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I have been looking at several of Mikhail Tal's games recently, as they are exciting to analyze. Often Tal makes sacrifices in surprising positions, and almost always the computer finds that the positions are quite close despite the massive material deficit.

How can one accurately judge the compensation from a sacrifice? Are there any techniques to try to figure out whether or not it was worth it, and what should I be looking for here? How can I try to evaluate "king safety" and "weaknesses," and determine whether or not they are worth the material loss?

Thanks,

Note: I am not just asking about a combination which returns the material.

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I will assume here that you are talking about sacrificing pieces and not just pawns, which require less compensation.

For a sacrifice to work out you generally need to either 1) checkmate the opposing king, or 2) eventually gain the material back. There are other scenarios, but they're less common and I will ignore them here. Often goal 2 occurs because Black is required to give up material to avoid goal 1.

Either of those goals can be accomplished in the short term by a forced sequence that you can calculate, or in the longer term, where you don't calculate an exact sequence that will lead to your win but are confident that your good position will lead you there eventually.

Unfortunately there are not reliable quantitative compensation metrics such as "an exposed king is worth two pawns". Long-term sacrifices are generally made on intuition, which improves as one gains experience. There are some decent rules of thumb, though. Here are some good signs that your attack may succeed:

  • If you have two more pieces attacking his king than he has defending it.
  • If you are down a piece, but currently have more pieces actually participating in the game than he does (for example, both your rooks are active, but he still has one on a8 blocked in by a knight on b8).
  • If it is difficult for him to route his pieces back to defend his king.
  • If it is difficult for his king to run away (often the bottleneck is at f8).
  • If you have certain configurations of pieces (e.g., a knight on the sixth rank supported by a pawn, or doubled rooks on an open file) that are more valuable than you'd expect purely from their "point value".
  • If the king's pawn cover is broken, or if he is stuck in the middle of the board.

If, say, you can sacrifice a piece for a pawn and three of the above conditions apply, that is a good sign that the attack will succeed at one of its two general aims. But it all depends on the actual position.

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    Unfortunately there are not reliable quantitative compensation metrics such as "an exposed king is worth two pawns". Long-term sacrifices are generally made on intuition, which improves as one gains experience. Very well said. – chubbycantorset May 7 '13 at 23:44
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One of the better books on "judging" a sacrifice is "The Art of Sacrifice" by Rudolph Spielmann, even though it is a bit dated (1935).

In a "non-gain" sacrifice, one doesn't really think of compensation. Instead, the mentality is, "can I get a winning attack?" Normally, the sacrifice is large enough so that if you don't win by attack, you will lose the game on material. So most sacrifices involve either an immediate "mating attack," or at least a "king hunt," whereby the king is forced into the center of the board. In one notable game (against Rubenstein), Spielmann sacrificed a whole rook for a king hunt, and won with a mating attack of queen and rook against queen and two rooks, because the enemy king was totally exposed.

Other sacrifices can occur when your opponent has most of his pieces on say, the queen side, and you outnumber in the vicinity of his king say, five pieces to two. In that case, the priority is to remove the two pieces so that the remaining three can administer checkmate. In that case, it might be worth sacrificing a rook, or even a queen for a key defending knight; after that loss, the opposing king is helpless against your remaining pieces.

An exception to the rule about compensation occurred after a "classic" sacrifice of a bishop for the h pawn. Spielmann also captured the g pawn, and eventually the f pawn fell, so he had three pawns for the piece, enough "compensation." Eventually the Black king was driven to the queen side, where it was in the way of his other pieces, and Spielmann could win by queening his "passed" h pawn; with the "crowding" of Black's pieces on the opposite side more than compensating for his piece advantage.

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