So I started writing this taking my shot in more general lines. But then I thought it was an intersting problem, so I couldn't help myself and I brought in the maths.
My shot:
River check - call:
he bets:
-better sets? I would tend to think better sets raise turn, but if he failed to do so. However I rather think he would check them behind after that river. He might sometimes bet it too. KK I suppose is 3betting pre. 77 is not staying flop. So it is JJ. Could he play AA like that? Pretty horrible imo if he is that solid. (2 combos of KK, 1 of JJ) -> (3/6)
-str8s: always. Would he raise the flop with QsTs? Also, would he stay after flop with other straight combos apart of QsTs? If you think he only stays with that one, then there is one combo. (1/1)
-busted nutflush draw: probably not, but sometimes as a bluff. Would he raise the flop with AsKs? KsTs, KsQs are the other possible fds I see there. However, after you check having gone that strong during the hand, he might think it is for value, so I don't think he likes trying a bluff there. He might bet AK though. (1/3)
-2 pair hands (apart from AK): probably not. This could be KJ. Him being solid, I doubt he plays it non suited. I very much doubt he is staying 2 streets with AJ, and anyway, I would rather think AJ 3bets (as KJs probably too, but more times AJ than KJs). 2 suited KJ and the extremely odd AJ just to account for random stuff. (0/3)
(In the hands above, I think many are playing differently on the flop, but I leave them like that to balance with the odd time he does that plus some random bluffs)
So he is betting (5/13). Of those hands you beat 2, and you are beat 3 times. You win 2/5 (40% of the time, 1.5 to 1). If he bets $84 into $87, you get odds of 2 to 1.
He is checking behind 8 hands. Of those 3 are better sets, 3 are 2 pair hands, 2 are busted flushes. You win 5/8, ie., 62%.
0.38*(0.4*171 - 0.6*84) + 0.62*(0.62*87 - 0.38*87)= 19.7
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River bet:
What hands would he call your pot size river bet with given that river? Here you know best, but just to give some numbers, I will fill it like this:
calls:
- better sets: yes (again, I doubt they will play like that, so I wont count a total of 6 combos as we are calculating calling hands) (3/3)
- busted draw: nope. (0/3)
- AsKs: probably yes. (1/1)
- KJ: not even a crying call I think. (0/3)
raises:
- He raises QsTs and the odd missed fd. AsKs was counted as a calling hand, so I would just include KsQs as a possible bluffing raising hand as he has the Q blocker. (2/2) However I would be inclined to think he bluffes with a lower frequency. I would set this to the standard 1/3. However you are playing check/fold to shove, then here you are losing all the time.
calls (4/12), raises (2/12).
So when you bet, you win the pot straight away 6/12 times. Of the times he calls, you beat him (1/4) of the time.
0.5*87 + 0.33*(0.25*171 - 0.75*84) +0.17*(-84)= 9.97
(BTW, If you elect to call any shove, then it doesnt work out for you: 0.17*(0.33*436.82 - 0.67*307 ) = -10.5, so you have to fold to his shove here, good fold)
(I don't think the way he plays QQ changes much?)
So unless I went it wrong somewhere (which is very much possible), you are better off check/callin. However, I think you can analyse this more accurately as you know him better, but from the above it looks that the optimal play here is to check/calling river.
Now, however: I think all this work quite positively because we are assuming he is staying with stuff like KJ. If you can be bothered, you can play it around and see that it is very easy for this to break down as soon as we make him tighter...it is a starting point though, and you have all the formulas above so you can quickly tweak the numbers (do check the win/lose amounts though).