Difficult (I think) Q I posted over in the poker theory forum

[microbet]

Actually, just based on feel, in push/fold mode I like having 7 BBs HU.

Maybe 8.

[eastbay]

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Edit: I mean 10 Big blinds.

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I think this is still too large.

Edit: I think we're pushing a lot here and the BB is calling a lot. I can see that it's over the original 1100 or so that I thought in my first case, but I think we get to the heart of the main strategy pretty quickly when we get past the 23o=fold point.

I'm going for 2400 chips each.

Lori

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I guess there's two questions in here. One is what stack size gives us the max edge, and then what is the critical stack size where we slip into being a dog.

I don't know how long I should drag this out, but I will say you guys are getting pretty close to one or both of those answers.

eastbay

[eastbay]

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Actually, just based on feel, in push/fold mode I like having 7 BBs HU.

Maybe 8.

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A man of fairness. If you like breaking even, you're stunningly close (you've bracketed the answer).

For bastards like Lorinda who want max edges, that's too deep.

eastbay

[the shadow]

Thanks for posing the problem. Here're some random thoughts.

For the moment, I assume that the house's stack in the BB is greater than the SB's stack.

When the SB has a large stack (in terms of numbers of BBs), then the game is unfair to the SB. My speculation is that the BB's edge increases as to SB's stack increases, although not necessarily proportionately.

At some point, the game is fair to both players. Based on microbet's suggestion and eastbay's clue, that point seems to be when the SB's stack is between 7-8 BBs.

As the SB's stack decreases, the game becomes unfair to the BB.

However, at some point, as the SB's stack decreases further, the SB's edge decreases and ultimately the game becomes fair again to both players. For example, if the SB's stack is less than 0.5 BB, the game is a coin flip fair to both players.

I need to think further about the point at which the SB's edge is greatest, but in the meantime I've got a question back to eastbay.

Let's assume for the moment that the SB's stack is greater than the BB's stack. Does your simulation show that the game maps out the same way? If so, that would support the view that the optimal push-call strategy for both players depends solely on the small stack's size in terms of BBs.

The Shadow

[eastbay]

I'm not matching up with someone else's optimality solver, so I want to resolve that before making any more assertions about my solution.

eastbay

[eastbay]

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I'm not matching up with someone else's optimality solver, so I want to resolve that before making any more assertions about my solution.

eastbay

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Ok, looks like the other guy had a bug and he's reproducing my results exactly now, so I'm confident in my answer.

eastbay

[eastbay]

[ QUOTE ]
Thanks for posing the problem. Here're some random thoughts.

For the moment, I assume that the house's stack in the BB is greater than the SB's stack.


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Right, you're always covered, basically.

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When the SB has a large stack (in terms of numbers of BBs), then the game is unfair to the SB.


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Correct.

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My speculation is that the BB's edge increases as to SB's stack increases, although not necessarily proportionately.


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Yes.

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At some point, the game is fair to both players. Based on microbet's suggestion and eastbay's clue, that point seems to be when the SB's stack is between 7-8 BBs.


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Correct. Very near 7.8 BB, the game is fair.

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As the SB's stack decreases, the game becomes unfair to the BB.


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True.

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However, at some point, as the SB's stack decreases further, the SB's edge decreases and ultimately the game becomes fair again to both players. For example, if the SB's stack is less than 0.5 BB, the game is a coin flip fair to both players.


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Also true. Which means that there must be a maximum somewhere in between.

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I need to think further about the point at which the SB's edge is greatest, but in the meantime I've got a question back to eastbay.

Let's assume for the moment that the SB's stack is greater than the BB's stack. Does your simulation show that the game maps out the same way?


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The minimum stack defines the value of the game, since this is all that can get into the pot. If I have 4k chips and BB has 6k, this is no different than if I have 6k and he has 4k.

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If so, that would support the view that the optimal push-call strategy for both players depends solely on the small stack's size in terms of BBs.

The Shadow

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Note that this calculation does not factor in any tournament-esque considerations. That is, we are viewing each hand of this game independently. So there may be issues relating to "survival value", etc., that aren't considered here that are relevant to SnG play. I am simply valuing the game by the expectation of each hand by itself.

I have a way brewing to address that problem pretty rigorously, though.

eastbay

[eastbay]

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I'm not even sure I have the right answer, but I do have an answer.

NL, HU game theory Q

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For deep money, this game is unfair to the SB. Below a critical stack size, the advantage shifts to the SB. This critical stack size is about 7.8 BB. The edge for the SB then increases to a maximum at about 4 BB, and then sharply decreases again.

#BB +/-chips for SB
12 -33.0
11 -25.9
10 -18.1
9 -10.1
8 -1.6
7 +6.7
6 +15.3
5 +22.5
4 +26.4
3 +20.6
2.5 +11.8
2 +3.8

At the breakeven point, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q4o+,Q2s+,J7o+,J2s+,T7o+,T4s+,97o+,95s +,87o,84s+,76o,74s+,64s+,53s+,43s (0.631976)

bb call hands: 22+,A2+,K2+,Q7o+,Q3s+,J9o+,J7s+,T9o,T8s+,98s (0.457014)

There are no mixed stratgies (hands that you play some of the time but not always) at this point, interestingly enough.

At the point of maximum edge for the small blind, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,54s (0.737557)

bb call hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,53s+ (0.740573)

Again, there are no mixed strategies, and moreover, within one hand they are playing the same strategy! I suspect this may point to a deeper principle of some kind, but I'm not sure what it might be.

So, my answer to the original question is: "I would insist that my stack be less than 3120 chips, and if I get to choose my stack, I choose 1600 chips."

eastbay

[the shadow]

eastbay, thanks to you and Marv for your work on these simulations.

The Shadow

[the shadow]

[ QUOTE ]
At the point of maximum edge for the small blind, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,54s (0.737557)

bb call hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,53s+ (0.740573)


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It's interesting that the gap disappears (and is even just fractionally negative) at this point.

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So, my answer to the original question is: "I would insist that my stack be less than 3120 chips, and if I get to choose my stack, I choose 1600 chips."


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If you push and lose, the game's over. Even if you push and win, it's time to cash out (at least half your chips) since the game's now unfair to you.

The Shadow