If you know the answer already... please don't post yet. Yes, you're smart or read it in its myriad of formats in the past. blah blah blah. this is for fresh minds... and I am posting a problem originated by Chris "Jesus" Ferguson.

Suppose you and I are playing NL $1/2 and each of us has a stack of $50K behind us. Yep, $50,ooo. It's folded all around to me on the small blind, and I make it $5 to go. However, before you look at your cards, I accidentally expose my holding - two black Aces.

You know my cards, and I know you know my cards. What are you going to do? What hands will you play against me? How will you play them.

That's the question... don't worry if you're wrong. I was. No shame. But, I'm sure you can google the answer.

and for those of you who find this old hat, despite all its wonderful implications and revelations about how to view poker, please just help me and give me other examples that you have seen - URLs would be helpful too.

Remember, I have two black aces.
What are you going to do in the BB?
What will you fold with? Call? or what?

You can exclude raising any hand. Because just look at what you would do if someone raised you when you had AA? You'd go all-in.

EDIT, I thought more

Reverse Implied Odds

When we hit, we're not getting much
When we miss, we're paying him off trying to chase with more than likely improper odds to chase it.

Anything and everything. It's a classic you know that I know what you have, and any bluffs (or big hits) I make on the flop you HAVE to respect I know what you hold.

I can't seem to google the solution, a link would be nice

Good question... Lemme take a shot at it. I'm probably way off. But what the heck.

Let's start with the basics. Even with 7-6 suited or 5-6 suited which is your best bet to crack Aces, I'm not above 25% to win with even those hands(according to cardplayers preflop calculations). I'd be putting in 3 bucks to get 7 (your initial 2 plus his 5)... when you really need the pot to be 12 or higher to make the call reasonable.

Now the fact that we both have 50k is supposed to change the situation a bit... by forcing me to consider implied odds since this is NL.

BUT. If I end up on a draw, there's a reasonable chance that you aren't going to give me the right pot odds to make a call. So I'd be forced to make a second bad call here, hoping to hit my hand.

If I do hit my hand, what are the odds that you are going to pay me off? Well, if you are considering that I'd only call preflop with a hand that has a reasonable chance of cracking your Aces, and a turn card now makes it appear that I have done so, I don't think I will get much more out of you.

Therefore, I would probably fold just about any hand preflop, if I knew the villain had pocket Aces.

Ok, for simplicity's sake, say I have 55. I tell you I have a PP and will bet on the river anytime my hand wins, about 1 in five times, without bluffing. Since I'm not bluffing, you fold when I bet. That gives me my pot equity of $2 or so. I have to fold pre-flop.

Now, say I tell you I will bet not the 20% my hand draws out but will bluff an extra 10% of the time. When I bet, there's a 33% chance that I'm bluffing. I can then bet $10.50 on the river. You are risking 10.50 to win 20.50, not the odds you need to call so you still fold. Now, I win 30% of the time, my pot equity before the river is $3 and I can call pre-flop and break even.

You take this as far as it goes. I have $49995 with which to work on the river. I can make my opponent risk that much to win $50005 if I bet it all on the river, giving him 1.0002-1 odds to call the river. If I can arrange it just so, I want to bluff just enough that he can't call the river bet. If I tell him I will bluff just shy of 20% of the time, when I make my river bet, I will be bluffing 49.9996 of the time and have a winning hand 50.0004 of the time. He can't call. This gives me a pot equity of just less than 40% or $4. By calling pre-flop, I win a dollar on average by adopting this strategy.

Of course, you have bluff exactly the %ages you say and your opponent has to believe you. I'm not sure how to get over that.

This works with any hand. Figure out your %ages of winning against aces and double to figure out your pot equity. As long as you're above 15% to win, you can call profitably. If you are under 15% (AK say), it's better to fold outright because even with bluffing, you can't push your pot equity above the $3 it costs to play.

Call with anything here. The reason being is that you know exactly what he has and HE IS AWARE THAT YOU KNOW. As long as an Ace doesnt come on the flop, bet/raise about the size of the pot. The guy with AA might fold. This is because he thinks to himself, "He knows I have Aces. Why is he betting? Either I'm beat or he's on a stone cold bluff." In his mind, the AA guy will be running around in circles and might think its not worth it to continue further.

I haven't read all of the replies here, but since he knows that you know he has aces, then you should call without looking at your cards and bet out P percent of the time, where P is the percent that two random cards beat aces on the flop. It doesn't matter what the flop is, (assuming it doesn't help the AA hand) you just bet out with a certain probability so that Chris is indifferent between calling and folding.

I don't know if this is right, but I think the answer is somewhere along these lines.

If he had to respect your bet just because he knew you had aces, then you should bet on any flop (without even looking at your cards).

But if your strategy is to bet every flop, then the villain, I'll call him Chris, can increase his EV by calling, say, half of your bets. Since AA are a favorite over any starting hand, then he will win money on average and you will lose money.

So now we see that we can't bet every flop, and Chris shouldn't respect all of our raises. In order for both players to be making the best moves given their opponent's strategy, they will have to randomize their play.

BTW, this is all assuming that calling is correct.

I've been thinking about it, and if you fold preflop, you lose $0 or $2 depending on your reference point. I prefer to say you lose $0 by folding, since anything you put in the pot is no longer yours. And I am pretty sure that the optimal strategy if you call preflop can't make a profit, so at best it could make $0. So folding could be as good as calling, and it would be better if you disliked risk.

Anyway, maybe we can work out the solution together.

So what's the "official" answer?

Yeah, I'm kind of curious too, but not enough to google it.

Here's one more thing I figured out. The AA player can never bet the flop, because we know his hand so we would only call when we are ahead. In fact, he can't bet any of the streets for this reason.

This means so far what happens is, AA makes it $5 to go, we call, the flop comes, AA checks, we ... ?

I think the answer is one of the following:

Fold ALL hands

or

Call ALL hands

If you call, as soon as you bet the guy with aces will (most likely)fold, so you win the $5. If you don't bet, he will so you will (most likely) have to fold, losing $5.

If you fold, you lose $2.

Is it better to have a chance to either win or lose $5, or to automatically lose $2? For me, I think the answer is easy - fold, putting nothing at risk. :wink:

He's my answer, which of course is distilled from other much smarter players. You always call. Except for a few exceptions when the Aces flop the nuts (in which case the Aces should go all in and end the matter) if your cards have a 3/70 chance in winning, you should play. Any two random cards has a 15% chance to win, but every 2-card hand has those odds. Calling is a +EV move! So, of course you do it unless you don't have the heart.

At this point there is a lot of discussion of strategy and bluffing which I am not fit to discuss on this thread. I will try to get some links up though...

The point of this problem that I see is this: Poker is more about what information you have and not what cards you have. Your opponent has Aces, but you have that knowledge and that gives you the advantage.

More and more I realize as my ability to read people's hands I recognize how much more powerful that is than having a great starting hand. And the reverse... all those people that just play the cards, are just so missing the boat.

Hmm... can you post/link to the full analysis? It's not that I don't believe you; I fully believe Jesus knows more about this than I do, but I want to reconcile this with what I had figured.

GRUNCHING

play any two cards...

Any two cards have a HUGE potential pay off. We know when our hand is good, when our outs our live, and that our opponent has a hand he will pay off with. What more do you want?

You flop bottom two pair with 94s, PIZOWNORED!!

You can play the hand knowing he has an overpair.

Yeah, but he knows you know. It's not about a potential big payoff, because if he picks off a big bluff you lose tons. It's about EV. Assuming perfect play on both sides, it seems there should be a perfect solution.

There is a perfect solution... if you both play optimally, and I think ECON TIM hit on it.

You call the 5, and there is now 10 in the pot. He knows you'll call with any two cards, so you have what amounts to two random cards.

What percent of the time will we be ahead on river? Let's say it's 1/10. Let's also say that the aces never bet the flop or the turn because we'll only call when we win.

If we're ahead one in ten times, and decide to make a half-size bet, how often should we be betting this flop so he can have no positive expectation to call?

Well, if we're betting 5 into a 10 dollar pot, he's getting 3-1 on his call. If he's getting 3-1 on his call, we should be bluffing 1/4 times we bet. (If he always calls, he loses thrice totaling fifteen dollars in losing calls and wins once, totaling fifteen dollars that are in the pot. If he never calls, he loses nothing.) If we do this, he can make no play that is a longterm winner... he can only try and "outguess" us, which is impossible in the longrun.

Since we hit the flop 1/10 times, and want to bluff 25% of the time (assuming we bet half the pot of course), we're going to end up betting about 2/15 rivers in the long run, making sure to keep our real-bet:bluff ratio at exactly 25%.

(shrug) Fun, no?


Ice

Ok... this may be a silly question... but wouldn't he bet post flop if he puts us on a possible drawing hand? Doesn't he still not want to give us a free card?

I'm no good at this game theory stuff. But it's very interesting.

Seems like kind of a fun problem, but very complicated overall I would think.

I'm not sure, but I'm guessing that the optimal strategy would be to fold some percentage of your most dominated hands (like A7o) preflop, instead of calling with any two. This would increase the cumulative winning percentage of the hands you do call with,which I think would allow you to bet more frequently postflop than you could otherwise.

Whatever the optimal preflop strategy is, I'm sure the optimal postflop strategy will depend a lot on the texture of the board, since for example a flop of 765 all hearts would need to be bet a lot more often than a flop of K94 rainbow, since a random (or semi-random) hand is much more likely to be ahead of aces postflop with the first flop than with the second one.

Yes, to work out the entire problem mathematically would take alot of work. But intuitively, as many people have guessed, you would reduce much of the negative expectations in an optimal strategy, if you knew his cards, so you would almost always have a +EV for calling.

But, I'll show the guide lines on how to do it, and maybe you can do guys can figure out the entire expected value for player one calling preflop with a random hand.

Lets just say for now, we called whether it was a good idea or not. This presents us with a game on the flop.

In this game, I have several decisions to make as the first to act:

1. bet as a bluff
2. check and fold
3. check and call a bet from the aces
4. check and raise as a bluff
5. check and raise for value
6. bet for value

Player two has a bunch of decisions to make as responses to my decisions..

Lets take Player 2 response to my betting:

Well there is 10 dollars in the pot, and lets say, as econ_tim suggested, we leave out pot equity so when someone folds their value is zero rather than negative.

if player 1's probability of bluffing is, pb, and your probability have having flopped and actual better hand is 15%(as a random hand will against aces). And you have a bet size B.

value of calling: (pb/(pb+.15))*($10+ - (.15/(pb+.15))*(
value of folding: 0

Now this might seem interesting because you could set the expected values of folding = to the expected value of calling a bet on the flop, and you would have a point where player II wouldn't care if he called or folded.

pb*($10+ = .15B or pb = .15B/($10+

This might see interesting, because you could plug in a bet size or a bluffing probability and get out and bet size or bluffing probability, but really its essentially meaningless. First of all, we just made up a strategy that player II would decide between calling and folding. What if the better strategy had a decision between folding and raising instead? So you would first have to work out one strategy all the way to the end, and then prove its an optimal strategy. Plus, the bet sizes shouldn't and bluffing frequency shouldn't just be arbitrary, if you solved the whole strategy, you could find the optimal bet size, and you would have your bluffing
frequency tied to your hand strength as well.

But.. this is kind of the ideas you use to solve game theory problems. Get the values of each decision and compare and come up with the optimal strategy. You can see how complex this would get by adding in the later streets.

Maybe if I have time later I'll try to actually work it all out.