Game Theory: working model for limit holde'm?

[_D&L_]

Thought I'd ask if anyone knows of any working game theory model's (that are published or on the Internet) for approximating a nash equilibrium for limit holdem. Even a 2 person game is fine.

[almostbusto]

Lee Jones published one for short stack HU NL holdem. thats as close as you will get

[gull]

No.

[justkevin]

Two University of Pittsburgh researchers have come up with a Nash Equilibrium for limit hold 'em.

Of course, you probably meant "texas" limit hold 'em-- theirs is for a special simplified form of limit hold 'em they appropriately have named "rhode island hold 'em."

Still, it's a start and they've also proposed ways in which the solution for the simplified game can be used to approximate the real one.

Links:
http://www.cs.cmu.edu/~sandholm/RIHo...roceedings.pdf

http://delivery.acm.org/10.1145/1140000/...CFTOKEN=6184618

[Shandrax]

I beat GSI in Rhode Island HE almost every time and so often that I lost interest fast. So what does it tell us? Luck outperforms perfect play in the short run easily.

The funny thing is that GSI is randomizing over every decision. The perfect player doesn't know clear cut solutions unless he got the absolute nuts. This is quite interesting, because most people usually want to find the single correct play in a specific situation. GSI proves, that it usually doesn't exist.

Here is the data from the GSI thought process during a sample session where I just won 245 chips, which is one quarter of the starting stacks:

randomizing over: { 0.838331 0.161669}
randomizing over: { 0.595678 0.404322}
randomizing over: { 1.0 5.83824E-11}
randomizing over: { 1.0 1.7296E-10 3.25074E-11}
randomizing over: { 0.866325 0.133675}
randomizing over: { 0.664656 0.335344}
randomizing over: { 1.0 3.41467E-10}
randomizing over: { 0.210927 0.784943 0.00412945}
randomizing over: { 0.0281602 0.97184}
randomizing over: { 0.910428 0.0895722}
randomizing over: { 0.716853 0.214779 0.068368}
randomizing over: { 1.0 1.13397E-9}
randomizing over: { 1.0 1.26685E-9 7.10307E-10}
randomizing over: { 0.408898 0.591102}
randomizing over: { 0.259316 0.740684}
randomizing over: { 2.59441E-12 1.0}
randomizing over: { 0.709331 0.290669}
randomizing over: { 0.337878 0.662122}
randomizing over: { 0.441767 0.558233}
randomizing over: { 1.0 1.56352E-11 1.13013E-11}
randomizing over: { 0.327085 0.672915}
randomizing over: { 0.735231 0.264769}
randomizing over: { 9.05347E-13 0.840855 0.159145}
randomizing over: { 1.0 1.63432E-10}
randomizing over: { 0.857912 0.142088}
randomizing over: { 0.948296 0.0378176 0.0138867}
randomizing over: { 0.866325 0.133675}
randomizing over: { 1.0 9.20872E-11}
randomizing over: { 0.763469 0.233409 0.00312186}
randomizing over: { 0.566018 0.433982}
randomizing over: { 0.139537 0.860463}
randomizing over: { 0.0400841 0.959916}
randomizing over: { 0.0487961 0.951204 1.63846E-11}
randomizing over: { 0.403789 0.596211}
randomizing over: { 0.658084 0.341916}
randomizing over: { 1.0 1.69578E-12 2.56873E-11}
randomizing over: { 0.710259 0.289741}
randomizing over: { 0.671484 0.328516}
randomizing over: { 1.0 6.59776E-10}
randomizing over: { 0.467552 0.395435 0.137014}
randomizing over: { 0.717759 0.282241}
randomizing over: { 1.0 1.35084E-11}
randomizing over: { 0.910428 0.0895722}
randomizing over: { 1.0 3.51845E-11}
randomizing over: { 1.0 8.18084E-12}
randomizing over: { 1.0 1.08642E-11 1.71402E-11}
randomizing over: { 0.568279 0.431721}
randomizing over: { 1.0 2.75511E-12}
randomizing over: { 0.951689 0.0483115}
randomizing over: { 0.838331 0.161669}
randomizing over: { 0.611748 0.388252}
randomizing over: { 1.0 4.14511E-10}
randomizing over: { 1.0 8.87148E-11 8.41871E-11}
randomizing over: { 0.0 0.218108 0.781892}
randomizing over: { 4.17208E-15 5.76241E-11 1.0}
randomizing over: { 3.96114E-13 1.0}
randomizing over: { 0.399335 0.600665}
randomizing over: { 0.676277 0.323723}
randomizing over: { 1.0 1.59815E-12 2.53853E-12}
randomizing over: { 0.566018 0.433982}
randomizing over: { 0.408898 0.591102}
randomizing over: { 0.953209 0.046791}
randomizing over: { 0.731294 0.204964 0.063742}
randomizing over: { 1.21026E-14 0.708122 0.291878}
randomizing over: { 1.56851E-14 0.961972 0.0380275}
randomizing over: { 1.55191E-12 1.0}
randomizing over: { 6.06445E-11 1.0 7.42482E-11}
randomizing over: { 0.399335 0.600665}
randomizing over: { 1.93472E-14 0.905488 0.0945116}
randomizing over: { 1.0 7.41221E-9}
randomizing over: { 6.00423E-10 1.0}
randomizing over: { 6.06288E-16 0.781054 0.218946}
randomizing over: { 0.172996 0.827004}
randomizing over: { 6.28489E-14 0.700917 0.299083}
randomizing over: { 0.710259 0.289741}
randomizing over: { 1.0 2.57472E-10}
randomizing over: { 5.93038E-11 1.0}
randomizing over: { 0.0281602 0.97184}
randomizing over: { 0.664656 0.335344}
randomizing over: { 1.0 1.84763E-10}
randomizing over: { 5.43636E-12 1.0 4.06861E-11}
randomizing over: { 0.996326 0.00367383}
randomizing over: { 0.467552 0.395435 0.137014}
randomizing over: { 0.47156 0.52844}
randomizing over: { 0.992352 0.00764812}
randomizing over: { 0.735231 0.264769}
randomizing over: { 3.21745E-13 0.814655 0.185345}
randomizing over: { 1.0 2.92219E-10}
randomizing over: { 4.84492E-13 0.999986 1.39473E-5}
randomizing over: { 1.0 1.79359E-11}
randomizing over: { 0.30123 0.679896 0.0188733}
randomizing over: { 1.33882E-11 1.0}
randomizing over: { 0.664656 0.335344}
randomizing over: { 1.0 3.41467E-10}
randomizing over: { 0.944571 0.0554286}
randomizing over: { 1.29095E-11 1.0 4.07226E-11}
randomizing over: { 0.676277 0.323723}
randomizing over: { 1.0 1.08688E-12}
randomizing over: { 1.0 9.90443E-14}
randomizing over: { 0.664656 0.335344}
randomizing over: { 0.448255 0.551745}
randomizing over: { 9.61043E-12 0.932169 0.0678311}
randomizing over: { 1.0 4.45337E-8}
randomizing over: { 0.0281602 0.97184}
randomizing over: { 0.399335 0.600665}
randomizing over: { 0.300355 0.699645}
randomizing over: { 9.34622E-12 1.0}
randomizing over: { 0.408898 0.591102}
randomizing over: { 0.3159 0.6841}
randomizing over: { 0.910428 0.0895722}
randomizing over: { 1.0 1.12962E-10}
randomizing over: { 1.0 8.24387E-11 5.7219E-11}
randomizing over: { 0.676277 0.323723}
randomizing over: { 1.0 1.90339E-11 6.94639E-12}
randomizing over: { 0.953209 0.046791}
randomizing over: { 0.963642 0.0363578}
randomizing over: { 1.0 2.14703E-10 1.22451E-10}
randomizing over: { 0.676277 0.323723}
randomizing over: { 0.996519 0.00348146}
randomizing over: { 0.0 0.0 1.0}
randomizing over: { 0.0 1.0}
randomizing over: { 0.885121 0.114879}
randomizing over: { 1.0 5.28436E-10}
randomizing over: { 1.0 2.71564E-11}
randomizing over: { 0.921126 0.0788738}
randomizing over: { 1.0 1.71492E-12 1.84855E-12}
randomizing over: { 0.399335 0.600665}
randomizing over: { 0.0048833 0.995117}
randomizing over: { 8.11853E-10 1.0}
randomizing over: { 1.03072E-12 9.2892E-12 1.0}
randomizing over: { 0.871221 0.128779}
randomizing over: { 1.0 8.10192E-13}
randomizing over: { 1.0 5.03214E-13}
randomizing over: { 0.735231 0.264769}
randomizing over: { 0.0312152 0.968785}
randomizing over: { 0.866325 0.133675}
randomizing over: { 1.0 8.08003E-12 7.82855E-12}
randomizing over: { 0.910428 0.0895722}
randomizing over: { 0.884969 0.115031}
randomizing over: { 0.868617 0.131383}
randomizing over: { 0.860768 0.139232 2.79562E-10}
randomizing over: { 0.735231 0.264769}
randomizing over: { 0.368039 0.631961}
randomizing over: { 7.74517E-12 2.28952E-8 1.0}
randomizing over: { 7.57444E-12 1.0}
randomizing over: { 0.879844 0.120156}
randomizing over: { 1.0 5.64068E-10 4.90645E-10}
randomizing over: { 0.921126 0.0788738}
randomizing over: { 1.0 1.31375E-12 1.52784E-12}
randomizing over: { 0.399335 0.600665}
randomizing over: { 0.866325 0.133675}
randomizing over: { 1.0 1.1264E-12}
randomizing over: { 1.0 3.36388E-12 4.22776E-13}
randomizing over: { 0.951638 0.0483617}
randomizing over: { 0.563508 0.313637 0.122855}
randomizing over: { 0.903679 0.0963214}
randomizing over: { 0.674453 0.322692 0.00285519}
randomizing over: { 2.69441E-10 1.0 1.41813E-10}
randomizing over: { 0.735231 0.264769}
randomizing over: { 3.21745E-13 0.814655 0.185345}
randomizing over: { 1.0 5.59803E-10}
randomizing over: { 1.0 1.23702E-10}
randomizing over: { 0.921126 0.0788738}
randomizing over: { 1.0 6.33142E-12 1.09634E-11}
randomizing over: { 0.885121 0.114879}
randomizing over: { 0.171509 0.694283 0.134208}
randomizing over: { 1.0 1.88617E-11}
randomizing over: { 1.0 4.79993E-11 1.60126E-11}
randomizing over: { 1.33882E-11 1.0}
randomizing over: { 0.696704 0.303296}
randomizing over: { 0.985428 0.00981261 0.00475984}
randomizing over: { 0.921126 0.0788738}
randomizing over: { 1.0 2.27209E-11}
randomizing over: { 0.856481 0.143519 9.57043E-13}
randomizing over: { 0.709331 0.290669}
randomizing over: { 0.563491 0.436509}
randomizing over: { 1.0 1.07363E-9}
randomizing over: { 0.908446 3.01174E-8 0.0915536}
randomizing over: { 0.81677 0.179924 0.00330574}
randomizing over: { 0.710259 0.289741}
randomizing over: { 0.99631 0.00369047 3.64656E-10}
randomizing over: { 0.0382743 0.796769 0.164957}
randomizing over: { 2.8186E-13 1.0}
randomizing over: { 1.0 1.00717E-11 1.67362E-11}
randomizing over: { 0.664656 0.335344}
randomizing over: { 1.98665E-14 0.811764 0.188236}
randomizing over: { 1.0 9.34101E-11}
randomizing over: { 4.01486E-13 1.0 8.8806E-11}
randomizing over: { 1.0 1.47957E-11}
randomizing over: { 8.99969E-13 0.990926 0.00907401}
randomizing over: { 0.467552 0.395435 0.137014}
randomizing over: { 5.1267E-13 1.0}
randomizing over: { 1.0 8.07117E-11 1.44418E-10}
randomizing over: { 0.710259 0.289741}
randomizing over: { 0.99631 0.00369047 3.64656E-10}
randomizing over: { 0.81677 0.179924 0.00330574}
randomizing over: { 0.399335 0.600665}
randomizing over: { 1.5052E-14 0.646606 0.353394}
randomizing over: { 8.9307E-14 1.0}
randomizing over: { 1.0 1.86192E-9}
randomizing over: { 7.68396E-12 1.0 4.24743E-10}
randomizing over: { 1.0 3.07476E-11}
randomizing over: { 0.062838 0.937162 4.43073E-12}
randomizing over: { 0.0382743 0.796769 0.164957}
randomizing over: { 1.75691E-13 1.0 3.437E-10}
randomizing over: { 0.590993 0.265289 0.143718}

[pmiranda.pt]

Hi,

you can find some papers about it by Tom Ferguson (father) and Chris "Jesus" Ferguson at:
Tom Ferguson UCLA Papers

[_D&L_]

Thank you for the links. Pmiranda and Justkevin, the links are quite interesting. I have been working on my own for a while on a program to approximate a Nash EQ for limit poker, and just wanted to spend some time reading about other approaches or methods I could use in my program.

And Shandrax, I agree absolutely that the concept of a Nash EQ differs from most players approach to poker. Players tend to think single-mindedly, should I always do A or B when I have Y. Rather than thinking what proportion of A to B creates an optimal strategy.

One of the drawbacks to a NashEQ is that it could be a weak NashEQ - that is, one that breaks even with every other strategy. For instance, rock-paper-scissors, the NashEQ is 1/3 rock, 1/3 paper, 1/3 scissors. No strategy statisticaly beats it, but it doesn't beat any other strategy either. Might explain your luck, Shandrax, in Rhode Island limit. Again, I'm not familiar with Rhode Island poker - but just my understanding of NashEQ tells me that you aren't likely to break-even or beat the NashEQ for a sustained period unless its a weak NashEQ.

From the time I've spent analyazing Texas limit poker, I can tell you the Nash EQ appears fairly strong. Some non-nash EQ strategies can break even with it, but they are an infinitely small slice of all possible strategies. Making the likelihood of playing a break-even strategy by luck, virtually 0. So in other words, I think there is a pot of gold at the end of the rainbow for solving (or much more likely, approximating) a Nash EQ for Texas limit.

[marv]

[ QUOTE ]
Thank you for the links. Pmiranda and Justkevin, the links are quite interesting. I have been working on my own for a while on a program to approximate a Nash EQ for limit poker, and just wanted to spend some time reading about other approaches or methods I could use in my program.

And Shandrax, I agree absolutely that the concept of a Nash EQ differs from most players approach to poker. Players tend to think single-mindedly, should I always do A or B when I have Y. Rather than thinking what proportion of A to B creates an optimal strategy.

One of the drawbacks to a NashEQ is that it could be a weak NashEQ - that is, one that breaks even with every other strategy. For instance, rock-paper-scissors, the NashEQ is 1/3 rock, 1/3 paper, 1/3 scissors. No strategy statisticaly beats it, but it doesn't beat any other strategy either. Might explain your luck, Shandrax, in Rhode Island limit. Again, I'm not familiar with Rhode Island poker - but just my understanding of NashEQ tells me that you aren't likely to break-even or beat the NashEQ for a sustained period unless its a weak NashEQ.

From the time I've spent analyazing Texas limit poker, I can tell you the Nash EQ appears fairly strong. Some non-nash EQ strategies can break even with it, but they are an infinitely small slice of all possible strategies. Making the likelihood of playing a break-even strategy by luck, virtually 0. So in other words, I think there is a pot of gold at the end of the rainbow for solving (or much more likely, approximating) a Nash EQ for Texas limit.

[/ QUOTE ]

The two obvious opponents to play your bot against are vexbot and sparbot from Poker Academy. (If you aren't familiar with the work of the University of Alberta GAMES group, you should check then out - but I'm sure you are).

It would be interesting to hear how you do as other good nash bots (my own, and GSI) have published results against those two.

(I'm not sure I believe that a perfect nash bot will beat most opponents as this would imply that no nash equilibria is very mixed, while the results for Rhode Island Holdem suggest just the opposite.)

Marv