[Scuba Chuck]

<font color="blue"> I was chatting about SNGs with someone, and talking in particular about bigstack play. I ended up writing this long theoretical explanation why 2nd stack should fold AK to a bigstack push. The question is, should I believe it? Are there flaws? What are your thoughts? </font>

(BTW, we are discussing whether 2nd stack calling with AQ (and frankly AK) against a push-bot bigstack is +EV on the bubble

So, let’s instead look at the PREFLOP &amp; POSTFLOP $EV values of challenging the bigstack, and compare the results that way. As this more accurately depicts the outcomes.

Also, I’m going to just take a few random samples of chip stacks to discuss this point.

Parenthesis Set 1: Preflop $EV values
Parenthesis Set 2: Postflop $EV values if 2nd stack wins a showdown with bigstack (not including forced bets for analysis)
Parenthesis Set 3: Postflop $EV values if bigstack wins a showdown with 2nd stack (again, not including forced bets for analysis)

Situation 1
UTG 4200 (34.9%) (15%) (45.1%)
Button 800 (11.9%) (16.8%) (25.2%)
SB 1500 (20.8%) (25.0%) (29.7%)
BB 3500 (32.4%) (43.2%) (0%)

Situation 2
UTG 4600 (36.2%) (22.6%) (45.1%)
Button 800 (12.0%) (13.8%) (25.2%)
SB 1500 (20.9%) (22.6%) (29.7%)
BB 3100 (30.9%) (41.0%) (0%)

Situation 3
UTG 5000 (37.4%) (27.3%) (45.1%)
Button 800 (12.2%) (12.6%) (25.2%)
SB 1500 (21.1%) (21.4%) (29.7%)
BB 2700 (29.2%) (38.7%) (0%)

Situation 1: The combined $EV of UTG and BB here is 67.3%. If they clash, and SB wins, the new combined $EV is 58.2%. Now, remember, $EV represents actual money. And we know that money doesn’t just disappear, so where does it go? Well, in this case, 9.1% of the equity prize pool was transferred to the small stacks. This is a large reason why you shouldn’t challenge the other bigstack at this stage, you are losing money. Furthermore, by challenging bigstack, when you win, you gain 10.8% of the equity prize pool, but when you lose, you lose 32.4% of the equity prize pool. This means that you need to win 3 times as often as you lose for this to be breakeven. (Does anyone see why folding AK now feels good?).

Situation 2: The combined $EV of UTG and BB here is 67.3%. If they clash now, and SB wins, they new combined $EV is 63.6%. We now see the disparate stack sizes beginning to make calling potentially more palatable. Thus, here, only 3.7% of the equity prize pool is transferred to the small stacks. But, there’s still a huge flaw in the ointment. When SB calls and wins, he gains 10.1% of the equity prize pool, but if he calls and loses he loses 30.9% of the equity prize pool. Again, his call must be a 3:1 favorite for this to be breakeven.

Situation 3: The combined $EV is again 67.3%. If they clash now, $EV is no longer significantly transferred to the other small stacks (implying this is a better play), as the combined $EV afterwards if SB wins is 66%. When SB calls and wins, he gains 9.5% of the equity prize pool, but when he loses, he loses 29.2% of the equity prize pool. Again, his call must be a 3:1 favorite for this to be breakeven.

So, am I saying that 2nd stack should never call with AK/AQ? The closer together the two bigstacks are in magnitude, the more unprofitable a call it is – when you are shorter in stack. But as the stack sizes begin to adjust, and the blinds come into effect, now what happens? Let’s review situation 3, when the blinds are 200/400. This time, the first parenthesis reflect $EVs of the stacks assuming UTG pushes, and the everybody folds. The 2nd parenthesis reflects calling, and BB wins.

UTG 5000 (39.3%) (27.7%)
Button 800 (13.1%) (13.1%)
SB 1500 (19.9%) (19.9%)
BB 2700 (27.7%) (39.3%)

Now, calling and winning, gives you an 11.6% additional $EV increase, and calling costs you 27.7%. Now you only need to win 2.4 times to every one. Or, just 70% (as opposed to 75% before ~ 3:1 odds). So no matter how you slice it, calling sucks. The only positive that you can take away from this analysis is that when the stacks are aligned like in situation 3, at least calling isn’t giving $EV away to the smaller stacks when you win.

Let’s take the flip side: What happens when bigstack pushes, and shorty calls and wins?

Again, assume the 200/400 blinds.
UTG 5000 (39.3%) (34.1%) (42.1%)
Button 800 (13.1%) (24.4%) (0%)
SB 1500 (19.9%) (16.4%) (26.6%)
BB 2700 (27.7%) (25.1%) (31.3%)

I am demonstrating an important off topic point here.
When you are the shorty, and you’re playing against a pushbot bigstack, this is the hand and situation you want to call with, do not challenge the other two. His hand range is any two cards. If the other guys move in, they have a hand. Anyhow, you can see here that shorty gains 11.3% of the equity prize pool if he wins, and loses only 13.1% if he doesn’t. Therefore he’s getting 1.16:1 on his money, or, in reality, only needs to win this hand 46% of the time. Erego, shorty should be calling here with any two cards. This is the most profitable decision!

But back to the bigstack (‘cos we’re discussing proper bubble play). As you can see, in theory, bigstack doesn’t want to win this hand against shorty, as he transfers 21.6% of $EV to the other two stacks. (You can see that shorty has 24.4% of equity if he wins, but if big stack wins, he only gains 2.8% of equity) So in theory, again, bigstack really does not want to have a good hand here.