Chris Ferguson (Game theory problem)
 

I recently read a book which had a game theory related problem originatly presentet by Chris Ferguson...

NL HE 1$/2$

Everyone folds to SB (50.000$) who makes it 5$ to go.
Before BB (50.000$) looks at his hand SB accidently reveals two black aces...

Well apparently game theory somehow dictates that BB can call with any hand and then bet any flop that gives a random hand 15% or higher of winning and the aces will have to fold...
If you know this problem and the reasoning behind the solution then plz try to explain it to me in a way I can understand [img]/images/graemlins/confused.gif[/img]

Re: Chris Ferguson (Game theory problem)
 

I think it would be because now that he knows you'll be making a move X% of the time and actually have a hand that beats him X% of the time (since he knows you have two red Aces), but whether he calls or folds doesn't change your EV from the hand and he can do nothing better than guess if you're ahead of him.

I'm not sure on how to do the math on this (even though I understand it as I read in TOP) but I believe the 15% is just the number thats needed so that he can bluff in such a way that makes what I stated above true (that mathmatically the SB can not change the BB's EV on the hand without a total guess)

Re: Chris Ferguson (Game theory problem)
 

I have not seen this problem before, but it doesn't add up the way you describe it. I'm assuming you mean there is $10 in the pot preflop if BB calls, and SB and BB each have $50 left after the preflop betting.

If BB calls preflop and goes all in postflop, SB has to put up $50 to win $60. He'll do this if he thinks his chance of winning is $50/($50+$60) = 5/11 or better. He'll always better than 5/11 chance against a random hand. So if BB goes all-in all the time on any flop, SB calls him all the time. The worst flop for black AA is triple red suited connectors between 6 and 9, AA is 60% to win against a random hand. Even with a perfect flop, BB loses money with this strategy.

To make this work, BB must consider not only the chance of AA against a random hand, but the chance of AA against the hand BB actually holds. Let's say he goes all in on any hand for which he has a 30% chance of winning, and his average chance of winning on these hands is 46% (I just made these numbers up). If he does this, then SB will fold whenever he goes all in. So he makes $7 when this happens. If he folds with less than a 30% chance of winning, he loses $3 on those hands. If there is 1 of the first for two of the second (again, I made those numbers up), he has positive expected value in calling.

While it's possible to compute the numbrs in the last paragraph, you'd need a computer to go through all the calculations. It's not a simple game theory example. I think there must be something you left out.

Re: Chris Ferguson (Game theory problem)
 

It was Matt Matros's Book right? I don't remember the explanation exactly but I think it was included in the book.

Re: Chris Ferguson (Game theory problem)
 

It's actually a much simpler problem than it first appears. The answer is:you can call with any hand.
The reason?
Because after the flop, if you bet out at it, the guy with the Aces has to fold. He has no idea if you are bluffing or have the goods, whereas you have complete information. It's hard to play poker against someone who knows what cards you hold.

Tommi

Re: Chris Ferguson (Game theory problem)
 

[ QUOTE ]
It's actually a much simpler problem than it first appears. The answer is:you can call with any hand.
The reason?
Because after the flop, if you bet out at it, the guy with the Aces has to fold. He has no idea if you are bluffing or have the goods, whereas you have complete information. It's hard to play poker against someone who knows what cards you hold.

Tommi

[/ QUOTE ]

It's not that simple. You have to use that the stacks are both $50,000, while the pot is only $10 (if BB calls).

The complete information is more valuable than the preflop nuts only because the stacks are deep enough.

Re: Chris Ferguson (Game theory problem)
 

Thank you all for the response!

[ QUOTE ]
I think it would be because now that he knows you'll be making a move X% of the time and actually have a hand that beats him X% of the time (since he knows you have two red Aces), but whether he calls or folds doesn't change your EV from the hand and he can do nothing better than guess if you're ahead of him.

I'm not sure on how to do the math on this (even though I understand it as I read in TOP) but I believe the 15% is just the number thats needed so that he can bluff in such a way that makes what I stated above true (that mathmatically the SB can not change the BB's EV on the hand without a total guess)

[/ QUOTE ]

Well I can follow this far, but I don´t see how it adds up to giving BB a positive EV... Or at least better than EV -2$ per hand. Otherwise it would be a better play to just fold preflop.

a 15% hand, (which a random hand according to the book should be) can make a large bet offering close to 1-1 odds for SB, a little less than 30% of the time, and that bet would, no matter what the choice, have EV 0 for both players.
Well what about the last 70%??? Somehow big bets on the later streets are part of the answer, but I don´t see how... [img]/images/graemlins/confused.gif[/img]

Re: Chris Ferguson (Game theory problem)
 

Question about the hypo- does the SB know that the BB saw his hand?

Re: Chris Ferguson (Game theory problem)
 

i think there's some confusion. if they are left with $50,000 stacks, then he should fold because there's no sense in calling to win $10.
however if they only have $50 stacks, then $10 is a significant percentage of that so he should call.

Re: Chris Ferguson (Game theory problem)
 

If I'm the SB, and the stacks are both $50, I'm going all-in in the dark, on any flop.

Re: Chris Ferguson (Game theory problem)
 

I'd love to see the original problem. I don't think you got it entirely right in the particulars, or maybe the original was flawed - here is my take on it, though.

In order for the call by the BB to be correct, he will have to win the pot 30% of the time (assuming, as we'll see later, that he breaks even on any subsequent action). If he wins 30% of the time, his result for this call is 0.3 x $7 - 0.7 x $3 -- i.e., he will break even.

I'll assume this is HU so the BB acts first, for simplicity. Then after the flop comes, the BB can either bet 50k or concede the pot (Again, i'm ignoring the millions of other things that could happen). As explained in TOP, the BB should bluff with such frequency that the SB is indifferent between calling and folding. In this case, we can ignore the pot relative to the $50,000 bet, so the BB sbould be 'bluffing' half the time that he bets. If he bluffs less, the SB will always fold, and if he bets more often, the SB will always call. (There is obviously another complication here, to wit, suck-outs - i'm trying to simplify this to the point where it makes sense [img]/images/graemlins/smile.gif[/img]).

So now we know that:
1) the BB needs to take 30% of these pots, and
2) the BB should bet twice as often as he is ahead

and i think this is where the 15% comes in...if the BB is ahead on 15% of flops, he can bet 30% of the flops, and break even on the whole deal.

So the question is, does the SB flop a hand that beats AA >15% of the time? If so, he can call (in theory) and make money.

Strictly speaking, i think the SB may need to do better than 15%. Something like 5-8% of the time, AA will flop the best possible hand, and the SB can't bluff at those pots.

Re: Chris Ferguson (Game theory problem)
 

I think you misread the problem. You can call with any hand if the AA player goes all in on every flop. Which means you only call him ($45) when you got him beat. Thus giving even 72o pot odds to call.

Re: Chris Ferguson (Game theory problem)
 

[ QUOTE ]
i think there's some confusion. if they are left with $50,000 stacks, then he should fold because there's no sense in calling to win $10.
however if they only have $50 stacks, then $10 is a significant percentage of that so he should call.

[/ QUOTE ]

If you only have $50 stacks, you can't profitably call, as the guy with AA can push all-in on every flop and still show a profit. (You won't be ahead of AA very often, and you'll often be vulnerable to redraws (when you make the most likely hand to beat AA -- two pair).)

With $50,000 stacks, the BB can bet about 30% of rivers (15% of the time he has the best hand on the river plus another 15% of the time it is a bluff). Since SB can't profitably call the final river bet (say you bluff just a tad less than you value bet), you can now bet about 60% of turns (since if the SB called, he'd be forced to fold half of the time on the river). And with the same logic you can now bet 100% of the flops. The reason deep stacks are important -- on each street you can make a bet that is significantly bigger than the size of the pot, thus essentially giving your opponent only 1:1 odds on a call.

Also, you have to take into account flops in which AA is the nuts (e.g. A [img]/images/graemlins/heart.gif[/img] 9 [img]/images/graemlins/diamond.gif[/img] 6 [img]/images/graemlins/club.gif[/img]) as well as flops in which AA still crushes a random hand (e.g. A [img]/images/graemlins/diamond.gif[/img] K [img]/images/graemlins/spade.gif[/img] T [img]/images/graemlins/heart.gif[/img] and K [img]/images/graemlins/spade.gif[/img] K [img]/images/graemlins/heart.gif[/img] K [img]/images/graemlins/club.gif[/img]). On the latter flops, the BB cannot bet 100% of the time.

Nevertheless, after all the math works out, the BB can profitably call the raise with the deep stacks, since he can (in principle) play the hand in an optimal manner that will generate positive EV. Of course, there would be tremendous variance in such a play if SB decided to play nonoptimally and call some/all of the BB's bets. If the SB calls the flop and turn, say, then BB must give up about half of the time. Even so, he still has positive EV on the play no matter what SB does.

So while the result is interesting in some ways that perfect information can be more useful than the best possible starting hand, it only happens once the stacks get big enough.