Applying [0,1] game #6

[Aisthesis]

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.

[Aisthesis]

What is the optimal way to counter the following suboptimal strategy for A, in which A plays too tight?

A calls a raise [3/4,1]
value-raises [3/4,1]
never bluffs

[Aisthesis]

Here's the surprizing best counter-strategy that I get:

B bluffs [0,5/8]
limp-folds [5/8,17/20]
limp-calls [17/20,1]

I get a value for B here of 783/800, or almost $1 (=SB) per hand.

B never value-raises here! His bluffs actually make a profit, so he only limps when his expected profit exceeds the profit of bluffing. If he were to value-raise at a value below 3/4, he would be losing more when A raises. And if he value-raises above 3/4, he can just let A raise for him.

Translated into poker hands, we already know those in the top 3/8 and 1/4. The only ones we don't know are those above 17/20 = 85%. Those are: 33-AA, A4s-AKs, A7o-AKo, KTs-KQs

Has anyone here ever tried a strategy like that against a moderately tight opponent heads-up?

[Aisthesis]

Some general considerations on where I'm trying to go with this:

First, the current games seem to me to allow some conclusions only with regard to the first few actions in a heads-up match. We haven't yet explored what happens when re-raises are allowed but just hand selection and raising quantities for the first raise on the part of each player.

Second, I'm assuming that the way to begin a heads-up match against an unknown opponent is to play the optimal strategy until reads allow you to define the opponent's suboptimal strategy. Then you can switch to optimal exploitation of that particular strategy. (just as a note, I actually did try this yesterday in a heads-up match on Stars and won it in 10 hands! Admittedly, the cards also fell nicely on the decisive hand, which was a pre-flop all-in with my 99 against my opponent's AQs which failed to improve, and one match obviously isn't anything on which to base any solid conclusions... but still, I was already ahead before the decisive hand, although not enough for adequate recovery had I lost that one)

However, third, when switching away from the optimal strategy to exploit an opponent's suboptimal play, I think you need to be a bit careful about covering your tracks.

For example, in 6a), where the conclusion was that the SB starts raising all of his bad hands against an overly tight opponent and limping on all others with the intention of trapping on his very best ones, A is eventually going to catch on to this. For the moment, I'm thinking the best way to deal with that is simply to switch back to "optimal play," probably also eliminating the bluffs at first, since A is going to be calling raises too often until he realizes that you've switched strategies again.

But with a fairly small repertoire of strategies based on these [0,1] games, I think it's going to be effectively impossible for an opponent unfamiliar with these strategies to adjust adequately in the course of a game. In my opinion, even the simple optimal strategies of games 6 and 7 (with the exact figures for raising, bluffing, and limping hands) goes well beyond the thinking of most players. And an optimal strategy is going to be the only way to gain immunity from strategy changes.

[Aisthesis]

In game #7, where the raise is set at $6 for both players, the result showed that only the top 1/8 of SB's hands can handle a raise of that size. So, let's suppose B isn't aware of this and raises the top 1/4 of his hands as in game #6. Let's also assume that B fails to bluff at all. What's the optimal strategy for A here?

I'll assume the following strategy for B in game #7:

B: limp-folds [3/8,5/8]
limp-calls [5/8, 3/4]
value-raises [3/4,1]

[Aisthesis]

Given this suboptimal strategy for B in game #7:

B: limp-folds [3/8,5/8]
limp-calls [5/8,3/4]
value-raises [3/4,1]

A adopts the following strategy for an EVa of 59/192 (=$.31):

A: calls a raise [5/6,1]
bluffs [0,1/2]
value-raises [11/16,1]

Again, since B isn't bluffing, A can be fairly picky about calling the raise, and since B is folding too often, A should bluff a lot.