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Lets consider a NL HU Match 100BB deep (no rake). Both players are playing GTO (game theoretical optimum).

Lets assume in the GTO Strategy the SB open limps or open raises r times and open folds f times, with f + r = 1. r_ev is the part of the initial pot of 1.5BB the sb wins when r.

r_ev <= 1.

Furthermore I guess it is commonly excepted, that the SB has an positiv expectation. Hence: 0 < ev_sb = -ev_bb = -0.5 * f + r * r_ev * 1.5 ev_sb <= 1 (since the small blind can win from a GTO perspective at the most the 1bb from BB)

We get: r >= 0.3333; r_ev >= 0.3333

Are there more known boundaries for the introduced values?

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3 Answers

I think there is a problem with your definition of ev_sb; you've included a recursive definition that doesn't make sense to me.

Also, your definition of r_ev has possibly complicated the problem definition. Instead of

r_ev is the part of the initial pot of 1.5BB the sb wins when r

I redefine r_ev to be the EV to the SB when SB open-limps or open-raises, which is -0.5 * %lose + 1.0 * %win. When doing EV calculations for the SB, you only count SB's contributions when SB loses (ie -0.5), but only count BB's contributions when SB wins (ie 1.0, not 1.5).

The following is known about these variables in GTO play:

  • 0 < ev_sb < 1
  • ev_sb = -ev_bb
  • ev_sb = -0.5 * f + r * r_ev

which implies that

  • 0.5 < r_ev < 1

but r is unbounded (except that 0 < r < 1, by definition). So, r_ev does have a tighter bound, while r has a looser bound, than what you've asserted.

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I don't think anything is proven, not even that ev_sb >= 0, so the only bounds we have are trivial: -0.5 <= ev_sb <= 1.

An easier question is "What ev_sb do people find solving abstracted versions of HUNLHE". It would be be interesting to know the sorts of values people are getting, i.e.

1) The value of the (abstract) game from the SB's point-of-view, and state the accuracy of your solution - so if your solution is actually exploitable (within the abstract game) for 0.001BB/hand, say so.

2) The number of information sets in the game (give the combined total for both players).

3) The effectively stack size. The OP only mentioned in 100BB but data for other effective stack sizes would be interesting too.

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You can get some tighter bounds by calculating the expected value of some specific strategy (call it S1), knowing that the GTO strategy will be >= S1. For instance let S1 be the strategy where the small blind goes all-in with AA, and folds all other hands. It is a pretty simple probability problem to calculate the expected values in that situation. There is a broad class of "all-in preflop" strategies that you can calculate similarly, and the GTO solution is at least as good as the best of those.

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