Head Up Theory Question
 

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?

Re: Head Up Theory Question
 

21

Re: Head Up Theory Question
 

Assuming that you bet or raise if you have a better hand than opponents average hand, you reraise 8 times. (when the opportunity to reraise time number 9 gets to you, his average hand is 999,998.1 without rounding.)

For pot limit, I would guess you bet the pot when you figure to be ahead of 66% of the hands he could have, since you are always offering him 2:1 (not sure about the first bet, probably still 500K or higher). That gets you to 5 reraises (all raises assumed to be full pot) before you stop if I did the math right.

In pot limit, though, I think you can make your bets more exactly, but I would really have to think about that and there are information considerations to not betting the size of the pot.

Edit - If your initial bet line is 666,667, it is still 5 reraises if I did the math correctly.

Re: Head Up Theory Question
 

499,998

Re: Head Up Theory Question
 

You should reraise 9 times in the fixed limit game.

Re: Head Up Theory Question
 

Since I have unlimited money in front of me I do not fear the risk of ruin against my opponent, so I bet and raise until he calls.

Also, since we both have unlimited money the amount won or lost ends up being inconsequential anyway.

Re: Head Up Theory Question
 

[ 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.

[/ QUOTE ]
Glad you have finally found the game to which you want to challenge Negreanu.

Re: Head Up Theory Question
 

Just to keep it close, Daniel won't look at his cards

Re: Head Up Theory Question
 

In limit, I would put in fifteen bets (a bet and seven raises by me total) and call the sixteenth bet.

EDIT: actually, I think maybe I should put in 13 and call the 14th.

Re: Head Up Theory Question
 

Actually, my answer is probably wrong since I did not take the initial pot into account correctly, and a propper strategy will definitely have some degree of bluffing in it. The only way my answers can be right is if you assume after a reraise or two that the liklihood of bluffing has been dimished to a negligible number. I am not sure though that, for instance, in the first case 500,000 is the line of demarkation to bet out with, so the average hand you reveal by betting might not be 750,000. In the end this wont impact the answer more than one reraise I am pretty sure, but it seems, nonetheless, important.

Re: Head Up Theory Question
 

i never stop raising, if he has the nuts he will never stop raising either so there will never be a showdown. plus no matter how much i lose i still have an infinite amount left.

i'm rich! let's gambool

Re: Head Up Theory Question
 

I would be willing to wager about $900,000 on this, give or take a few thousand depending on if he is wearing sunglasses or not.

Please explain to me people why you would only be willing to raise 13 or 14 times when you only have a .000001 chance of being beat. I think you guys are playing scared. You have a huge edge here and need to push it to the max.

Re: Head Up Theory Question
 

[ 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 ]

Can I assume at the end, you call or fold with a game theoretically optimal frequency?

Re: Head Up Theory Question
 

[ QUOTE ]
Actually, my answer is probably wrong since I did not take the initial pot into account correctly, and a propper strategy will definitely have some degree of bluffing in it. The only way my answers can be right is if you assume after a reraise or two that the liklihood of bluffing has been dimished to a negligible number. I am not sure though that, for instance, in the first case 500,000 is the line of demarkation to bet out with, so the average hand you reveal by betting might not be 750,000. In the end this wont impact the answer more than one reraise I am pretty sure, but it seems, nonetheless, important.

[/ QUOTE ]
Assuming there is nobluffing

Are you basing your reply on limiting his range by a fraction every time he reraises.

When would you raise if your oppnent leads out (no bluffing) And when would you simply call?

This is what I made of it.

Worst case, you open with 500000, he raises with atleast 750,000, you raise with atleast 875,000, and so on.

Is this correct?

Do pot-odds matter at all here? Would the raising range get progressively smaller (more than half) since there is more money in the pot?

Re: Head Up Theory Question
 

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. (edit: obvioulsy for fixed limit only).

Re: Head Up Theory Question
 

If the answer is as few as 9 as fnord and tipperdog say it's a good lesson and I happen to have a story to illustrate the point.

The one time I played at a table with David was a 10-20 game at the Mirage years ago. I had KK and the guy next to David had AA. This may have been my first experience with KK vs AA. Unlimited raising heads up and I reraised the hell out of my KK - I'm sure I went way too far with them. When I finally just called I heard David on the other side of the table comment something like, "gee I wonder what they've got?"

With unlimited reraises, How many reraises should you put in preflop with KK?

PairTheBoard

Re: Head Up Theory Question
 

[ QUOTE ]
With unlimited reraises, How many reraises should you put in preflop with KK?


[/ QUOTE ]

many less than with 999,999 vs. 1,000,000. Do you see why?

Re: Head Up Theory Question
 

Normally, if he bets the pot, you figure he has a hand that is better than 50% of all possible hands. Of course, bluffing is possible early on, but if he doesn't have a legit hand, he will eventually grow tired of bluffing, and call or fold. If he continues to bet the pot, we must assume he has a hand.

My guess is, each time the pot is bet, you can narrow your opponents possible hands by half. So with his first bet, you can assume he has a hand higher than 500,000. When you re-raise, he can assume that you have a hand better than 750,000. Eventually, it will get to the point where he would not bet without the nuts. You call one bet before it gets to that point.

Re: Head Up Theory Question
 

[ QUOTE ]
My guess is, each time the pot is bet, you can narrow your opponents possible hands by half. So with his first bet, you can assume he has a hand higher than 500,000.

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So you think the right strategy is to open for a pot-sized bet with anything 500,000 or up? I'm not sure I see why that is right. When you bet there, you are offering your opponent 2-to-1 on a call. So he should call (or reraise) if his hand is good a third of the time. This means that if he think you will bet with anything, he should call (or reraise) with 333,333 or better.

This means that your bet would be a loser because more than half the hands in the range [333,333 - 1,000,000] beat your 500,000. Further, his hand range is probably stronger than that since he knows you hold better than a random card since you chose to bet. And you will sometimes be reraised and then have to either call and lose or fold and possibly get bluffed out. So it looks to me like opening for the pot with 500,000 or up can't be the right strategy and you should need a better hand than that.

Re: Head Up Theory Question
 

How many less?

If you are dealt #1,000,000 you're not going to be worried until you put $1,000,000 in the pot. Then you're going to be very worried and say "Did I miss a rule or something?"

With hand number #999,999 you're not worried until you put $999,999 then you're going to say "It's one in a million!"

anyway it seems I don't understand the concept.

Re: Head Up Theory Question
 

I made a post and it was a unanimous vote that you play "HeadS up", not "Head Up"

Re: Head Up Theory Question
 

Actually, I don't think it is nearly as simple as I thought. Even if you get raised after your 8th reraise you would have to mix your strategy to call 20/21 times and fold 1/21 times.

Re: Head Up Theory Question
 

[ QUOTE ]
21

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That looks like log2 rather than log4.

Not that I'm saying log4 is the correct answer.

Re: Head Up Theory Question
 

I would reraise indefinitely. I would probably do this as well if I was dealt 999,998 even though my opponent is now twice as likely to beat me. [img]/images/graemlins/smile.gif[/img]

Not sure about 999,997...

I think I see the point in this excersise though. At what point in the cards do you start to worry you're beat? Well, it's somewhere between 500,000 and 999,999, I suppose... But where?

Re: Head Up Theory Question
 

[ QUOTE ]
Actually, I don't think it is nearly as simple as I thought. Even if you get raised after your 8th reraise you would have to mix your strategy to call 20/21 times and fold 1/21 times.

[/ QUOTE ]

Are you saying you would fold 999,999 1/21 times just to mix up your strategy? This is clearly wrong. But maybe I am misunderstanding your post. Also, an expert would not play so predictably in such a way that he/she would always have 1 million after raising 8 times.

Re: Head Up Theory Question
 

[ QUOTE ]
[ QUOTE ]
With unlimited reraises, How many reraises should you put in preflop with KK?


[/ QUOTE ]

many less than with 999,999 vs. 1,000,000. Do you see why?

[/ QUOTE ]

Of course, and exactly my point if the number for 999,999 vs 1,000,000 is as few as 9.

PairTheBoard

Re: Head Up Theory Question
 

I disagree that the number for 999,999 vs 1,000,000 is 9. I think that is way too weak-tight. I would put the number in the hundreds of thousands.

However, I anxiously await more info from David, explaining why (if no one gets the correct answer first).

Re: Head Up Theory Question
 

You should just call, to trick the other expert.

Re: Head Up Theory Question
 

I don't know. This question is hard. It's a bit more advanced than anything in the [0,1] game although clearly related to it.

Optimally, we should bluff some small percentage of the time (a fraction of the worst hands). The bluff raise would be this same percentage of the bluffing set. And so on, each successive raise reducing the bluffing hands set from the previous raise by the same fraction. Assume we bluff with the first bet with the bottom 20% of our hands, a guess but I think a reasonable one. The reraise by an expert will be with the bottom 4% of hands. Obviously this series quickly approaches 0.

We call at the ratio of pot size over pot size +1 of our set of hands that can beat a bluff. So on the first bet if it is right for the opponent to bluff with 20% of his hands then 80% of our hands beat all his bluffs. 3 (pot size) / 4 (pot size +1) of 80% is 60% we call with 60% of our bluff beating hands. As with each succesive raise the set of bluffing hands gets smaller then the smaller the set of bluff catching hands. This applies to our opponent just as much. So how many times can we bet this hand for value?

A guess is 8 or 9. As the chances of bluffing descrease the number of calling hands gets smaller. Both of these happen fast.

With pot limit betting it happens even faster. I'm assuming the only bet size is the pot. In limit the ratio of bluff catching hands gets larger, but in pot limit it stays the same, 2/3. The number of bluffing hands is smaller because the rish reward ratio is worse. So the set of bluff catching calling hands is smaller and the group of value betting hands decreases as well. I'm thinking you can only bet it for value somewhere around 4 times.

Conclusion.

limit - 8 or 9 times
pot limit (must bet pot) - 4 times

regards,

raisins

Re: Head Up Theory Question
 

I think the answer is the quickest algorithm to find 999,999

IE start at 500,000 higher go to 750,000 higher go to 875,000

etc etc etc etc

Re: Head Up Theory Question
 

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.

Re: Head Up Theory Question
 

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

Re: Head Up Theory Question
 

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.

Re: Head Up Theory Question
 

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

Re: Head Up Theory Question
 

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.

Re: Head Up Theory Question
 

[ 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.

Re: Head Up Theory Question
 

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.

Re: Head Up Theory Question
 

[ 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

Re: Head Up Theory Question
 

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?

Re: Head Up Theory Question
 

[ 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.