# Game Theory Optimal

Heads-Up Calling Strategy

Here are the "Game Theory Optimal" heads-up calling ranges for defending against an opponent that is playing strictly push-fold on his button hands. These ranges were derived from information found in the book: The Mathematics of Poker.

20BB) 25% - 22,A5,A2s,KT,K9s,QTs

15BB) 31% - 22,A2,K9,K7s,QT,Q9s,JTs

12BB) 36% - 22,A2,K7,K4s,QT,Q8s,JT,J9s

10BB) 41% - 22,A2,K5,K2s,Q9,Q7s,JT,J8s,T9s

09BB) 44% - 22,A2,K4,K2s,Q8,Q6s,J9,J8s,T8s

08BB) 49% - 22,A2,K2,Q7,Q4s,J9,J7s,T9,T8s,98s

07BB) 53% - 22,A2,K2,Q6,Q2s,J8,J6s,T9,T7s,97s

06BB) 58% - 22,A2,K2,Q4,Q2s,J7,J4s,T8,T6s,98,97s,87s

05BB) 66% - 22,A2,K2,Q2,J5,J2s,T7,T4s,97,95s,86s,76s

04BB) 78% - 22,A2,K2,Q2,J2,T5,T2s,96,94s,86,84s,76,74s,64s,53s

03BB) 95% - 22,A2,K2,Q2,J2,T2,92,84,82s,74,72s,64,62s,53,52s,43,42s,32s

02.6BB) 100% - call everything for 2.6BBs or less

Note: "Game theory optimal" and the dictionary definition of optimal are not the same. A "game theory optimal" calling range for a certain number of BBs is a range of hands that cannot be exploited by an opponent (as long as he is truly playing push-fold). Unexploitable in this case means if such an opponent is not pushing with the "game theory optimal" range of pushing hands for that number of BBs, you'll automatically have an edge on him* whether he is pushing too loosely or too tightly (* an edge on him for the hands when he has the button that is - for your button hands where you act first preflop, that's another matter. See
Heads-Up Push-Fold Strategy.

You should not be calling this loosely against most opponents though, since most players push tighter than optimal, so unless you have a strong read otherwise, a good default plan should be to exploit this by tightening up on your calling.

Also, most people won't be playing strictly push-fold. If an unknown player pushes all-in, he could indeed be playing push-fold or he could be playing limp-minraise-push-fold, and this just happens to be one of his pushing hands. The more often a player limps, the stronger his average shoving hand tends to be.

When you're up against players not playing push-fold, you need to do some hand reading and know what adjustments to make. Here's an example.

Inexperienced players often have the following pattern when the stacks are about 7-12 BBs heads-up: They tend to limp or raise small with a lot of medium strength or weak hands as well as limp or raise small with their big hands. And when they actually push all-in, they show up with a lot of hands like AJ-A2 and 99-22 which is a pretty strong range (despite it not including AA,KK, etc).

Against this type of player, calling with the "Game Theory Optimal" hands will be calling way too loose. You'll get slaughtered. You need to call tighter versus his all-ins and shove all-in yourself quite a bit when he limps. And also you should reshove against his small raises with a slightly looser range than you are willing to call his all-ins with.

The game theory optimal calling ranges are more appropriate the smaller the stacks are, because that makes it more likely the opponent is playing push-fold (such as when you're up against most multi-tabling sit-n-go grinders who will be playing straight push-fold when the effective stacks get below about 12BBs).

The effective stack is equal to the shorter stack of the two. So if one player has 20BBs and the other player has only 7 BBs, then the players' decisions should be based on 7 BB stacks (regardless of who is the bigger or the smaller stack).

What if he's playing push-fold, but he's pushing tighter than the "game theory optimal" pushing range? If you know he's pushing tight, you should tighten up your calling to exploit his tightness. By how much, you'll have to be the judge.

But as long as he's truly playing push-fold, then calling him with the "Game Theory Optimal" calling hands is unexploitable, meaning he can gain no edge on you. If he's pushing tighter than optimal, you'll be getting all-in with less equity (on average) the times you do get all-in, but that will be more than made up for by the extra times he folds and gives you the blinds.

Similarly, if your opponent is shoving every single button, you should loosen up on your calling to exploit him, but you would still have an edge on him if you only called with the optimal calling hands. Yes, he would steal the blinds more often, but that would be more than made up for by the extra pot equity you would have (on average) the times you do call.

Game theory is a beautiful thing.

related article:
Heads-Up Push-Fold Strategy

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