What is wrong with this statement? - Poker Theory

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What is wrong with this statement?

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05-04-2015 , 10:38 PM

XXIV

centurion

Join Date: Dec 2014 Posts: 162

3 player limit game. Players are dealt a hand from a [0,1] distribution where the lowest card wins at showdown. BB and SB post 1 and 0.5BB respectively, effectively making a 1.5 pot preflop. If a player enters the hand he must raise to 2BB. Once the first player raises no other raises are allowed.
If a player raises and there is a call then there is a showdown.

Is it correct to assume the following? "Button should open with a frequency that makes both BB and SB indifferent to calling or folding combined". Assuming we are trying to find an equilibrium solution for the game.

If not, what is wrong? What would be the correct way to approach this problem?

Last edited by XXIV; 05-04-2015 at 10:43 PM.

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05-04-2015 , 11:47 PM

SiQ

Carpal \'Tunnel

Join Date: Jan 2008 Posts: 7,248

well there's huge flaws with the game you've setup for what I think you're trying to figure out.
is 0,1 from a finite deck or does everyone always have 50% chance of having 0 or 50% chance having 1? huge difference between the two cases.

if we're talking about an unlimited deck, there is a 75% chance that at least one player in the blinds will have a 0 - so there is no sense in bluffing with a 1 because it would need to work 57% of the time but it only works 25% and you have 0 equity when called. They won't ever fold when they have a 0 and they shouldn't ever call with a 1.

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05-05-2015 , 12:08 AM

XXIV

centurion

Join Date: Dec 2014 Posts: 162

This is a typical one full street limit game as explained in The Mathematics of Poker, but instead of 2 players I'm trying to figure out for 3 players.
"In [0,1] games both players are dealt a uniformly random number between 0 and 1. This means that each player has an equal chance of being dealt any number between 0 and 1."
The deck does not have 0 and 1 only. It is actually a continuous infinite distribution.

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05-05-2015 , 12:15 AM

SiQ

Carpal \'Tunnel

Join Date: Jan 2008 Posts: 7,248

Yeah, it's an unlimited deck of 0's and 1's, right?

The NE solution for button is to fold when he has a 1 and raise when he has a 0.
After the BTN folds the game is the same as the MOP book.

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05-05-2015 , 12:29 AM

XXIV

centurion

Join Date: Dec 2014 Posts: 162

The "deck" contains an infinite amount of numbers between 0 and 1. That is 0.0023123, 0.13111124, 0.4444444441, etc. I'd like to know if it's correct to model BB and SB as a single player when considering opening the button.

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05-05-2015 , 12:51 AM

uberkuber

veteran

Join Date: Jul 2009 Posts: 3,129

I guess you can see SB and BB as one player being dealt 2 hands and playing the best (which means lowest in this case) of the 2, since there's no reraise allowed. It could be a good approximation, although sometimes it might happen if you limp the button and SB raises that BB will fold a better (lower) hand than SB's.

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05-05-2015 , 07:46 AM

just_grindin

Pooh-Bah

Join Date: Dec 2007 Posts: 5,263

Quote:

Originally Posted by uberkuber

There's only 1 raise which is the open. Everyone else can only call.

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05-05-2015 , 09:10 AM

pasita

adept

Join Date: Mar 2007 Posts: 1,101

Modelling the blinds as one player might not work too well... they're getting different odds, and depending on which one of them calls (or both), the pot size will be different too.

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Button should open with a frequency that makes both BB and SB indifferent to calling or folding

Button will only make one hand for each blind player indifferent. Finding that hand is part of the solution, imo.

Must say it's been a while since I've thought about these, so I might be off. My first attack at this problem would be to write a learning algorithm and see what turns out.

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05-05-2015 , 10:31 AM

uberkuber

veteran

Join Date: Jul 2009 Posts: 3,129

Quote:

Originally Posted by just_grindin

There's only 1 raise which is the open. Everyone else can only call.

Oh I see. Button can only raise or fold then. If button folds, SB can only raise or fold. BB will always call or fold (or win 0.5 BB if folded to him), he will never be in a position to check behind.

Rules are stricter than I had understood.

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05-05-2015 , 10:37 AM

#10

NMcNasty

Pooh-Bah

Join Date: Feb 2004 Posts: 3,990

Quote:

Originally Posted by XXIV

I'd like to know if it's correct to model BB and SB as a single player when considering opening the button.

I'm pretty sure the consensus was for multiplayer games is that there are at least two different types of solutions. For this game there's a solution where both blinds are working together to maximize their EV, and another solution (for a game state) where if only one player deviates their strategy they cannot gain the advantage (the Nash equilibrium). So by counting the blinds as a single entity you're finding something, but it isn't Nash.

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05-05-2015 , 12:46 PM

#11

just_grindin

Pooh-Bah

Join Date: Dec 2007 Posts: 5,263

Quote:

Originally Posted by uberkuber

That's the way I understood it.

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