Applying [0,1] game #6
Applying [0,1] game #6 directly to a NLHE heads-up situation, I get the following optimal strategies (with A in BB and B in SB): (a raise is always to 2xBB)
B bluff-raises: J2o-J3o, T2s-T3s, T4o-T5o, 94s-95s, 96o, 85s, 87o, 75s-76s, 65s
B limp-folds: J2s-J4s, J4o-J7o, T4s-T6s, T6o-T8o, 96s-97s, 97o-98o, 86s-87s
B limp-calls: KX, QXs, Q4o-QJo, J5s-JTs, J8o-JTo, T7s-T9s, T9o, 98s
B value-raises: 22-AA, AX, K8s-KQs, KTo-KQo, QTs-QJs
A calls a raise: 22-AA, AX, KX, QXs, Q4o-QJo, J5s-JTs, J8o-JTo, T8s-T9s, T9o, 98s
A bluff-raises: 82o-83o, 72o-73o, 62s, 62o-64o, 52s, 52o-53o, 42s, 42o-43o, 32s, 32o
A value-raises: 22-AA, AX, KXs, K4o-KQo, Q7s-QJs, Q9o-QJo, J9s,JTs, JTo, T9s
Actually, this doesn't look too far off from "reasonable" heads-up play in practice. The reason I chose game #6 is that in game #7 the value-raises occur too rarely (only top 12.5%). In my own practical experience, that's just not enough raising. So, comparing the 2 models speaks for keeping the raise small.
As for the hand-ranking behind this, I just used Karlson-Sklansky. While created for a different purpose, I think it reflects the relative strength of heads-up hands quite accurately. I simply ordered the hands on this list and related the percentile ranking of each hand to the respective number between 0 and 1 (AA is 100%, hence 1, TT is 96.7%, hence .967, etc.)
There are some obvious limitations here. First, the rules of game #6 exclude check-raising (for B) as well as re-raises. Actually, a game allowing those components is one that I'd like to analyse, although it gets a lot more complicated (allowing one pot-size re-raise as well as a check-raise should still be doable, I think, although difficult--for me, anyway). But I thought we had sufficient results already to take a first stab at what this would mean if the game can be applied directly to a specific poker context.
There are obviously some other shortcomings in this model. For one thing, in a heads-up situation subsequent events will significantly change hand values: If you have AA, it is true that you have the best hand pre-flop with a probability of 100%. But if your opponent sees the flop, it's actually rare that you will still have "the nuts." So, this model can make no statement at all concerning play on the flop or beyond.
For me, anyway, B's raising hands are actually pretty close to the hands I will generally raise heads-up. But I bluff a lot less than "recommended," and in BB, my criteria for raising don't go down. I also typically call raises less frequently than this model would suggest.
Anyhow, I'd be very interested in hearing whether others think there is any sense in applying this model to pre-flop heads-up play. Despite certain qualifications, even this simplified model doesn't seem to me to do all that badly.
And since very few opponents in my experience are playing optimally according to this strategy, I'd also like to explore optimal adjustments to a few typical suboptimal strategies on the part of an opponent. I'm planning on doing that in some separate posts over the next few days.
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