Model for short-stack all-ins

[Jerrod Ankenman]

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Finally, to translate into poker hands, we need a hand ranking system. But the top ones should stay pretty much the same. The 95% should translate roughly to 99-AA and AQ/AK--unless one wants to quibble over whether 99 is better than AJs. That's pretty re-assuring to me, as those are exactly the hands I'd move in with as shortstack even UTG.

As one gets to around 90% (5-player game), one can start thinking about hands like 55/66, A9s, ATo and KQs.

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Sorry, Aisthesis; I've been busy and haven't had a chance to read or respond to this, and I'm departing soon for WPF, so it'll probably have to wait until after.

Two things though:

- Multi-player games always have Nash equilibria, but these are not "optimal" solutions in the sense that if you play your strategy from the Nash equilibrium you can't be exploited. You can normally find coalition strategies that exploit the Nash equilibrium strategy. You might want to google a post of mine on rgp from a few years ago about a three player rock maniac game.

- The matchup between .95 and .93 in the high-number wins [0,1] game is 100% win for .95. (I don't know why you all don't play the lowball version, it makes the algebra much easier). The matchup between AK and TT in holdem isn't. Be careful in trying to map the [0,1] game to holdem; the equities aren't the same or even similar.

Jerrod

[Aisthesis]

Thanks for taking a look at this, Jerrod!

I'll see if I can find the link mentioned and post if I can locate it. Anyhow, when you get back and if you have the time and inclination sometime, I'd be most interested in exploring any (preferably simple to start with) multi-player games you might have just to get a feel for how it works. The collusion part of it sounds like an interesting dimension--although hopefully isn't the case in most practical HE contexts.

Might take another stab at this one with the lowball version in the meantime. Doesn't the simplicity really depend on whether you end up with more "1-x"s or just plain "x"s in the indifference equations? Anyhow, I'm definitely down for any strategy that makes the math easier (noticed you preferred the lowball version in your introductory webpages), as some of these simultaneous equations can get rather nasty pretty quickly.

Definitely in agreement regarding the mapping to HE. My general attitude toward the [0,1] game (applied to non-river issues) is generally just that it's a helpful tool, but always has to be taken with a grain of salt. The extreme case, along lines you mention, is 22 vs. AK, where 22 is likely to be way down there on most hand ranking systems. But if you can somehow exclude a bigger pair (big "if" there!) from the range of hands your opponent has and are down for a coinflip, even 22 can suddenly become pretty decent even if it's not a case where you're drawing to the set.

Actually, that's another topic that I think would be very much worth exploring sometime, as my "grain of salt" translation is obviously pretty vague. I will have to admit that I really like applying the [0,1] game more to pre-flop issues simply because the range of holdings is completely open prior to betting--hence at least allowing the assumption of completely even hand distributions. Once you get to the river, or even the flop, the distribution of possible hands makes for some major complications, as I see it.

[Jerrod Ankenman]

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Might take another stab at this one with the lowball version in the meantime. Doesn't the simplicity really depend on whether you end up with more "1-x"s or just plain "x"s in the indifference equations?

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Yes, that is the reason.

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Actually, that's another topic that I think would be very much worth exploring sometime, as my "grain of salt" translation is obviously pretty vague. I will have to admit that I really like applying the [0,1] game more to pre-flop issues simply because the range of holdings is completely open prior to betting--hence at least allowing the assumption of completely even hand distributions. Once you get to the river, or even the flop, the distribution of possible hands makes for some major complications, as I see it.

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Well, what you say about unequal distributions on later streets is absolutely true. What you might look at are games like the [0,1] vs [0,1]^2 game, where player B has hand X with probability density function X^2. This actually resembles (to some arbitrary extent such that it makes sense to Bill and I) one-card vs two-card draw situations in lowball. I think this game has pretty strong insights about playing in the blind against a field raiser.

Another game you might look at, which I think I'm going to explore in our book, is the straight-flush vs nut flush vs aces hand on the flop in holdem (ie, flop of 9h 6h 3c and the hands are Ah2h, 8h7h, and AcAd). Try to figure out collusive strategies for the various pairs of hands--it's interesting.

Jerrod