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Bishops and Knights are both traditionally considered to have a similar value, worth roughly three pawns each. However, I have heard arguments that under certain circumstances they are not so closely comparable. Are there situations where Bishops are more valuable than Knights, or vice versa? What factors should I weigh when considering a trade-off, for example?

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There's a lot that can be said on this topic, but here are some of the factors that enter into my decision making process:

  1. I have a personal preference for Bishops over Knights. This has nothing to do with their inherent value, but is simply because I tend to utilize Bishops better. It's worth considering whether you have a penchant for one over the other, because your personal play style may determine which is more valuable for you.
  2. In general, Knights are better early in the game and Bishops are better late in the game. This is because, in the early game pawns tend to obstruct Bishops, while Knights can move more freely.
  3. There can also be a difference in value between Bishops (the so-called Good Bishop and Bad Bishop). If most of your pawns are on black squares, your "white" Bishop (the one on white squares) has greater freedom of movement (Good Bishop), while your "black" Bishop's movement will be hindered by your pawns (Bad Bishop). Your Good Bishop probably has a higher value than your Bad Bishop at that point. A common tactic is to trade your Bad Bishop for your opponent's Good Bishop. This can create a slight advantage. You might also trade your Bad Bishop for one of your opponent's Knights.
  4. Two Bishops is generally better than two Knights or one of each. Engineering exchanges that result in you keeping two Bishops and leaving your opponent with two Knights or one of each, may result in a small advantage for you. It's debatable whether two Knights are better than a Knight and Bishop. I tend to think so, but it's very situational.

Obviously, these are generalizations that may not always apply. Chess is a complex game and the situation on the board dictates what makes the most sense, but they are reasonable rules-of-thumb and worth keeping in mind. The real answer depends on your strategy, your opponent's strategy, and the current state of the board. That is what really determines the relative value of the pieces at that moment.

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"Two Bishops" are generally thought to confer a small advantage over "two knights" or "one of each." The reasoning is that the disadvantage of the bishop is that it can only ever control one color, but a player with both bishops doesn't have that problem. I've never heard that two knights is generally preferable to one knight and one bishop. –  BlueRaja - Danny Pflughoeft May 4 '11 at 5:35
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@BlueRaja I remember reading that somewhere, but perhaps it's just one author's opinion. It would certainly be situational. 2 Knights is definitely not better in all circumstances, as 2 Knights vs. lone King cannot force checkmate, while Bishop and Knight can. –  Todd May 4 '11 at 15:33
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A simple reason why 2 knights could be considered better than 1 of each is that the knights can simultaneously protect each other whereas any form of protection for a mixed pair would be one-way only. –  Brian Schroth Jun 20 '11 at 20:15
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@Todd: No, it's a generally held consensus that having both bishops when your opponent has bishop-knight or two-knights is an advantage. That is why, in annotations of grandmaster games, you'll often hear them speak of "the double bishops." It's also why, when a grandmaster gives up his double-bishops by trading a bishop for a knight, it is surprising, and nearly always deserves a explanation. It's also probably where lower-level players/teachers got the (false) idea that "bishops are worth slightly more than knights." –  BlueRaja - Danny Pflughoeft Jan 22 '12 at 20:05
    
@BlueRaja-DannyPflughoeft Sure, but all I was suggesting is that B+B > K+K > K+B. B+B is the strongest option, but I tend to think (and I also remember reading somewhere) that K+K is stronger than K+B. Were you disagreeing with that? Not sure. –  Todd Jan 22 '12 at 22:23
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Todd wrote up a great answer, and points 2, 3, and 4, of his response are spot on.

My only addition would be to state that my preference for knights is based on the following:

  1. For whatever reason, even at low-to-mid level play, you can often "surprise" your opponents with a knight. Their non-linear movement pattern means that in one to two turns your knight can be wreaking havoc in ways your opponent was not anticipating. This is less true at higher levels of play, but in general I find that setups with bishops are usually involving pins and get telegraphed pretty easily.
  2. In late game bishops do have the ability to travel the board faster but if you are stuck with one bishop the king merely has to stay off the same colored squares ("hot lava!"). The bishop will have to be the "assisting" piece while some other pawn or piece actually performs the mate.
  3. I'm personally partial to the ability of a knight to fork pieces without being in the line of fire for immediate retaliation. That is, bishops have less utility to me when it comes to forking queens or other bishops in that they have to put themselves in danger to perform the fork. This is, of course, entirely dependent on the context of the situation.

I'm upvoting Todd's post, but just wanted to offer some counterpoint regarding my personal bias towards knights.

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It should be emphasized that this is just your opinion, not a widely-held bias by any means... in fact, in my experience, most low-to-mid level players tend to believe (or are taught) just the opposite: that the bishop is slightly more valuable. –  BlueRaja - Danny Pflughoeft May 4 '11 at 22:37
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I'm not sure how I could have made it any more clear that it was only my opinion. ;) –  Michael May 5 '11 at 1:59
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As a general rule of thumb, Knights are better in closed positions, and Bishops are better in open ones. Bishops are usually considered slightly better than Knights because they move faster, and you can force mate with 2 Bishops and the lone King vs opponent's lone King; something you cannot force with 2 Knights. It is really situational. With as many times as I have missed Knight forks and Knight moves made by my opponents over the years, I never underestimate Knights. If I am up the exchange, I would rather it be Rook vs. Bishop than Rook vs. Knight. It really is all situational. I think the value of 3 points for Bishop and Knight is correct. Although I have seen various books list the Bishop at 3.25 points; 3.5 points; and even as high as 4 points. Most have Bishop and Knight as 3 points each.

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Connected to this, the fact that closed positions can develop into open positions, while open positions generally remain open, tends to give bishops a slight edge. –  Chris Dodd Nov 17 '11 at 0:13
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Great answers here, however there are a couple of factors that I have not seen addressed.

The concept of "Good bishop" vs. "Bad bishop" has been mentioned, but it is worth clarifying that having most of your pawns on one color is not sufficient to make that color bishop a "bad bishop". If the bishop is outside of your pawn perimeter, it can still be quite effective, although its avenues of escape may be somewhat limited. A "bad bishop" typically refers to one that is trapped within your pawn perimeter, and therefore cannot threaten any of your opponents pieces, and therefore is forced to remain as a very limited defensive capacity.

Another key difference between knights and bishops is pinning versus forking. As has been mentioned, knights are more effective than bishops at forking. Bishops can fork, but it is far less common than knights forking.

However, knights are completely unable to pin, or skewer, another piece. At higher levels of play, where you can safely expect that your opponent will be able to spot potential knight forks, the pin becomes more valuable.

Finally, the overall long-range capability of the bishop makes is more valuable in many end-games. A knight will take several turns to move from one side of the board to another, whereas a bishop can do this in one or two moves. This puts the knight at a severe disadvantage any time you are trying to put pressure on the king on one side, while trying to cope with an advancing pawn threatening promotion on the other side of board (or pawns on either side of the board, as Steven describes).

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One reason for trading a bishop for a knight is to inflict "doubled pawns" on an opponent. (It's possible but less likely the other way around.)

Former World Champion Jose R Capablanca felt that knights were more valuable with queens on the board (because their moves are "different"), and bishops more valuable after queens were exchanged. In one epic game (against Marshall), he had a bishop versus knight, and traded his queen plus a pawn for Marshall's queen, just to reverse the relationship.

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I have found that there definitely are cases in which doubled pawns are only as good as a single pawn - meaning that you would, in effect, be trading your bishop for a knight and a pawn, in an exchange on your opponent's third rank. One case in which this is especially and vividly true is after your opponent has castled; if you can manage to double pawns near his king, you can often make good use of the opened file. –  Daniel δ Jun 24 '11 at 16:12
    
@drm65: Removing a knight pawn (in front of an enemy king) is HUGE. It is sometimes worth a piece by itself: N (or B) x g7, k recaptures. Or Rx f6, g7 x R. –  Tom Au Jun 24 '11 at 16:17
    
I'd like to play against you. You sound good at chess. :] –  Daniel δ Jun 24 '11 at 16:25
    
@drm65: I USED to be good at chess. Some 35 years ago when I was the captain of my high school team. But haven't played much since then, am rusty. What is NOT rusty is my memory of concepts. –  Tom Au Jun 24 '11 at 16:42
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Michael mentioned this in passing, but it's worth making explicit: a lot of the bishop's endgame value comes from its range, and specifically in circumstances with pawns on either side of the board. For instance, pawns on b2 and g2 vs. a lone black knight should be a win for the pawns, whereas with a bishop black can straightforwardly capture both pawns.

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  • The value of the pieces depends of course on the position. I've heard from sources like NM Dan Heisman that knights are considered to be worth closer to 4 pawns right at the start of a game, and that both knights and bishops on average are worth a "tad" more than 3 pawns.
  • Knights usually rule closed positions (where pawns obscure most or all of the diagonals) while bishops usually are very powerful in wide open positions.
  • Having an untouchable outpost knight (i.e. it can't be threatened by opponent pawns, it is supported by a pawn, on a square of the opposite color of your opponent's only bishop where he has no knights left) deep in opponent territory or even in the center can be worth as much as a rook in some cases because it's very hard to get rid of without giving up the exchange (rook for knight).
  • Others have already mentioned the bishop pair (having two bishops while your opponent does not have two bishops is called "having the advantage of the bishop pair" and is worth on average about half a pawn).
  • And as another tidbit, remember that two bishops + king vs. king can force checkmate while two knights + king vs. king cannot force checkmate.
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