blitzing on third and long and the problem with poker forum advice

[Subfallen]

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I think you have this confusion exactly backwards. First, I am not necessarily talking about facing "maximally exploitive opponents" when I am talking about using mixed strategies in the OP or elsewhere in this thread. Second, even when playing "maximally exploitive opponents", I'm saying I believe that there are times a mixed strategy is called for even when the various parts of the mix have different expected values. In your earlier post you denied this assertion, and then claim that it all flows from a DEFINITION, of all things. And then you call me clueless.

I'm not going to get into a flame war with you, and maybe I'm wrong in my second assertion (though I doubt it), but you are showing an incredible lack of reading comprehension.

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Wow. Let me walk you through this, one step at a time.

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First, I am not necessarily talking about facing "maximally exploitive opponents" when I am talking about using mixed strategies in the OP or elsewhere in this thread.

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Yes, you're talking about using exploitive strategies, not optimal ones. And exploitive strategies are always pure strategies. It doesn't make sense to talk about mixing strategies against exploitable opponents, and I defy you to provide any example to the contrary. (Hint: none exist. Oscillating between different pure strategies can be indicated, but never mixing.)

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Second, even when playing "maximally exploitive opponents", I'm saying I believe that there are times a mixed strategy is called for even when the various parts of the mix have different expected values.

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Here you are also categorically wrong. The very definition of an optimal mixed strategy in game theory requires that the mixed strategy yield equal EV across all actions vs. a maximally exploitive opponent.

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In your earlier post you denied this assertion

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No.

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And then you call me clueless.

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Yes. If you ever swallow your pride enough to actually acquire even the most cursory familiarity with game theory, you will agree.

[RoundTower]

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WTF GUYS THIS IS NOT A POINT UP FOR DEBATE THIS IS A MOTHERF'ING DEFINITION WE'RE TALKING ABOUT HERE. MAYBE I SHOULD START A THREAD ARGUING THAT 6 IS A PRIME NUMBER. HUH? PROVOKE SOME GOOD DISCUSSION THERE, YES?

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hey I have a degree in game theory (among a lot of other things, lol), and this is not trivial to me.

we are not arguing over the definition of an optimal strategy. I want an example to the following, or a proof that it does not exist:

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[in a poker like game] a mixed strategy in which some of the elements have different expected value from the others is maximally profitable

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Also I think Pete is one of the best posters on the forum, and it's not close. He is not hard to understand though he may be wrong here.

[pete fabrizio]

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There are some situations where you can prove this is the case (see my example about dry-sidepot bluffing).

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I disagree, I don't think you have proved this. For a start you failed to consider the case where sometimes you value bet light into the dry sidepot on the river. I would like to see a better example (doesn't have to be a poker example, a contrived game might be even better). I'm not 100% sure if this is possible, a year ago I could have worked it out.

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My example is all about being able to thin-value bet into the dry sidepot, but it works either way. Say you know squarely that your opponent has top pair, and both you and your opponent know that the third player has middle pair. Now based on the action, your opponent knows (correctly) that you have either a missed straight draw (with undercards so you play the board) or a straight. If you don't ever dry-sidepot bluff when you missed, you will not ever be able to value-bet your hand when you hit, since he will correctly fold every time you bet. On the other hand, if you bet your missed straight a small percentage of the time (just enough to offer him enough pot-odds that he has to call against your range), then you extract value the times that you have the straight, which will be a large majority of the times that you bet the river overall, and you profit. So the times you don't have the straight, the correct strategy is majority check, minority bet, even though "check" has 0 e.v. and "bet" has negative e.v. Is there a mistake here?

[Subfallen]

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I want an example to the following, or a proof that it does not exist:

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[in a poker like game] a mixed strategy in which some of the elements have different expected value from the others is maximally profitable

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Try slapping yourself really hard and then reading it again.

If it still isn't obvious, observe that exploitive strategies are pure strategies for the simple reason that we never choose not to exploit our opponent when his strategy is exploitable. If he's exploitable, we exploit him 100% of the time or we give up EV.

[pete fabrizio]

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Yes. If you ever swallow your pride enough to actually acquire even the most cursory familiarity with game theory, you will agree.

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I have a more than cursory familiarity with game theory. Whatever your level of familiarity, your posts in this thread reveal that you greatly overestimate your understanding. But regardless, I'm through arguing with you.

[grizy]

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I want an example to the following, or a proof that it does not exist:

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[in a poker like game] a mixed strategy in which some of the elements have different expected value from the others is maximally profitable

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Try slapping yourself really hard and then reading it again.

If it still isn't obvious, observe that exploitive strategies are pure strategies for the simple reason that we never choose not to exploit our opponent when his strategy is exploitable. If he's exploitable, we exploit him 100% of the time or we give up EV.

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Uh... an exploitable mixed strategy often (not always) can only be exploited by another mixed strategy.

I think you misunderstand chen and akerman's argument... they are not arguing mixed strategy is not optimal; they are arguing against most opponents, who are not using a mixed strategy or even half way observant, the optimal strategy is almost always pure. This IS consistent with what Pete and Tower are saying.

[Truthiness24]

We listen, commiserate, and give our input, all in a vacuum. But so what? At least we are giving a "default" answer. The answer to any question of "what would you do?" is always "It depends." If you aren't astute enough to see that, to take it for what it is, then you shouldn't be here.

It is still helpful. Sometimes I say "aha .. i didn't think of it that way." Hopefully you still have those "aha" moments too. Isn't that why we do this?

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Isn't the optimal play, or mix of plays, wholly dependent on the makeup of your table and the hand? Game theory is nice, but all it does is analyze empirically certain truths that the best players know instinctively.

Leaning too much on empirical study of poker is like learning how to pick up girls by taking Mystery's class -- you can get functional, even skilled, but you're still lame. Lean too much on numbers and you miss out on a whole lot of the mindgame, which is what makes poker glorious.

I think we are all right -- kind of -- but dancing around the question, approaching it from different angles. I think that this is the great part of poker forum advice.

[grizy]

Post deleted by grizy

[grizy]

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Second, even when playing "maximally exploitive opponents", I'm saying I believe that there are times a mixed strategy is called for even when the various parts of the mix have different expected values.

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Here you are also categorically wrong. The very definition of an optimal mixed strategy in game theory requires that the mixed strategy yield equal EV across all actions vs. a maximally exploitive opponent.

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This is only true under specific conditions, and definitely not in Pete's scenario. Namely, in Pete's scenario, your opponent's "strategy" is not independent from Pete's mixed strategy. Figure this out yourself.

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Ultimately, as far as we're concerned as poker player, the optimal mixed strategy is the one that has the maximal EV in the payoff matrix, let's just use pete's example... x=probability of betting without straight, y=probability of you having a straight, and z=villain's probability of calling.

**** x **** 1-x-z ***** y
z ** -50 ***** 0 ***** 50
1-z * 0 ****** 0 ****** 0

That's the payoff matrix, and as long as y is dependent on x (the only variable you can manipulate), you cannot definitively say there is no mixed strategy that is better than a pure strategy. Let's just say z=x, and y=2x (this implies 1 out of 3 times you bet, you won't have the straight), your net payoff in this mix strategy would be 100x^2-50x^2... for a net 50x^2 profit in this mixed strategy.

Obviously this is an overly simple game, but it does demonstrate Pete's point.

Note if z is a fixed value (aka your opponent isn't even half way observant or adjusting to your game) then it's never profitable to bet without the straight.

[Subfallen]

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So the times you don't have the straight, the correct strategy is majority check, minority bet, even though "check" has 0 e.v. and "bet" has negative e.v. Is there a mistake here?

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I have a more than cursory familiarity with game theory.

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= does not compute.

Against an maximally exploitive opponent in a clairvoyant situation like the one you describe, bluffing does not have -EV. It has 0EV because his calling frequency makes you indifferent to bluffing.

It gets no more basic than this.

If your opponent is not maximally exploitive, and will be either folding or calling too often, you would not want to use a mixed strategy. You would either always bluff (because bluffing is +EV) or never bluff (because bluffing is -EV.)

What part of this do you not understand?