heads up theory

[catlover]

David Sklansky and I are playing the following game. We each ante a million dollars. We are each dealt a single card with a number on it between 1 and 100, 100 being the best. We then play a single round of limit poker, with betting in $1 increments.

David has the button. I look down and see that I have been dealt a 40. What is my play?

[JoshuaD]

Don't fold.

[kitaristi0]

If i've understood this correctly, each number from 1 to 100 is equally likely to come up. If you have the number 40, there are 39 numbers you beat (1-39) and 60 numbers you lose to (41-100). Thus if/when David bets, call and hope.

[FNHinVA]

Leave it to others to elaborate.

[randomstumbl]

This is actually a pretty interesting decision.

Obviously, it depends what he will call and raise with. I don't think he can possibly fold anything. Even folding the nut low is bad because he could still split. If our opponent will fold a 1 we should bet everytime.

If we check, I'm confident he will bet 50 or higher, probabably more. Meaning, we lose at least .5 of a bet by checking - probably more. We're in a dominated situation though, so this could easily still be the best outcome.

If we bet out we win 1 bet 40% of the time and lose either one or two bets 60% of the time (again, we can't really fold to a raise because of the size of the pot).

So, through some quick and dirty logic I think that means if he raises with the top 30% we have a break even decision.

This is where it gets complicated though. What range will we 3 bet with. What range will he 4 bet with. How often will we check-raise and how will that affect his strategy. I know this is a simplified situation, but it's still really complex because we have to solve the entire game to answer any question for certain. Without doing all that work, my guess is that this is a really close decision.

[Pog0]

considering the ante size to bet size ratio, you can just take any bluffing assumptions out of the equation.

Your best play would be to follow these simple guidelines: don't bluff and don't fold.

Somehow I think the variance in this game might be a little high.

[randomstumbl]

[ QUOTE ]
Your best play would be to follow these simple guidelines: don't bluff and don't fold.

[/ QUOTE ]

Your statement isn't entirely correct. You should certainly raise if you have 100 or 99 or 98.

The real question is where is the cutoff? Should you raise with 50; should you raise with 40?

[K C]

First of all if David has the button and it's heads up it's his move [img]/images/graemlins/smile.gif[/img]

With a million dollar ante and 1 dollar limit, it's of course going to be correct to never fold regardless. Say you only have a 1 - there's a 1% chance you will split the pot, which means a million dollars if you do, which you'll want to see for just a few bucks for sure [img]/images/graemlins/smile.gif[/img]

There's no point in raising unless you think you're opponent is foolish enough to fold, which is hard to imagine [img]/images/graemlins/smile.gif[/img]

KC
kingcobrapoker.com

[randomstumbl]

[ QUOTE ]
There's no point in raising unless you think you're opponent is foolish enough to fold, which is hard to imagine


[/ QUOTE ]

Seriously, come on people, it's not that simple of a question. He will bet anytime he has anything higher than 50 when checked to (probably even slightly lower than that).

If we take the initiative and bet he will not raise with a 50; he probably wouldn't even raise with a 60. Since the pot is a million and one dollars he will call with a 1 though.

This problem illustrates the "gap concept" pretty clearly. You can be less than 50% to win and have zero fold equity and still be right in betting because of this concept.

[bobman0330]

This is a very interesting hypo.

I think we can ignore the technical requirement that a player bluff about .000001 of the time.

Now, if it is checked to David, he should bet everything higher than 50, check the rest, and bluff epsilon of the time. (This is a first approximation, assuming you check-call every hand. Actually, he should bet hands that are better than half your checking range, minus some to account for the possibility of a check-raise)

You should check-call any hand worse than 50, except 1 millionth of the time when you bluff by betting or check-raising. (I said we should disregard bluffing, but I'm not going to because I enjoy referring to the infinitesimal frequency with which you should do it.)

Hands better than 50 should be played either by betting or check-raising. This is where it gets interesting. The goal is to prevent Sklansky from value-betting his weak hands. If you check-call everything below 50 and bet everything else, he can value-bet 25 and up. At the same time, it's important to get value from your hands by betting a lot.

I have no idea how to solve this right now, but I'll think about it some more.