Using Game Theory in SNG Tournaments (Part 3): Semi-Bluff Analysis

Moving right along with our study of game theory in the SNG environment, we’re going to be looking at some bluffing and semi-bluffing scenarios this week. The point here is to introduce players to incorporating ICM into their EV calculations and to give you some ideas to think about when it comes to playing post-flop poker with short stacks in the late tournament environment. The point of this is to see that there’s a difference between how you have to think about tournament scenarios and cash game scenarios.

Continuation Betting in a Hyper-Turbo

Suppose we’re down to three players on the bubble in a hyper-turbo with a payout structure of 0.65-0.35. The blinds are at 100/200, and before the blinds are posted Hero has 2800 chips, the SB has 1000 chips and the BB has 2200 chips. We open to 600 from the button, the SB folds, and the BB calls.

The flop comes whatever it comes, and the BB checks to us. At this point, the pot is 1300 and we have 2200 behind while the BB has 1600 behind. Any bet is effectively going to be a shove for 1700 chips. Suppose we continuation bet all-in expecting to have 20 percent equity when our opponent calls. How often do we need him to fold for our bet to be profitable?

Note that if we were to open fold right now, our stack would be worth 0.3736 of the prize pool. Our actual value of checking would be slightly higher than that because sometimes we’ll win the pot or get more bets out of him. We’re going to call it a worth of 0.4000 total for the sake of estimation and modeling this scenario.

To calculate this, we need to know what our stack size will be worth in each of the three different possible outcomes. We can use an ICM calculator like this one to find the values we need based on the stack sizes of the remaining players in these three situations:

The big blind folds. (0.4801)
The big blind calls and we win (busting him out). (0.6000)
The big blind calls and we lose. (0.1475)

Now let’s call the percentage of the time that he folds F. The chances of each of the possible outcomes are as follows:

The big blind folds F percent of the time.
The big blind calls (1-F) percent of the time, and we win 0.20 of the time. Total: (0.20)(1-F)
The big blind calls (1-F) percent of the time, and we lose 0.80 of the time. Total: (0.80)(1-F)

We just have to pair these percentage chances together with their respective values in the tournament to get the EV equation for shoving.

EV of shoving = (0.4801)(F) + (0.6000)(0.20)(1-F) + (0.1475)(0.80)(1-F)

Note that earlier we said the value of checking is going to be worth about 0.4000. We want to know when the EV of shoving is greater than the EV of checking, so we need to solve the following inequality:

EV of shoving > EV of checking
(0.4801)(F) + (0.6000)(0.20)(1-F) + (0.1475)(0.80)(1-F) > 0.4000

We will do that using basic algebra as follows:

(0.4801)(F) + (0.6000)(0.20)(1-F) + (0.1475)(0.80)(1-F) > 0.4000
0.4801F + 0.12 – 0.12F + 0.118 – 0.118F > 0.4
0.2421F + 0.238 > 0.4
0.2421F > 0.162
F > 0.6691

As you can see, we need him to fold more than 66.91 percent of the time.

What If This Was a Cash Game?

If this was a cash game, the EV of open folding would just be 0, so the EV of checking would probably be in the range of 300 as an estimate of how many chips on average we would profit from that options. Notice that we’re talking about chip totals now instead of our equity in the prize pool which is a key difference in how the calculations are done. Our resulting EV equation for the value of shoving would be based on the same three possible outcomes from a cash perspective:

BB folds, we win the 1300 pot.
BB calls, we win at showdown, we win the 1300 pot + his 1600 remaining stack.
BB calls, we lose at showdown, we lose 1600 chips.

The resulting EV equation for shoving would be:

EV of shoving = (1300)(F) + (2900)(0.20)(1-F) + (-1600)(0.80)(1-F)

Again, we would say that we want the EV of shoving to be greater than the EV of checking, and we would solve the resulting inequality accordingly. Remember that we approximated the EV of checking to be about 300:

EV of shoving > EV of checking

(1300)(F) + (2900)(0.20)(1-F) + (-1600)(0.80)(1-F) > 300
1300F + 580 – 580F – 1280 + 1280F > 300
2000F – 700 > 300
2000F > 1000
F > 0.5

In the cash game equilvalent scenario, we would need our opponent to fold just 50 percent of the time for this shove to be profitable. This is a far cry from the 67 percent we would need him to fold on the bubble in the SNG situation we looked at above. If you study enough of these situations, you’ll see a similar type of thing happen over and over again.

How to Study This Even Further

Instead of giving you a ton of examples to look at, I’m going to guide you along what types of patterns you’ll want to look for when studying post-flop push/fold scenarios on or near the bubble. Look at what happens when losing makes you go bust, and look at what happens when it just takes you from first or second place and drops you to third place. You’ll also want to look at what happens if you’re already in third place.

On the other hand, you need to also evaluate scenarios like this from the perspective of the person who would be calling. This will give you a more in-depth idea of what’s going on in these situations.