[eastbay]

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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