Thursday, December 20, 2012

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.

Sunday, December 09, 2012

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.

Wednesday, December 05, 2012

"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.