Game Theory: Unusual Question #3 and #4

[Jerrod Ankenman]

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If you looked at the assumption made in my previous post

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If A's strategy is:
[0,x] : raise(bluff)
[x,y] : fold
[y,z] : call
[z,1] : raise
And B's strategy is:
[0,a] : fold if raised
[a,1] : call if raised,
where 0<x<y<a<z<1

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you see that it doesn't matter which hands in [0,y] will be bluffed with,
since z>y anyway!

But if you don't assume A would have thought of that, this would not even
necesserily be B's (co-)optimal strategy I think.

Next Time.

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Actually, it would. The only requirement that a strategy pair be optimal is that neither player can unilaterally increase his equity by changing strategy. So your solution is in fact optimal.

What's also true is that all strategies where B bluffs the right number of hands in the range below calling are all co-optimal.

The fact that a strategy <A> is dominated doesn't preclude it from being optimal; it simply means that there exists another strategy <A'> which performs better than or equal to <A> against all counterstrategies. The strategy where you bluff [0,z] does worse against some suboptimal strategies from A (like where he calls with some bad hands) and is dominated by the strategy where B raise-bluffs his best folding hands rather than his worst.

Jerrod

[PairTheBoard]

This is news to me Jerrod. From the little I've picked up here I had thought that it was always best to bluff with your worst hands. So that's not true? Which works best in actual Poker?

Also, why is it better for B to bluff with his BEST folding hands while it's best for A to bluff with his WORST folding hands in this game.

PairTheBoard

[karlson]

Since A has the option to check, his non-worst hands have some chance to win if he doesn't bluff - hence the "bluff with your worst hands" type of solution common to these types of problems.

However, B is going to fold all hands below X that he's not bluffing with. Since they never have a chance to win if you decide not to bluff with them, it doesn't matter if you bluff with your worst hands or not. So, you might as well bluff with your best hands (below your limping cutoff) just in case A deviates from optimal strategy and calls with a bad hand.

[Jerrod Ankenman]

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Since A has the option to check, his non-worst hands have some chance to win if he doesn't bluff - hence the "bluff with your worst hands" type of solution common to these types of problems.

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In this game (particularly), it also doesn't matter for A either because once he has raised, he can no longer fold. But what you're saying is the right concept; when you are folding a range of hands and some bluffs need to come out of that range, it's best to bluff with the BEST hands you would have otherwise folded, just in case your opponent loses his mind and calls with something stupid.

Jerrod

[PairTheBoard]

ok I see what you guys are saying. But why can't you look at B's bluff raise similarly to A's by way of B's limping hands. B has a range of hands he will limp with. Once in that range he is not folding, similiar to A having the check option. Why shouldn't B bluff raise with his worst limping hands?

And how should I apply this to Poker? I'm on the river and bet into. I have a range of hands to fold,call,and raise. Should I bluff Raise with my worst folding hands, my best folding hands, or my worst calling hands?

It seems to me that it depends on my assesment of my opponents likely holdings. Am I more likely to push out a better hand by bluff raising with my worst, or to get a worst hand to call by bluff raising with better hands. Also, I'm usually going to gain a lot more with a successful bluff than inducing a bad call. Maybe that's the point Aisthesis was getting at in #4 when he talked about A getting more Bang from his Buck when he bluffs.

PairTheBoard

[Jerrod Ankenman]

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ok I see what you guys are saying. But why can't you look at B's bluff raise similarly to A's by way of B's limping hands. B has a range of hands he will limp with. Once in that range he is not folding, similiar to A having the check option. Why shouldn't B bluff raise with his worst limping hands?

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Because he does better by limping with those hands than by bluff-raising with them.

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And how should I apply this to Poker? I'm on the river and bet into. I have a range of hands to fold,call,and raise. Should I bluff Raise with my worst folding hands, my best folding hands, or my worst calling hands?

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Well, in real poker, you should bluff-raise hands that you would otherwise fold that block your opponent from having very strong hands, like hands that contain the ace of the three-flush suit.

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It seems to me that it depends on my assesment of my opponents likely holdings. Am I more likely to push out a better hand by bluff raising with my worst, or to get a worst hand to call by bluff raising with better hands. Also, I'm usually going to gain a lot more with a successful bluff than inducing a bad call. Maybe that's the point Aisthesis was getting at in #4 when he talked about A getting more Bang from his Buck when he bluffs.

PairTheBoard

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What's important to understand are a couple of things:

--When you bluff-raise, you expect to lose the pot 100% of the time when you are called. You bluff-raise your best hands in preference to your worst ones in case your opponent does something stupid, but against an optimal-strategy-playing opponent, your EV is not changed at all by which hands you bluff-raise with. You bluff raise because you want your opponent to call when you value raise. If you never bluff-raise, he can simply fold hands worse than 5/6 to your raises and do better.
--You bluff-raise with hands that you would otherwise fold. You're not trying to induce worse hands to call you. That would be a value raise.

[PairTheBoard]

Thanks Jerrod. I appreciate that explanation.

PairTheBoard