Anyone have any thoughts on this game, how well my opponent was playing and what we should've done better?
Your opponent played fairly well until move 13.
3. Nf3 isn't bad, but 3. cxd5 is preferable because it allows white to gain a tempo when black recaptures. (Qxd5 allows Nc3, and Nxd5 allows e4)
Black's plan of 8...Qe7 and 10...O-O-O is inaccurate in my opinion because white can push with a3, b4 and b5 and attack quicker than black can counterattack on the kingside. The pawn on a6 creates a weakness on the dark squares and allows white to quickly threaten to open up either the a or b-file after pushing the b-pawn to b5.
9. Nb5 is a useless move which doesn't threaten anything and allows 9...a6, and white has to immediately retreat the knight and lose a move.
11. Bd2 allows black to exchange off the awkward bishop on b4. I prefer 11. Nc2, preparing to attack the bishop with a3 and b4 with gain of tempo. (Gaining tempos on the queenside is especially important at this point because black has castled queenside and white wants to push pawns and attack as quickly as possible.
13. Nb3 blocks the b-pawn and hinders white's ability to attack on the queenside. The knights on a3 and b3 are unproductive. White should go Nc2 and then push pawns to c5, a4, b4, etc.
13...b6? is where your opponent's position started to fall apart. It severely weakenes the a6 pawn and exposes the king to attack on the files and light squares. A simple h4 and h3 is perfectly fine for black.
15. Nxb5, although not accurate, is still a powerful piece sacrifice because it opens up the a and b files and the queen on b5 is extremely powerful. Practically speaking, it isn't that bad because it forces black to play extremely accurately in order to not get overwhelmed by white's attack.
On move 16, what would be black's best move? ...Qd7 looks like it stops the immediate threats, so that would be my choice unless I'm missing something.
6 Bd3 trades the good bishop Bf1 not on the color of the pawns b2 d4 e3 f2 h2 for the bad bishop Nf5 on the same color as the pawns b7 d5 e6 f7 h7. Better 6 a3, Qa4, Be2, cxd5 or c5.
9 Nb5 is a blow in the air: ...a6 kicks it back. Better 9 Ne2 threatening 10 a3.
10...O-O-O is shaky. Safer and better 10...O-O
13...b6 just weakens the position of the king.
15 Nxb5 is unnecessary and incorrect. Just 15 Nc2 and then 16 a4 launch an attack without giving up material yet.
16...Kd7 walks the king into the pin. 16...Qe8 holds.
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