Helmuth - Ferguson Head Up Match Theory Question

[David Sklansky]

Bingo

[FortunaMaximus]

[ QUOTE ]
No adjustments.

You want to do whatever you can to maximize your % chance of winning any given HU match.

Once you accept that, it is clear that the current "score" of the match, or number of games left to be played is irrelevent.

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Pretty much. The rest is assessing the proper attack strategy to counter his. And that's the early microadjustments.

Otherwise, it's still attack, attack, attack.

[Shandrax]

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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.

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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.

[Shandrax]

[ QUOTE ]
Bingo

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So moving all-in if the SC numbers indicate it is not an optimal strategy by definition? Not being able to make a mistake looks very optimal to me.

This brings up the question "What is the goal of Poker?" (don't say "to win money" now). Botvinnik defined the goal of Chess as trying to win material (not to mate the king). Is the goal of Poker trying to exploit the mistakes of your opponent or trying to avoid making mistakes yourself?

[CityFan]

[ QUOTE ]
Bingo

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Give almostbusto some credit though.

From reading this board and the SMP board it's clear that to convince some people that you are right, it's necessary not only to give a logical argument for why you are right, but also to explain why their arguments to the contrary are fallacious.

Otherwise, they can't distinguish between the sound argument and the flawed one, so they stick to their original point of view.

I know I'm not the only one who shook their head when they read "increase variance", but are you going to explain to that poster why he is mistaken, or not?

[almostbusto]

yea definitely give almostbusto some credit.

you guys who say it only matters that each leg of the match are independent are WRONG. each leg must be independent and identically distributed. thats why i declared the variable is binomial. by virtue of being binomial it is IID.

yes that is a hugely nitty point, but to say my answer is a poor one is a nitty point [img]/images/graemlins/tongue.gif[/img]

[CityFan]

[ QUOTE ]
yea definitely give almostbusto some credit.

you guys who says its only matters that each leg of the match are independent are WRONG. each leg must be independent and identically distributed. thats why i declared the variable is binomial. by virtue of being binomial it is IID.

yes that is a hugely nitty point, but to say my answer is a poor one is a nitty point [img]/images/graemlins/tongue.gif[/img]

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I don't think the term "identically distributed" is meaningful in this context though.

I know why DS hates your answer. If I were to hate it, it would be because the argument is garbled. A garbled argument in maths is usually a sign of incomplete understanding.

(which is better than no understanding at all...) [img]/images/graemlins/grin.gif[/img]

[almostbusto]

[ QUOTE ]

I don't think the term "identically distributed" is meaningful in this context though.

I know why DS hates your answer. If I were to hate it, it would be because the argument is garbled. A garbled argument in maths is usually a sign of incomplete understanding.

(which is better than no understanding at all...) [img]/images/graemlins/grin.gif[/img]

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no identically distributed is of crucial importance. if for example odd match numbers followed one distribution and even match # followed another, there is no guarantee that this is true. do you see why? if you don't i could come up with a crazy example.

BTW, i still see nothing garbled. alanwhoever said he didn't understand my statement that maximizing your winrate minimizes variance. if you read the other posts its obvious.

tons and tons of posters were saying 'i would play more agg to increase the variaiton' or 'i would constantly want him playing for all his chips'. they were effectively saying 'i would play less than otherwise optimal in order to increase variance' and i was just pointing that out to them.

[alanbrown]

I'll take a crack at trying to surmise Busto's thought process and where it was astray...

Busto states that variance is maximized when your likeliehood of victory is 50%. Agreed. It's clear that the more likely you are to win or lose, the more you will tend toward winning or losing. Nothing too controversial here.

However, he focuses on variance as being part of the problem when it's not. Variance is the short term ability to stray from long term expectation. Variance is what happens despite your best efforts. And the only thing you can do to alter it is to make smaller bets for smaller rewards. But this strategy would already be rolled into your optimal strategy for winning a single game. It wouldn't be affected by previously won games.

Here's what's wrong with Busto's argument in short

Predicate 1:
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the result of a heads up match is a binary variable (1 = win, 0 = loss)

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Conclusion:
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the results of a heads up match follows the binomial distribution

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The conclusion doesn't follow from the premise. The result of many HU matches will follow the binomial distribution if the probability of winning each individual game is 'independent' of other games and is constant.

predicate
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variance is maximized when your winrate is 50%

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conclusion
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people who think they can lower variance by playing less optimal are mistaken

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The conclusion is completely unrelated to the premise. It doesn't follow. In fact it's wrong. I can indeed lower my variance by playing less optimally. If I play badly enough, I'll lose almost every game, and my variance will be virtually zero.

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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.

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This is all he needed to say for his post to be valid.

[almostbusto]

alan you are wrong. i made my point about variance to show how other posters were off.

[ QUOTE ]
The conclusion is completely unrelated to the premise. It doesn't follow. In fact it's wrong. I can indeed lower my variance by playing less optimally. If I play badly enough, I'll lose almost every game, and my variance will be virtually zero.


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particularly when you say this you are mangling my words. you CAN'T LOWER VARIANCE BY PLAYING LESS THAN OPTIMAL WHILE STILL MANTAINING A >= 50% WINRATE. that was very obviously implied, infact i included that completely trivial caveat in a later sentence.