Typing from my phone so hope formatting works.
I don't agree with your thoughts on checking dark. If villain is planning to fold to any bet, then how did he get to the point he's seeing a flop? And the alternative is he's airway decided to check raise any board? You'd better have very specific reads to plan to c/r any board vs a guy who just 3! you. We can discuss those ideas too, but really I want to give some perspective on how to critically analyze a poker hand.
I hope that by typing my thoughts basically verbatim here, I can help some of you better understand the type of thought process you should engage in at the table in order to make sound decisions.
Hero is late position (2.2M)
MP covers
Effective stacks (2,200,000)
Dealt to Hero: K:club: K:spade:
MP raises to 85k, Hero 3! to 200k, MP asks how much is behind and calls 115k.
So first we need to figure out what range villain continues with here. Unfortunately we're in a vacuum, I don't know his tendencies to call 3! OOP, his MP raising range, or any of that. Heck I don't even know how deep he is, what the blinds are right now, or if ESPN's definition of MP matches mine! So with a lack of villain info here, I'm going to put his original range at {22+, ATo+, A9s+, some suited broadways, and some 87s
hands}.
I really like Hero's 3!. Sizing is fairly standard for tournament poker, and sets up a pot that plays well for a big pocket pair. Hero manages to get ~10% of his stack in preflop, good place to be while holding a big pair.
So what is villain continuing with? Again, not having been present to observe play makes this hard. Smaller PP's aren't getting value to set mine, esp OOP, and weakish aces now have to worry about domination. At this point I think villain's range is something like {99+, AQs, maybe KQs, and some 87s type hands}. Note that I would discount hands I expect villain to 4! with. Therefore, I don't think it as likely he holds AA or AK.
My plan moving forward would be to try and get stacks in on basically any flop without an ace. Monotone red will give me pause, but that's about it.
Flop (505,000)
MP checks dark,
8:club: 6:club: Q:heart:
Hero bets 250k, MP raises to 635k, Hero???
So again we need to determine what villain's range is. Given what I'd previously assigned, I now think villain's range is something like {99, TT, JJ, QQ, AQ, KQ, flush draws, straight draws, and maybe an 88 or 86s}. The 88 is more probable as I imagined villain could perhaps continue with 99, and 88 isn't too much a stretch. 86s depends on previous observation and knowing if suited gappers are part of villain's MP raising range.
Based on this, I'm happy taking my kings in against that range. I further believe a good part of that range will give me action right now. AQ/KQ probably feel pretty good, 99-JJ could convince themselves we're trying to buy it with AK kind of hand, and draws I honestly don't care what they do. If they fold, I'm good with that, and if they wanna throw more money in with 8 or 9 outs, let's do this! So my answer is to jam over villain's c/R.
Actual play, Hero calls (385k)
Turn (1,775,000)
Effective stacks are now (1.75M)
8:club: 6:club: Q:heart: [A:spade:]
Oh, look, an ace! Perfect! So now AQ pulls ahead, but more importantly those 99+ holdings are shutting down. This ace crushes our 3! range, so the only hands that will call a jam on this card are either convinced were full of crap, or aren't worried about the ace. Since we ourselves are concerned with that ace (read it'd beat us) notice that betting here effectively induces hands we beat to fold and hands that beat us to call. Let's chalk this card up as another reason I like jamming the flop.
MP checks
We gain no info from this check. Villain would check basically his whole range here. I check behind for above listed reasons. Additionally. we're sitting with one pot sized bet remaining so a bet of any size might as well be jamming.
Hero checks
River (1,775,000)
8:club: 6:club: Q:heart: A:spade: [4:diamond:]
MP bets 820,000, Hero???
Hero has to fold here. Look at this from villain's perspective. Your range includes plenty of aces, and is in general quite strong. Unless villain is going into a while bunch of levels, he expects you to have a hand that can make this call fairly often. Because of this, I don't think villain bluffs often here, he's saying the ace doesn't worry him, and I like a fold.