Head Up Theory Question

[a361lbs]

Here is how I would "solve" the game. It is classic game theory for those in the know. In each hand there are a finite number of "pure strategies". A pure strategy 'X' is a list of one million numbers (X(1), X(2), ...) indicating how many bets of your opponent's you will be willing to call for each possible card you may be dealt, you reraise if that number has not yet been reached. Of course X(one million) = infinity, but to see that there are only a finite number of pure strategies we have to admit that it is unreasonable to raise without bound if you don't have the nuts. So for some N X(999,999)<N and we need not consider strategies with X(1,000,000)>N. We can choose N to be the largest number that will allow our computer (possibly the worlds fastest super computer) to solve the following linear programming problem in a reasonable amount of time.

Thus if two players were to use pure strategies X and Y exclusively the profit of player X would be proportional to
Sum(i=1 to million)[(Sum(j=1 to i-1) min(X(i), Y(j)) -
(Sum(j=i+1 to million) min(X(i), Y(j))]

If two expert players play the game then, they will play the "optimal mixed strategy" which is simply a probability density of how frequently they should play each pure strategy. Note: the optimal mixed strategy may not be unique, but the optimal value of the game will be.

Given this data it is well known how to solve for the optimal mixed strategy, although difficult to write down the algorithm in text editor.

Suffice to say, it is likely that there will not be a single number associated to how many bets you should call, but rather a probability distribution ranging from a low of around 20 to a high of several hundred, if not thousands.

If I had to play one hand only and got dealt the 999,999 I would think something was fishy and fold. If I was playing a long session I would rely on my read of my opponent, but put in no fewer than 100 bets.

-pounder

[reubenf]

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I don't think you can deduce "expert" play here without first making an assumption about the initial bettor's betting strategy.

If the initial bettor will bet anything over 500,001, that will yield a different reaction strategy from the initial bettor betting anything over 667,001, for example. I don't think there's enough information presented yet to really solve the problem mathematically.

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The expert who knows nothing at all about his opponent does not try to exploit his opponent's weaknesses since nothing is known about them. He just plays an unexploitable strategy. The question is "How do you play 999,999 against someone whose strategy cannot be exploited."

This is of course not applicable to any non-tournament poker you would play, because you should only play non-tournament poker with an edge.

[cocked&locked]

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Agreed, heck, I would love to play any no-limit game against such a person. This question is just like a no-limit game since there is no cap. Although I do think you could actually milk someone out of $1,000,000 easier doing it $1 at a time instead of making an up front $1,000,000 bet from the get go.

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Consider this -

You are playing heads up and have $1000 in front of you. Is the same amount of money going in preflop if you are playing 1)$1 limit or 2) no-limit.

You might well push all-in if you think you can get a call w KK in a no-limit game, but surely you aren't suggesting that you would raise 999 times with KK, are you?

[reubenf]

I'm not really defending that guy by attacking your attack on him, but here goes.

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You might well push all-in if you think you can get a call w KK in a no-limit game

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The minimal requirement here is that I think I can get a call with a couple hands I have beat (because obviously AA will call). And I'm still not saying it's optimal if QQ and JJ are calling, just that that's the minimal requirement.

The problem is clearly someone's doing something wrong if all-ins are being called preflop with QQ and JJ.

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but surely you aren't suggesting that you would raise 999 times with KK, are you?

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Well if this is the same opponent who is calling my no-limit raise of 500 times the pot with QQ, JJ, then the same answer still seems to apply.

[jthegreat]

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The expert who knows nothing at all about his opponent does not try to exploit his opponent's weaknesses since nothing is known about them. He just plays an unexploitable strategy. The question is "How do you play 999,999 against someone whose strategy cannot be exploited."


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I'm very curious to see you present the unexploitable strategy.

[reubenf]

[ QUOTE ]
[ QUOTE ]
The expert who knows nothing at all about his opponent does not try to exploit his opponent's weaknesses since nothing is known about them. He just plays an unexploitable strategy. The question is "How do you play 999,999 against someone whose strategy cannot be exploited."


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I'm very curious to see you present the unexploitable strategy.

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Well an unexploitable strategy will almost certainly be a mixed strategy rather than a pure strategy, but we can probably get pretty close with a pure strategy (which is the type of strategy the original question is worded to elicit).

Not that I really have to produce an unexploitable strategy to know one exists, and I definitely don't have to do it to know that the strategies that put huge numbers of bets in are extremely exploitable.

However, try exploiting any strategy that decreases the number of hands your opponent is on by a percentage each raise, only raising when your own hand beats a certain percentage of those hands. Such a strategy has already been posted in this forum. If you can exploit such a strategy, you will almost certainly only be able to do so with a similar strategy that uses better percentages and is more difficult to exploit. Keep refining until you either get an unexploitable strategy, or you start cycling between the same strategies (in which case the unexploitable strategy will be a mixture of these strategies).

[cocked&locked]

I agree that in my haste, the stack size is substantially out of whack for the example to hold water (or that fairly high no-limit blinds needed to be included).

The point I was trying to make was the difference between making many incremental bets and an all-in bet. Each incremental bet tells us something new about the opponent's hand that an all-in bet doesn't necessarily tell.

Point taken though - and thanks for the constructive critisism.

[PairTheBoard]

I suggest first coming up with the best suboptimal strategy in which all bets are at least called. Give sound game theory reasons for exactly why the betting ranges for each raising decision are what they are.

Is it possible the difference between the 9 Raise people and the 20 raise people that the 20 raises are including the raises by both players and the 9 raises are only counting those of the Hero?

PairTheBoard

[reubenf]

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Is it possible the difference between the 9 Raise people and the 20 raise people that the 20 raises are including the raises by both players and the 9 raises are only counting those of the Hero?

PairTheBoard

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I think SOME of the 9 raise people and SOME of the 20 raise people are the same because I think SOME of the 20 raise people actually mean 20-bet.

[BobK]

Since both players are experts and have unlimited resources, it seems to be a straightforward question of logic.

2nd nut knows that if opponent holds nut, opponent will consistently raise to maximize their profit. This being the case, if the hand never ends due to unlimited resources on both sides, 2nd nut will never lose if they keep raising. No showdown, no loss, and they might even have the best hand because...

Taken to the next level a 3rd nut hand should never stop raising either since neither 2nd nut or nut should logically stop raising if they really are expert at the game.

You can see where this ends up going as the level of logic becomes greater.