the blinds go up quite quickly, though, so it's not long before the skill factor starts to diminish.

Very early in the tournament, I had top pair and called a skilled player's multi-street bluff to go up to about 20K. A little later I made a silly mistake that ended up confusing my opponent and earning me a big pot. With pocket tens, I flopped a set on a board of KJT. My opponent bet small and I tried to raise to 1600. but I accidentally put in a 5K chip instead of the 1K chip, so my bet was 5600 (into a pot of around 2000). I knew I couldn't take it back, so the best thing to do was play it off as if I meant to do that. This way, I get the benefit of giving my opponent a wrong impression of my hand, plus (perhaps more importantly), a big bet like that is likely to be noticed by the rest of the table, giving them all a false impression of my playing style. The enormous bet was a mistake, but it has lots of pleasant side effects if I don't let on that I didn't do it on purpose. My opponent had about 12,500 left and called the bet.

A small card came on the turn. We both checked. If my opponent had a straight draw, I wanted to give him a chance to bluff me on the river after he missed his straight. The river was another King. My opponent checked and I put him all in for his last 7500. He made what I think was a very bad call with Kx and I was suddently up to about 35K in chips. Another call of a multi-round bluff (this time I had AK high) put me up over 40K. I had quite a few lucky hands (eg someone called a big semibluff and I made my straight) after that and cruised into the final table with about 180,000 in chips. There were 1,050,000 total chips in play, so I had about 17% of the chips at that point. I shared the chip lead with one other player.

Third place paid around $1700 and seventh place paid almost $1000. This means that the correct strategy for a short-stacked player at the final table is to try to hang on for dear life in order to get at least that $1000 prize for seventh place. Players certainly adjusted their style for this consideration, but in my opinion they did not adjust it nearly enough. Players were still far too reluctant to fold hands that they knew were strong pre-flop, even though "strong" can just mean "60% chance of winning." It's not worth a 40% chance of getting knocked out just for a 60% chance of doubling your stack at that point in the tournament. That the players almost all made this mistake is important information to keep in mind for the next tournament, because it has significant strategic implications: getting to the final table with a short stack has a higher than normal EV, because they players in this tournament will make it unusually easy for me to simply fold all my hands and get into the money. Had I known my opponents would have this bias against folding on the bubble, my strategy yesterday would have been quite different when I had about 150K chips with twenty players left. At that point, the correct strategy would have been to start playing extremely conservatively and cruised into the money. I was fortunate to build up my chips a little bit more before the final table by playing a few more hands, but in retrospect it was not worth the risk of getting knocked out when I had such an easy $1k win waiting for me.

The first deal offered was a "proportional" payout deal that was offered when I was up to about 400k in chips and there were six players left. I asked what this meant and was told that whatever proportion of the total chips we had in our stack would be the proportion of payout pool we would each get. This meant I would get about 40% of $14,000, or $5600, which is almost first place money. I didn't bother to calculate this at the table, but I already knew from some casual analysis that such a deal would greatly favor those who had the most chips, so I quickly agreed, as did everyone else. We had stopped playing and were all counting our chips before someone finally backed out and we continued playing.

The mistake I noticed players making of being too reluctant to fold their strong hands in order to move up in payment was even more acute when it got down to three players. The difference between second and third place was $1400. The older guy who eventually won the tournament had complete disregard for the strategic implications of this and played as if second and third place were equivalent and all that mattered was coming in first. Seeing this, the younger third player should not have played any hands until I had knocked out the older guy. Instead, the third place player only tightened up slightly and I eventually knocked him out. At one point, the older guy caught up to me in chips after I made a call getting 3 to 1 odds on what I thought was about a 50% shot. I should probably have folded, but I made the mistake in failing to tighten up my game enough. I should have just waited until the younger guy was knocked out before playing any more hands. To be fair to myself, I had a lot of things to keep track of after making it so deep into a tournament for the first time in a few years. Fortunately, it worked out okay, and now I know one extra thing I can focus on improving.

When we got down to three players was also when the second deal was offered. I had a big chip lead (I was at about 600k I think), and the younger player offered to give me first place money and split the second and third place money with the other guy. Obviously, this would have been a fabulous deal for me, but the older guy didn't seem interested. I probably should have tried negotiating around that incredible offer, but the older guy was a bit hard to communicate with and seemed entirely uninterested in what we were talking about. In retrospect, I think he might literally not have heard that there was an offer being made at all, and I may have cost myself quite a bit of money by not making certain that he understood the offer. One reason was that the casino did not officially sanction the deals, and the dealer would not stop dealing to allow us to discuss it. When the earlier offer had been made, we had all stopped playing, but this time the older player immediately kept playing and I took that as a cue to drop the subject.

By the time the tournament ended, the older guy and I had been playing heads up for almost an hour and the blinds were 25k and 50k with 5k antes. His style was pretty conservative pre-flop, rarely raising and sometimes even folding on the small blind. He would bet most flops, however. I was up about 800k to 200k at one point and had him all-in, but I lost on a 60% shot. On the hand that crippled me (bringing me down to 35k), I had 66 and my opponent raised to 150. He had been betting most flops, so I figured I would just call him pre-flop and let him bluff the flop. The flop came QQ4 and my opponent checked out of turn. (He often acted out of turn, blaming his hearing. He never took back an action that was made out-of-turn, so I'm inclined to believe he was not being manipulative.) This is an excellent flop for 66, and I did not want to give him a chance to hit a pair on one of what were likely 6 outs for him on the turn, so I bet out 140. He pushed all-in for his last 200 or so. I think his most likely hand here is Ace-high, but it was suspicious that he was planning to check the flop but then decided to raise when I bet. In any case I was getting about 4-1 odds and called. He had Q8 suited; I guess he had been planning either to check-raise (if he thought he was first to act) or slow play until the turn. Anyway, no 6 came. I got may stack back up to 140K, but then lost with K5 when he paired his 4 with a K4.

Certainly, this is a nice and encouraging result for me, but I am also very disappointed not to have won. Also, I got very lucky to get to the final table. However, I am happy to be solidly in the black for my new tournament career. Most importantly, I learned quite a bit about my weaknesses, and I'm eager to return to the felt to try to exploit the weaknesses I saw in many of my opponents. These are exactly the sort of motivations and progressions that inspired my professional poker career when I started seven and a half years ago. I don't have as much time to pursue that passion as a did back then, but it's an exciting development. I now think that playing in the 2013 WSOP is a real possibility.

*****

I got a remarkable comment to my recent post about Poker Strategy by Nesmith Ankeny. The comment was really quite stunning to me because I had never heard of this book before I got it from the library and I certainly had no inkling that anyone would feel passionately about it. My impression was that this was an entirely overlooked book in the poker world, but then someone showed up out of the blue on my little-known blog just to defend the book's honor. Just now I looked at the book review section of Gambling Theory and Other Topics by Mason Malmuth, and indeed Malmuth reviewed the book briefly and gave it a 9/10. I did not remember it among the 124 he reviews there. Here is the entirety of Malmuth's review (p 348):

90. Poker Strategy, Winning With Game Theory (9) by Nesmith Ankeny.  This is an excellent book on draw poker based on a game-theory approach. Many strategic concepts are discussed, and I think this book is must reading for serious players.

I wish I had taken this seriously back in 2005 when I got Malmuth's book. Ankeny's book is now a little redundant if you read The Mathematics of Poker by Chen and Ankenman, but back in 2005 I think the concepts in Ankeny's book could have given me a head start of a few years in terms of how I now think about poker. I think I will search Malmuth's reviews for more books that he deems "must reading for serious players." Come to think of it, I think Malmuth was still doing reviews on the 2+2 website as of a few years ago, so I should check over there, too. I'll let you know what I find.

"Poker Strategy" by Nesmith C. Ankeny and Transferring Poker Skills Across Varieties

There is a wide range of skills that are useful regardless of what variety of poker you want to play, but for any particular game certain skills could be much more relevant than others. A classic example is the ability to successfully determine when a player is bluffing, which is useful in limit holdem but much more important in no-limit holdem. Employing squeeze plays, estimating implied odds, raising effectively for secondary reasons (and recognizing when you opponents might be doing this), and semi-bluffing are a just a few of the many other skills that differ in importance from one poker variety to another.

Sometimes, if I only play one type of poker for any stretch of time, I will completely neglect the skills that are less important for that game. By playing or studying other varieties of poker (or toy games), I often manage to remind myself of some of these other considerations. Not only does this provide an opportunity to make subtle improvements in my primary poker variety, but, for me, it's also a good way to rekindle some excitement about the game by opening up a new direction to explore. This was the main benefit I found to reading a book analyzing Jacks-or-Better poker. A side benefit was that I gained some historical perspective on the history of analyzing poker using game theory.

The book I finished reading was Poker Strategy by Nesmith C. Ankeny. Well, I skimmed it, anyway. That is, I skimmed the parts I didn't skip over. In some ways, the book is horribly dated - imagine if someone today tried giving an unspecific title like "Poker Strategy" to a book theoretically analyzing Jacks-or-better draw poker, which nobody even plays anymore - but much of it was timeless. The game theory analysis looked accurate and went deep enough to provide specific results related to draw poker that I imagine would be extremely useful if anyone still played it.

As I alluded to, there was one skill in particular that was clearly of utmost importance in Jacks-or-Better that is not as central to no-limit holdem: the ability to pin down exactly what sort of hand your opponents are looking for. Given a player's bets before the draw and the number of cards he draws, it is possible to significantly narrow down his range of hands with a high degree of confidence. In particular, you can usually tell if he has a flush draw as opposed to a made hand (or at least a bluff of a made hand). In holdem, it is much easier to disguise flush draws as made hands, and so figuring the probability your opponent holds one type of hand or the other is not nearly as determinative to holdem strategy. (It is relatively useful in stud games, however, and this was my most glaring weakness when I played that form of poker.) As a result, I have not spent much time thinking through the intricate tactics of playing when you have various opponents whose hand types are reasonably well known.

Let me give you an example. In any form of poker, if you strongly suspect your opponent either has a very strong hand or nothing (a drawing hand), it is usually correct to check to him, assuming your hand strength falls in between those extremes. Now suppose you have two opponents, the first of whom likely has a moderately strong hand and the second of whom has one of these drawing hands. This is an interesting situation that I hadn't considered carefully before because it simply doesn't conform to the way I usually think about no-limit holdem, my preferred form of poker. However, Ankeny has a nice, thorough analysis of this situation in his book. It turns out that this situation presents a strong bluffing opportunity if you are betting first, because the player with the moderately strong hand will find it very difficult to call, while the player drawing to the flush or straight will usually miss. I don't know if this situation will come up in any future game I play, but I might find myself in a similar situation, and at the very least it got me to think about the game in a way I might not have otherwise.

Much in the same way the Chen and Ankenman's Mathematics of Poker was useful to me even though it mostly analyzed toy games, the fact that Ankeny's analysis in Poker Strategy was sound and thorough made it somewhat worthwhile even though it did not cover any of the specific games I play.

Should poker players be willing to play in short-handed games?

Absolutely. Not only should you be willing to play short handed, but short-handed games are actually preferable to full-ring games.

Many players refuse to play unless a game is nearly full. Still more will play if there are five or six players but not if there are fewer than that. Sometimes, while sitting with three or four other players who are unwilling to play until more players arrive, I ask them why they don't like to play shorthanded. One such player told me that short-handed games are too expensive because the blinds come around too often. Sometimes it's simply that they don't want to play shorthanded in a game with me in it. Some players come for the social atmosphere and find that the competition gets too personal with fewer players. Other players seem to just like to sit around folding most of the time while waiting for AA or KK. Still other players probably have other concerns.

So, why do I think short-handed games are not only worthwhile but are actually better for poker players? Let's look at the three main concerns of a poker player: winnings, social experience, and game purity.

Effect on winnings: The old adage that "you need to play more hands when you play short-handed" is overly vague. The truth is that there is a much better way to transfer your strategy from a full-ring game to a short-handed game: When playing X-handed, play exactly as you would play in the same position relative to the button in a 9 handed game after the first (9-X) players fold. As I will now demonstrate, the two scenarios are very similar.

Suppose you are playing in the cutoff or button of a 9-handed game and the first five players after the big blind fold before the flop. If you have an idea of how to play in this situation, then you can play in a 4-handed game. There are only three very minor differences in the two situations. First, the 4-handed scenario would move faster because the deal would be quicker and you wouldn't need to wait for the first five players to fold. Second, the hand in the 4-handed game would be cheaper in most casinos because in most casinos the short-handed game would have a smaller rake and maybe no jackpot drop. (Both scenarios have four players left, so you are about equally likely to win the hand and end up paying the drop.) Third, in the four-handed game, you wouldn't need to worry about any bunching effects, and so the game is slightly easier if you are trying to combinatorially guess your opponents' hands. (As we will see presently, this is probably a negligible concern anyway.) So, the game is faster, cheaper, and easier. Faster and cheaper are clear advantages. Easier is a wash because it applies to your opponents as well as to you.

So, the strategy in a short-handed game has a nearly direct parallel to a situation most full-ring players are already familiar with: the situation where the first several players fold. Consider the implications of this. The first five hands after the blind are the least-often-played hands. They are the spots where you are most often sitting and doing nothing except waiting for your opponents to finish playing (and maybe gathering information to use against them). If you could just eliminate these five spots, you would vastly increase the rate at which you would have decisions to make. That is exactly what happens when you play short-handed. Winning poker players make their money by making better decisions than their opponents, so having more decisions in any given period of time translates directly to a higher win rate.

In addition, short-handed games require players to quickly gain a good sense of their opponent's strategies, because they will be going up against the same opponents in each hand. Also, the nature of short handed games is that they tend to fluctuate in size, requiring an ability to adjust to changing circumstances. These are skills that good players can put to better use in short-handed games than full-ring games.

I anticipate two main objections to my argument that short-handed games are better for your win rate. First, the situations I've described are not exactly the same. After all, in a 9-handed game where the first five players have folded, there is a "bunching" effect: because folded hands are slightly more likely to have contained low cards, the remaining cards are slightly more likely to be high (and thus strong). Just for a quick example, suppose the first five players will each play if and only if they are dealt an ace. Suppose they all fold this hand. If you are in the cutoff and also don't have an ace, then all four aces are still left and there are only 40 cards left (52 minus 12). For each of your opponents, the chance of having pocket aces is now 1 in 130 (4/40 times 3/39), much better than the usual 1 in 221. However, this is an extreme example. In practice, players do play some low cards and they do fold with hands like A2 (most players, anyway). The "bunching" effect is just not all that big a factor. Maybe I'll do a deeper analysis of this later. For now, I'll just say that I don't believe this is a big enough effect to contradict my claim that you can view them as being essentially the same. I concede that it makes sense to be tiny bit more wary of your opponents in the 9-handed game after five folds than in the 4-handed scenario, but making a big adjustment would be a mistake. You would be safe if you made no adjustment at all.

Second, you may argue something like:

Sure, those scenarios are essentially the same, but you are overlooking some real advantages to having those early-position hands that exist only in full-ring games. These afford me important opportunities to apply my superior poker skills. For one thing, I can use the time after I fold these hands to watch my opponents play. Since I'm more observant than my opponents and since I'm better able to apply new information, this represents a real advantage to me. Even ignoring this, my basic strategy in these EP hands is far superior to some of my opponents. The biggest weakness of many of my opponents is that they play too loosely, and this weakness is most pronounced in EP. Why remove these opportunities for my opponents to play badly?

These are reasonably strong arguments. It's true that playing short-handed eliminates some opportunities for strategic advantage, and for a thorough analysis, I would need to consider the value of these strategic advantages. For now, let me just give an intuitive explanation for why I don't think these arguments are strong enough contradict my claim that short-handed games are better. Yes, we are giving up some opportunities to gain value over our opponents through superior use of observation and superior play in EP. However, there is an opportunity cost to these advantages. For the benefit of getting to sit and observe your opponents, you are losing the opportunity to actually be playing more hands and applying your strategic advantages; if you really want extra time to observe your opponents, you can achieve that in a short-handed game by simply sitting out a few rounds and watching. As for your advantage over your opponents who play terribly in EP, this comes at the cost of the advantage you have over your opponents' LP and blinds strategy. Every hand we would get to play in EP is replaced in short-handed games by extra hands in LP and the blinds. Since these are hands that you are much more likely to be playing (rather than folding), it seems to me that your advantage over your opponents here is likely even greater than the advantage you have over your opponents in early position. I agree that it is much easier for your opponents to play terribly in EP, but it is also much easier for you to play great in LP and the blinds, because they often involve lots of decision making deep into hands. Playing "great" in EP basically just means having the discipline to fold a lot.

There are some rational explanations for why a winning player would have a higher EV in full-ring games. Perhaps they really aren't good at playing in the cutoff and on the button compared to their proficiency at playing in early position. Perhaps they thrive on post-flop situations that are multi-handed. Perhaps they don't have the ability to concentrate in the way that is required when you need to make many decisions each hour. Perhaps they make their profit by preying on players who play poorly in early position, or perhaps they play in a venue where such players are particularly prevalent. Perhaps their game does not lower the fees for short-handed games. If these describe your situation, you may have a strong case that you have a better EV when playing full-ring games. However, I don't think these are common characteristics of poker players or venues, and, in any case, any advantage you may have in EV per hand when playing full-ring games would have to outweigh the increased number of hands you are able to play in the faster short-handed games. A 4-handed game likely has twice as many hands per hour than a 9-handed game. Asking a player to make twice as much per hand in a 9-handed game than a 4-handed game is a very tall order.

In the end, I think that with a smaller rake and a greatly increased decision-rate, short handed games make for much more efficient winnings.

Big advantage to short-handed games, assuming you are better than your opponents in late postion and on the blinds.

Effect on social atmostphere: Full-ring games afford players the opportunity to sit back and relax. They chat with their opponents and even discuss poker strategy without worrying too much that the player they are speaking with will play many hands with them. Basically, the game is less contentious. On the other hand, they can sometimes get boring and players get impatient with each other for playing slowly.

Small advantage to full-ring games.


Effect on game purity: Smaller rake means slightly less negative effect on the game. Fluctuating game size requires a flexible strategy, which I view as a slight positive. However, some would argue that a full-ring game, with its slower pace and emphasis on patience and observation, is the way "pure" poker was meant to be played.

Very small advantage to short-handed games.


Unless you come mostly for the social experience, you should prefer short-handed games.

What is the best fee structure for poker players?

(Another in my series of posts addressing what I think poker players should want from a card room.)

In Los Angeles, the rake tends to be $1 taken from the pot before the flop and another $5 after the flop, including $1 for the jackpot drop. In Las Vegas, it used to be as low as $4 with the fourth dollar not being taken out until the pot reached $80 or even $120. In higher-limit games (10-20 NL and above or 100-200 limit or above), the players usually pay "time," which means they pay something like $5-$15 to the house every half hour, but the pots are left unadulterated.

Nobody likes paying these fees, but we all acknowledge that the casino needs money to for all sorts of expenses and would like to make a profit on top of that. This does not mean that players should just accept whatever fees are levied and by whatever means the casino pleases. Not all fee structures are equally bad for players, and, whatever the structure, a smaller fee is clearly better. So, what structure is best for casino poker players? Can we do better? How much does the size of the fee really matter? Should we be "comparison shopping" and pressing the casinos to keep their fees low?

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First, let's do a comparison of paying rake versus "time."

Effect on winnings: If you are playing in a 9-handed NLHE game, you will likely play about 30 hands per hour and win about 3. If you play tight and take a break once in a while, this is probably closer to 2.5 hands won per hour. Time games cost about $10-$25 per hour plus maybe $3 in tips. In a raked game, it costs $7 per hand won, including tip. Multiplying by 2.5, this gives $17.50 per hour. This corresponds to a $14.50/hour time game (assuming $3 of tipping per hour). This means that if you are a tight player, most time games are slightly more expensive.

Most players do not chop in time games, but most players do chop in raked games. It's probably better for you if your opponents chop because the game will go faster. This probably adds one or two hands per hour. The downside is that you don't get to watch your opponent's heads-up strategy as often if they are chopping, but I think an extra hand or two should be worth more than this extra information.

As I've said before, taking money from a pot rewards and encourages tighter play, which is bad for the game, especially NLHE. Adding money to the pot has the reverse effect. You are effectively playing higher stakes when you pay time instead of rake.

The advantage here really depends on the size of the rake or time fee. Time fees are usually slightly higher, but this is offset by chopping and increased pot sizes.

Effect on social experience: The time games encourage players to stay at the table, lest they waste hands they already paid for. Most players like this, but I find it annoying. I like taking a few leisurely breaks during a session, and I don't mind if the game becomes shorthanded.

Rake games encourage chopping the blinds, which helps the game go faster. On the other hand, if you don't chop, you might have some upset neighbors in a rake game.

Slight advantage to time games.


Effect on game purity: Time fees preserve the purity of the game. Rake fees alter the pot size and encourage chopping. This is especially problematic in low-limit games where the rake is a large percentage of the pot.

Advantage to time games.


Time games often cost a couple extra dollars per hour to play. I think it's worth it for game purity and social experience, but only if it's not too expensive. In higher-limit games, the rake does not disrupt the game too much and I would only pay an extra $3 or so per hour to play a time game. Even this might be too extravagant. That's an extra $3000 per year if you are a pro playing 1000 hours.

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How important is the size of the fee?

If you do some comparison shopping, you can likely find a place that takes $1 or $2 less in rake ($2.50 to $7 or so per hour) or charges $5 or so less per hour in time fees. An amateur player might play 100-200 hours a year, which comes to around $500-$1000. A pro might pay an extra $5000-$10,000 per year. I'd say it's a very significant but not overwhelming concern.

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What is the ideal fee structure? Well, I identified problems with both rake and time fees. The rake affects the purity of the game and rewards tighter play. The time fees unduly discourage taking breaks. The following idea fixes both problems:

Make the button pay the fee each hand. Like a "hand fee," only it's paid just by the player who has the best position. The one drawback here is that you can't really adjust the fee to the size of the pot like you can with a rake, and if there is no flop it's still only fair to charge the button the full amount. If this results in the game being too expensive, the casinos can start charging $4/hand instead of $5 (for example).

If someone pays his blinds but skips his button, treat it as if he missed a blind -- that is, he must post or wait for the blinds to reach him before he gets another hand. This fixes the problems with both the rake and time fees.

Are any new problems likely to arise with this solution? Any way players will try to "game the system" other than skipping the button, which I already addressed? I can't think of any.

*****

I am considering playing in the WSOP main event this year. (I might look into selling up to fifty 1% shares in my winnings. Is that legal?) It's something that I almost feel I should do; playing the main event would be a great way to cap off my poker career, and it's something that people always ask me about when they hear I played professionally. Considering that I haven't played recently, my EV is surely lower than ever, but my enjoyment would probably be higher than ever.

Still, Noon to 12:30 am for days on end? That is too much poker!

Should No Limit Holdem Players Want Antes in Their Games?

Two of the main triggers of the poker boom in the 2000's, the movie Rounders and ESPN's well-produced broadcast of the WSOP (especially the Moneymaker victory), both specifically glorified the game of no-limit holdem. Rounders famously quoted Doyle Brunson calling NLHE the "Cadillac of poker games." Brand-new poker players came to the casinos looking for holdem games, especially no-limit holdem games. Before long, veteran stud players migrated to the no-limit games hoping to prey upon the newbies. The result is that, compared to holdem games, stud games have plummeted in popularity since the 1990's. Today, stud games feel downright antiquated.

One distinguishing feature of stud games is that they use antes, and, as a result, antes now seem as antiquated as the stud games they're used for. Actually, while antes are almost never used in NLHE cash games, they do usually come into play in the later rounds of tournaments. This is cause for consternation among some tournament players, mostly because the antes increase the effect of luck as compared to skill. Perhaps ironically, then, one place you can occasionally see antes in NLHE cash games is in the very high-stakes games, where you might expect the players to want to maximize the importance of skill. The reason they like antes is that many of these top players have recognized the problem I mentioned midway through my previous post on jackpots: standard NLHE has very little action if everyone in the game is playing a strong, solid, tight and aggressive game. To fix this problem, top pros have sometimes decided to add antes to their games. So, should the rest of us be advocating for the introduction of antes into our NLHE games? Probably, but let's look at the major considerations.

1. Maximizing winnings.
2. Having an enjoyable social experience.
3. Having a pure, intellectually stimulating game of poker.

All three factors have opposing forces that I will attempt to weigh against each other.

1. Maximizing winnings: If you are a strong player with a given set of opponents, your EV will probably be slightly improved if you are using an ante. With more action, you will be correct to play more hands, and this will give you more opportunities to use your superior skill against your opponents. Also, there will be more money in each pot, which means more potential profit. There are two major drawbacks, however. First, your expected volatility will certainly rise substantially because the antes directly increase the luck factor and, with more money being bet, you will effectively be playing higher stakes. This will possibly force you to move down to lower stakes. Second, your typical opponent's biggest weakness, playing too many hands, will suddenly become much less of a liability. Your weak opponents will be playing better even if they fail to adjust for the introduction of the ante. This is all subjective, but my guess is that these factors give a bit of a net boost to EV.

The previous paragraph assumes "a given set of opponents," but we must  also consider the effect on the player supply in an ante game versus a no-ante game. In the short run, only good players seem likely to want to use antes. The first players who will want to make the move to "ante NLHE" are those who are confident in their ability to adjust to the new strategic challenges posed by the introduction of an ante and those who recognize the problem I just described with NLHE losing action if you are at a table of solid players. We can expect that these players are better than average, and they will certainly not be novices. Despite this, I think that in the long run, bad players will really enjoy the antes. Having more action and higher volatility is way more fun. If we can get to the point where antes are no longer a novel concept, novices will be happy to play with them. In the long run, which is what I'm primarily concerned with in these blog posts, I think the antes will attract more bad players than good players because of the increased luck factor. Another small net positive for EV.

2. Social experience: For most people, getting to play more hands is more fun. Sure, there will be the odd player who likes to sit back and wait for AA or KK, but who needs those guys anyway? Others will object to the increased luck factor, but just as many will enjoy it. In the short run, some players will be put off by the novelty of antes, but others (such as myself) like thinking about novel strategic situations, and in any case the novelty will wear off. A net positive.

3. Pure, stimulating game of poker: Some purists, including yours truly, are just concerned with having a robust, stimulating, fairly-played game, unencumbered by factors introduced by the casino such as jackpots, rakes, or confusing rules or promotions. The antes affect none of these factors except that they have the potential to engender a much more stimulating game by discouraging tight, nitty play. Some purists additionally want a traditional or low-luck game of poker. For these players, the antes may be a problem. From my perspective, however, the antes represent a major boon to the game itself.

Many of the factors above relate to my personal preferences, which may make it hard to convince other players (or my readers) to come to my side if they are not already so inclined. For example, if you understand the importance of having luck in a poker game (this is what attracts the bad players) but you still think it's better to try to minimize it, there's not much I can do to convince you that antes would be a good thing. Also, my approximations on the effect on EV are certainly up for debate. However, if you agree with me on these factors (and you should!) then you'll agree that introducing antes is a really good idea.

I wonder whether we can ever really make the jump to adding antes in lower stakes games. Perhaps it's like the QWERTY keyboard: research shows it's not optimal, but it's just not worth making the switch. It's not surprising that the high-stakes NLHE games were the first to adopt antes. Nobody playing those stakes is going to have an issue with getting outside their comfort zone to play a slightly novel form of poker. (Well, maybe Andy Beal.) They're not worried about scaring away new players, and they are looking to maximize the amount of action in their games. It's also possible that these high stakes games have fewer action players and are thus more prone to going cold.

All of these factors that encourage the use of antes diminish as you go down in stakes. What I think needs to happen for antes to become prevalent is for enough regular players to realize that the top pros are playing with antes. When that happens, the game will no longer look as weird and novel as a Dvorak keyboard. In the meantime, regular players can advocate for antes by convincing their fellow poker players that antes are a good thing and by asking their floor men to make "interest" lists for ante games.

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.

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.

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].

My Blocking Bet is Disrespected

Generally, I'm not a big proponent of the idea that you should try to think one "level" higher than your opponent, as is advocated in a lot of NLHE poker books, including a chapter called "Multiple Level Thinking" in No Limit Hold 'em: Theory and Practice by David Sklansky and Ed Miller. Game theory takes care of this because it solves for the infinite-level. It cannot be out-thought; there is no higher level. Anyway, I had an interesting hand this week in which I ended up needing to think one level higher than my opponent. This is unusual. It only came up because my opponent and I have a rich history of hands against each other and because I made a weak play at one point in the hand that my opponent attempted to exploit. In fact, this opponent is the same player I named "X" in an earlier post, "Raising as a Bluff." Player X is a very good player. I know he has been paying special attention to my game for the past year, and he seems to pride himself on acting on his reads. I play a relatively predictable style; he knows this, and I know he knows this.

The reason my strategy is relatively predictable is that I usually try to play close to "optimally" in a game-theoretical sense. I don't go as far out of my way as most players do to mix up my play. If I truly played optimally, my strategy would not be exploitable (by definition), but, in practice, my play has many deviations from optimal, but intentional and (mostly) unintentional. One play I like to make that's probably not optimal is to make a small blocking bet when I'm out of position on the river and my opponent is likely to successfully bluff me out of the pot if I check. Once in a while I will make this same play with the near nuts. This makes it very expensive for my opponent to raise me as a bluff, and most people will just call me unless they have the nuts. I don't think player X has ever seen me make this type of blocking bet with the nuts, though. I know he has seen me do it a few times with medium-strong hands that cannot stand a raise.

On to the hand in question: Player X had about $1700 and I had him covered. I raised in early position with AsQh to $30 and got two callers, including player X.

Flop($100 pot): QcJh6c. I bet $65, X called.
Turn($230 pot): 4c. I checked, X bet $120, I called.
River($470 pot): 6h. I bet $160. My intention here is to force a cheap showdown. In my mind, X cannot be sure I didn't flop a set or two pair and then make a full house on the river. This should make it very dangerous for X to raise me here if he just has a flush. In truth, I probably would have bet a set on the turn even though the flush came in, but how could Player X be sure of this?

After I bet, player X says, "Really, Keith?" He thinks for 30 seconds and then raises to $660. My plan was to fold to a raise, but I stop to consider the circumstances. I have to call $500 more to win a pot that will be almost $1800 all told. If there's a 28% chance that X is bluffing, a call is profitable. I thought back over my play and realized that it was highly unlikely I would have a full house on the river here. Player X knows I couldn't have a hand like Q6 because I don't raise with that preflop, and he knows that if I had a set, I would have bet out on the turn. This means that my blocking bet was very transparently weak. My range here doesn't really include anything to scare my opponent. The truth is that he realized this before I did, and he had the guts to capitalize on it. However, I still had a chance to rectify the situation. I just had to think one level beyond X. When he raised me to $500, he could not have expected me to call. If he had a flush, he would have raised less or just called. I've seen him bluff with small pocket pairs before, and that is what I kind of expected him to have. I called and he showed me Td9d, a broken straight draw.

It occurs to me that most of my posts about hands involve my explaining how clever I was. Let me just acknowledge at this point that this is a result of severe selection bias. It's more fun for me to write about such hands, and I imagine it's more fun for you to read about them.

Philosophies on "Running it Twice"

In the $500 NL game I play in, players are allowed to make a specific type of deal if they go all-in before the river: they can run the rest of the cards either two or three times and then split the pot in halves or thirds, respectively. The pieces of the pot are distributed according to who won each of the two or three iterations. (I gave an example in this post.)

It is strange and fascinating to listen to other players' philosophies on whether to make a deal and whether to run the board twice or three times. As far as I can see, there are only a few things that deserve any consideration:

1. What is my EV for each option? This can be answered mathematically: my EV is the same for all three options. This follows from the fact that EV(A+B) = EV(A)+EV(B), regardless of whether A and B are correlated. If we take A to be the first running of the cards, and B to be subsequent runnings of the cards, we can see that the EV does not depend on what type of deal we make.

2. What happens to my volatility for each option? This can also be answered mathematically: the volatility is lower if you run the board more times. This is the reason why running it more times might be worthwhile.

3. How much time is this going to take? If you are a winning player, you can lose money if too much time is wasted. Also, I would consider it bad table etiquette to needlessly waste the other players' time. Splitting the pot three ways can take a while. In fact, last week I won five-sixths of a pot because we ran the board three times and we chopped the last one, and this took a minute or two to sort out. Another thing that can waste time is the possibility that your opponent might take a while to decide whether to make a deal. I've seen such a decision take over five minutes.

4. How will this affect my table image? Being willing to run the board twice can change the way people play against me. The most significant difference is probably that some players may try to get all-in earlier in the hand in order to give themselves a chance to make a deal and chop the pot. They also may be more willing to call my all-in bets before the river. This is probably a good thing, since players are deviating away from trying to maximize their EV, but for certain players, it likely makes them play better (if they will now call with +EV hands that they would have otherwise folded).

Points 1 and 2 seem to me to be by far the most significant, but I also often consider point 3 and choose not to make a deal if the pot is small. I do this out of respect for the other players' time and because I figure it won't add too much volatility to my winnings. Point 4 is one I don't worry too much about, but it seems like it could be a reasonable consideration.

Most players know points 1 and 2 (minus the math), yet they often take longer than a minute to decide whether to make a deal. What could they be thinking about? Clearly it's not point 3 (time considerations), or they wouldn't waste everyone's time with their slow decisions. For most players, it's not point 4, either. The answer? Whim. This wouldn't be so bad if it weren't for the fact that they clearly consider these decisions to be excruciatingly important, and once they form their opinions on which option is best, they smugly ridicule players who make a different decision. Listening to them argue about it is like trying to watch one of those shows where pundits discuss politics. Nobody has any evidence to support their claims, but they are all quite satisfied in their certitude. For example:

"If I call a big bet with a draw, I'm trying to win the whole pot. Why run it twice? To get back half the pot? It's better to just fold and save your money than call and make a deal."

"If you initiate the thing, it's three times," meaning that if you put in the bet and are called, you should always run it three times, rather than twice.

Maybe I'll try to document more of these, but the point is that the decisions are made on a whim, and then supported with whatever "evidence" seems to fit.