optimizing calling all-in in a heads-up, all-in or fold poker model

[eastbay]

Warning: probably only potentially interesting to math geeks

At the end of a tournament, it's common to be in a (mostly) all-in or fold situation. When you're on the big blind, what should you be calling all-in with? It seems clear that the answer depends on the size of the blinds relative to the stacks (how long can you wait?), and also to a great extent what your opponent is pushing with.

To make things real simple as a first experiment so that we can work some numbers, what if we presume heads-up, initially equal stacks, a big blind that is 10% of the stack, and a maniacal opponent who is going to push all-in on every single hand.

Also assume that when you're on the button, your play is dead even, so we're only going to look at a series of plays with you on the big blind, deciding whether or not to call.

What hands would you call with?

I wrote a computer model that uses something like a genetic algorithm to search out the best set of calling hands for this situation. The answers are approximate, taken from samples of 20,000 tournaments played out until somebody wins, using trial sets of calling hands (a population), choosing the sets that give the best results, discarding the rest (survival of the fittest), and using those best sets to generate some new alternatives (mating and mutation), and repeating until changing your strategy tends to make the results worse and not better.

Before I share the results, anyone care to make a guess at the answer? Maybe expressed as a percentage of hands ranked in order of their win rates against a randomly chosen hand. Another interesting question is: how much of an edge can you get over your opponent when he uses this strategy, if you make the best choice for calling hands? Can you destroy him by making the best choices, or is he actually playing decently to push in from the button every single time?

Results to come.

These calculations are expensive. It's basically an overnight run to look at one set of parameters. Once I am satisfied with the "push on every hand" answers, I would like to try some different opponent strategies, and apply the same model to compute the best counter strategy. If your opponent is only going to push, say, Sklansky hand groups 1&2, and fold everything else, what changes in the optimal set of calling hands? Groups 1-4?

Looking at this from the other direction, if the big blind is playing an optimal calling counter strategy, how tight should the small blind be with pushing? What push hands give the smallest edge to the big blind if he is going to play a perfect counter strategy?

Before you launch into some big rant about how this is at best loosely related to real poker, save your effort. I understand that. But to me, this kind of exploration is fun and also potentially useful in that some basic concepts and trendlines might come out of it that might be worth keeping in mind.

eastbay

[Mergualdo]

Why is it necessary to simulate this? For given stack sizes and blinds, one can directly compute which hands have positive EV for the optimizing player, right? Ah, no, I see. Given the non-optimizing opponent is giving away EV every hand, it may be optimal to pass on certain low EV situations. But then the set of calling hands must optimally be contingent on relative stack sizes and blinds. I get the impression from your post that you are trying to find the optimal set of hands given you are going to suboptimally play the same set of hands regardless of the particular situation.

And, oh yes, this has nothing to do with real poker. Shame on you.

[eastbay]

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Why is it necessary to simulate this? For given stack sizes and blinds, one can directly compute which hands have positive EV for the optimizing player, right?


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Because chipEV != $EV, amongst many other reasons. And even were the simulator and optimizer overkill for this particular problem, I mostly wrote them because they extrapolate to more sophisticated problems where direct methods would be impossible.

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Ah, no, I see. Given the non-optimizing opponent is giving away EV every hand, it may be optimal to pass on certain low EV situations.


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Sure. The classic "why are tournaments different than cash games" thing.

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But then the set of calling hands must optimally be contingent on relative stack sizes and blinds. I get the impression from your post that you are trying to find the optimal set of hands given you are going to suboptimally play the same set of hands regardless of the particular situation.


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In my initial formulation, the strategy remains fixed for the tournament - that's just currently a constraint of the strategy. You're right that this is clearly not perfect; if you've got pot odds after almost taking someone out, you call with more hands. This important effect is ignored. I'll think about how to incorporate it after I get the fixed strategy results giving good answers.

If you can think of a good way to approach it, I'm certainly interested in it.

If you remember my first set of simulations that you dismissed [img]/images/graemlins/wink.gif[/img], I made the game cards-up. Then the criterion for calling was based on a one-parameter combinration of chipEV (known) and "risk" to your stack in the amount of the call.

So I wanted to take a next step with that idea that had a little more applicability. tournament endgame poker is mostly all-in or fold, which is nice, because it is potentially computationally tractable (trying to simulate flop play and beyond is extremely difficult with a zillion variables).

So, I'm certainly open to a better model which is still computable. Maybe you could assume that you know the range of hands that your opponent will push with, and compute a chipEV based on that, and make the counter strategy a function of chipEV and risk once again. That might be a better approach.

The nice thing is, I now have the simulator and the optimizer that I could plug this problem into pretty easily.

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And, oh yes, this has nothing to do with real poker. Shame on you.

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Thanks!

eastbay

[Mergualdo]

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Because chipEV != $EV, amongst many other reasons.


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I've thought about this a fair bit and have a hunch that deviations of chipEV from $EV are rare and probably not that important in the grand scheme of things. Still, I'd love to know the shape of the chip EV/$EV curve--where are the nonlinearities, etc.
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In my initial formulation, the strategy remains fixed for the tournament - that's just currently a constraint of the strategy.

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Tsk tsk I do not approve.

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If you can think of a good way to approach it, I'm certainly interested in it.

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I think you have to work backwards from a static end point. Say, the opponent is down to his last chip. No, that's not static because to solve the one-chip problem you have to know what to do in the two-chip problem in case he doubles up. So it is still dynamic, but finite at least. So what you have to do is come up with strategies for each of the possible chip combinations and then figure out how to vary them so that they eventually converge. I.e., given the one-chip strategy, the two-chip action is optimal, and given the two-chips strategy, the one-chip strategy is optimal, and so on and so forth for every possible combination. Because it is finite (chips are integers) you could conceivably do it, but man would probably walk on Mars before the computation was complete. Still, it would be interesting to look how the equlibrium strategy changes conditional on the dead money and relative chip count. Probably the optimal strategy isn't very much less maniacal than the maniac strategy.

It is moments like this that I am glad I have very limited programming skills because I would have wasted a lot of time on this kind of stuff.

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If you remember my first set of simulations that you dismissed [img]/images/graemlins/wink.gif[/img],


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That must have been a different Mergualdo. This is the first I've ever posted here.

Good luck

[PrayingMantis]

eastbay,

I have two questions/observations, regarding your post. I'm not sure what I'm saying is completely true or anything, so I'll be happy to read any counter-criticism.


1. This refers to how your model actualy works. I understand you'll try different strategies on the blinds, and play them, each, for a whole HU tourney. However, you state:

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Also assume that when you're on the button, your play is dead even, so we're only going to look at a series of plays with you on the big blind, deciding whether or not to call.

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If so, how will your simulation work up those "button" situation? Is it going simply to skip them, and leave Hero on the blind for the whole game, constantly puting a fixed amount of his stack in the pot? This is, obviously, very bad for him. Or is it that each time hero is on the button, both opponents push (that's a good simulation of "even play")? I think the way you are treating this problem might have significant affects on the outcome.

I guess you can solve it by stating that both players are paying equal blinds on each hand. However, this is *very* different from real HU tourney situations, when BB puts two times what button puts. This can be extremely important with high blinds and equal stacks.


2. When you ask: "what is the optimal play against this maniac?", and on the other hand state that both of you play dead even (when Hero is on the button), you are obviously saying that your only "advantage" against the opponent is in the range of hands you choose to call his all-ins with. And this goes back to a variation on the eternal "System" debate, i.e., it is a question of how high is the blind comparing to your stack.

If the blind is very high (approaching 100% of your stack) you cannot, by definition, have any "optimal play" against this guy. It's a crap shoot, and his play is optimal as yours (if you call). As your stack grows in proportion to the blind, you can wait for better and better (range of) hands to call. In the opposite example, when the blind is approaching 0% of your stack, your "optimal" play will be simply to wait for AA. His play, in this case, is far from optimal as possible.

Computing the range of hands to call with, for each proportion of blinds/stack, should be a fairly easy mathematic problem, IMO. I don't see why a simulation is needed here. Please elaborate.

PrayingMantis

[Mergualdo]

The way I suggested can be solved, right? No need for simulation. Yes, that's right. Try the simplest case of each player has two chips, one chip for a single blind, and hands can be only high, medium or low. 1/3 probability of each. That is a problem you can write down and solve, probably by hand, certainly with Mathematica or something. Then extrapolate that to deeper stacks and more types of hands. There should be some algorithm to efficently solve that type of huge problem somewhere.

[citanul]

This sounds much like a cash prize question DS asked just about 2 months ago.

Karlson made computations of which hands to go all in with NL heads up as the optimal strategy with a stack of X, and which hands to call with with a bet to you of X, or however you want to put it.

Do a search for Karlson - Sklansky hand rankings, I think it was in the General Theory section. There should be a link somewhere to a web based version of the results which are enlightening.

citanul

[Bozeman]

My guess: best 40% of hands.

[Moonsugar]

[ QUOTE ]
I've thought about this a fair bit and have a hunch that deviations of chipEV from $EV are rare and probably not that important in the grand scheme of things. Still, I'd love to know the shape of the chip EV/$EV curve--where are the nonlinearities, etc.

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They happen all the time. I can't tell you how many times I have made money being on the bubble with a short stack by folding in a pos EV situation.

[eastbay]

[ QUOTE ]
This sounds much like a cash prize question DS asked just about 2 months ago.

Karlson made computations of which hands to go all in with NL heads up as the optimal strategy with a stack of X, and which hands to call with with a bet to you of X, or however you want to put it.

Do a search for Karlson - Sklansky hand rankings, I think it was in the General Theory section. There should be a link somewhere to a web based version of the results which are enlightening.

citanul

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I remember it. That was a related, but different question.

Thanks, though.

eastbay