Head Up Theory Question

[FishAndChips]

Well I'll take my shot at a guess...

Assuming no bluffs, and given the pot odds, when you first bet, you are offering your opponent 3-1 odds to call. Thus, you would value bet with a hand that's a favorite when called. Assuming your opponent calls with a range from 250,000- 1,000,000, you could bet hands greater than 625,000 for value. Knowing this, your opponent would raise for value against you, if his hand was a favorite against your range of 625,000-1,000,000. Therefore he could raise with hands from 812,500- 1,000,000. Once again, you would raise with a hand that was favored over that range etc.

By halfing the range of hands each time, and assuming all raises are for value, it would seem you should call on the 20th bet.

For pot-limit, you offer your opponent 2-1 on a call when you bet and therefore you should bet for value if you are a favorite over a range of hands from 333,333- 1,000,000. Thus it seems you could value bet any hand from 666,666-1,000,000. Knowing this, your opponent could raise for value with hands 833,334-1,000,000 etc. Using this logic, you would call on the 20th bet.

I'm not sure if any of this is correct. I'm tired and it's time for bed. [img]/images/graemlins/smile.gif[/img] I look forward to David's analysis.

[PairTheBoard]

I think the idea of hand ranges being halved each time is wrong.

With your first bet your opponent has 3-1 pot odds to call.
With your first reraise he has 7-1 pot odds to call. This continues, 11-1,15-1,19-1, etc.

As the pot odds get larger he will be inclined to call one more bet with MORE hands not less. He knows this is also true for you which makes him even more inclined to make his own reraise for value. It seems to me that this observation should get the order of magnitude for the solution up near Six Figures at any rate. I know it's a fuzzy inexact idea. It also doesn't seem right that the hand range would increase rather than decrease. Still, I think it points to a much higher number than most people are giving.

PairTheBoard

[uphigh_downlow]

Interesting. This is what i was thinking as well. Pot-odds certainly matter, and the way we halve the range seems to ignore any pot odds.

The first question, that noone really has answered is, "What is expert strategy here?"

I tried to work s strat into a mrkov chain, but am finding it difficult to use just information of the last action by one player to define it. Guess I have to get off my lazy as and find a paper and pen.

Can people post their strategies for expert play?

As in
When do you call?
When do you raise?
When do you fold?

You can assume that n actions that have taken place.

[NYCNative]

I think a better question is how one would handle betting if they had 2...

[uphigh_downlow]

I assumed that a bet can be treated as a raise on a 0 bet.

Course there is no information at this stage.

Presently I'm stuck on Bettor bets random hands. So opponent calls enough to break even. Now bettor knows the calling range, so he moifies his betting range, So caller modifies his calling range. and it keeps going.

But it has to converge at a certain point which might be considered optimal, and I'm trying to determine that.

But you are right, once we know what to open with, things can become simpler. Or atleast I hope so.

here are my thoughts:

Bettor bets randomly with the entire spectrum of cards.
Caller can call all but bottom 25% of his cards to break even

Now bettor knows this
So 25% of the time he will pick up $2
and 75% of the times he will either win $3 or lose $1 on being called.
I'm ignoring the raise possibility here for the time being.

so for this to be break -evn play
Bettor can bet with all but 5/6th of his cards
Or mebbe its slightly different.

Can anyone tell me P(X>Y) Xand Y in [a,1] and [b,1] respectively. Is it just the difference of means?

Am I far off or does this seem correct.

Then I could start of with, if bettor bets top X% of his cards, what would be the correct calling strategy, which would change the betting strtegy, and maybe I can try to see where it converges.

[Shoe]

[ QUOTE ]
Interesting. This is what i was thinking as well. Pot-odds certainly matter, and the way we halve the range seems to ignore any pot odds.

The first question, that noone really has answered is, "What is expert strategy here?"


[/ QUOTE ]

It would not be linear.

[uphigh_downlow]

Just finished my tournaments, while reading this.
too sleepy to finish it off.

So here is the general idea. Hopefully by tommorow, someone would have rubbished it off or it might be the correct track. I guess then I can start or stop wasting more time with this [img]/images/graemlins/laugh.gif[/img]

lets look at the game slightly differently. There is no raising. You can call/fold. After you call you can open again or check. ie:Once a guy opens, if the other guy calls, then he gets a chance to open again. The action isnt over until someone checks.

So options are fold/call and check / call and open again

Special case. The first to act always just opens.

Now each action constitutes a step or a state

Lets say that on the nth action ( Assume that for the opener n=1), the correct top% of hands as compared to the opponent, that the opener should lead out with is X(n)

So opener leads out for X(1)*100%(special case to be determined along the lines of my previous post. This is because he currently believes the opponents hand to be ranomly distibuted in 100% of the spectrum.

Then if the caller decides to call, he is given the choice to bet/open now (same as a raise) as if we had just started. As if he is first to act and becomes the opener.

In the first case the opener had to imagine that the holdings of his opponent covered the entire range, but now the second opener has extra information and should open for top X(n)th of the range of the opponents holding

This is X(n)*X(n-1).

It seems to come out like a factorial. I'll just use the notation.

So X(n)! is the top% of hands you can raise with. If it becomes less that 2/million. you should just call.

This strategy applies equally to both players.

[MtDon]

[ QUOTE ]
You and another player ante a dollar and both have unlimited money in front of you. You are both dealt one card from a deck where the cards are numbered from one to a million. You are dealt 999,999. You bet and get raised. How many times should you reraise, a dollar at a time before you just call against an expert? What about in Pot Limit?

[/ QUOTE ]

I think I'd raise for about 5 minutes. Either limit or pot limit.

However, in pot limit, if we can raise verbally or using a mouse click on a computer, raising the pot each time we would almost certainly run out of the world's supply of money before 5 minutes is up. [img]/images/graemlins/tongue.gif[/img]

Since I have unlimited money already in front of me, my net worth won't change, or at least not significantly, so how much I win or lose doesn't realy matter from a financial piont of view. But I don't want to waste too much time on the betting either - so 5 minutes seems about right to me.

The reason to bet is to either make more money or to cause the other player to fold a better hand. In the limit game, we know for certain that the expert player will not fold to a $1 bet once the pot is over a few dollars. So the only reason to bet is to get more money from the other player.

A question to ask is what is the chance that the other player misread his hand? After a while you've got to figure that the expert thinks he has 1,000,000, but that doesn't necessarily mean he has it. Could the expert have misread 100,000 for 1,000,000? If time isn't a concern, I'd keep raising until he looked at his hand at least once more, or as some others have said after the pot is in six figures. And I'd look at my hand again after a few minutes also.

Another question to ask is: How likely the expert thinks that you will raise several thousand dollars with 999,997 or even 999,500? What are the chances the expert has 999,998 and will raise several thousand, based on his read on you?

But as I said, I'd base my raises on time, rather than dollars bet.


-- Don

[FNG]

For another $1 each time your pot odds are phenomenal. Never stop raising. If a bet or a raise is correct at the beginning, it is always correct afterward.

Pot limit, always raise. The pot is laying you 4:3 on calling a reraise, so if you've got better than 750,000 you're correct to always raise, aren't you? If you both have better than 750,000, the game should never end.

This answer seems uncomfortably trivial. Is there something I'm missing?

[MicroBob]

[ QUOTE ]
After the first bet I belive you should reraise 8 times, and you should fold to his/her reraise after your 8th reraise since you know he/she cannot be holding the 999,999th card.

[/ QUOTE ]


I must have missed the part where my opponent was evidently not allowed to reraise 8 times with card #999,998 or
999,923 or whatever.

I know many many people who would put in more than 8 reraises with a hand inferior to 999,999.


Folding after 8 re-raises seems REALLY stupid to me.

Calling after this many re-raises seems kind of stupid.


Why are so many saying 'assuming no bluffing?'


If you were going to fold after my 8 re-raises every time just on the assumption that I was holding the nuts then I could just re-raise it constantly on every single hand and make a fortune (on top of my already infinite bankroll supposedly).
The one time you actually are holding the nuts I can just call on the 9th re-raise and let you take that pot....while constantly re-raising you out of each and every single pot other than that one.


So folding to X number of re-raises on the assumption that 'well...he must have me beat' is clearly a terrible strategy.


The halving thing that seems popular is not appropriate imo but I have no idea why.
But I agree with shoe that just calling or folding after 8 or 9 re-raises is missing a lot of EV here.



I will be VERY interested to read David's ideas on this.