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Hand is played by famous player Rui "Gohanounet" Cao at 25/50. He description of opponent : a good regular, but too tight for heads up, have trouble bluffing enough, is solid when calling and when betting outside continuation bet.

We hold K⋄Q⋄T♥2♠ on the BB.

Opponent raises 150, we call.

Board comes T♠8♣4⋄

What should be our line here ? We have several choices :

  • Betting, folding to a raise
  • Checking, calling a bet
  • Check-raising

What is theorically the best option and why ?

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We have 52% equity against his range. Let's do a brief and rough ev calculation of all scenarios:

Supposing bet size at 200$ in the 300$ pot.

Donking

  • 12% of the time we will be raised by a strong hand and have to fold: -200$
  • 40% of the time we will be called. Our raw equity is then around 45% of a 700$ pot. It will be hard to realize our equity as we are OOP, and with a hard-to-balance range. We will suppose we realize 40% equity. +80$
  • Remaining 48% are folds. +150$

Total Average EV: 83$

Check

We will consider two betting frequencies, either our opponent bets with the strongest 70% of his range; or he bets with the strongest 50%, and compare the EV of the two scenarios.

  1. If he bets 70%, when our opponent checks behind, we will know we have the best hand and/or good fold equity on the turn. While we may only be ahead by ~60%, because our opponent will fold hands that have equity against us we can obtain around 65% pot equity total. 30% * 65% * 300 = 58.5$

  2. If he bets 50%, we estimate around 55% equity when he checks behind. 50% * 55% * 300 = 82.5$

Calling

  1. Against his strongest 70%, we have around 45% raw equity, but will be hard to realize it fully. While his range here is looser than against a donkbet, we can possibly play against his top range (TT 88 etc.), and it's possible our opponent will barrel again. Luckily, we can rely on hitting top two pair some amount of time. I suppose we will realize around 40% equity. 70% * (40% * 700 - 200) = 56$

Total Average EV: 114.5$

  1. On the alternate scenario, we will be a bit more behind, and it's closing a fold, but we will suppose we realize 35% equity. 50% * (35% * 700 - 200) = 22.5$.

Total Average Alternate EV: 104.9$

Raising

We will suppose we raise the 200 bet to 700. We suppose the strongest 10% of his initial range will shove the pot, that he will call roughly 30% of his betting range as he is a solid opponent.

Our equity when called is perhaps the most difficult thing to estimate : on one hand, we are always behind, on the other hand, we setup a good situation for further bluffing. If our opponent may call once with some JJ+ or J9 on the turn, he will miss to improve most of the time, and we may have two pair or better, allowing us to barrel a proportion of turns and especially some that gives us a draw.

  1. We will suppose we realize 30% equity, to represent the fact we are after all against a much stronger range. 70% * (55% * 500 + 30% * (30% * 1700 - 700) - 15% * 700) = 79.1$

Total Average EV: 137.6$

  1. On the alternate scenario he will be much more prone to shove. Even if he could fold a little more initial range on raise if we adapt our raise frequency to his betting, our equity drops hard : we would have 50% * (49% * 500 + 35% * (28% * 1700 - 700) - 21% * 700) = -73.5$

Total Average Alternate EV: 9$

Summing up

Although check-raise is situationally the most profitable play against a 70% continuation bet frequency its equity drops really hard against a solid continuation bet range, meanwhile check/calling is a really solid option overall. While not bad, donk betting misses some EV due to the fact our opponent can raise us out of the hand at fair frequency.

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    Where are you getting all of this data on the opponent's range percentage, fold percentages and our equity against all of those ranges? How do you know that you have 52% equity against his range?
    – Michael d
    May 31, 2019 at 16:19
  • @Michaeld Simulation from propokertools says our hand is at 52% against top 80% hands. I also used it to count combinations. Much of the remaining are rough estimates and educated guesses.
    – Diane M
    Jun 1, 2019 at 0:34
  • Don't forget playability in later steets! This analysis would be correct if the hand ended here, but what are we going to do on the turn/river? We can only be comfortable on a king or a ten
    – David
    Oct 29, 2019 at 7:24
  • @David The equity is taking into account the board and the backdoor equity of the hand. Its playable because it has a pair and two backdoor draws, i.e. the flush draw and the straight draw. Hero also has a backdoor boat draw. Also, there can be more hidden outs. For example a 2 is a great card as it takes hero in the lead vs overpairs.
    – Jonast92
    Oct 29, 2019 at 13:40
  • @Jonast92 Equity is not the same as playability. Just because you have a, say, 52% equity, it doesn't mean you'll get on average 52% of the money except in an all-in situation. In Hold'em A2o has 50.77% equity against 87s, but 87s is easier to play with since you know more often where your hand stands with regards to the opponent's range
    – David
    Oct 29, 2019 at 15:50

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