top set with position (theory)

[Aisthesis]

I'm first going to assume rational opponents. If they can't lay down a 9-outer or KK, then a raise is definitely the best play here (in my live game, I think this actually may apply to some).

Case 1: You call, but are going to slam any "blank" (more later) turn. For easy calculation, I'm going to assume a $10 flop pot and plenty of stack-depth.

You know 4 cards from your hand, plus 3 from one opponent, and let's just say that KK is the other hand in, so there are 10 known cards and hence 42 unknowns in the deck.

Of these, 9 are going to fold you out, and you lose $10.

Another 29 are going to go as follows: KK again bets pot of 30, call, then you raise and win. So, you win a total of 80 here.

Then there are the paired boards (1 A, 1 K, and 3 eights for 4 instances), where things get a little tricky. These fold out the straight draw for sure, and the cases are all rather different.

Once you have quads. KK probably half-pots the 30, which you flat call, and the river is then 60. I think you can probably average maybe 20 on this river, which is what I'd bet to a check. KK also might easily make a mistake and bet the river into you, though, so the 20 may be conservative. Anyhow, you win 20+15+20 = 55 total (definitely more against some players).

Once KK presumably has quads. I think KK checks here, and I think AA should, too, played this way. I'm for calling any river bet. And, since we assumed that your quads get a call for 20, let's assume KK also bets 20, and you call. I'm calling full pot here, too, but note that the same player who bets his quads at full pot (seems rare) is also likely to bet full pot on the turn when an A hits. So, the extra should be more or less a wash. Let's just say you lose 20 plus your flop call. So, it's -30.

3 times KK has a big full. It's also kind of tricky here since 88 is just so much less usual than KK. 88 is in any case capable of winning a pretty big pot off of me here. Let's say KK bets the turn at 15, I call, then half pots the river again for 20, where I mini-raise, and he flat calls. So, you win 20+15+40 = 75.

EV for the whole thing: 57.14

Case 2: You just take it down. EV here is only 20. So, calling is vastly superior, even though you get outdrawn sometimes.

Hmmmm... I think this raises some other interesting issues, too. While I wouldn't suggest it in a 3-way pot, I wonder if it's not a good idea to routinely raise AQJT HU in this spot, simply because now AA is much less common than KK as betting hand due to your own holdings. And that should theoretically buy you calls from KK sometimes on an AA raise (eventually, a raise like this will be shown).