Head Up Theory Question

[Slade]

On the issue of if you should ever fold 999,999, I think it is dependant on if you are raising 999,998 the same amount of times as 999,999. If you are then you should never fold 999,999. If not then some small amount dictated by the pot size should be correct. Note that this is for two reasons. One being that you would rather choose to fold something lower than 999,999 if you could have something smaller in the same situation. The second being if your last value raise with 999,999 is something you would only make with 999,999+ then your opponent couldnt be value raising with a lower range of numbers. (If an opponent is making a value raise with 999,998+ then you should always respond with a call when you have 999,999. However this situation would be impossible if your last value raise came with 999,999+)

[marv]

[ QUOTE ]
You have 9. You bet.

I am acting second. I have an unexploitable strategy for the second player which involves raising the first bet with 8,9, 10 every time.

How is folding a 9 ever part of a correct strategy?

Enlighten me

[/ QUOTE ]

It's 'correct' in the usual game-theory sense:
There is a pair of 'strategies' (which specify how to act in every possible situation, and may involve randomly choosen actions) call them P1 (first player's strategy) and P2 (the second) such that:

1) If I'm first player and use P1, you never get more than 8% whatever strategy you play
2) If I'm second player and use P2, I never get less than 8% whatever strategy you play

It just happens that P1 does something rather odd with the second highest card.

I can put the pair P1, P2 on the web so people can look at them/try to beat them if really necessary (but it wouldn't be very illuminating).

P1 and P2 aren't unique. Maybe there's a P2 which behaves as you specified, I don't know.

Marv

[xpsyuvz]

marv,

(If you have the time to post it) I would be interested in seeing an unexploitable P1 and P2 strategy. (You were talking about a ten card game, but if you can illustrate it with even less cards, it might be even easier to understand.)

I suppose it is possible to have an unexploitable strategy, but I don't quite see what it would like yet.

[parappa]

[ QUOTE ]
[ QUOTE ]
just because a strategy is game theoretic optimal doesn't mean the strategy has to have stupid crazy [censored] in it like folding the 2nd nuts for one more bet in a big pot in limit poker

in this thread, people seem to be posting nonsense and then using big words like "game theory" "unexploitable" "optimal" "Nash rebluffing equilibrium douching" without understanding the concepts.

[/ QUOTE ]

Are you claiming to understand Nash rebluffing equilibrium douching better than I do?

[/ QUOTE ]

I laughed out loud at this. The problem is, these kinds of things are why my wife no longer asks me what I'm laughing about. [img]/images/graemlins/smile.gif[/img] The "Nash rebluffing equilibrium douching" punchline just doesn't seem to work offline. [img]/images/graemlins/smile.gif[/img]

[locutus2002]

10 times.

Each player should advertise the max by raising when their number exceeds 50% of the remaining range.
There is no point in overvaluing your hand (bluffing) as opponent will not fold and will call once his advertised range exceeds his number.
It's a power of 2s problem, 2^^20th = ~1,000,000 so ~20 raises. Hero puts in 10 of them.

I see three major implications for pot limit hold em
1) given 1000 or 2^^10 possible hold em hands you should never put in the 5th reraise without the absolute nuts (AA heads up).
2) Each raiser is advertising an expected range of hands greater than the expected range of hands of the previous bettor.
3) Conservative players will avoid losing chips from overvaluing their hands as they will not put in a reraise that they cannot call without exceeding the value of their hand. Aggressive players will gain chips from the conservative players undervaluing their hand, and thus leaving money on the table.

[SossMan]

I would keep raising until he decides to slowplay because I can feel that board coming:

[ 999,995 , 999,996 , 999,997 , 999,998 , x ]

[Underlord]

[ QUOTE ]
If the game allows no folding and no checkraises, this is exactly game #3, with the result that you should raise with the top sqrt(2)-1 percent hands (Labeled the "Golden Mean of Poker" in the [0,1] posts) that you put your opponent on. Working out a chart, I get that with the top 2 cards in the game, you should be willing to put in the 15th bet; for the 16th bet you need the top card.

[/ QUOTE ]

The $1 ante also deviates from the [0,1] game.

Assuming no ante, no folding, and no checkraises it's exactly the Bill Chen [0,1] 3rd game (part 3).

http://tinyurl.com/auj5

"What this means is that when infinite bets are allowed, you should generally put in another raise with about 41% of the hands your opponent can have to have raised you this far."

From the indifference equation:
x_n = r^(n-1)/(1+2r)
r = 0.414 "Golden Mean of Poker"
x_n = point where opponent is indifferent to checking or betting
n = number of bets to be put in before stopping

This gives the optimal betting strategy for any given hand in this game:
h = hand # (ie 999,999)
n = number of raises where u should stop
ln (1- (1.828 * h/1000000)) / ln(0.414) + 1 = n

Since this game only has 1,000,000 values, the chart is short:
(how many bets should be put in with top x hands)
1 = 547,000
3 = 93,000
5 = 16,000
7 = 2000
9 = 470
11 = 80
13 = 13
15 = 2
17 = 0.4

Only odd numbers because check-raises are not allowed and you act first.

You put in the 15th bet with your top 2 hands (999,998 and 999,999). A 16th bet requires you having the top hand of 1,000,000. This assumes your opponent is also playing using optimal strategy (expert).

Answer: Put in the 15th bet and then just call.

[cheapsuit]

bet/raise straight through to the 499,999th bet. call on the 500,000th bet if your opponent raises instead of calls.

say you are playing in a limit game that you can bet-raise until you are out of money. preflop you get dealt KK and raise. everyone folds accept one other opponent who raises, you reraise, he reraises... so the question here is how long do you continue to rereaise until you assume he has aces. being that there is a 220:1 chance of him having pocket As, you can confidently make 110 bets until you start to breech the terroritory that, from a probability standpoint, assumes your opponent has As. once you past the 110 mark, if your opponent raises it probably means that he has the aces because otherwise he would just call.

now, in this particular question it is something of a name-that-tune game. so you bet, saying that you have at least a 2, your opponent raises saying he has a 3, you reraise saying you have a 4, etc etc. now, the question is, at what point does it become unprofitable to bet because your opponent raising now means that he most likely has the million-card. at the 499,999 bet made by you, you are stating that you have the 999,999. if youre opponent bets, you can assume he has the million and only call hoping that he either does not have it and played incorrectly or that he calls and you have him beat. if you lose, you lose only the minimal amount you would have while going for the maximum amount of profit.

from a theoretical level, you are saying with that 499,999th bet that even if you have me beat because you have the million-card, that original dollar ante still rightfully belongs to me (as part of your equity).

i noticed alot of posts stating to stop raising after the 10th or 11th bet or something like that, but that almost seems to be ridiculous. its like one of those things that makes sense theoretically, but if you were to actually be in that situation, the practice of the theory would be borderline silly. why would you only place 11 bets on a hand that you have a 99.9999% chance of winning? you would certainly pay more than 11 bets for a chance to win a million dollars.

so thats my two cents, let me know what you all think

--cheap

[reubenf]

I polled a bunch of my friends about this question. I just wanted their intuition. Most of them don't play poker, and the ones that do play it for fun on an intuitive level. They may or may not be winning players but they have large leaks which are obvious to anyone with a hint of a mathematical understanding of the game.

Anyway, 23 out of 25 said to put in a million bets. They are almost all quite intelligent people. The two who got it close have never played poker in their life. (My wife said to put in 20 bets, and another friend said to put in something like some constant multiple of some log of a million). The math grad students of the bunch, and those who I think could if they wanted get a graduate degree in math, all understood immediately why their intuition was wrong after I explained it. The intuitive poker players of the bunch all had a very, very hard time accepting that their answer was extremely wrong. There is an overlap of two people in those groups, and those people had a difficult time understanding at first but could see why they were wrong after some effort.

[cheapsuit]

ok, wait a second. i was thinking about the example i used in my previous post with KK vs possible AA and i think im a dumbass. i would not in practice raise KK 110 times because of the 220:1 chance my opponent has of having AA. i was trying to think of myself at the table with a load of chips in front of me, raising 110 times and then calling and saying 'im such an idiot' to myself... so that was a stupid thought-experiement. the raising 110 times with KK is only profitable if the cards in your opponents hand are random and he is betting them blind. then you can bet up to 110 times or 109 rather with a call on the 110th. but if he is not betting blind it is far far less than that.

i do, however, stick to my answer of 499,999/8 bets (not sure which) and a call on the 500,000th provided your opponent doesnt call, but am only doing it for the time being. i have yet to see an answer with somethine like 20-21 bets that i like. why would you only want to win 42 dollars plus the antes if you have the second-best card out of a potential million. i think my original answer still makes sense to some degree but now think i am missing something/not considering something.

oh wise david, please tell us the answer!