Is this a difficult choice for anyone else?

Or is it just me?  My knight can take a rook or it can take a pawn.  If it takes the rook, I lose my beloved knight.  If it takes the pawn, it lives to fight another day.  I take the pawn!  Is my choice justifiable?

A rook's worth five points, so it's best to take it, and lose the knight.

The goal of chess is not to preserve your pieces,  it's to get to checkmate.   There's nothing wrong with spending your precious pieces, as long as they're taking even more precious pieces of the enemy's with them.   Making good trades is a great way to win the game.

In the absence of other information:

Taking a rook and losing the knight => 5-3 = 2 point advantage for you.   Taking a pawn for nothing => 1 point advantage for you.  Taking the rook is better, by 1 point.

But, if you can also take another pawn for free... 2 point advantage for you.  Just as good as taking the rook and losing the knight; do whichever one will annoy your opponent more. 

Or, if you can take the pawn, and then also take the rook before your knight is taken: 3 point advantage for you.  Do it (or try it if it has at least a 50% chance of success).

Finally, the 1 point difference between your two options might be compensated by position.  Will taking the pawn totally mess up his center?  Leave his king wide open?  Is this your last good chance to stop the pawn from promoting? (On the other hand, will taking the rook prevent him from castling?)

Of course, if you're asking this question, you probably shouldn't have a whole lot of confidence in your ability to estimate your chances of capturing two pieces in a row, or of converting positional advantages.  So just take the rook.

a knight is a knight but a rook is a rook

I would probably take the pawn.

There's nothing wrong with presenting your opponent an opportunity to make a mistake, if it doesn't cost you anything.  Here it does cost you one point (opportunity cost of letting the rook escape), so you need to have a little more than hope.  Maybe you're on tight time control and you haven't calculated it out completely... but presumably, neither has your opponent.  If it works in all the lines you did try, give it a shot.  You'll probably learn something. 

"50% chance of success" is the correct game-theoretic measure in this situation (it's the point where your expected reward for taking this risk is equal to your expected loss from subjecting yourself to the risk)  but it's actually a really really hard thing to evaluate in chess.  You're usually either in a state of certainty, or mistaken certainty, or near-total ignorance, as far as your chances of success are concerned.  Ultimately, we play on hope most of the time.

I think the only "correct" answer here is that it depends on the position.  A knight can be worth at least the value of a rook in some positions (super-closed, or with an unassailable outpost).

Likewise, an opponent's pawn may be so valuable to him that he'd much rather lose the exchange than the pawn, and you'd be much better off nabbing it than blindly assuming the material difference attributed to the rook exchange will win the day.

h3!

Well if we are talking about a knight on the 6th( or third) rank, in most situations it's probably not wise to exchange it for an inactive rook. But it all depends on the practical situation.

You guys are overanalyzing this.  The theory was that "my knight lives another day" is, *in the usual case*, enough compensation for letting the rook live.  That's as close to clearly wrong as you're going to get in chess.  

FTQ: 1) I never mentioned a third pawn.  2) According to you, what is the correct indifference point if it's not 50% ?  (The choice is: trade knight for rook, or take free pawn and then, with X% chance of success, also trade knight for rook.)   

I agree

>Your whole argument was based around the possibility of being 3 pawns up.

Nope, one of the possibilities I mentioned was the possibility of being 3 net *points* up, with 2 of those points coming from a rook (after subtracting the value of the lost knight).   If I'd actually advocated chasing three pawns, that would have been at lease a little wierd and unlikely.  But letting your knight live when there's a good chance of using it for something good within a move or two? That's exactly the sort of thing the original questioner should be thinking about.  

Haven't you ever learned and improved by making a move and then seeing what happens?  Call it "hope chess" if you want.  You don't have to play or learn this way. But it actually is possible to combine informed guesswork with good chances of winning, if your guesses aren't too costly.  That's what game theory is all about.  There actually is a correct game-theoretic answer to the question of when a player should make this trade in the absence of perfect information, and your personal preferences have nothing to do with that answer.

no u wouldn't

The exchange (ie rook for minor piece) up equals a win in most positions. A pawn up is half way to a win in most positions.

FTQ's idea that the first point counts more, and subequent gains increasingly count for less, is an intriguing one, but seems to go against the grain of assigning points in the first place. 

Are you saying one should start applying a discount after the first *point*, the first *piece*, or the first *exchange*?   These are all different - as an obvious example, a discount starting after the first point would make a queen worth less than nine points at any time; the others wouldn't.  I could calculate how big you think the discount should be by working backwards from your 62% guess in the earlier example, but only if you can stop being condescending long enough to actually express yourself clearly.

Also, if material changes from being very unequal to more equal, do we somehow undo the discount?  Say I lose my queen early, then do some even trades, then take my opponent's queen.  If all those trades/points/pieces were progressively discounted, I'd be considered to be losing, even though material is now exactly equal.  Is that the result you wanted?

Depends what it leaves on the board.

It's highly situational.  A rook alone can mate a king whereas even two knights cannot unless the opponent is willing (this assumes no pawns are on the board).  Sometimes it may not be wise to take the rook, like giving the opponent connected passed pawns on the sixth.  A knight on the sixth is usually worth exchanging a rook for, and sometimes for an exchange sac one can play a rook to the sixth and establish a strong there that would really cramp the opponent.  Petrosian and Botvinnik have a couple of such games where they did that. 

Here's an example of Botvinnik willingly giving up the exchange for a strong initiative:

Edit: wait, the example exchange with Petrosian was rook for bishop with the idea of making way to place a knight on an excellent square so goes beyond the scope of this post.