Simplified heads up NLHE toy game - Page 6 - Poker Theory

CalamityMarcus · 12-09-2009, 03:22 AM

calcul time !!

Jean · 12-29-2009, 11:21 AM

Hi there,

I've been thinking a bit about this equilibrium for BvB preflop NLHE play:

Quote:

Originally Posted by DeuceBuster

Code:

>>> printpokersol(jean,xjean,yjean)
root;FOLD
PureSet Size = 0.2986   MixSet Size = 0.0017   Total Size = 0.3004
                                                   
                                                   
                                                   
                                                   
                                                   
                                        94s 93s 92s
                                            83s 82s
                                            73s 72s
                    96o 86o                 63s 62s
                T5o 95o 85o 75o 65o             52s
                T4o 94o 84o 74o 64o 54o         42s
                T3o 93o 83o 73o 63o 53o 43o     32s
            J2o T2o 92o 82o 72o 62o 52o 42o 32o    
[ T2s 0.5670 ]


root;JAM
PureSet Size = 0.5475   MixSet Size = 0.0192   Total Size = 0.5667
    AKs AQs AJs ATs A9s A8s A7s A6s A5s A4s A3s A2s
AKo     KQs KJs KTs K9s K8s K7s K6s K5s K4s K3s K2s
AQo KQo     QJs QTs Q9s Q8s Q7s     Q5s Q4s Q3s Q2s
AJo KJo QJo     JTs             J6s J5s J4s J3s J2s
ATo KTo QTo JTo     T9s T8s T7s T6s T5s T4s        
A9o K9o Q9o J9o T9o     98s 97s 96s 95s            
A8o K8o Q8o J8o T8o 98o     87s 86s 85s 84s        
A7o K7o Q7o         97o 87o     76s 75s 74s        
A6o K6o Q6o                 76o 66  65s 64s        
A5o K5o                             55  54s 53s    
A4o K4o                                 44  43s    
A3o K3o                                     33     
A2o K2o                                         22 
[ AA 0.5349 ; Q6s 0.8162 ; J9s 0.9163 ; J7s 0.4506 ; 99 0.4245 ; 88 0.8830 ; 77 0.9448 ]


root;LIMP
PureSet Size = 0.1237   MixSet Size = 0.0093   Total Size = 0.1329
                                                   
    KK                                             
        QQ                                         
            JJ          J8s                        
                TT                          T3s    
                                                   
                                                   
            J7o T7o                                
            J6o T6o                                
        Q5o J5o                                    
        Q4o J4o                                    
        Q3o J3o                                    
        Q2o                                        
[ AA 0.4651 ; Q6s 0.1838 ; J9s 0.0837 ; J7s 0.5494 ; T2s 0.4330 ; 99 0.5755 ; 88 0.1170 ; 77 0.0552 ]


root;LIMP;jam;FOLD
PureSet Size = 0.1026   MixSet Size = 0.0013   Total Size = 0.1039
                                                   
                                                   
                                                   
                                                   
                                            T3s    
                                                   
                                                   
            J7o T7o                                
            J6o T6o                                
        Q5o J5o                                    
        Q4o J4o                                    
        Q3o J3o                                    
        Q2o                                        
[ T2s 0.4330 ]


root;LIMP;jam;CALL
PureSet Size = 0.0211   MixSet Size = 0.0080   Total Size = 0.0291
                                                   
    KK                                             
        QQ                                         
            JJ          J8s                        
                TT                                 
                                                   
                                                   
                                                   
                                                   
                                                   
                                                   
                                                   
                                                   
[ AA 0.4651 ; Q6s 0.1838 ; J9s 0.0837 ; J7s 0.5494 ; 99 0.5755 ; 88 0.1170 ; 77 0.0552 ]


_______________________________________

root;JAM;fold
PureSet Size = 0.5520   MixSet Size = 0.0000   Total Size = 0.5520
                                                   
                                                   
                                            Q3s Q2s
                                J6s J5s J4s J3s J2s
                                T6s T5s T4s T3s T2s
                            97s 96s 95s 94s 93s 92s
            J8o T8o 98o     87s 86s 85s 84s 83s 82s
        Q7o J7o T7o 97o 87o     76s 75s 74s 73s 72s
        Q6o J6o T6o 96o 86o 76o     65s 64s 63s 62s
        Q5o J5o T5o 95o 85o 75o 65o     54s 53s 52s
        Q4o J4o T4o 94o 84o 74o 64o 54o     43s 42s
        Q3o J3o T3o 93o 83o 73o 63o 53o 43o     32s
        Q2o J2o T2o 92o 82o 72o 62o 52o 42o 32o    
[ ]


root;JAM;call
PureSet Size = 0.4480   MixSet Size = 0.0000   Total Size = 0.4480
AA  AKs AQs AJs ATs A9s A8s A7s A6s A5s A4s A3s A2s
AKo KK  KQs KJs KTs K9s K8s K7s K6s K5s K4s K3s K2s
AQo KQo QQ  QJs QTs Q9s Q8s Q7s Q6s Q5s Q4s        
AJo KJo QJo JJ  JTs J9s J8s J7s                    
ATo KTo QTo JTo TT  T9s T8s T7s                    
A9o K9o Q9o J9o T9o 99  98s                        
A8o K8o Q8o             88                         
A7o K7o                     77                     
A6o K6o                         66                 
A5o K5o                             55             
A4o K4o                                 44         
A3o K3o                                     33     
A2o K2o                                         22 
[ ]


root;LIMP;check
PureSet Size = 0.4857   MixSet Size = 0.0264   Total Size = 0.5121
                                                   
                                                   
                            Q7s Q6s Q5s            
                        J8s J7s J6s J5s J4s J3s    
        QTo                                        
        Q9o         99                             
        Q8o J8o T8o     88                         
        Q7o J7o T7o 97o 87o 77              73s 72s
    K6o Q6o J6o T6o     86o 76o 66                 
        Q5o J5o T5o     85o 75o 65o 55             
        Q4o J4o     94o 84o 74o 64o     44         
    K3o Q3o J3o T3o 93o 83o 73o 63o 53o 43o 33     
        Q2o J2o         82o 72o 62o 52o 42o 32o 22 
[ K7o 0.1428 ; K5o 0.1349 ; Q3s 0.5914 ; J9o 0.5552 ; T4o 0.8987 ; 96o 0.7876 ; 95o 0.1437 ; 83s 0.1724 ]


root;LIMP;jam
PureSet Size = 0.4540   MixSet Size = 0.0339   Total Size = 0.4879
AA  AKs AQs AJs ATs A9s A8s A7s A6s A5s A4s A3s A2s
AKo KK  KQs KJs KTs K9s K8s K7s K6s K5s K4s K3s K2s
AQo KQo QQ  QJs QTs Q9s Q8s             Q4s     Q2s
AJo KJo QJo JJ  JTs J9s                         J2s
ATo KTo     JTo TT  T9s T8s T7s T6s T5s T4s T3s T2s
A9o K9o         T9o     98s 97s 96s 95s 94s 93s 92s
A8o K8o             98o     87s 86s 85s 84s     82s
A7o                             76s 75s 74s        
A6o                                 65s 64s 63s 62s
A5o                                     54s 53s 52s
A4o K4o                             54o     43s 42s
A3o                                             32s
A2o K2o         T2o 92o                            
[ K7o 0.8572 ; K5o 0.8651 ; Q3s 0.4086 ; J9o 0.4448 ; T4o 0.1013 ; 96o 0.2124 ; 95o 0.8563 ; 83s 0.8276 ]


_______________________________________

I'm wondering whether slowplaying big pairs is mandatory, if so why? From my naive point of view, it seems to me that strengthening the limp range with high pairs could be replaced by strengthening it with (more numerous) other strong hands.

Do you think you could compute a quasi-perfect equilibrium for this game, as you did for the 2-street [0,1]-game, Travis? If it's a lot of work don't bother, I'll statisfy myself with an intuitive answer :-) It would not give us the uniqueness of the equilibrium, but at least tell us that only big pairs are slowplayed for instance.

The nice thing with real poker in view, is that slowplaying big pairs makes even more sense with postflop play being allowed when the hand gets limp-checked. The limper gets some kind of "nuts or air" range of hands on the flop, which is pretty much the optimal way of using big pairs (rather than pushing them and slowplaying bigs cards, which would be terrible with postflop play since the opponent could tell when our slowplays have "missed").

Quote:

Originally Posted by DeuceBuster

Here's my computed Nash equilibrium. This solution could easily be wrong. If anyone could verify, that would be wonderful. It also appears to be a perfect equilibrium, so I would not be surprised to learn that it is the unique Nash equilibrium for this game.

What do you mean by perfect equilibrium? I didn't find a definition for this in wikipedia (only subgame perfect equilibrium, which usually can't exist in poker I guess?). The only thing I found was here: http://folk.uio.no/gasheim/5200f9t3.pdf, Definition 248.1, but I can't make sense out of this definition (the last few lines could be replaced by saying that each sigma_i is a Nash equilibrium??). And under this definition a perfect equilibrium in not unique if I understand correctly.

DeuceBuster · 01-05-2010, 12:59 PM

Quote:

Do you think you could compute a quasi-perfect equilibrium for this game

OK, finally got to this. The quasi-strict is the same as the equilibrium originally posted in post #29. While not conclusive, this evidence strongly supports my assertion that the equilibrium is likely unique. This would not be surprising ... the jam/fold game appears to have a unique equilibrium (I say 'appears' bc I don't know if it has been proven, but I have certainly never heard of alternate equilibria from the ones that have been independently published publicly and privately verified many times).

Quote:

What do you mean by perfect equilibrium?

Many equilibria can include dominated strategies. A relevant discussion is here:

http://www.daimi.au.dk/~bromille/Papers/seqeqsoda.pdf

The relevant part begins with the last three lines of 2nd page and continues through the first column and a half of the next page. In the example given, check-calling with the Ace is an equilibrium, but is dominated by check-raising. I was referring to the "normal-form perfect equilibrium", which excludes dominated strategies, making it a more sensible equilibrium to use against an opponent who may depart from GTO play.

Jean · 01-06-2010, 06:08 AM

Ok thanks, that's great.

I've browsed through several papers without finding a definition for normal form perfect equilibrium. I hate life

So the definition would be: "An equilibrium where both strategies are not dominated?"

Nichlemn · 01-11-2011, 07:41 AM

Bump.

I've been playing backgammon recently and have been impressed at the heuristics and computer programs used to approximate game/match equity. It's made me think about how we could do a better job in poker.

A decision to call or fold heads-up on the river is at least theoretically easy - estimate your opponent's range and whether your hand has sufficient equity against that range to call given the pot odds. Before the river, you have to consider what your hand's EV will be given future betting rounds, which is quite complicated (especially on the fly).

I've wondered if option pricing from finance might be of assistance. For instance, calling preflop against a player who will shove all-in on every flop is very similar to a call option. You pay a "premium" (the amount you have to call) for the right but not the obligation to get all-in on the flop. The "strike price" is the minimum equity required to call given the pot odds. The average equity is not important, what is important is your total expected profit when you "exercise" the option. Like for long financial options, variance increases the value of the "option".

Now, it's rare that you'll get someone who will shove all-in every time on the next street, so the actual option pricing is a lot more complicated. But you could still get a first approximation this way. Perhaps we could come up with a more complex model in which characteristics of the opponent can vary, e.g. looseness and aggression.

Nichlemn · 02-04-2012, 06:24 PM

bump