[TNixon]

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tnixon, there is no third level thinking here because we're solving for game theory optimal solutions.

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If your opponent deviates from equilibrium strategy, then you can improve your results by making the appropriate adjustment.

The unexploitable equilibrium strategy in not optimal in every situation, and if your opponent deviates from equilibrium, you can (and should) exploit that fact.

Jay, here's what I believe to be an EV calculation for betting 67+ against somebody who calls 34+, that is equivalent to yours (in that it ignores the posted small blind).

*2/3 you fold: 0
1/3 you bet 67+:
*1/3 of 1/3 BB folds: +1.5
2/3 of 1/3 BB calls
*1/2 of 2/3 of 1/3 BB is under 67, you win: +3.5
1/2 of 2/3 of 1/3 BB is 67+
*1/2 of 1/2 of 2/3 of 1/3, you win: +3.5
*1/2 of 1/2 of 2/3 of 1/3, BB wins: -2.5


2/3 0
1/9 +1.5 (.167)
1/9 +3.5 (.389)
1/18 +3.5 (.194)
1/18 -2.5 (-.138)


Probability check: .666 + .111 + .111 + .056 + .056 = 1, so we've accounted for everything.

.167 + 3.89 + .194 - .138 = .612

If folding 66- and betting 67+ is higher EV than betting 100% against this particular opponent, then this statement:

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This shows that raising with any number is better than folding

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Is false.

Should your opponent be calling anything 34+ there? Absolutely not. But that's why this goes to third (and possibly fourth) level thinking. If he thinks you're betting only 67+, he should call somewhere around 78 or 79 or better, and he's making a profit, because you're not betting enough hands.

**EDIT IN PROGRESS**
I just realized that I may have got something wrong in the equity calculation above. Still, it appears that the end result is almost certain to be higher than .167. More in a minute.

***EDIT COMPLETE***

Ok, yeah, I got it wrong. I figured you would win 2/3 of the time when the big blind called, but you actually win 3/4 of the time (all the time when he is lower than 67, and half the time when he is higher than 67). But that makes this *higher* EV than what I had figured.

I substituted the correct values above.