Heads Up Game Theory exercise

[jay_shark]

ahhh I see the problem .

The simulator assumes that if you fold then you're down 0.5 in EV but this is not true . Once you post the sb , it's not yours anymore , so your EV for folding is 0 and NOT -0.5 .
Notice that -0.5 + 0.1666666 = -0.3333333333

[TNixon]

The simulator is not calculating EV.

The simulator is telling you how much you win or lose over a session of a million hands. The total profit or loss has to include the posted blinds.

Not taking the posted blind into account gives you a calculation that is +EV, but where you will actually lose money over time.

Which means the calculation is misleading at best, and flat out incorrect at worst.

*Edit*

The whole point of trying to figure out the EV of betting 100% against an opponent who will call with any number 34 or higher is to figure out if that is a profitable play, is it not?

Simulating 1 million hands shows pretty clearly that you will lose money in that situation, to the tune of 1/3BB per hand.

Therefore, any sort of math that tells you it is profitable to bet 100% *has* to be wrong, does it not?

[jay_shark]

Let me ask you this .

Lets say I post the sb and I can only call instead of raising and villain can only check . I have card # 26 , what is my EV for this game ?

It should be 1.5*25/99 - 0.5*74/99 = 0.00505....

[TNixon]

Simulator says:

Final result: p1 = -471660.000000, -0.471660/hand
Final result: p2 = 471660.000000, 0.471660/hand

But, again, this is not calculating EV. It's calculating total profit and loss.

And it should be fairly obvious that if you're flipping a weighted coin that is only 26% in your favor, and 74% in your opponents favor, that you're going to lose money in the long run.

Yet your EV calculation shows this to be a profitable situation?

Ignoring the posted blind gives you a very misleading (incorrect, IMO) result.

Which would indicate to me that you cannot ignore the posted blind.

Where math doesn't match reality, the math is wrong.

Try this:

Calculate the EV in your original game of folding 52-, and betting 53+, against somebody who will still call with 34+. This *should* end up being higher EV than betting 100% (if it doesn't, then I'm not sure what to say except that the math does not match the reality here).

But if betting 54+ and folding everything else *is* higher EV, that should throw a serious wrinkle into the "if betting 100% is +EV, then folding can never be the correct play" assumption.

[omgwtfnoway]

jay_shark, in mop the games are solved where sb folding is -.5bb in ev, can you explain why this is not the case here.

tnixon, there is no third level thinking here because we're solving for game theory optimal solutions. a strategy is not game theory optimal unless you could tell your opponent your whole strategy and he could still not exploit you with this information. hence, third level thinking is fruitless.

fwiw i also don't think it's correct to raise with all hands here but can you see how an optimal strategy might still be -ev? for very small stack sizes this game may favor the big blind so if both players play perfectly the bb will be a favorite.

[jay_shark]

Maybe this will clear things up .

If you have to decide before the game begins , whether or not you wish to play , then you would safely decline . You save yourself -0.33333333.

BUT , if you are coerced into playing this game; maybe some guy has a gun to your head , and post the sb , then the correct strategy is to move all in 100% of the time . If you employ this strategy then you will be ahead 0.166666 .

Your overall profit is 0.16666666-0.5 = -0.33333333

Hopefully the confusion is cleared . My assumption was that you were in the game which was outside of your control . I probably should have stipulated this but I thought it was understood .

[omgwtfnoway]

yes that clears it up, thanks

[TNixon]

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

[/ QUOTE ]
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:

[ QUOTE ]
This shows that raising with any number is better than folding

[/ QUOTE ]
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.

[jay_shark]

Tnixon , if you bet 67+ only then why the heck would your opponent be calling with 34+ . Villain is getting 2:1 on his call so he should be calling with 78+ .
This checks out since there are 11 numbers between 67 and 78(excluding 78) , and 22 numbers from 79-100 .

Game theory does not apply if your opponent is not playing optimally .

[omgwtfnoway]

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

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


[/ QUOTE ]
you can adjust from an optimal strategy to more effectively exploit an opponent but your strategy is no longer optimal. in order to solve these games we've made the assumption that both players are playing optimally. hence, the maximally exploitive strategy and the optimal strategy are one and the same.
[ QUOTE ]
The unexploitable equilibrium strategy in not optimal in every situation, and if your opponent deviates from equilibrium, you can (and should) exploit that fact.



[/ QUOTE ]the working definition of "optimal" here is "unexploitable" not "maximally exploitive." if you deviate to take advantage of an opponent's poor play your strategy has become more exploitive and is no longer optimal.