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Hi guys,

Thanks for the input. Note that I am not claiming that this result is either original or practical ... just a poker theory idea that interested me.

AaronBrown, excellent example with AAAA2, you hold 22, impossible even to split. Solid point. So let's ignore splits for now, as you suggest.

But when you write:

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I think you define P as the probability that you will win or split, Q as the probability you will win and S as the minimum of the two stack sizes after you call bet B.

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I actually defined Q to be the conditional probability that you will stack your opponent given that your hand ends up being best by showdown. So regardless of your opponent's actions and/or other factors, Q will be a positive number, even if exceedingly small in certain situations. E.g., you call the $25 bet in the $1-$2 example, flop quads, and your opponent flops ace-high. Then Q > 0 regardless of S since your opponent may make a series of crazy bluffs, turn an ace and refuse to believe you're sitting on the fourth deuce, etc.

So now let us re-define P as per your guidance such that P is the probability that your hand would prove best if dealt to showdown (considered at the time of the call/fold decision), and add on the hypothesis that the situation is one where there is some chance of an outright win, i.e., P > 0.

With these clarifications and added assumptions, I believe the simple S > B / (P Q) is still valid.

Best Regards,
Collin

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Aside from reverse implied odds considerations (as in somehow you'd never pay him off if he gets a better hand than yours), you're ignoring that Q is a function of S. HTH.