Simulated heads-up endgame in all-in or fold mode

[dana33]

After lurking for a couple of months, this is my first post to the Two Plus Two forums.

Inspired by eastbay's optimizing calling all-in in a heads-up, all-in or fold poker model posting, I decided to do my own simulations along the same general lines, but with a somewhat different approach. Here is what I have done so far:

1. Got Steve Brecher's C code off the Web for computing heads-up showdown probabilities for winning outright (Pwin) and splitting the pot (Ptie) based on two-card starting hold 'em hands. Hacked the code to run in batch mode and compute the 169x169 matrices for Pwin and Ptie for all possible combinations of hands. I had to do some combinatorics for this as well. E.g., to compute Pwin for Q9s versus KJo, you need to properly weight the different cases according to whether there is a shared suit between the two hands, and if so whether the K or J shares the suit with the first hand.

2. Computed P(hand), the probability of getting a particular two-card starting hand, and P(hand2|hand1), the conditional probability that a single opponent has hand2 given that you have hand1. Computing these is a matter of simple combinatorics.

3. Used these probabilities to drive simulations of a tournament heads-up endgame where both players are in "all-in or fold" mode due to the size of the blinds. Each player has a set of hands he will move in with from the SB, and another (typically smaller) set of hands with which he will call from the BB if the first player has not folded. If the SB player does not move in, he folds.

In eastbay's simulations, he assumed an opponent who always moved in, and optimized the specific set of hands with which one should call. I simplified things by ordering the 169 starting hands according to Pwin against a random hand, and then introducing parameters Pmovein and Pcall representing the percentage of these hands with which to move in or call. E.g., if Pmovein is 80% and Pcall is 60%, a player will move in with the top 80% of hands in this list, and call with the top 60%. In my simulations, both the "hero" and "villain" have their own values for these parameters, each of which I varied in 20% increments, but always keeping Pmovein >= Pcall.

The results so far for the case where both players start the endgame with equal sized stacks which are 10 times the size of the BB:

If "par" is winning the tournament half the time, it is easy to do much worse than par by being too tight. E.g., if your opponent only moves in with the top 40% of hands and calls with the top 20%, you can win up to 69% of the time with optimum counter-tactics. But it is hard to do much better than par if your opponent is sufficiently loose. It appears that moving in with the top 80% of hands and calling with the top 60% is a virtually unbeatable combination, in that your opponent can do no better than par in this case (within the statistical error of my simulations). This seems surprisingly loose, but that's what the simulation results say. (I would be interested in any game theoretic argument anyone has regarding whether these results are right or wrong.)

Against a complete maniac who pushes and calls with anything, my results are fairly flat from Pcall=20% to 80% (with a win percentage that varies only between 50% and 54%), using the best Pmovein in each case. But there does appear to be a maximum in the curve close to 40%, which is consistent with the results that eastbay reported. I'll need to increase the number of tournaments I simulate for each set of parameters in order to be more confident in these particular results.

As discussed in eastbay's thread, there are more sophisticated hand selection strategies one could use, such as making Pmovein and Pcall functions of the stack-to-blind ratio. I plan to try these once I find the time.

Comments welcome.

Dana

P.S. I want to specifically thank Steve Brecher for posting his code to the Web and eastbay for getting me started on this effort. Whether or not I learn anything about poker from the results, I had a lot of fun writing the code.

[eastbay]

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After lurking for a couple of months, this is my first post to the Two Plus Two forums.


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Excellent.

[ QUOTE ]

Inspired by eastbay's optimizing calling all-in in a heads-up, all-in or fold poker model posting, I decided to do my own simulations along the same general lines, but with a somewhat different approach. Here is what I have done so far:

1. Got Steve Brecher's C code off the Web for computing heads-up showdown probabilities for winning outright (Pwin) and splitting the pot (Ptie) based on two-card starting hold 'em hands. Hacked the code to run in batch mode and compute the 169x169 matrices for Pwin and Ptie for all possible combinations of hands. I had to do some combinatorics for this as well. E.g., to compute Pwin for Q9s versus KJo, you need to properly weight the different cases according to whether there is a shared suit between the two hands, and if so whether the K or J shares the suit with the first hand.

2. Computed P(hand), the probability of getting a particular two-card starting hand, and P(hand2|hand1), the conditional probability that a single opponent has hand2 given that you have hand1. Computing these is a matter of simple combinatorics.


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Very nice. I spent some time doing the same things a few days ago. We should compare numbers. We can do that on or off-list, whichever.

I did mine at 3am so I'm sure I've made a mistake or two, but getting our numbers to agree should be good for both of our codes.

[ QUOTE ]

3. Used these probabilities to drive simulations of a tournament heads-up endgame where both players are in "all-in or fold" mode due to the size of the blinds. Each player has a set of hands he will move in with from the SB, and another (typically smaller) set of hands with which he will call from the BB if the first player has not folded. If the SB player does not move in, he folds.

In eastbay's simulations, he assumed an opponent who always moved in, and optimized the specific set of hands with which one should call. I simplified things by ordering the 169 starting hands according to Pwin against a random hand, and then introducing parameters Pmovein and Pcall representing the percentage of these hands with which to move in or call. E.g., if Pmovein is 80% and Pcall is 60%, a player will move in with the top 80% of hands in this list, and call with the top 60%. In my simulations, both the "hero" and "villain" have their own values for these parameters, each of which I varied in 20% increments, but always keeping Pmovein >= Pcall.


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Ok. I'll see your simulation and raise you one.

My idea is stick with the two-parameter strategy of a top N% to push, top M% to call, but to re-order the hands as they rank against typical "all-in" hands. It will be interesting to see if this makes much of a difference in the results. The results of the rank order itself might be of some interest.

[ QUOTE ]

The results so far for the case where both players start the endgame with equal sized stacks which are 10 times the size of the BB:

If "par" is winning the tournament half the time, it is easy to do much worse than par by being too tight. E.g., if your opponent only moves in with the top 40% of hands and calls with the top 20%, you can win up to 69% of the time with optimum counter-tactics. But it is hard to do much better than par if your opponent is sufficiently loose.


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If there's anything I've learned from these kinds of heads-up analyses, this is it. When blinds reach 10% of stack or greater, heads-up, it's always better to push too much than too little. Almost without bound this is true. Waiting for a "premium hand" is a total disaster. Pushing every single hand, no matter what, is a much stronger strategy for winning the tournament. Possibly counter-intuitive, but true.

[ QUOTE ]

It appears that moving in with the top 80% of hands and calling with the top 60% is a virtually unbeatable combination, in that your opponent can do no better than par in this case (within the statistical error of my simulations). This seems surprisingly loose, but that's what the simulation results say. (I would be interested in any game theoretic argument anyone has regarding whether these results are right or wrong.)

Against a complete maniac who pushes and calls with anything, my results are fairly flat from Pcall=20% to 80% (with a win percentage that varies only between 50% and 54%), using the best Pmovein in each case. But there does appear to be a maximum in the curve close to 40%, which is consistent with the results that eastbay reported. I'll need to increase the number of tournaments I simulate for each set of parameters in order to be more confident in these particular results.

As discussed in eastbay's thread, there are more sophisticated hand selection strategies one could use, such as making Pmovein and Pcall functions of the stack-to-blind ratio. I plan to try these once I find the time.

Comments welcome.

Dana

P.S. I want to specifically thank Steve Brecher for posting his code to the Web and eastbay for getting me started on this effort. Whether or not I learn anything about poker from the results, I had a lot of fun writing the code.

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I think we're just getting warmed up with this line of inquiry, and I do think the results mean something qualitatively, at least.

Your results would take on a whole new meaning if you made some graphs. If you don't have a good graphing program, send me the numbers and I'll throw them in Excel and post the results, if you want.

eastbay

[dana33]

Thanks for the great feedback. We seem to be thinking along the same lines -- e.g., reordering the hands based on how they do against other legitimate hands as opposed to random hands.

I need to run some more simulations in order to get the noise down before I can get any meaningful graphs, but I agree that it's a good idea and will do it once I get better data.

A newbie technical question: I don't have my own Web site where I could post a graph and link from here. Once I create a graph, is there any way to upload a GIF or Excel file to the bulletin boards here and embed it in a posting?

[eastbay]

[ QUOTE ]
Thanks for the great feedback. We seem to be thinking along the same lines -- e.g., reordering the hands based on how they do against other legitimate hands as opposed to random hands.


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See my results for any ace, any pair rankings, freshly posted.

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I need to run some more simulations in order to get the noise down before I can get any meaningful graphs, but I agree that it's a good idea and will do it once I get better data.


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Isn't it sick how many samples are required for answers good to three decimal places? It's kind of disconcerting about the luck factor in poker in general, IMO.

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A newbie technical question: I don't have my own Web site where I could post a graph and link from here. Once I create a graph, is there any way to upload a GIF or Excel file to the bulletin boards here and embed it in a posting?

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I think you'd have to find a free webhost that would allow it - there aren't too many of those left.

If you're gung ho, I have a webserver that we could use to collect some of these results in one place - I could give you an account on it.

eastbay

[allenciox]

I have done some related tests --- figuring out what are the best hands for calling all-in bets based on what percentage (5%,10%,15%,25%,40%,67%,100%) of bets an opponent has raised all-in with. I used a matrix of 169x169 matchups, but only had the ev of each matchup. Would it be possible for me to get your matrix that includes Pwin and Ptie? Then I can collaborate in this effort.

Thanks!

[dana33]

eastbay,

Please send me your email address in a private message and I will send you my matrices of probabilities so we can cross-check.

I have done some more extensive simulations and the conclusion that "push with top 80%; call with top 60%" is a hard to beat strategy (with a 10:1 stack:BB ratio) is holding up. Given this, I wonder how important the exact hand-ranking scheme is. Even though the top hands clearly change depending on whether you are up against a random versus a strong hand, do the bottom 20-40% of hands change that much?

[eastbay]

I'm interested on how rapidly this optimal strategy changes with blind/stack ratio.

If you make it 5%, is it a little different? A lot different?

I'm going to try to write a lean/mean simulator for this that can crank large samples in a short amount of time. My full scale NLHE simulator is far too slow (due to the generality) for really exploring this space.

eastbay

[eastbay]

[ QUOTE ]
eastbay,

Please send me your email address in a private message and I will send you my matrices of probabilities so we can cross-check.

I have done some more extensive simulations and the conclusion that "push with top 80%; call with top 60%" is a hard to beat strategy (with a 10:1 stack:BB ratio) is holding up. Given this, I wonder how important the exact hand-ranking scheme is.

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I have a hunch it will result in optimal strategy being a bit tighter, since you'll be facing "stronger" hands when you call, I think you're going to have to tighten up your calling standards.

That's a guess, though.

eastbay

[lorinda]

This old thread may be of interest.

Note I haven't seen the thread that this post relates to.

Heads up vs blind man

Lori

[dana33]

[ QUOTE ]
I'm interested on how rapidly this optimal strategy changes with blind/stack ratio.

If you make it 5%, is it a little different? A lot different?


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It's quite a bit different. While 80%/60% push/call is nearly unbeatable for a 10% BB:stack ratio, the nearly unbeatable strategy for a 5% ratio is closer to 60%/40%. However, even here, 80%/60% is not all that bad -- about the best you can do against it is a 54% win rate (with equal starting stacks).

Of course, when the BB is only 5% of your stack size, you're probably not yet to the point that "move in or fold" is a winning approach, so the assumptions of the simulation start to break down.

Note that in my current simulations, the strategy remains fixed throughout. That is, the players do not change their strategies as their individual stacks grow or shrink. On my list of 57 other things to try with my simulator is your previous suggestion of making the percentage of hands played a function of a player's current stack size.