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inlemur
06-08-2005, 12:06 AM
I am crossposting this from the poker theory forum because it seems like it could be a fun problem for some of the mathier folks. If this is an out of line thing to do, please let me know and I'll contact a mod to lock the thread.

I apologize if this is an old question, because it seems like the kind of thing that has probably been posed several times, but it can't really be searched for.

Suppose I design a robot that plays poker, and you are going to play a heads-up NL hold-em tournament with it. The robot's strategy is to push all in with any two cards without even looking at them.

What strategy should you adopt to maximize your equity? That is, what hands should you call with? How does this change as a function of the blind structure of the tournament? Finally, what is your equity against the robot assuming perfect play?

Calling with any favored hand does not seem correct unless each player has a very small number of BBs. For example, many hands are slightly +EV against a random hand but if it costs very little to wait for a hand with a larger +EV vs random, then it seems correct to do so. But how can you relatively accurately calculate your equity as a function of the structure?

This is a summarization of a question a friend asked while I was teaching them how to play poker. I was trying to explain that unlike say, basketball, where I would lose to LeBron 100% of the time, in poker, even a two line program could win a heads up game against Phil Ivey some percent of the time. I could not, however, come up with that percent chance of winning figure, though I suspect it's around 20% for most reasonable blind structures.

Thanks for any replies.

Orpheus
06-09-2005, 06:13 AM
I don't believe that this is readily calculable. Though you are correct that it might beat him some percentage of the time, YOU could beat LeBron... if he slipped and broke his leg. It'd be bad luck, nothing more.

But still, let's take a shot at it.

Early in the match, when bothe players had nearly even stakes, Phil Ivey would obviously fold unless he believed he probably had the best of it--and since your bot uses little or no judgement, Phil would almost always be right.

If both players played "the closed eyes all-in" strategy, we would expect most matches to be exactly 1 game long (unless there was a push) and rarely would a match exceed two hands (two consecutive pushes).

EARLY PHASE:
Ignoring psychology (since the bot has none), I'd expect Phil to surrender his blind until he had a "good enough" hand. Forget strategy and reads: they both start with identical stakes, which means a loss would be instand death, which Phil isn't going to risk without sufficient cause.

The Early phase calculation depends on the %age of hands Phil considers "good enough". this would give us a basis for characterizing how many blinds he'll give up before getting a callable hand (and indirectly, his chances of being beaten by dumb luck when he finally calls)

This "callable percentage" will increase as his stack is nibbled bown by the blinds. It will become increasingly necessary for him to take a stand with a hand that might not be as good as he likes. Since the all-in bot is bluffable, and Phil wouldn't have any chips left after a call for a raise, I doubt the callable %age would exceed 50% (i.e. Phil would might call with "better than a random flop", but never "worse than the average random flop").

In that extreme case, Phil would wind against the "closed eyes all-in" 75% of the time +/- an SD ca. 6.25% [it's too early for me to be doing real math) if we assume that chances of winning are uniformly [linearly] distributed over the top 50% of hands. This assumption may not be correct: there may be "clumps": e.g. the 75th percentile might only have a slightly greater chance of winning than the 50th percentile, but (only?) the top 10% of hands might have a very appreciable advantage.

As a general rule: Phil either folds (losing just the blind) or calls (and either wins or loses). Assuming Phil does nopt call the first hand, The bot has the advantage of winning the initial blind, and can therefore survive (by the skin of its teeth) losing any hand but the first hand played, while Phil would be immediately eliminated by losing ANY hand (until he wins one)

This advantage is rather small, however, until "the midgame" because the Bot would only survive with several blinds worth of chips, which overwhelming chipleader Phil could more readily call with a wider range of hands (and expect to win)

MIDDLE GAME
From these metrics we could guess at he chances that Phil would lose by bad luck in in the first several hands, but what I call "the middle game" starts when the accumulated blinds become substantial

a) If Phil has not yet called, his dwindling stack pressures him to take progressively more risk and bet with progressively less desirable hands

b) if Phil has called and won (if he lost, he'd already be out) the bot might lucky going all-in with its few surviving chips, leaving it an underdog, but steadily nibbling at Phil's stack until he is dealt another good hand

(a) is pure personal psychology, which varies from player to player. I'm not sure that there is any reliable metric that accurately predicts how quickly his "urgency to call" would build. Unless Phil gets a good hand soon, (b) will eventually approximate the first hand, where both players had approximately equal chips. This time, however, Phil will know what he is up against, and be less discriminating about the hands where he "takes his stand" Instead of calling with premium or near-premium hands, he might call the all-in with any good hand (e.g. top 33% or more)

END GAME
I'm not sure there would be any distinct endgame strategy. The ubiquitous all-in makes it a knife-fight between balloon animals. Sometimes one might be twisted tight enough to be "partly popped" but generally it'd be all over pretty quick, one way of the other.

I like Ivey's chances. A lot.