Dave Computes Playing No-Limit Poker against an All-In Maniac

You're heads-up (or may as well be) in a no-limit poker game against a maniac who goes all-in every hand.  How often should you call him down?

This scenario is quite common in play-chip poker games, where no money is on the line.  Clearly we should be able to exploit the maniac's play, and we can.

I wrote a computer simulation of poker games like this, beginning with various stack sizes and using various betting and calling thresholds.  In each scenario, I simulated 100,000 games in which one player eventually ran out of chips.  Here's what I learned.

Stack sizes make a big difference in correct calling frequency.  In particular, what really counts is the size of the smaller stack relative to the size of the ante.

Size Of Smaller Stack	Optimal Calling %	Sample Hold'em Hands	Caller Win %
100 × ante	7%	77, AK, ATs	95%
50 × ante	10%	66, AT, KJs	92%
20 × ante	23%	K9, K7s, A5	84%
10 × ante	34%	A2, K4s, K6	74%
5 × ante	55%	J3s, Q3, 87s	63%
2 × ante	76%	T3, 92s, 76	53%

against a maniac who goes all-in 100% of the time

In other words, if one of you has a stack size of 50 times the ante and the other has a larger stack, then you should call down the constant-all-in-maniac 10% of the time (which corresponds to hands like 66, AT, and KJs in Hold'em), and you can expect to finish with all the chips 92% of the time.  For me, the real take-home point here is not to loosen up too much when the stack sizes are big.

What if the maniac "only" goes all-in half of the time (and folds the other half)?  Here's what my simulation found for that scenario.

Size Of Smaller Stack	Optimal Calling %	Sample Hold'em Hands	Caller Win %
100 × ante	1%	AA, KK	99%
50 × ante	1%	AA, KK	98%
20 × ante	5%	77, AQs, AJs	95%
10 × ante	10%	66, AT, KJs	90%
5 × ante	17%	A8, KT, A6s	81%
2 × ante	34%	A2, K4s, K6	69%

against a maniac who goes all-in 50% of the time

Not surprisingly, when the maniac tightens up, so do we.  But notice we should now play much tighter against the 50% maniac than we did against the 100% maniac.  In a Hold'em game with stack sizes of just 50 times the ante, the maniac is raising hands as poor as J5s, and we're correct in folding QQ!  Why?  Because the simulation tells us so, presumably because our stack size (along with the 50% of the time that the maniac folds his ante to us) lets us wait for a nearly sure-thing with AA or KK, rather than risk going bankrupt with an unlucky loss with QQ.

How about if the maniac goes all-in with just 20% of his hands (and folds the rest)?  Now even a stack size of just 5 times the ante is sufficient to let us call with only the top 1% of hands.

Of course, many all-in maniacs will limp into a hand instead of folding, so these aren't very realistic scenarios.  Nonetheless, it's clear that when your opponents are making disproportionately huge raises and going all-in after tiny pots, your best strategy is to be patient with your large stack and play tight, knowing that your opponents will pay you off when you make huge hands.