Home game + a hand analysis

I've been playing a little $3/$6 6-Max and doing well. Nothing interesting going on there. I played in my home game last night and got 5th of 6. We were playing limit holdem.

The guy who got 6th is a decent player and a regular winner in the $3/$6 6-max Party games. In other words, he's who the only one who plays more shorthanded limit holdem than I do. So we have this other guy who basically likes to raise every pot and keeps the heat on. Easy enough, I play against these types of guys all the time. You just wait till you have a hand and pound away at them. Here are a couple of hands that describe how well this line went tonight.

I'm UTG with KJo. I raise, donkey 3-bets, all fold, I call. Flop comes AK7 rainbow (1 heart). I bet, he calls. Turn Q, I bet, he calls. River Q, I check-call and lose to Qh5h.

Later on, CO limps, I have AcKc in the SB (donkey is the BB and will almost certainly raise) and limp. Donkey doesn't disappoint and raises, CO calls, I 3-bet, all call. Flop comes Kh6h6s. I bet, donkey raises, CO folds, I 3-bet, donkey calls. Turn 7h, I bet and get called. River 7, I bet, get raised, call, and lose to 79o, no heart. Huh? Yes that's right, 79o. After the hand he says "I almost folded on the flop too." I respond with "How can you fold 79 on the K66 flop?"

Either of these can be considered the play of the day. You choose.

Actually, there's a better play of the day from a hand I wasn't in. Donkey is playing heads up against an inexperienced but ok player (let's call him IP). IP is in the SB and raises, Donkey calls. Flop is 752 rainbow, 3 bets go in. Turn 2 (still rainbow), 2 bets go in. River 5. IP check-raises with KJ on the 75522 river and gets called by donkey who has... you guessed it, T6. This guy called the river with ten high! I don't know what to think of this river raise. On the one hand, it's a really bad bet against most players, who definitely won't call with worse hands, will probably re-raise their monsters, and will call with anything in between. But when the guy's range is any two and he'll still call you with T high? Maybe it's a good bet then at that point -- I still don't think so, but let's figure it out.

A few assumptions --
1) Donkey's range is precisely any two cards that do not include the Kh, Jh (our cards) or the 7h, 5s, 5c, 2d, 2h (the board cards).
2a) Donkey will re-raise with a full house or better, which we are going to pay off.
2b) Donkey will fold if he cannot beat the board.
2c) Donkey will call everything else.
3) There are 10 big bets in the pot after the villian has bet.

Let's categorize his possible starting hands now. There are (45 * 44) / 2 = 990 possible hands for him to have (remember -- we have KhJh and the board is 7h5s5c2d2h, leaving 45 cards in the deck), and we will categorize them based on how they play on the river.
1) Hands that make a full house. He needs to have 77, any 5, or any 2 (we include 55 and 22 here). He can have 77 exactly 3 ways, 5x exactly 88 ways, and 2x exactly 88 ways for a total of 179 hands. We lose 3 bets in this scenario.
2) Hands that don't beat the board. Here he must have 34, 36 or 46. There are 16 ways to have each, for a total of 48 hands. We win 1 more (11 total) bet against these hands.
3) Hands that are somewhere between full house and K-high. This includes any pocket pair that does not make a full house (33, 44, 66, 88-AA), 7x (but not 77, 75 or 72), and Ax (but not A5 or A2, or A7 or AA). Here are the pocket pairs -- 33, 44, 66, 88-QQ, AA (6 ways each) and KK (3 ways) for 57 hands. 7x can be made 3*38=114 ways. Ax can be made 3*34=102 ways. There are 273 hands here, and we lose 2 bets to each of them.
4) Hands that are exactly K-high. This is any hand that is Kx, where x is not an A, K, 7, 5, or 2. There are 93 hands that satisfy this. (1st card -- one of 3 Ks, 2nd card -- one of 33 cards: Q*4, J*3, T*4, 9*4, 8*4, 6*4, 4*4, 3*4). We win 5 bets here (1/2 of the 10 bet pot).
5) Hands that are between 8-high and Q-high. There are 990 - 93 - 273 - 48 - 179 = 397 hands here. We win 12 bets against each of them.

So, our EV by raising here is (-3)(179/990) + (11)(48/990) + (-2)(273/990) + (5)(93/990) + (12)(397/990) = -0.54 + 0.53 - 0.55 + 0.47 + 4.81 = 4.72. If we are able to fold the river when he 3-bets, it's still about 4.90.

If we just call, we simply win 12 bets when we're ahead and lose 2 bets when we're behind. We're ahead 397 + 48 = 445 times, we're behind 179 + 273 = 452 times, and we chop the other 93 times. The EV of calling is then (12)(445/990) + (-2)(452/990) + (5)(93/990) = 5.40 - 0.91 + 0.47 = 4.96. So calling is definitely the best option, and folding after you make a value raise sucks anyway.

Let's look at another set of assumptions, namely that the villian will 3-bet his full houses and his hands that don't beat the board, but do everything else the same. If we pay off the 3rd bet, our EV by raising is (-3)(179/990) + (13)(48/990) + (-2)(273/990) + (5)(93/990) + (12)(397/990) = -0.54 + 0.63 - 0.55 + 0.47 + 4.81 = 4.82. If we don't pay off the 3rd bet (and get bluffed sometimes), our EV is (-2)(179/990) + (-2)(48/990) + (-2)(273/990) + (5)(93/990) + (12)(397/990) = -0.36 - 0.10 - 0.55 + 0.47 + 4.81 = 4.27.

If we call and win slightly less than 1/2 the time, the money in the pot provides a nice overlay for us to call on the river. It's just that calling is better than raising, even in this most exteme scenario. The whole point -- if you're not raising for value and you're not raising as a bluff but you can't fold -- just call! This should seem obvious, but people put in raises in these spots all the time and it just doesn't make any sense.

Have a nice day, I have work to do.