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 Poker Theory General poker theory

 06-12-2012, 12:24 AM #1 newbie   Join Date: Mar 2011 Posts: 32 Solving a hand Can someone explain to me how to calculate GTO for a hand. You can use an example if its easier, make whatever assumptions, etc just curious about how to do this
 06-12-2012, 08:21 AM #2 veteran     Join Date: Aug 2009 Location: Stanford, CA USA Posts: 3,399 Re: Solving a hand Here is an example; You are SB at 100bb depth vs BB also 100bb and your hand is KK face up. Opponent has random hand and can see your KK. Find a way to play your KK to maximize your profit from it. This means design an approach to play it that the opponent cannot exploit. Obviously you know open shoving is profitable because he needs to have AA,KK to call you and they will have it only 0.531% of the time so you will then lose 54.76bb but the rest of the time you will be gaining 1bb so overall open shoving is taking your stack to 0.00531*45.24+(1-0.00531)*101=100.7bb for a profit of 0.7bb by shoving; This is not a strategy that can be exploited. But is it optimal? We do know if it exists (and its not open shoving) it has to be leading to larger than 0.7bb profit and clearly its definitely below 1bb since if it werent it makes no sense because opponent can be folding all hands but KK,AA and pushing with them when you raise and end up with a result better than folding everything= -1 or +1 for us. So clearly we know the solution is less than 1bb and higher or equal to 0.7bb. If one can find a strategy that is different than open shoving eg raising to 30bb or something and then playing post flop that results in a better outcome regardless how opponent reacts then we have improved on the open shove. Another practical problem is to solve how to play say a freezeout of 3bb vs 3bb (winner take all tournament both hands hidden). Determining what hands to play from sb and how is the solution. How to solve for the GTO solution is not a trivial problem and depends on the situation. We could try to solve the above 2 examples to see the process. Last edited by masque de Z; 06-12-2012 at 08:26 AM.
 06-12-2012, 08:52 AM #3 journeyman     Join Date: Mar 2012 Location: Mexico Posts: 256 Re: Solving a hand what is GTO? looks like an interesting thread
06-12-2012, 08:53 AM   #4
Pooh-Bah

Join Date: Aug 2007
Location: ithaca
Posts: 5,049
Re: Solving a hand

Quote:
 Originally Posted by CarliC what is GTO? looks like an interesting thread
check out the theory terminology sticky

06-12-2012, 09:15 AM   #5
veteran

Join Date: Aug 2009
Location: Stanford, CA USA
Posts: 3,399
Re: Solving a hand

Quote:
 Originally Posted by CarliC what is GTO? looks like an interesting thread
http://en.wikipedia.org/wiki/Game_theory

http://en.wikipedia.org/wiki/Strategy_%28game_theory%29

http://en.wikipedia.org/wiki/Nash_equilibrium

Basically its a solution (ie a well defined strategy of how to play given what is happening from opponent and your own data) that no player can deviate from without ending up in a worse result on average.

Think tic tac toe and how you need to play so that nobody wins. Knowing how to play regardless how opponent chooses to play prevents them from winning and if he/she knows what to do also you cant win either. So at this point both players are in equilibrium and they cant win unless one deviates from that solution.

In the poker case every well defined game situation will have some strategy how to play it that yields a result that cannot be reduced by the opponent doing anything. The opponent can only make things worse (for them) by not knowing hot to respond to such strategy.

Such strategy becomes valuable if one assumes perfectly rational knowledgeable opponents so it doesnt necessarily become the best choice in real life where people play making many errors as if they are not rational or knowledgeable, either because the problem itself is ultra complex or because they are bad at choosing ways that are not easily identifiable as naive after brief study. To exploit those errors you can either try to create a strategy that is fit to manipulate opponents exploiting their bad choices assuming those choices persist or select to play GTO if known (usually in the most general kinds of poker situations its not known but may be approximated). However in doing so you open the door to becoming yourself exploitable (by the very definition of GTO) because that strategy is generally different than Game theory Optimal (GTO). So you choose yourself an exploitable strategy to exploit your opponents better than what you would do if you chose a GTO strategy that would still lead to decent result but not as great in general. It is exploitable but if nobody realizes how to exploit it, its a better alternative!

So you exploit opponents 2 ways when they play non GTO themselves. You either play GTO if you know it or you create an exploitable strategy that capitalizes on their mistakes which itself is an exploitable strategy if these opponents changed and decided to exploit yourself too by adjusting properly. Eg if you steal too often against a tight player you become yourself exploitable if that player changes gears and counterattacks with a better frequency than their tight nature had them do before. If you had tried to steal optimally they wouldn't be able to exploit you but you would be losing profit (opportunity loss) because in most cases people make errors they fail to notice easily or they notice too late.

In any case knowing for a situation the Nash equilibrium solution (GTO here) allows you to play in a way that is not exploitable, basically you secure the minimum value that your situation guarantees if your opponent knew how to play ideally.

Here is an example for a 2 player game with 1 prize when their stacks are 10bb each in the push or fold mode (usually good enough for small stacks). The suggested behavior in pushing or calling ranges constitutes the Nash Equilibrium of that problem. Deviating from it will lead to worse results by the one doing it if the other remain in it.

http://www.holdemresources.net/hr/sn...6=&s7=&s8=&s9=

What this means is that if the SB pushes wider like say 80% of hands or tighter like 10% of hands the BB by maintaining the calling ranges or the pushing ranges when they themselves become SB they will do better on average.
Clearly such strategy changes as the stacks change so it needs to be updated from one move to another.

Notice also the concepts suggested above. If you know the SB is very tight you can modify the calling range and make it tighter , much tighter actually (why call a tight range with semi good hands?) . In doing so you can be exploited if the SB turned to Nash solution and pushed wider. If they cant adjust though and continue to be tight your best plan is to call tighter than Nash suggests when they push and to push as Nash suggests when its your turn to be SB. This is a different strategy than the equilibrium one that will be exploited if the opponent stopped being so tight. But until that happens its a strategy that is superior to the Nash solution. Against an unknown opponent until you establish more information you can always play according to Nash solution and this secures that you cannot be denied what your situation deserves if both played perfect.

Last edited by masque de Z; 06-12-2012 at 09:36 AM.

06-12-2012, 09:19 AM   #6
journeyman

Join Date: Mar 2012
Location: Mexico
Posts: 256
Re: Solving a hand

Quote:
 Originally Posted by yaqh check out the theory terminology sticky
nice work sr very informative ty

Quote:
 Originally Posted by masque de Z How to solve for the GTO solution is not a trivial problem and depends on the situation. We could try to solve the above 2 examples to see the process.
it would be nice to see the process

edit. masque just saw your other post ty for your great explanation

 06-12-2012, 09:21 AM #7 Pooh-Bah     Join Date: Mar 2009 Location: Villain Posts: 4,881 Re: Solving a hand nice posts masque
06-13-2012, 01:59 PM   #8

Join Date: Jul 2009
Posts: 1,026
Re: Solving a hand

Quote:
 Originally Posted by Paul Valente nice posts masque
+1

To add to the discussion, here's an interesting COTW thread (Concept Of The Week) from the "Micro Full Ring" section:
http://forumserver.twoplustwo.com/78...egies-1148105/

 06-13-2012, 09:46 PM #9 Pooh-Bah     Join Date: Mar 2009 Location: Villain Posts: 4,881 Re: Solving a hand Here's a good article by Phill Newall that shows some application of game-theory. http://www.twoplustwo.com/magazine/i...s-with-gto.php I recomend all of his 2+2 magazine articles. And his book is awesome too.

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