Calling a non-all-in bet is not an option in preflop-only games, so we can only make these situations more complicated by adding re-raises or allow- ing players to use different bet sizings. After a while, looking at games with more preflop bets is also no longer useful, since these games will not represent real play at any stack sizes. A SB can realistically play jam-or-fold with a particularly short stack, and a BB can play shove-or-fold facing open raises with a somewhat deeper one, but once a BB starts making non-all-in 3-bets, the SB will almost always want to have a flat-calling range because of his postflop positional advantage. So, we will not chart out detailed so- lutions for more complicated preflop-only games. However, we will look at one more in order to motivate and present an important concept, the In- difference Principle.
A - SB B - BB Minraise Fold Fold C - SB 3-bet to 5 BB Fold D - BB 4-bet to 12 BB Fold E - SB All-in Fold Call
Figure 4.1: The minraise/3-bet/4-bet/shove decision tree.
Consider the minraise/3-bet/4-bet/shove game shown in Figure 4.1. The SB can raise or fold first to act at decision point A. If he raises, the play moves to point B where the BB can 3-bet or fold. At point C, the SB can 4-bet or fold. Finally, facing a 4-bet, the BB can shove or fold at D, and the SB can then call or fold at E.
Suppose the players begin 50 BB deep. Fill in the total ex- pected stack size for each player at each leaf of the decision tree in Figure 4.1. We will use these later.
What does GTO play of this game look like? As you might guess, at many stack sizes, the SB will start out raising a lot of hands and folding some bad ones. The BB will then 3-bet with good hands and fold some worse ones. The SB will 4-bet a tighter range of good hands and fold others, and so on.
However, it turns out that we can do a lot better than this without resort- ing to the brute-force methods which we have used so far.
Recall something we learned in Chapter 2 about equilibrium strategies. By definition, they are maximally exploitative strategies when facing an op- ponent who is playing his own GTO strategy – in this case, every hand is played as profitably as possible. So, the only way a hand can be played in two different ways at the equilibrium is if both of those actions have ex- actly the same EV, that is, if the player is indifferent to his choice of actions. This is known as the Indifference Principle: if a player plays a mixed strat-
egy with a hand at the equilibrium, then it must be that all of the actions he takes with a nonzero frequency have the same EV. This is a powerful state-
ment, because it tells us something about the GTO strategy of the player’s opponent. If a player is indifferent between two options, it must be that his opponent is playing in such a way as to a make him so. We will be able to leverage this constraint on the opponent’s strategy to learn a lot about GTO play in many situations.
This principle is intimately connected with the idea of balance. For exam- ple, think back to our discussion of balance in Section 2.3.1. We talked about the play of small pocket pairs from the BB when facing a raise 35 BB deep. We saw that there is good reason to think that just going all-in with these hands would be part of a GTO strategy. Those hands have good hot- and-cold equity versus most calling ranges preflop but can be difficult to play profitably postflop. We saw, however, that if our entire jamming range in that situation consisted of small pocket pairs, Villain’s counter- strategy would involve calling all-in with a wide variety of rather mediocre holdings, over a third of all hands in total. At this point, Hero’s counter- counter-strategy involved jamming over open raises with all strong hands – this would almost certainly be the most profitable way to play premium holdings since the SB is calling all-in so often. But, if Hero is jamming pre- flop with all of these, Villain’s re-adjustment is to tighten up his calling range, probably to the point that all-in is no longer Hero’s best option with many of his strong hands.
Here comes the important point. Assume we know for a fact that unex- ploitable play with small pocket pairs from the BB at 35 BB is to go all-in.
Focus on Hero's play with the high-card hands (such as, perhaps, A-Ko) that might also be jammed for balance-related reasons. If Hero is always shov- ing these high-card hands, then Villain’s counter-strategy incentivizes him to stop. So, his equilibrium strategy can not be to always shove. However, if he is not shoving at all, then Villain’s counter-strategy incentivizes him to start. Thus his equilibrium play can not be to never shove. This is the sort of argument that tells us that Hero must be playing a mixed strategy at equilibrium. Since he can not be always shoving nor can he be never shov- ing, his equilibrium play must involve doing both with non-zero probabil- ity. Since he is taking both actions, we know from the Indifference Principle that they must have the same EV at the equilibrium. To make this as clear as possible, imagine that Villain was playing so that shoving had slightly higher EV than not-shoving. In this case, Hero would start shoving more, but then Villain’s adjustment would cause shoving to become less profit- able, thus incentivizing Hero to move back towards his unexploitable mixed strategy. And vice-versa. The only stable situation is where the two actions have the same EV. This observation gives us some information about Villain's equilibrium play – his equilibrium strategy, whatever it is, must make the EVs of Hero’s two options equal.
To begin to see how we can use these ideas to address the minraise/3- bet/4- bet/shove game, recall some observations we made about the min- raise/shove preflop-only game in Section 3.2. We found that the SB’s equi- librium strategy at many stack sizes involved both open-folding and raise- folding some hands. Since neither of these actions results in going to showdown, the particular hands the SB chose to make these plays with were not important. (This is not precisely true due to card removal effects, but these are relatively small and will be neglected in the following discus- sion.) So, hands which were open-folded could just as well have been raise- folded, and vice versa – all that mattered were the overall frequencies of open-folding and raise-folding. Thus, in the case of the minraise/shove game, at stack sizes where SB is sometimes open-folding and sometimes raise-folding, the SB is indifferent between the two actions with his weak hands at the equilibrium. Furthermore, the BB’s equilibrium strategy must involve shoving over the SB’s opens with just the right frequency to make it so.
Write down the SB’s EVs of raise-folding and open-folding in the minraise/shove game at 20 BB as a function of the BB’s 3- betting frequency. Find the BB jamming frequency that makes them equal. Is this re-raising frequency the one which is used at the equilibrium?
We can use reasoning based on this principle to essentially solve the min- raise/3-bet/4-bet/shove game shown in Figure 4.1. There are multiple situations in this game where one of the players is indifferent between two actions with some hands, and the other player must be playing in such a way as to make him indifferent. We will list them now and work out the consequences which each indifference has for the GTO strategies. Suppose, for concreteness, that starting stacks are 50 BB, the 3-bet is to 5 BB and the 4-bet is to 12 BB.
To see where the following indifferences come from, again think back to the SB’s situation in the raise/shove game. When he played his open- folding and raise-folding lines, his holding was unimportant because he never got to showdown. And so, with any hand which was not good enough to take to showdown, he would take both lines with some fre- quency. Knowing this, we applied the Indifference Principle to conclude that the EVs of both actions were equal. In this game, there are a number of lines that both players can take with hands they have no intention of getting all-in. The SB can open-fold, raise-fold, or raise-4bet-fold. If he is indeed taking all of these actions with non-zero frequencies, then the EVs of all three must be equal, since the actual holdings used to take these lines are interchangeable. As for the BB, he can fold or 3-bet-fold with the hands he is unwilling to get all-in, and if he is doing both at the equilib- rium, the SB’s strategy must make him indifferent to the choice.
Of course, at sufficiently deep stack sizes in the raise/shove game, the BB was not able to shove often enough to discourage the SB from opening all his buttons. Then, the indifference was not satisfied, and the SB’s open- folding frequency was 0. However, this was not really the case at any stack sizes where it made sense to think in terms of the raise/shove decision tree in real play. We will find some similar breakdowns here at extreme stack sizes, but we will start out by assuming that the players are indeed taking all their non-showdown lines with some non-zero frequencies.
The SB is indifferent between raise-folding and open-folding.
It must be the case that the BB is 3-betting enough that those two ranges have the same EV. We can set those two EVs equal in order to solve for the 3-bet frequency, X, that makes this happen. We have
EVSB(open-fold) = EVSB(raise-fold) 49.5 = 48X + 51(1 − X)
since we necessarily end up with 49.5 BB if we open-fold, but if we raise- fold, we end up with 48 BB the X of the time the BB 3-bets and 51 BB the (1-X) of the time he does not. This equation is satisfied by X=0.5. That is, the BB will 3-bet half of the hands with which he reaches decision point B, and fold the other half. This is the same as in the minraise/shove game since the calculation really depends only on the SB’s open raise sizing. The BB is indifferent between 3-bet-folding and folding to the SB’s first raise.
It must be the case that the SB is 4-betting with a frequency such that those two plays have the same EV. We can solve for the SB’s 4-bet frequency, X, that makes this possible. We have
EVBB(fold) = EVBB(3-bet-fold)
49 = 45X + 52(1 − X)
which is satisfied by X=3/7. So, the SB will 4-bet with 3/7 of the hands with which he gets to decision point C and fold the other 4/7.
After getting to decision point C, the SB is indifferent between raise-folding and raise-4-bet-folding.
It must be the case that the BB is 5-bet shoving enough that these two actions have the same EV. Let us solve for the shove frequency, X, that makes this happen.
EVSB(raise-fold) = EVSB(4-bet-fold) 48 = 38X + 55(1 − X)
which is satisfied by X=7/17. Thus, it must be that the BB is shoving with 7/17 of the hands with which he gets to decision point D.
What have we found so far of the solution to this game? Consider the BB’s strategy. First, facing a SB open, he 3-bets 50% of the time. Then, when fac- ing a 4-bet, he 5-bet shoves 7/17 of that 50% so that his total 5-bet shoving range includes 7/34 or about 20.6% of hands. We see that the constraints arising from the Indifference Principle completely specify the BB’s frequen- cies in this game!
What about the SB’s strategy? At the beginning of the hand, the SB can ba- sically divide his holdings into two categories: hands which are able to to call all-in profitably once they get to decision point E and those which are not. If a hand is more profitably played by raise-4-bet-calling than raise-4-bet- folding, then raise-4-bet-calling is also better than open-folding and raise- folding, since the EVs of the lines ending in a fold are all equal at equilib- rium. So, those which are able to call all-in at point E will constitute his raise- 4-bet-calling range. Finding these hands, at least approximately, is easy. At point E, the SB will have to call a raise of 38 BB to win a pot which will total 100 BB, so his calling range will just be all hands that have at least 38% eq- uity versus the BB’s shoving range. We know that the BB’s shoving frequency is 20.6%. Assuming the BB’s shoving range is the top 20.6% of hands, the range of SB holdings which have 38% equity is {22+, A2s+, K9s+, QTs+, JTs, T9s, A2o+, KTo+} which is about 25.5% of hands.
The other 74.5% of hands held by the SB will be folded at some point, either by open-folding, raise-folding, or raise-4-bet-folding. We are still neglecting card removal effects, so if a hand is a folding hand, it does not matter where it is folded as long as we get the correct folding frequencies at each point. So we know how many hands the SB is calling all-in with at point E. Let us find his frequency of folding to the BB’s all-in. This requires us to make one last application of the Indifference Principle. Consider the last BB decision: to shove or fold to the SB’s 4-bet. The weakest hands in the BB’s shoving range are not profitable if they have to get it all-in – shoving these is only
profitable since the SB sometimes folds to the shove. Furthermore, even with the SB sometimes folding, the very worst hand in the BB’s shoving range is essentially breaking even. That is, the EV of shoving it is more or less the same as the EV of folding it. The same can be said of the strongest hands in the BB’s folding range at point D.
The idea here is this: both the worst hand in the BB’s shoving range and the best hand in his folding range are effectively indifferent between shov- ing and folding. In fact, there is often a real indifference here – the hand on the borderline between a shove and a fold is often played with a mixed strategy at the equilibrium. The BB’s shoving range is comprised of 20.6% of all hands, so his range might look like {55+, A3s+, K8s+, QTs+, A7o+, K9o+, QJo} and A-6o is a good representative of the bottom of his shoving range and the top of his folding range. (It turns out that A-6o is actually played with a mixed strategy in this case.)
Thus, the BB must be indifferent between jamming and folding A-6o, and the SB must be folding enough to the shove to make it so. What SB folding frequency does this imply? A-6o has equity 0.4214 versus the SB calling range. Let the SB’s folding frequency be X. Then, at the equilibrium:
EVBB(fold A6o at point D) = EVBB(shove A6o at point D) 45 = 62X + (2 · 50 · 0.4214)(1 − X)
which is satisfied by X=0.144. So, at decision point E, the SB will be folding about 14.4% of the time and calling the shove the other 85.6% of the time. Let us find the rest of the SB’s frequencies. We have already estimated the range with which the SB is calling the shove (25.5% of all hands), and taken together with his last folding frequency, we can find how many hands in total the SB brings to point E where he faces the shove, that is, the total fraction of hands he raises and then 4-bets. If this total number of hands is X, then 85.6% of X is 25.5% of all hands, so he must be getting to decision point E with 29.8% of all hands.
Working the SB’s frequencies back up the tree, we see that he raise-4-bets with 3/7 of the hands he raises, so he must raise with 7/3 x 29.8% = 69.5% of all hands. This means that he open-raises 69.5% of hands in total, and
he must open-fold the remaining 30.5% of hands at the equilibrium. We have now, to good approximation, found the equilibrium frequencies of the game. To find the BB’s frequencies, we made no reference to things like individual hand strengths or equities. We just found the bulk volume of hands necessary to make the SB indifferent between the various lines he takes with his weak hands.
There is an additional important point to be made about the solutions to the minraise/3-bet/4-bet/shove game. Remember how we found the SB ranges. We found what hands he is actually willing to get it all-in with. Then, we found how many other hands he had to reach each decision point with in order to be unexploitable. What are these other hands? Bluffs, of course! With 50 BB stacks and the bet sizings given, the SB is willing to get all-in with 25.5% of holdings. So, at the beginning of the hand, he raises with those 25.5% and also another 44% of hands which are bluffs. If he gets 3-bet, he continues by re-raising all of the 25.5% as well as a smaller chunk of bluffs which comprises about 4.3% of all hands.
These frequencies are the exact ones needed to make the BB indifferent to having bluff-raising ranges himself. For example, if the SB neglected to in- clude the 44% of bluff hands in his opening range, then the BB would no longer be indifferent between folding at point B and 3-bet-folding. He would never 3-bet-fold – since the SB is never raise-folding, 3-bet-folding would be burning money. In fact, if the SB included even slightly less than the GTO number of bluffs in his opening range (but kept his other ranges the same), the BB’s best response 3-bet-folding frequency would still be zero.
What would be the consequences to the BB’s exploitative strategy if the SB played decision point A with GTO frequen- cies, but neglected the 4.3% of bluffing hands when he 4-bet at point C?
As we will see, when designing equilibrium and exploitative ranges, it is often best to think in terms of frequencies first and then find hands to add