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

[eastbay]

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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.


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Good point.

Maybe more interesting is this: what happens 3-way? If you say that you will always fold if there are two people in the pot, and both your opponents are playing strategy X, what is the best counter strategy Y? Is it looser or tighter than heads-up? I think it's got to be quite a bit tighter, to account for the probability that your opponent might get knocked out first.

We'd have to assume a payout structure to do this sensibly based on EV. 50/30/20 seems standard enough.

eastbay

[dana33]

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Maybe more interesting is this: what happens 3-way? If you say that you will always fold if there are two people in the pot, and both your opponents are playing strategy X, what is the best counter strategy Y? Is it looser or tighter than heads-up? I think it's got to be quite a bit tighter, to account for the probability that your opponent might get knocked out first.


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Oh man, I don't even want to think about generating the conditional probabilities I would need for 3-way action! Your simulator is probably better equipped than mine to handle this case.

On the stack:BB ratio and its effect on optimal strategy, I have done a bit more. I collected data for stack:BB ratios of 20:1, 10:1, and 5:1, and fitted a cubic model to each data set. Using each model, I ran a numerical optimization to determine the percent of hands to move in and call with, given an opponent's move in and calling percentages. By trial and error, I varied the opponent's strategy until the best you could do for counter-strategy would be to match his strategy -- meaning each of you would win half the time, and with any deviation from the strategy, you would lose more than half the time.

Based on this analysis, and still using the ranking of the hands against random hands, the unbeatable strategy in each case is:

5:1 stack:BB -- Move in with anything (100%), call with top 75%.

10:1 stack:BB -- Move in with top 70%, call with top 50%. (Note that this differs from the 80%/60% result I posted previously, which was based only on inspection of the data.)

20:1 stack:BB -- Move in with top 60%, call with top 30%.

I trust the trends in the above numbers more than the exact numbers themselves, and they certainly make qualitative sense. The bad news is that, unless your opponent plays much too loose or much too tight, the edge you get from optimal strategy is not that great. Once the blinds get big, it's a bit of a crapshoot no matter how you slice it.

[Im Just A Bill]

I must say, you guys have inspired me. I am relatively new to hold 'em and am just getting back into stats. This is my first post here. If you get some of this stuff posted, I would love to take a look. I could also provide some web space if needed.

Bill

[eastbay]

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Maybe more interesting is this: what happens 3-way? If you say that you will always fold if there are two people in the pot, and both your opponents are playing strategy X, what is the best counter strategy Y? Is it looser or tighter than heads-up? I think it's got to be quite a bit tighter, to account for the probability that your opponent might get knocked out first.


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Oh man, I don't even want to think about generating the conditional probabilities I would need for 3-way action! Your simulator is probably better equipped than mine to handle this case.


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It's probably easiest to just shuffle and deal a real deck, and work from there. Or, for speed's sake, just draw randomly from an unshuffled deck.

eastbay

[ctv1116]

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Oh man, I don't even want to think about generating the conditional probabilities I would need for 3-way action! Your simulator is probably better equipped than mine to handle this case.

On the stack:BB ratio and its effect on optimal strategy, I have done a bit more. I collected data for stack:BB ratios of 20:1, 10:1, and 5:1, and fitted a cubic model to each data set. Using each model, I ran a numerical optimization to determine the percent of hands to move in and call with, given an opponent's move in and calling percentages. By trial and error, I varied the opponent's strategy until the best you could do for counter-strategy would be to match his strategy -- meaning each of you would win half the time, and with any deviation from the strategy, you would lose more than half the time.

Based on this analysis, and still using the ranking of the hands against random hands, the unbeatable strategy in each case is:

5:1 stack:BB -- Move in with anything (100%), call with top 75%.

10:1 stack:BB -- Move in with top 70%, call with top 50%. (Note that this differs from the 80%/60% result I posted previously, which was based only on inspection of the data.)

20:1 stack:BB -- Move in with top 60%, call with top 30%.

I trust the trends in the above numbers more than the exact numbers themselves, and they certainly make qualitative sense. The bad news is that, unless your opponent plays much too loose or much too tight, the edge you get from optimal strategy is not that great. Once the blinds get big, it's a bit of a crapshoot no matter how you slice it.


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A few questions: when you say stack/BB ratio, do you mean total chips out there BB. For example in a 10 person SNG where everyone starts with 1000 chips, and blinds are 500/1000, is that 10:1 ratio? Or are you referring to BB in relation to your stack?

Does your stack position as compared to the other person affect your strategy at all?

[eastbay]

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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.


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I tried a Sklansky/Chubukov type criterion for pushing in:

http://www.decf.berkeley.edu/~chubukov/rankings.html

and it did slightly worse than a fixed strategy that was more aggressive, no matter what fixed percentage calling strategy I attached to it.

Hmmm.

eastbay

[eastbay]

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A few questions: when you say stack/BB ratio, do you mean total chips out there BB.
For example in a 10 person SNG where everyone starts with 1000 chips, and blinds are 500/1000, is that 10:1 ratio? Or are you referring to BB in relation to your stack?


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stack/BB means just that. stack over big blind.

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Does your stack position as compared to the other person affect your strategy at all?

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No. These strategies are fixed for the tournament and are not a function of relative chip position.

I have tried using a push strategy which follows the Sklansky/Chubukov (Karlson) idea for positive EV plays, but it did worse and not better than some other strategies which are independent of stack size.

eastbay

[dana33]

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A few questions: when you say stack/BB ratio, do you mean total chips out there BB. For example in a 10 person SNG where everyone starts with 1000 chips, and blinds are 500/1000, is that 10:1 ratio? Or are you referring to BB in relation to your stack?

Does your stack position as compared to the other person affect your strategy at all?

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So far, I have only looked in detail at the case where the stacks are equal when the heads-up match begins, in which case the ratio is of either player's stack to the BB. I.e., if the BB is 100 and both players have stacks of 1000, that is a 10:1 ratio.

I have run some simulations where the stack sizes differ -- e.g., your stack is 1000 and your opponent's is 2000 (with BB=100). I still need to do the full analysis of the data. But from inspecting the raw results, it appears that if your opponent uses the 60/30 strategy, which would be optimal for equal stacks of 2000 each, then the best counter-strategy is looser than the 70/50 optimal strategy for equal stacks of 1000 each. (I hope that sentence is intelligible!) Short answer: it appears that your stack position relative to your opponent does indeed affect your optimal strategy.

I'll post more once I find the time to analyze my results for unequal stack sizes.

[dana33]

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I have run some simulations where the stack sizes differ -- e.g., your stack is 1000 and your opponent's is 2000 (with BB=100). I still need to do the full analysis of the data. But from inspecting the raw results, it appears that if your opponent uses the 60/30 strategy, which would be optimal for equal stacks of 2000 each, then the best counter-strategy is looser than the 70/50 optimal strategy for equal stacks of 1000 each. (I hope that sentence is intelligible!) Short answer: it appears that your stack position relative to your opponent does indeed affect your optimal strategy.


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Just to clarify: In the simulations I have run, the strategy is not dynamic. That is, when I say a strategy is optimal for this or that ratio of stack to BB size, I mean that it is the best FIXED strategy given the INITIAL ratio once heads-up play begins.

A dynamic strategy makes more sense in practice -- e.g., even if you start with a stack:BB ratio of 20:1, if your opponent steals a bunch of blinds such that you get down to a 5:1 ratio, you will surely need to loosen up at that point. At some point, I plan to enhance my simulation code to support such a dynamic strategy.

[dana33]

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Does your stack position as compared to the other person affect your strategy at all?

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I have now analyzed my results where stack1:BB is 10:1 and stack2:BB is 20:1 at the start of the simulation. It turns out that the optimal fixed strategy for both players is identical, and is equal to the average of the optimal strategies for equal stacks for the 10:1 and 20:1 stack:BB cases.

That is, my prior results for equal stacks say that 70/50 push/call is the optimal strategy for a 10:1 ratio, and 60/30 is the optimal strategy for a 20:1 ratio. For the unequal stack case above, the optimal strategy is 65/40 for both players.

So, given the assumption of a fixed strategy and these results, it appears that the optimal strategy depends only on the mean stack size relative to the BB, not on the stack sizes relative to each other. I'll have to see how these results change once I relax the fixed strategy assumption.