Head Up Theory Question

[HelloGoodbye]

Well, with AA I prefer to just call then fold to the flop, so without the nuts, open folding would be correct.

[M.B.E.]

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Even if I think there is only a 35% chance that he would take it to 12 bets with EITHER 999,998 OR 999,997 that still gives value to it because the chances he has ONE of those hands are greater than the chances that he has 1-million.

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No, I don't think so.

If I understand you right, you're supposing that with 1,000,000 he would always take it to 12 bets; with either 999,998 or 999,997 there is a 35% probability he would take it to 12 bets, and with any other hand he would never take it to 12 bets.

Then by Bayes' theorem, if he takes it to 12 bets then the probability he has 1,000,000 is 10/17 = 0.588.

But even if the probability he had 1,000,000 was a bit less than 0.5 we still couldn't reraise because when he does have it, it will cost us $2 more since he'll reraise again.

[M.B.E.]

Can we stipulate that checkraising is not permitted in this game? That would make the solution simpler.

[MicroBob]

I don't have time right now....but I'm going to check out that Bayes theorem link in a feeble attempt to better understand what the hell you are talking about.


I will defer to your obvious general expertise on the calculations that got you to 10/17 here.


I was just trying to find an example where a lesser than 50% chance at 999,997 would still be worth re-raising. Obviously I did it incorrectly and put the number too low.

This really isn't the precise way to do it...just a very simple approximation to try to prove a point.


More specificaly...I think with your 999,999 after 8 bets are in you could think in terms of something like:

chances of going to 12 bets with 999,998: 40%
chances of going to 12 bets with 999,997: 35%
chances of going to 12 bets with 999,996: 30%
chances of going to 12 bets with 999,995: 20%

something like that anyway.
Whether these approximations are even possible with out opponent to such specificity (and therefore have any relevance AT ALL) seems pretty unrealistic to me the more I think about it.

[M.B.E.]

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I was just trying to find an example where a lesser than 50% chance at 999,997 would still be worth re-raising. Obviously I did it incorrectly and put the number too low.

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Okay, but 50% is the wrong number entirely to be looking at.

Consider this example. Playing limit holdem, heads-up on the river the board is Qh-Js-9h-6h-2h. You have Kh-7s, the second nuts. Your opponent bets into you, and based on the action so far, you figure there is about a 45% he has the Ah. Hence you probably have the better hand. Do you raise there? If so you are making a mistake (your correct play is to call).

EDIT: Actually, in that example you could raise IF you knew your opponent will never threebet without the Ah, and he will never fold to your raise. Then it would be correct to raise and fold to a threebet.

[the shadow]

Good question. I've skimmed a few but not all of the responses. Here's my thoughts on the 1st part of David's question.

1. I would first make sure that high card wins. As I recall, the classic [0, 1] game was initially written with the low number winning. I'd hate to make that mistake here.

2. The pot odds just keep get better and better for a raise. As long as the villian is raising, there's no reason to stop at 5, 11, 28 or some other seemingly random number of raises, given that I'm holding the next to best card. Frankly, if the raises are $1 at a time, I don't see how such a relatively small number of raises reduces the range of hands that the villian might be holding.

3. Even so, at some point, it makes sense to call. For example, I'm pretty sure that somewhere long before the pot hit $1 billion, even I might put the villian on the nut card and would just call. (Heck, by then, I might be so embarassed about overbetting my hand that I'd fold, look at my watch, and tell the villian that I had to run to the Wynn to play Daniel.)

4. So the question is when to call? It seems to me that the optimal strategy is to raise until the pot size equals or exceeds the number of my card and then call. In this case, I would raise until the pot is at least $999,999 and would then call.

If I follow this strategy, it seems to me that villian cannot exploit it. Since the odds of him having a higher card than I is 50/50, the strategy should assure me in the long run of winning at least half the pots for any given card number.

Let's say that instead of 999,999, I'm dealt 314,159 or maybe 271,828. I keep raising til the pot hits my card and then call. After playing a few million hands, the villian's going to figure out my strategy pretty quickly. In fact, I'll save him the trouble and tell him upfront.

What's he gonna do about it? If he holds a higher card, he'll win the pot when I just call. But if he holds a lower card, is he really going to keep raising beyond the value of his card when he knows that I'll call at some point and he's throwing money away? I suggest the answer is no. If he does that consistently, over time I'll come out ahread, but since he's an expert, I can assume he won't do that.

Now maybe this approach is entirely wrong, and I should be breaking out my albegra or calculus textbooks, but it seems to me that this game boils down to this -- whoever draws the low card has to pay that amount of money to the other player. In the long run, that's a breakeven game if both players use the asserted optimal strategy.

The Shadow (who's still thinking about whether bluffing plays more of a role once the game's pot limit)

[brick]

Thanks for breaking this down. Nice work.

[the shadow]

As to the second half of David's question dealing with a pot-limit game, my thought so far is that bluffing is a factor here. It really wasn't in the first game, because, given the pot odds, the betting would end with a call, rather than a fold. Maybe if the pot's $400,000, I might just call sometimes, rather than raise, which opens me up to a possible reraise for either value or a bluff.

But in a pot-limit game, let's say that I'm dealt 500,001 and the pot is $250,000. Do I want to call or raise? If I raise by the amount of the pot, it's now $500,000. That means that even if the villian holds a lower card, there will be some fraction of time that he'll want to re-raise by another $500,000 to a total pot of $1 million. Some fraction of the time I'd want to call that raise (thinking he's bluffing) and some fraction I'd want to fold. I might even want to reraise so the pot grows bigger than $1 million.

I haven't come to a conclusion about what the optimum bluffing strategy is yet, but any optimum strategy will have to include bluffing in the pot-limit version of this game.

The Shadow

[David Sklansky]

I'm glad I asked this question. Some people need to do some serious rethinking.

[M.B.E.]

[ QUOTE ]
More specificaly...I think with your 999,999 after 8 bets are in you could think in terms of something like:

chances of going to 12 bets with 999,998: 40%
chances of going to 12 bets with 999,997: 35%
chances of going to 12 bets with 999,996: 30%
chances of going to 12 bets with 999,995: 20%

something like that anyway.

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I don't think it would look like that. After 12 bets I'm pretty sure that we and our opponent will both be playing either a pure strategy or something very close to it. So for example, it won't be anything like a 30% chance that he would have gone to 12 bets with 999,996; rather it would either be something like 0.000001% or something like 99.999999% (if it is a mixed strategy at all, which I'm not sure of).