Expected Value of Postflop-Shoves

Introduction

In this article

• Estimating villain's range in a postflop push situation
  
• Determining EV using the estimated range
• General model for postflop EV calculation with variable Fold Equity

Players often discuss the EV of preflop shoves. However, we may also apply the EV equation to postflop decisions. Assuming there are N possible outcomes, EV is given by:

Suppose we have a heads-up pot with two outcomes – our opponent folds, or he calls. To determine the probability of each outcome, we must dissect his range to determine how many hand combinations it currently contains, and how many of these hands we will fold out with a shove. The reward of him folding is the current pot size, and the “reward” of him calling is itself an expected value calculation that will depend on our equity against his calling range.

Let’s now get specific, and look at such a calculation in a recently played hand. We will then analyze the function relating expected value to fold equity.

$35 180-man SNG - Blinds $25-$50

Preflop: Hero is on the button with
4 folds, MP2 calls $50, 2 folds, Hero raises to $175, 2 folds, MP2 calls $125

Flop: ($425) (2 players)
MP2 checks, Hero bets $220, MP2 calls $220

Turn: ($865) (2 players)
MP2 checks, Hero?

Villain has $1200 left in his stack, and we have him covered. Our question is: What is the EV of semi-bluff shoving?

Narrowing his Range

We’ll begin answering this question by narrowing Villain’s range street by street.

Preflop he is likely calling with a wide range, perhaps any pair (besides premium pairs, which we assume he would raise), any suited ace, offsuit aces with a nine or higher kicker, any broadway hand, and mid suited connectors.

When he calls our flop bet, we may further narrow his range to the following hand categories.

Sets: 22, 55 (He may check/raise these, but we’ll leave them in his range to be conservative.) (6 Hand Combos)

One pair: A2s, A5s, 65s, A9, T9s, 98s, 33, 44, 66-88 (57 Hand Combos)

Draws: 87s, JTs (Which he may fold, but we’ll exclude other gutshot and overcard-only hands), As3s, As4s, As6s-As8s, AsTs, AsKs, 7s6s, KsTs (16 Hand Combos)

When we shove the turn, his calling range may be approximately: any set, nine, king, combo draw, and nut flush draws. Specifically, this makes his calling range the following hands:

22, 55, A9, 98s, T9s, 8s7s, As3s, As4s As6s-As8s, AsTs, AsKs, and KsTs (33 Hand Combos)

Based on these assumptions, Villain is calling our turn shove with 33 of the 79 hand combos in his range. Therefore he is calling 41.8%, and folding the remaining 58.2%.

The EV Calculation

Now that we have Villain’s ranges, let’s calculate our EV given that he calls.

According to Equilab, we have 31.6% equity against his calling range. So when he calls, 68.4% of the time we lose the $1200 chips left in the effective remaining stack. The remaining 31.6%, we win the $865 in the middle, in addition to $1200 more chips, for a total of $2065 chips.

Therefore, our EV given that we’re called is:

EV = 31.6% x $2065 + 68.4% x (-$1200) = -$168.2

We now have everything we need to plug into the equation for the EV of a turn shove:

EV = Probability (He Folds) x Reward (He Folds) + Probability (He Calls) x Reward (He Calls)

= 58.2% x $865 + 41.8% x (-$168.2) = $433.1

Therefore the EV of semi-bluff shoving this turn, given our assumptions, is +$433 chips.

Our goal in the next sections will be to generalize this result by determining the function relating the EV of shoving to our fold equity.

Calculating “Reward” of Getting Called as a Function of our Fold Equity

Before we can determine the shove’s EV as a function of our fold equity, we must first write out the reward of getting called as a function of fold equity.

Let:

Pf = Probability that Villain folds

Pc = Probability that Villain calls

Rf = Reward of Villain folding

Rc = Probability of Villain calling

Since the outcome space is binary, Pc = (1 – Pf). Rf is a constant. If we additionally let Villain’s range heading into the turn be a constant, then Rc is indeed also a function of Pf as our equity when called is a function of Villain’s calling range (and therefore the fold probability).

What can we discover about this function relating Rc to Pf? Suppose the order Villain discards his range to our turn shove is:

JTs, 87s, A2s, 33-44, 66-88, 65s, A5s, 7s6s, AsXs, 9X, KsTs, AsKs, 22, 55

We can then determine Rc as a function of Pf by plugging into the Rc EV equation:

(Please see the Appendix for all data points.)

As we expect, Rc tends to decrease as Pf increases: the narrower Villain’s calling, the lower our equity against his progressively tighter and stronger range. Interestingly, however, the function is not strictly decreasing; we see a temporary rise as Villain discards higher flush draws, against which we have particularly low equity.

EV as a Function of the Fold Probability

We now have every term of the EV equation for semi-bluff shoving the turn as either a constant or a function of Pf, so we may analyze the EV of semi-bluffing shoving the turn as a function of our fold equity:

Our first observation is that the EV function is strictly non-negative. This result means that either Villain is folding often enough that our immediate reward of the current pot size make this shove profitable, or he is calling so wide that our equity against his range is sufficiently high that the overlay justifies getting it in.

Caveats to the Results

There are two important caveats to these results.

First, the range assumptions are vital. If you were playing this hand, you may put your opponent on a much narrower range on one of the earlier streets. Clearly such changes will significantly impact the profitability of shoving.

Second, just because shoving is profitable doesn’t strictly mean that other options aren’t more profitable. It requires many more assumptions to calculate the EV of betting smaller or checking behind. Against a very weak-passive opponent, for example, it could be more profitable to check back if you think he will play very literally at the river. Nonetheless, you would need a strong reason not to seize a high positive chip expectation by pulling the trigger on a turn semi-bluff shove.

A final question I want to address is the usefulness of this process, given how many assumptions and calculations it requires. Will you be able to estimate ranges, count hand combinations, and determine equities in the middle of a 16-tabling session? Unless you’re a savant, the answer is probably no.

However, just going through the process of calculating the EV of post-flop shoves will be highly beneficial, and you’ll become much more familiar with the important factors. In the example from this article, the biggest reason this shove is so profitable given our assumptions is because the pot is very large relative to the effective remaining stack at the turn, and the king is a great bluffing card that forces Villain to discard many of the hand combos in his range. And when he does call, we still have good equity with our combination draw.

In conclusion, if you make a habit of estimating the equity of post-flop shoves away from the table, your ability to recognize profitable post-flop shove opportunities in-game will quickly improve.

Appendix: EV Equation Data Points

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