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.


Anonymous said...

These last two posts are great, really excellently written and really interesting. My only quibble would be that in this post it doesn't seem like you've fully learned the lesson from your last post, namely that it takes a lot longer for people to get knocked out than one might expect. I think your percentages of people knocked out here are all about double what I would say.

Keith said...

Thanks, Rick. These truly are wild guesses, although I did put a few minutes of thought into it. I will take your opinion into account tomorrow when I try the tournament again.

After I published the post, I realized that it would have been useful to have more than one entry for each row to represent different percentiles -- or at least a second row to indicate the 95%ile, which might be more relevant than the "average" that I was guessing at. Without the full distribution (or an approximation to it), I can't calculate the EV of the fold strategy. Obviously, I neglected to update it. Frankly, I have some doubts in my ability to put all this to good use.

I do have some good raw data to include from this past Sunday, as I recorded all relevant info at the end of each round after the first break. Believe it or not, I got "bubble money" again. This time, what went wrong was a lack of discipline, as I lost patience and played hands when I could easily have folded into the money. Perhaps this was also a symptom of being overconfident, as you suggest, because I thought I could afford to lose a few more chips and still make the money by adopting the fold strategy later. Oh well.

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