Helmuth - Ferguson Head Up Match Theory Question

[David Sklansky]

Who knows me well enough to answer this fellow for me?

[DrVanNostrin]

[img]/images/graemlins/confused.gif[/img] [img]/images/graemlins/frown.gif[/img]

(I realized I was wrong too)

[BobK]

I wrote this up before I read the replies. Seems logical to me.

Dave said the opponent will continue to play the same way and you know them very well. Since you know them well, you must already be using your best strategy against them.

It is unlikely for you to be less skilled than your opponent if you're 5 matches up.
Worst case is your about equal(+-very little). That makes each of the following matches a 50-50 coin flip.

.5 ^ 5 = .03125. So you have a 3.125% chance of losing the tournament. If you are actually the better player, that figure would be even smaller.

I don't think I'm going to alter my strategy in an attempt to gain part of about 3.125% or less. I'll stick with the one that got me there.

[Shandrax]

[ QUOTE ]
this means that variance is maximized when your winrate is 50%. so people who think they can lower variance by playing less optimal are mistaken. play as optimal as you can and you will maximize the probability that you win that leg of the match and the match itself.

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So this is the solution? I guess my idea of maximizing variance wasn't far off.

The question remains, how do you play close to optimal? Move all-in if you got the SC number for it? That may only be half of the answer, what if you are in the BB and he limps or moves all-in himself?

[alanbrown]

That is Indeed the question. But the answer is the same for game 10 as it was for game 1. Pushing with the right SC number is never wrong, but it may not be optimal. As to what to do if he moves in on you, well, what you need there is the Reverse SC Number [patent pending! [img]/images/graemlins/wink.gif[/img]]. This number would say that if you have holecards H and for an opponent that was pushing the top x% of his hands, you should call any all in up to $Y. Turns out this numbers isn't hard to calculate.

As an example, let's say you hold 23o and you're the BB for 2k and the button goes all in 2/3 of the time. You should call if the call requires less than 4.5K. If it's more then you should fold.

If you have Q5s in the same situation, then you should call any AI raise of up to 23.5k.

Of course, these numbers aren't the answer to optimal strategy. Just a way to find and plug some of the more severe leaks.

[alanbrown]

I'm not sure I can claim to know you well but I think I know why you didn't like Busto's answer...

Busto's reference to the binomial distribution was unnecessary. It doesn't enter into the logic required here and therefore only served to confuse. Throwing in unnecessary equations changes the landscape of the discussion to Why Is This Equation Unnecessary, rather than What Is The Answer To The Solution. That is what is to be disliked about his answer.

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variance is maximized when your winrate is 50%. so people who think they can lower variance by playing less optimal are mistaken.

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I have no idea what this is trying to communicate.

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play as optimal as you can and you will maximize the probability that you win that leg of the match and the match itself.

[/ QUOTE ]

Agreed. Though the question we're trying to answer is does Play As Optimally As You Can change as you accumulate more wins.

The simple key here is 'independent events'. If each game is an independent event with the same starting conditions (chips, who deals, randomness of cards, rules) and the same goal (win the other's chips), then the optimal strategy must not have changed to succeed at it.

Let's imagine that the villain has amnesia and has no memory of previous games we've played or what the current scoreline is. His game is optimized to win This Game. Not a series of games. There is no 'series of games' to him. Just this one. And our strategy that is optimized to win This Game can not be improved upon by knowing we won't need to win another one afterwards.

All the points Busto made about variance and probability are unrelated to this issue. And unrelated points are harder to refute than false ones. I think that's why you hated his answer.

[StregaChess]

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The simple key here is 'independent events'......

Let's imagine that the villain has amnesia....

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The first part is true, the second never is....

This scenario has a real world corollary in Chess. In their first battle together for the world championship Karpov – Kasparov squared off in 1984, Karpov jumped out to huge lead of 4-0. What followed was 17 consecutive draws and Karpov pushed the score to 5-0, one more win and match is over. The match turned bizarre after the 5th win, strange early draws, and psychological pressure and finally after a total of 48 games the match was stopped with the score 5-3 and neither player declared the winner.

[alanbrown]

That's a fair point. The idea that the optimal strategy remains the same, assumes a rational opponent. Psychology is a much bigger part of poker than chess, so optimal strategy might change for that reason. But in that case it would change based on the specific opponent and no general advice could be relevant.

[Shandrax]

[ QUOTE ]
This scenario has a real world corollary in Chess. In their first battle together for the world championship Karpov – Kasparov squared off in 1984, Karpov jumped out to huge lead of 4-0. What followed was 17 consecutive draws and Karpov pushed the score to 5-0, one more win and match is over. The match turned bizarre after the 5th win, strange early draws, and psychological pressure and finally after a total of 48 games the match was stopped with the score 5-3 and neither player declared the winner.

[/ QUOTE ]

Very interesting example, but I am not sure if conclusions from Chess can be applied to Poker.

Chess is a game of complete information. During a game you always know if your opponent is trying to create an irrational position with lots of variance or trying to play for technical superiority in simple positions with less variance. That makes all the difference in the world because you can always chose the optimal defence.

In Poker you don't see your opponent's hole cards, so you don't know if he had the nuts or was just making a move on you. He can not only cheat you in every single hand, he can even trick into chosing an inferior strategy.

Another problem with chess is the drawish character of the game. Just like Capa said, it's very likely, that Chess is simply a draw. It's just very tough to force a win in Chess against an opponent of equal strength.

In Poker it's just a matter of time until both players get a strong hand at the same time and the clash is inevitable. If both opponents have equal strength then variance will make one a winner nevertheless.

Poker is closer related to Backgammon than to Chess. Comparisons should only be drawn to games of incomplete information.

Last but not least the specific situation was unique, because Karpov is player who is unable to "gamble it up". He only had one style to chose from.

[George Rice]

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Poker is closer related to Backgammon than to Chess. Comparisons should only be drawn to games of incomplete information.

[/ QUOTE ]

Backgammon is not a game of complete information?