26 March 2022

The fundamental importance of the relative value of the pieces

From Quora article "What are the values of each piece in chess?"

NM Dana Mackenzie's recent post "There's No Such Thing as an Even Exchange" - well worth the read - highlighted from a chess trainer's point of view the importance of evaluating all potential piece exchanges on their own merits. From time to time I've also mentioned this idea, which I assess is one of the major differences between master-level and amateur-level players. Below I'll gather together some related concepts and situations, which all reflect a core characteristic of master-level chess: the players recognize the fundamental importance of the relative, not absolute, value of the pieces.

Unfortunately the numeric "piece value" chart everyone learns as a beginner, reproduced at the top of this post, is...a lie. Like most beginner's concepts, it is quite helpful in the early stages of learning chess skill and playing opponents of similar skill. Once you get closer to the threshold of mastery, however, such concepts will have to be un-learned in order to make further progress. The problem is not that the beginner's rules aren't helpful. However, they are approximations and guidelines at best - only the actual situation on the board is real.

Chess strategy and tactics in fact depend on the relative value of the pieces. This is true in both a static sense (the current position's evaluation) and in a more dynamic evaluation, which takes into account the future potential of each piece. Tactical play involving sacrifices is the most obvious illustration of the idea of relative value, since by definition more material is given up during a sequence than it should be "worth". This is because the player initiating the tactical sequence calculates (rightly or wrongly) that the position at the end will be more favorable to them.

This type of tactical play ranges from forced mates, in which the amount of material sacrificed is truly irrelevant to the final result, to positional sacrifices where one person - perhaps even the strategic defender, not attacker - gives up material to reach an improved position. The latter case can be seen in a number of endgames where the defender finds a tactic to reach a fortress-type position with a sacrifice, or can simply leave the attacker with insufficient material to win.

In the absence of tactics on the board, maximizing the relative value of your pieces becomes the route to strategic victory - which of course can, in the process, produce new tactical opportunities. The primary goal here is to enhance the scope of your pieces (and pawns). This is directly and mathematically reflected by how many squares they influence. A corollary to this is how important those squares are. Naturally, being able to dominate the ones around your enemy's king is very valuable. For pawns the importance of the squares they can influence is especially significant, since once advanced, they can never again control the squares behind them.

The combination of the scope and importance of squares influenced by each side's pieces is therefore what drives specific positional situations and evaluations. Just a few examples: "good" knight vs. "bad" bishop; the value of a rook on the 7th and 8th ranks, or on open files; bishops on an open long diagonal; the strength of a centralized queen in an open position. Naturally the list could continue, but the point is that these should not be considered as special positional cases to be memorized; rather, they reflect the relative value of the pieces in each case.

For improving players, this then brings up the question of how to get better at making these relative evaluations and related decisions. When is trading bishop for knight a good idea? Should I simplify down material in an endgame? What about those mysterious-looking exchange sacrifices that masters make, without a forced win on the board?

Making a regular practice of reviewing annotated master-level games, ideally with the thinking process explained by one of the players involved, I have found to be the best method. Piece exchanges and other factors that directly involve the relative value of the pieces are then explained in the context of a specific board situation. The linked post at the top is a good example of this.

Some other particularly relevant examples from this blog:


  1. PART I:

    An interesting post - thank you!

    It is fashionable to "knock" the so-called "Reinfeld values" as being inaccurate in light of the research conducted by (then IM; now GM) Larry Kaufman. Regardless of the origin of these values, it is important to keep several factors in mind.

    First, there WAS an attempt (at least by Reinfeld) to demonstrate that the values are RELATIVE, not absolute. (It is often explicitly stated in older books that these are relative values.) A telltale relative indicator is that the values are based on an assigned average value of Pawn = 1; the other (piece) values are given in terms of Pawns. There is no such thing as a partial Pawn; ergo, the other values are whole-number multiples of the Pawn's value. Mathematically precise? No, not according to Mr. Kaufman's statistically derived numbers - nor from the Reinfeld values themselves. Anyone attempting to use these values slavishly (with using their own judgement) will quickly find anomalies. It is the same problem as figuring out the AVERAGE number of moves available to a specific piece type on a board that has only that single piece on it (without even Kings on board - an ILLEGAL position!). Mathematically, there will always be a strong correlation between the AVERAGE number of moves available to a specific piece type in isolation and the RELATIVE piece values (regardless of which scale is used).

    Second, providing more precise mathematical values obscures the fact that those values are still AVERAGES. Recall the story of the fellow who drowned crossing a river with an average depth of 2 feet. Why did he drown? Because the AVERAGE depth is NOT the same as the depth in specific places; he couldn't swim and fell into a hole that was too deep for him to walk across it. It does not matter how precise (1 decimal place? 2 decimal places? or more?) the values are. Approximating a half Pawn in value may already be too much mathematics: no one is going to take the time to perform the calculations in their head while playing a game of chess.

    Third, the Kaufman values are certainly biased statistically. Eliminating all games where the protagonists are below 2300 Elo masks the fact that the judgments of the best players has been formed in the crucible of actual games. In any specific position, those computer-based AVERAGE valuations may be as far off the mark as the Reinfeld values. It is the concrete position and the specific locations of every single piece that determine the relative value (between the two sides, not between the individual pieces). How many times have you seen games in which one side has an overwhelming advantage in terms of the Reinfeld (or Kaufman) values - and concurrently, a totally losing game?

    Fourth, GM Emanuel Lasker opined that the chess student must make valuations and judgements based on his own experience. The Reinfeld values are an attempt to substitute "something" for the lack of experience and judgement in beginners. GM Lasker states quite clearly that you should use your own judgement: if you believe that a Knight is more valuable than a Bishop IN A SPECIFIC POSITION, then valuate and make your move accordingly. I'm willing to bet that no one has calculated the difference between the various pieces still on the board in order to arrive at the next move. It is much more likely that the relative values we use are judgement-based, more of a "feeling" than the numbers would indicate, and NEVER that precise.

    In short, focus on refining your own judgement, not relying on a computer (or a GM) to do the work for you. I'm NOT knocking the work by GM Kaufman; it has its purpose and benefits. Basing moves on AVERAGE values is a losing proposition while playing concrete games. The very act of calculating or recalling the calculated values takes your mind away from what it should be focused on: what SPECIFICALLY is going on in THIS POSITION, and how to decide on the best move(s).

  2. PART 2:

    As an aside, I looked up Mr. Reinfeld's rating (using Wikipedia). Here's the section on his playing strength; he was NOT quite a "weakie".

    Although Reinfeld is remembered today mainly for his writing, he was also one of the strongest chess players in the United States from the early 1930s to the early 1940s, after which he withdrew from competition. He was ranked sixth in the country, with a rating of 2593, on the first rating list issued by the United States Chess Federation in 1950, after Reuben Fine, Samuel Reshevsky, Alexander Kevitz, Arthur Dake, and Albert Simonson.

    The chessmetrics group, which retrospectively rates chess events and players, endeavoring to estimate the playing strengths of players on an historical basis, places Reinfeld with a peak rating of 2532 in May 1942, good for #70 in the world. His highest world rank on this basis was #64, from March to April 1942.

    I could only wish I could get up to that "weakie" rank!!

    Crazy Bob


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