Heads Up Game Theory exercise

[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

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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.

[TNixon]

No, that doesn't clear it up. Because (I think) I just showed that even if you ignore the small blind, folding 66- and betting 67+ against an opponent who will call 34+ is higher EV than betting 100%.

And the simulator shows that betting 54+ is pretty close to optimal against somebody who will call 34+.

This is not really any different than a NL hand, where the small blind has an advantage. There's not a "jam or fold" system out there that will tell you to push any 2 at effective stacks of 3 big blinds.

[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.

[TNixon]

Optimal play would result in each player winning 50% of the time, just like optimal jam-or-fold does.

If you are betting any two cards, then I have trouble believing you are playing optimally, and I believe the fact that you're not taking the blind into account is leading you to believe that you are.

If MoP takes the blinds into account, then you should as well. In a jam-or-fold situation, you don't get to decide whether you're going to post the blind before you get dealt cards. Yet once you have cards, you have to decide whether to actually jam or fold. If those calculations take the small blind into account (and omgwtfnoway says they do), then you would have to take the small blind into account in your situation as well.

If betting 100% is not really optimal, then it should be easy to find a number where your opponent can still play optimally, but that results in a greater EV than betting every hand.

Searching for that right now, using the ways I know.

[img]/images/graemlins/smile.gif[/img]

[TNixon]

[ QUOTE ]
the working definition of "optimal" here is "unexploitable"

[/ QUOTE ]
Fair enough. This is probably just a confusion of terms.

Every time I've seen discussion of the strategy you're talking about (the unexploitable one), though, the term used was "unexploitable" or "equilibrium", not "optimal".

For example, the original SAGE article calls SAGE an equilibrium strategy, not an optimal one.

It only seems logical that "optimal" would vary as your opponent did. But whatever. Now you know what I meant, and I know what you meant. [img]/images/graemlins/smile.gif[/img]

[omgwtfnoway]

sorry for the confusion, most of my exposure to this stuff is from mathematics of poker and so i'm using optimal in the same way it's used in the book.

[TNixon]

According to an exhaustive search using my simulator (which is not exactly precise, because it's based on many random runs, and there is some small amount of variance between runs), optimal play appears to be somewhere in the neighborhood of having hero bets 46+, and villan calls 68+.

That's definitely not exactly right (I had to reduce from 1 million to 10k hands in order to complete the search this brute-force search this year), but it should be in the right neighborhood, and it should be easy to show that having hero bet 46+, with villain calling optimally, will be higher EV than betting 100%.

Do you want to do the EV math, or should I?

[img]/images/graemlins/smile.gif[/img]

[jay_shark]

I will show that there is not a better strategy than the one I mentioned . Again , you can make the assumption that your grandmother gave you 50 cents as a gift so it's not coming from your pockets .

Instead of choosing numbers from 1-100 , lets pick a random number from the interval [0,1]. Note that the two questions are the same when n approaches infinity . In the example , I've used , n=100 .

Let a be your optimal pushing range ; a>=0
Let x be your opponent's optimal calling range . 1/3<=x<=1

We can write a in terms of x . Notice that (1-x)/(x-a) = 2
x=(2a+1)/3

The probability that your opponent wins given that he calls is (x-a) + (1-x)/2 ; x-(3x-1)/2 + (1-x)/2 = 1-x

EV(x): (3x-1)/2*[1.5x + 3.5*x -2.5*(1-x)]

We wish to maximize this function using derivatives . Ughhh

EV(x) = 1/2 *[22.5x^2 -15x +2.5] after simplifying

EV'(x) = 1/2*[45x -15] =0
22.5x = 7.5
x=1/3

Woohoo !!

[TNixon]

Why change the problem, when it's easy to show that there are other solutions that are higher EV than your original calculation, even if we don't take the small blind into account? If there's a single higher EV solution where an opponent is calling optimally, then betting 100% of the time is not optimal.

Here's the math using 44/64 (hero bets 44+, villain calls 64+, which seems to be a little closer to optimal after narrowing the search range down a bit and bumping the hand count back up)

*.44 hero folds: 0
.56 hero bets
*.64 of .56 villain folds: +1.5
.36 of .56 villain calls
*.357 of .36 of .56 hero has < 64, villain wins: -2.5
*.321 of .36 of .56 hero has > 64 and wins: +3.5
*.321 of .36 of .56 hero has > 64 and loses: -2.5

.44: 0
.3584: +1.5
.0719: -2.5
.0647: +3.5
.0647: -2.5

total probability check: .9997, slightly off due to rounding, but pretty close

EV = .5376 - .17975 + .22645 - .16175

EV = .422

Still significantly higher than 0.167, indicating that pushing 100% is not at all optimal.

It appears that not taking the small blind into account led you to the wrong answer, since showing that the EV of betting 100% was positive led to the conclusion that betting 100% was better than folding, and that therefore it could never be correct to fold.

If you used the EV calculation that I originally did, it would have been immediately obvious that we might need to look for a better solution than betting 100%

Gonna go over your other algebra shortly, but I can't help but assume there's a mistake somewhere, because I just gave you a solution where your opponent can call optimally, but that is much higher EV than your original .167.

Seriously, though, this really isn't any different from an optimal jam-or-fold problem in NLHE, where you're not simply pushing with any 2 for 3BBs.

[ QUOTE ]
I will show that there is not a better strategy than the one I mentioned .

[/ QUOTE ]
First you'd have to show me that betting 44+ isn't a better strategy than the one you mentioned.

Which would be rather difficult using the same math that I just used to show that it *is*.

And if villain calling 64+ isn't the appropriate response to player betting 44+ (again, it might not be perfect, but it should be very very close), then tell me what villain's proper calling strategy would be, and compare the EV of that situation to both 44/64 and your original 0/34.