2.2.1 The Most Profitable way to Play a Hand

Given the above discussion, we can find a maximally exploitative strategy by just finding the most profitable way to play each particular pair of hole cards individually. To see how this works, we can just consider one arbi- trary hole card combination, call it H, and find how to play it most profita- bly given a decision tree and Villain’s ranges. Here we will address both how to play H to maximize our EV as well as how to find exactly what that EV is.

The first thing to notice is that we can actually talk about the expected value of holding H at any decision point P in the tree. Let us call this value EVH,P. As a quick example, suppose Hero is in the BB with 8♣-3♥, a hand he will fold if the SB raises. At the point after the SB raises, Hero’s EV is S−1 where S is his stack size at the beginning of the hand. That is, his total stack size at the end of the hand will be S−1, because he is just about to fold and surrender his blind. Hero should have a higher expectation at the point after SB limps. When the SB limps, we should be able to end up with more than S−1 on average, even if our strategy is to only put any money into the

pot when we happen to flop very well. If we call the decision point which is arrived at after Villain makes a raise A and that after he limps B, then in our notation, we have

EV_8♣3♥,B > EV8♣3♥,A= S − 1

A few other points should be made about hand values. First, notice that the value of Hero’s hand H depends strongly on Villain’s strategy. Also, recall that the EV of a hand at a decision point is different than its equity. Equity describes the fraction of the pot a hand would be entitled to on average if all betting stopped and hands were checked down. EVH,P refers to the expected total stack size at the end of the hand including future betting, folding, etc. Finally, notice that the EV of playing the whole hand with holding H is just the EV at the root decision point which we might write as EVH,root.

Let us start by considering the shove/fold game for an example. Shove/fold is the simplest approximate HUNL game. Its decision tree is shown in Fig- ure 2.1. The SB can either fold or shove all-in, and if he shoves, the BB can either call or fold. It is fairly common for players in the SB to restrict them- selves to a shove-or-fold strategy preflop at sufficiently short stack sizes. Suppose Hero is in the SB, and both players start with 10 BB.

Fold Call All-In Fold A SB — E BB — D B C

There are three different kinds of decision points in a decision tree. 1. Points which Hero controls

2. Points which someone else (either Villain or Nature) controls 3. Leaves which indicate that the hand is over and there is no choice

to be made

Let us first look at how to find the EV of having a hand at a leaf since this is the simplest case. There are no decisions to be made here, and the payoffs actually happen at the leaves, so it is very easy to find the EVs. There are two kinds of leaves.

1. Ones which follow a fold

2. Ones which indicate a showdown – either after the river or earlier if we have an all-in

There are two leaves following a fold in the shove/fold game. First, Hero can open-fold his button. This brings the state of the game to the point la- beled A in the figure. Hero’s EV is 9.5 BB at this point. That is, his average total stack size at the end of the hand is 9.5 BB after folding. In fact, the av- eraging is unnecessary in this case – his stack size at the end of the hand is exactly 9.5 BB no matter what hand he folded.

Now, if Villain folds to Hero’s shove, the game reaches the leaf labeled B. The value, EVH,B, of any of the SB’s holdings here is 11 BB since he wins Vil- lain’s BB without a showdown – he ends the hand with 11 BB regardless of his holding.

The game arrives at the final leaf, point C, if Hero shoves and Villain calls. Here, Hero’s expectation depends on his hand H. In particular, since both players are all-in, Hero’s EV is the frequency with which he wins the hand (and half the times they chop) times the size of the pot: 20BB x EQH where EQH is the equity of H versus the BB’s calling range. This is Hero’s total ex- pected stack size at the end of the hand. This expression comes straight

from the definition of equity. 20 BB is the total size of the pot after the players get all-in, and EQH is the fraction of the pot Hero wins on average when the players show down without any additional action.

That does it for the leaves. Now, what about those decision points which somebody else controls? Assuming that Hero is in the SB, there is only one of these in the shove/fold game – the point where Villain in the BB has the decision to call or fold. This is labeled D on the figure. What is Hero’s expec- tation with H at that point? Well, it depends on how often the BB folds to the shove and how often he calls. Suppose for example that BB always folded. It would be just as if his choice to call did not exist. The play would always move to point B, and so Hero’s EV with H at point D would be the same as that at point B: 11 BB. On the other hand, if Villain always called, then EVH,D would just be that at point C: 20BB x EQH.

However, point D is Villain’s decision point, and in general he will take both actions some of the time. Thus, to find Hero’s expectation at the point, we have to average over both probabilities according to Villain’s strategy. Suppose Villain calls CH of the time and folds FH of the time. Then, Hero’s EV of having hand H at point D is CH times his EV when Villain calls, plus FH times his EV when Villain folds. That is,

EV_H,D= FH· EVH,B+ CH· EVH,C

= FH· 11 BB + CH· (20 BB · EQH)

Notice, by the way, that CH and FH must sum to one. Also, they do depend slightly on H due to card removal effects. For example, if H contains an ace, it is slightly less likely that Villain holds an ace, and thus, assuming he plays a reasonable strategy, it is slightly less likely that he will call the shove.

Now, there are no decision points in this game which are controlled by Na- ture, i.e. where new cards are dealt. If there was such a point, then the value of having H there would just be the average of the values of having it after each specific new card comes. That is, we would find our expectation at that point by averaging over all the possible actions in Nature’s “strat- egy” just as we did for a decision point controlled by our opponent. This average must be weighted by the probability of each card’s being dealt. The likelihoods of any cards being dealt that are not already on board or

contained in H are usually approximately equal. However, they are not ex- actly so because they do depend slightly on Villain’s range. If his range is such that he is particularly likely to hold a certain card, then it is less likely that that card will be dealt. On the other hand, if his range does not in- clude a certain card, it is more likely that that card will be dealt. We will treat subtle effects like this correctly in our calculations in this book but will not dwell on them except in the rare cases they actually turn out to be significant.

Finally, there are the decision points we have control over. Point E, where Hero decides to shove or fold, is the only of these in the shove/fold game. What is his expected total stack size at the end of the hand given that he makes it to point E with a holding H? Well, this is a point we have control over, so the value depends on our choice of how to play H. Our choice will either take us to point A or to point D. So, the value of having the hand at point E will be that of having it at point A if we decide to fold, and it will be the value of having it at point D if we decide to shove. Since it is our deci- sion point, we get to choose which it will be. In practice, we are of course looking for the most profitable way to play each hand, so we just go with the bigger of those two. Thus, the value of a hand H at point E is

EV_H,E=max(EV_H,A, EVH,D)

where the max function gives whichever of its two arguments is bigger. Now we know how to find the value of a hand H for the Hero at any deci- sion point in terms of the values of the points below it. Remember that the value of H for the whole hand is its value at the root. If we figure out our most profitable plays with each of the 1,326 hold ’em hand combinations, we have found our maximally exploitative strategy. Finally, the value of the whole game is just the average over all hand combinations of the EVs of the individual hands.

Since these are such important ideas, we will go over a couple quick exam- ples. In the first, we will continue looking at the shove/fold game, but we will focus on the case when Hero is in the BB. In the second example, we will consider a slightly more complicated game and try to guide you to work through it yourself.

So, what is the EV of a hand in the shove/fold game when Hero is in the BB? That is, what is the value of a hand H at point E, EVH,E ? Since the BB does not control point E, it is just

EV_H,E = FH· EVH,A+ SH· EVH,D

where FH and SH are the proportion of times SB folds and shoves, respec-

tively. Now, we need to proceed recursively to find the unknown EVs refer- enced on the right-hand side of this equation. At point A, our total stack size is EVH,A = 10.5 BB with any hand. The action at point D is our choice, so our EV there is

EV_H,D=max(9 BB, 20 BB · EQ_H)

where EQH is the equity of H versus the SB’s shoving range. Hero’s two

choices are to fold and end up with 9 BB or to call and end up with 20BB x EQH, and he chooses the greater of those for any particular hand H. Thus, all in all, the BB’s total EV at the beginning of the hand is

EV_H,E = FH· EVH,A+ SH· EVH,D

= FH· 10.5 BB + SH· max(9 BB, 20 BB · EQH)

Now we look at a slightly more complicated game. The decision tree for the raise/shove game is shown in Figure 2.2. The SB starts out by either folding or raising. When he raises, the BB can fold or go all-in, and when BB goes all-in, the SB can call or fold.

Suppose that Hero is in the SB, his initial raise is a minraise (i.e. to 2 BB to- tal) and both players again start with 10 BB. We have labeled all the deci- sion points. We know the ranges with which Villain in the BB shoves or folds when facing a raise. Now, follow the series of questions to find how Hero should play each of his hands and his EVs when he does so. Try to an- swer them yourself before reading the solutions.

Fold All-In Raise Call Fold Fold B BB — C SB — A D F SB — E G

Figure 2.2: Decision tree for the raise/shove game.

Which point is the root of the tree? Point A, where play starts.

At which points do payoffs occur? At the leaves: B, D, F, and G.

Which of the leaves are led to by folds and what is Hero’s total expected stack size at the end of the hand if he has a hand H there?

♠

♠

♠

Which of the leaves involves a showdown and what is the value to Hero of having H there?

Point G. Here we saw a raise, shove, and call. The total pot size is 20 BB, and Hero’s EV is simply his equity share, 20BB x EQH,D.

What is the value to Hero of holding H at the root decision point A?

We have control over this decision, so we choose the maximum of the two options available to us: folding, which takes us to point B and a guaran- teed EV of 9.5 BB, and raising, and thus going to point C.

(2.1) EV_H,A=max(9.5 BB, EV_H,C)

This leads to the question: What is our value at point C?

Villain controls this point, so Hero’s EV is just a weighted average of the two possibilities.

EVH,C= 11 BB · FH+ EVH,E· SH

where FH and SH are Villain’s folding and shoving probabilities.

This leads to the next question: What is the value at point E?

Hero has control over this decision, so he just chooses the best of the two options given his particular hand:

EV_H,E =max(8 BB, 20 BB · EQ_H)

where EQH is the equity of hand H versus the BB’s shoving range.

So, what is the total EV for the SB of playing hand H? Now that we know the values at all the other points, we can just plug in the values for the other points into our earlier result, Equation 2.1.

EV_H,A=max(9.5 BB, EV_H,C)

=max(9.5 BB, [F_H· 11 BB + S_H· EV_H,E])

=max(9.5 BB, [F_H· 11 BB + S_H· max(8 BB, 20 BB · EQ_H)])

We have found the total EV of playing any hand in the shove/fold and raise/shove games against arbitrary opponent ranges, implicitly finding the best line for Hero with each hand as well. Let us take an example so as not to get too bogged down in the math.

Suppose we are playing the raise/shove game where both players start the hand with 10 BB. Hero is in the SB with 7-4o. Villain’s strategy facing a raise involves shoving with the range

33+, A2s+, K2s+, Q5s+, J8s+, T8s+, A2o+, K4o+, Q8o+, J9o+

and folding all other hands. What is Hero’s EV and maximally exploitative play with 7♥-4♣? We find this by plugging into the above equation. Given Hero’s hand and Villain’s strategy, we have Villain’s folding frequency FH = 0.5812, Villain’s shoving frequency SH = 0.4188, and the equity of Hero’s holding versus Villain’s shoving range EQH = 0.3267. Thus, we have

EVH,A=max(9.5 BB, [FH· 11 BB + SH· max(8 BB, 20 BB · EQH)])

=max(9.5 BB, [0.5812 · 11 BB + 0.4188 · max(8 BB, 20 BB · 0.3267)]) =max(9.5 BB, [0.5812 · 11 BB + 0.4188 · max(8 BB, 6.53 BB)]) =max(9.5 BB, [0.5812 · 11 BB + 0.4188 · 8 BB])

=max(9.5 BB, 9.744 BB]) = 9.744 BB

So, if Hero plays maximally exploitatively, he ends the hand with 9.744 BB on average. How exactly does he play his 7♥-4♣? Each of the spots where we evaluated one of the max functions in the above represents a spot where Hero chose to make one play over another. In particular, on the third line in the sequence of equations, we found that 8BB>6.53BB so that folding is better than calling if we get jammed on after we raise. Then, on the fifth line, we found 9.5BB<9.744BB so that raising initially is better than folding. So, Hero’s best line here is to raise and fold if Villain shoves. Finding maximally exploitative strategies is one of the most important

skills in poker. Remember that these strategies contain no loss-leader type plays – multiple lines are taken with the same hand only if the EVs of both choices are exactly the same. Since this is important, consider working through a few examples of your own for particular hands H or different decision trees.