Head Up Theory Question

[David Sklansky]

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?

[pheasant tail (no 18)]

21

[reubenf]

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21

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That looks like log2 rather than log4.

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

[pheasant tail (no 18)]

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That looks like log2 rather than log4.

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

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Not that there is anything wrong w/ using log4, of course. But I included metagame considerations in my decision to use log2. Considering the fact that I'll still have well over $999,000 in my bankroll after hand (should I lose), I should have plenty of opportunity to capitalize on it in our fine game.

I'd hate to have my opponent know that I will only put in the 10th raise w/ the nuts. I'd get no action [img]/images/graemlins/grin.gif[/img]

[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.

[fnord_too]

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.

[fnord_too]

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.

[uphigh_downlow]

[ 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.

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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?

[brick]

499,998

[tipperdog]

You should reraise 9 times in the fixed limit game.