Win an exchange (Rook for Knight) or win a free minor piece - which is better?

Suppose you have a knight that is forking your opponent's rook as well as a bishop. Now if you take the rook, then the knight gets captured, however the bishop is just hanging.

Which one would you take - The Rook or the bishop.

If you take the rook you are +2 and with the free bishop you are +3. However is it really that simple a calculation.

probably usually better to take the bishop but it depends on the situation... the one advantage of taking the rook is that it'll be your move again after you gain 2 points, whereas taking the bishop, although gaining 3 points, gives the opponent the next move.

It depends on the position. impossible to say for sure.

In general gaining a minor piece is better than R for minor piece.

In the 1990's, Larry Kaufmann's celebrated study of over 80,000,000 positions, published in Chess Life, gave the following average values: P=1, B = N = 3.25, R = 5, Q = 9.75.  Having 2 B's vs B + N or 2 N's is worth 0.5.

So going up a B is a 3.25 P advantage, where going up a R for an N is a 1.75 P advantage.

Of course, it all depends on the position: if getting rid of the Rook, for example, undermines your opponent's attack or defense it might be the better choice.

I just had a game where I had the choice of gaining a R for N with a good position or gaining a full N with a hard to defend position.  I went for the whole gain of an N and struggled to organize my pieces, maintained my material advantage by sac'ing a P, then finally pulled out the win.  I would have been smarter to do the R for N advantage, which Stockfish 8's 20-ply analysis confirmed, saying I would have been 4.32P ahead if I took the White R in the game below after White's 13th move, losing the N, instead of my move Nxb3 - which led to me being an whole N ahead and which put me only 2.71P ahead. My move with the Black Knight on move 14 was the one where I should have just gone R for N:

dependant on a lot of things. piece activity, which color squares are weaker etc. for instance i wouldn't take a rook for bishop exchange if the bishop was black and theyd have incredibly weak dark squares after i took it, or if my dark squares were at risk of being compromised, i would rather go further into the game without a blatant weakness like that.

Free piece is worth about 2X the amount the exchange is worth

Whenever I have a choice, I take the free minor piece.

The way I see it is this: if I move my rook to attack a pawn, his bishop defends it, then thats it, I have to forget about that pawn, and move onto something else.

But if I move my bishop to attack a pawn, his bishop defends it, then I move my knight to attack it again, I am going to win even more material!

And the snowball keeps getting bigger.