Hey, an actual NLHE game theory question/simulation.

[Nichomacheo]

I'm trying to make a program that solves simple jam/fold situations for me. Here's the situation:

SB vs BB. Both start with stacks of 10, SB posts 0.5 and BB posts 1. SB can shove/fold and BB can fold/call.

Currently I am doing this with a deck consisting of only aces king and queens. All-ins are evaluated by a program simular to PokerStove.

My preliminary results are that the SB should push with AA,KK,AKs,AKo and the BB should call with AA and KK only.

If anyone else could verify this, I would be thankful and impressed. If not, in general, how does this sound? I can provide some numbers for people that want to help me debug it.

/\/

[HP]

I am too lazy to verify this myself, but one way to verify this is too:

Use pokerstove. Double check that KK is at least a 0 EV call for BB. You can make jacks and lower dead cards i believe. Then, double check that AKs and QQ are at most a 0 EV call

After that, double check that AKo is at least a 0 EV push (where folding is considered 0 EV) for SB given that BB calls with AA/KK. Then, double check that pushing AQs and QQ are at most 0 EV

If all of the above are true, you've got yourself a Nash Equilibrium, and both players are playing in an un-exploitable fashion. I'd define that 'perfect poker'

[ QUOTE ]
If not, in general, how does this sound?

[/ QUOTE ]
Not good. I suspect you may be wrong. I suspect the un-exploitable strategy for the BB should be something like "call AA/KK 100% of the time, call with AKs 32.83% of the time, fold everything else"

I think it's way too much of a coincidence there are no 'boderline' hands (hands where they fold some non-zero percentage of the time, but don't fold some non-zero percentage also) in either of the player's ranges.

[HP]

Alright, you are lucky I am bored

I checked the EV of the SB pushing AQs:

If called by AA/KK, we have an equity of 31.846% if I've done it right

but according to my counting, we only get called 1 out of 5 times

so if we say the EV of folding is -0.5, the EV of pushing is:

0.8 + 0.2*(20*.31846 - 10)

which equals

0.0738

So unless I've made a mistake, pushing AQs from the SB is +EV if the BB is only calling AA/KK

[Nichomacheo]

Yeah, I'm crunching away at this... More time!

[HP]

While in the shower I thought of a way to find a Nash Equilibrium

First, suppose BB calls with AKs some percentage of the time. Call it X
Now, write out the EV of SB pushing with AQs in terms of X
Set this EV equal to the EV of SB folding. Now you can solve for X.
So now SB is indifferent to pushing or folding AQs

After you have done this, now suppose SB pushes AQs some percentage of the time. Call it Y
Write out the EV of BB calling with AKs in terms of Y
Set this EV equal to the EV of BB folding. Now you can solve for Y.
So now BB is indifferent to calling or folding AKs

And now you have a Nash Equilibrium

[jukofyork]

Download this: http://econweb.tamu.edu/gambit/

Then goto this page and follow the flash tutorial to have some idea what you're doing.

Then download and have a good look at "4cards.efg" (at the bottom of the same page).

You should be able then to work out how to get this working for your game using equities generated in poker-stove. Atm, I'm just trying to work out how to write some code to generate the ".efg" file automatically for the full push/fold game and then adapt it to allow for limps/checks using a simple heuristic instead of real post-flop play.

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

[Albert Moulton]

Isn't this what the section on Sklansky-Chubukov Numbers in No Limit Holdem Theory and Practice is all about?

Have you read that section of the book in relation to the problem that you are working on?

[ QUOTE ]
N_call and N_fold are the number of hands that will call/fold given that you move in with the maximum stack that you will do so with. P|call is the probability of winning given that you are called (plus 1/2 the probability of tieing). The last number is the original question.

S-C Numbers



[/ QUOTE ]

Armed with these numbers, NLHETAP gives a chart on opponent's pushing hand ranges (from any two cards, to just QQ-AA/AK), and then gives the hand ranges for +EV calls vs those ranges based on the S-C numbers.

Aren't you working on the same kind of thing?

[mojed]

I think the SC numbers are when we push with our hand face up, and hence are based purely on the probability of the opponent having a hand that has enough equity to call against our specific hand, not against our range.

The Jam/Fold numbers in The Mathematics of Poker consider our range in the opponents decision to call, and hence give the optimal solution.

[rufus]

[ QUOTE ]

If anyone else could verify this, I would be thankful and impressed. If not, in general, how does this sound? I can provide some numbers for people that want to help me debug it.

[/ QUOTE ]

You could compare to Alex Selby's optimal preflop which can be used to simulate pre-fold jam-fold by setting bet sizes.