[eastbay]

[ QUOTE ]
NL, HU hold'em, you're on the SB (every hand) and have only two options: push in or fold. Every hand starts with a specific stack size that both you and the BB have.

The "house" is on the BB, and the house plays optimally in the game theory sense.

The blinds are 200/400. What conditions do you require of the stack sizes to play this game against the house profitably?

eastbay

[/ QUOTE ]

This was a hard problem. Here's my answer:

For deep money, this game is unfair to the SB. Below a critical stack size, the advantage shifts to the SB. This critical stack size is about 7.8 BB. The edge for the SB then increases to a maximum at about 4 BB, and then sharply decreases again.

#BB +/-chips for SB
12 -33.0
11 -25.9
10 -18.1
9 -10.1
8 -1.6
7 +6.7
6 +15.3
5 +22.5
4 +26.4
3 +20.6
2.5 +11.8
2 +3.8

At the breakeven point, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q4o+,Q2s+,J7o+,J2s+,T7o+,T4s+,97o+,95s +,87o,84s+,76o,74s+,64s+,53s+,43s (0.631976)

bb call hands: 22+,A2+,K2+,Q7o+,Q3s+,J9o+,J7s+,T9o,T8s+,98s (0.457014)

There are no mixed stratgies at this point, interestingly enough.

At the point of maximum edge for the small blind, the optimal strategies are:

sb push hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,54s (0.737557)

bb call hands: 22+,A2+,K2+,Q2+,J2+,T5o+,T2s+,96o+,93s+,86o+,84s+, 76o,74s+,64s+,53s+ (0.740573)

Again, there are no mixed strategies, and moreover, within one hand they are playing the same strategy! I suspect this may point to a deeper principle of some kind, but I'm not sure what it might be.

So, my answer to the original question is: "I would insist that my stack be less than 3120 chips, and if I get to choose my stack, I choose 1600 chips."

eastbay