Thursday, August 01, 2013

The Horseshoe Cleveland Casino

Well, I haven't posted in six months, and now I find myself living outside of Cleveland and commuting into the city to play poker a few days a week at the Horseshoe Casino. Here is my review of the place after several cash sessions and tournament entries.

Saturday, February 02, 2013

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.

Thursday, January 31, 2013

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.