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.

Part-Time Poker Comeback

After a long hiatus from poker and a brief foray into academia, I've decided to return to playing poker part-time. (Most of the rest of my time will be spent watching my two year old and three month old sons, and I'm also doing some politics/skeptic blogging, not to mention my return to this blog.) For the next two months, I'll be playing one to two tournaments a week in Charles Town West Virginia and putting another three to five hours a week into poker research and blogging. That's not a whole lot of time, but my no-limit holdem game was very strong as of 2011, so I'm hoping it won't be too hard for me to translate that into tournament poker once I get the rust off.

One tentative goal I have is to be ready to play in the WSOP in 2013. I will probably only do this if I think I have positive winning expectation. I'm not sure how to determine that.

Live poker was legalized in Maryland casinos on Tuesday via a referendum. So, poker will be spread here quite soon, pending the result of a comically silly lawsuit in which the law's opponents are claiming that it required a majority of "qualified voters"as opposed to a majority of those who actually voted. (I guess if this gets overturned, then so will the gay marriage, gerrymander, and DREAM acts that also passed by referendum on Tuesday here in Maryland. Not to mention the electoral votes given to Obama!) The new law didn't figure much into my decision, though. Indeed, none of the Maryland casinos seem to be much closer to me than Charles Town WV anyway, and, besides, I made the decision to return to poker before the law was even passed.

The two tournaments I will be trying to play each week are at noon on Fridays and Saturdays. The Friday tournament is $100 plus a $25 bounty. I have begun doing some analysis of Bounty tournament strategy, and I hope to have a post up about that next week. The Saturday tournament is a $250 tournament. So far I have played four times and won $225 in bounties (in a single tournament). No cashes.

On my poker reading list:
Poker Strategy by Nesmith C. Ankeny. This is a book written by a mathematician in the early 1980's focusing on draw poker. I had not heard of this book before I found it in my local library system last month, and I'm very interested to see what the state of the art in poker game theory was thirty years ago. I've started it, and so far the author is on target with his poker philosophy, but he hasn't gotten into any real analysis. He has an unusual writing style that can interferes with getting to the point, but it can be amusing. For example (p 5):

A poker player considering luck in planning his moves at the poker table is like the Pope contemplating marriage. The thought is heresy and can only lead to trouble.

Poker Face: Mastering Body Language to Bluff, Read Tells, and Win. Also from the library, I thought this might be a more analytical version of Mike Caro's classic Book of Poker Tells, but after flipping through it, this one looks very fluffy and has no pictures. I'll take another brief look to see if there's anything worthwhile.

Kill Everyone. This book's peculiar name stems from its predecessor, Kill Phil, referring to Phil Hellmuth. One of the three authors is a professor at UMD College Park, and it looks reasonably analytical, so I think it will be worthwhile.

The Full Tilt Poker Strategy Guide: Tournament Edition. This is a book I bought a while back because I was interested in a few sections in particular. I really liked the analysis so far, and since it focuses on tournaments, I think it will be very worthwhile for me to read. (Full Tilt's legal issues notwithstanding.)

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?

------

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.

------

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.

------

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!

Has mathematical analysis changed the game a lot in the past few years?

This is similar to one of the last questions I got after my Mathematical Poker talk last month. Unless a commenter asks another question, I think this will be the last one I answer here on the blog.

The game has indeed changed a lot in the past several years, but it's hard to tell how much of that change is really attributable to mathematical analysis. Certainly, the games have gotten tougher in during the years I played, but I think the main factor may just be natural selection. I would guess that most of the players int the games I was playing in Los Angeles do not know the game theory results I presented in the talk. Ten years ago, I think there really was a Moneymaker-aided poker "craze", which was the inspiration for this blog's name. That brought in a lot of new players, and so games were easy to beat. The ensuing years weeded out the bad players, while the good players were more likely to stick around. This process is pretty hard to disentangle from the fact that the good players were also getting better with practice and study, so in order to guess at the influence of analytics, I need to rely upon conversations I've had with players. My impression from such conversations is that new players tend to have a good sense of the analytical fundamentals of the game, much better than almost anybody had ten years ago. Meanwhile, the older players tend to be mostly those who had a good intuitive feel for the game and were able to survive the natural selection process. Most of the best players today (especially the young players) seem to have a very strong analytical foundation, and that is where I think analytics have caused some of the biggest and most visible changes in poker. In light of that, the continued success of Doyle Brunson is all the more impressive to me.

Another area analytics seems to have made a big difference is in tournaments. I haven't been playing tournaments much, so I can't speak from experience, but tournaments seem more ripe for mathematical analysis. Also, my poker discussions with tournament players gave me the impression that they are more keen on the technical aspects of the game. 

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.

Victory

The past month or so, I've been playing the tournament at the Mirage during the week. The cash games at the Rio and elsewhere are still pretty juicy on the weekend, but it seems like during the week, the biggest fish are playing in these tournaments. I also like the change of pace of playing in a tournament. The Mirage tournament is $130 plus unlimited $100 rebuys for the first hour, except on Thursday, when it's $230 plus a single $200 rebuy. Most people do the rebuy, so on Thursday it's essentially a $430 tournament. I've played in six or seven of these in the past couple months, and have never managed to make it to the money, which is either top 5 or top 9 finishers (depending on whether there are at least 50 entrants). However, my tournament strategy has been improving and I just know I'm better than nearly all of the players who play in these things.

So, yesterday (Thursday), after I got up around 4pm, I decided I'd try to make it to the tournament. I've been reading Harrington on Holdem Volume II, which is an excellent book about tournament strategy, so I decided I would do some last-minute reading before heading out. I've reached the part on Heads-Up (1v1) play, which is only useful if I make it to the final two... little did I know this would be the case later on in the night. However, the tournament was over after only one heads-up hand, and I didn't play it quite as Harrington suggested anyway. So, I guess the last-minute studying was kind of useless after all.

There were 59 entrants and about 50 rebuys, for a prize pool of I guess $21,800. First place is supposed to get 26% but this didn't quite work out when I did the math, so I'm not quite sure what the deal is. Anyway, I received nearly what I think I was supposed to. Here are some highlights:

Tournament lasted exactly 6 hours, going from 7pm to 1am.

Dealt AA once, raised in early position (EP), won the blinds.

Never dealt KK or QQ.

Dealt JJ twice. The first time was in the first hour, and a short stack (550) had gone all-in in EP, and was called by the player to my right. Since my hand was quite strong and I like to play a risky style to accumulate a big stack (and I hadn't used my rebuy yet), I just called. I figured that someone behind me with a weaker hand would be more likely to play, possibly even raising all-in to try to isolate the original all-in. People sometimes do this with relatively weak hands like AJ or 99, and if my JJ held up, I would be easily the chip leader at the table. Being chip leader is a position I am quite capable of exploiting. By just calling, however, I'm inviting AQ to play, too... a hand I'd rather have fold. This was a risk I was willing to take. I'd rather have a player with AK fold, too, but I think AK would call even if I raised all-in (I had about 2600 total I think). Even though I had only called, everyone else folded. The flop came 346, and, to my surprise, the guy to my right went all-in, betting 1550 at a 1700 pot. This is an unorthodox move, and it caught me off guard. What could he have here? With one player all-in, I can rule out a complete bluff. My opponent must think he can beat the all-in player. Usually people will only bet in this situation if they have a very big hand. So, my initial thought was that he must have a set, probably 66. It was also possible he had AA or KK. With these hands, though, he should be betting only 500 or so, trying to draw more of my chips into the pot. I decided that a hand like 88 or 99 was much more likely. Also possibly AK. These are hands that are probably ahead of the all-in raiser, but are still quite vulnerable and thus would like to get me to fold. So, I called and was surprised to see that he had only AJo. The other player had 77. The turn was a 9 and the river a 5, giving the all-in player a straight, but I won the 3100 side pot and came out ahead.
The second time I had JJ I raised and just won the blinds and antes.

Never dealt TT or 99.

I was dealt a pair higher than 88 only three times in six hours! Perhaps this isn't all that surprising. If I played 221 hands, I should expect to get a pair higher than 8's six times on average, so getting them only three times isn't all that ridiculous. Also, I may have played fewer than 221 hands, I'm not sure.

By the time we were down to ten players, there were two tables of five players, and I had the second biggest stack at my table. Two or three players were super short-stacked, but ninth place paid out over $900 and tenth paid $0, so people were playing very tight to try to get into the money. Fortunately for me, the big stack at the table was sitting to my right, allowing me to fold whenever he entered a pot, and steal the blinds and antes whenever he folded (this player would eventually be the tournament's second-place finisher). We had ten players for quite a while. Finally, I was in the small blind and an extremely tight, somewhat passive old pro raised under the gun. I figured him for at least AQ or 99, probably better. The next two players folded to me. I raised all-in with QT. Since the raiser had the third biggest stack at the table, he could have just folded his hand here and coasted into the money easily. He had been playing so tight that I thought this was his main goal, so I figured he would likely fold unless he had AA or KK. Actually, he called with AK! Even if I had AQ, which is about the best he could hope for, he was only a 74% favorite. Against any pair he is at only around 45%. Against a complete bluff he was only about 65% to win. So he was risking a sure $900+ here unnecessarily. Anyway, my QTo was 36% to win, and I caught a queen and knocked him out. The short stacks thanked me, and I was the new chip leader.

We consolidated into one final table of nine players. After a few hands I realized that players were still hanging on and just trying to move up a spot or two, even though 8th, 7th, and 6th paid only $120 or so more than the next lower finish. This was a great situation for me, as the chip leader. I started going all-in about 75% of the time if the players had folded around to me. With a stack of about 40K, I was picking up 2K or 3K each time I did this. Before I knew it, I was up to 60K, out of 160K total on the table. People were noticing what I was doing and even grumbling about it, but I wasn't showing my hands, so they didn't realize just how indiscriminantly I was raising. At one point, though, a micro stack smaller than the small blind limped in. The players in the blinds were particularly tight so I just pushed all-in with J4o. I figured people would see that the all-in player was on the brink of elimination and not want to risk their stacks at this point. I was right, and despite losing the main pot, I still added a bit to my stack from the side pot, which consisted of most of the blinds. The downside was that I had to show my J4o, and now some people were clearly determined to catch me bluffing. One guy actually complained to me. He thought I should be letting more hands in to try to knock out the short stacks. I didn't respond, but obviously, I didn't want the short stacks knocked out at all, since they were the whole reason my strategy was working so well.

I kept the big stack until we were down to four players and I started losing some hands. My J9 lost to AT, my J8 lost to AK. I had the second biggest stack when I found ATs under the gun. Four handed, this is a big hand. I considered a normal-sized raise, but one of the players at the table had recently started raising me all-in whenever I did that. So, I just went all-in myself. The big stack called me with JJ, and I was on the verge of elimination. Fortunately, the dealer put two tens on the board, and I sucked out and became the chip leader again.

When it got to be heads-up, I had about 100K and my oppenent had about 65K. First place was about $1300 more than second, and the Mirage rules forbid deal making, so we had to play it out for high stakes. The blinds were 2K, 4K with antes of 300, and I had the small blind and the button. I looked down at A3o, a good hand heads-up. I raised to 12K, looking for a call because I would have position on the flop. My oppenent raised all-in for another 50K or so. This is the player who had been re-raising me all-in quite frequently in the past 45 minutes or so. As a result, I somewhat recklessly decided to call him. Fortunately, my hand held up against his KJs (I was about 54% to win), and, for the first time ever, I had won a live tournament. They even gave me a framed picture of myself sitting behind all the chips!