Chris Ferguson (Game theory problem)

[gomer]

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.

[winky51]

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.

[sweetjazz]

[ 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.