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Assume we have a standard chess set setup, but we make a slight change so that (for example) pawns can now move two squares forward on any move rather than just on their initial move. Even assuming that we can still use the established point values for the pieces (Knight=Bishop=3 Rook=5 Queen=9 or whatever system you wish to use), how would one go about finding out how much our modified pawn would be worth?

My initial thought would be to (re-)program a chess engine with the modified piece ability, change the internal values for it in several different ways, and then hold a series of engine tournaments until an approximate value can be narrowed down. This would work (as all point values are approximate and situational anyway), but point values for pieces have been around longer than computers, so there must be other methods available.

Any ideas?

Note: I am not actually looking for software recommendations, etc. to do this. I'm just curious what the best way of doing this would be.

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    The system used today has pawn=1, which means that a modified pawn will change the unit that is used to measure other pieces than pawns.
    – user2001
    Commented May 21, 2016 at 9:41
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    @RauanSagit Of course, but we can still assume a standard-pawn as the unit of measurment (or if you insist it must be on the board, give each side a mix of pawns and 'super pawns').
    – DTR
    Commented May 21, 2016 at 19:00
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    The values of all the pieces are interconnected, so changing the pawn's move would also change the relative values of all other pieces. Bishops would be useless in the endgame against your superpawns, and knights only slightly less useless, so their values would diminish more than the rook's or queen's would. All the pieces would need to be re-evaluated.
    – Kevin Suchlicki
    Commented May 27, 2016 at 17:24
  • @KevinSuchlicki If the bishop is the same color as the space the pawn needs to move to to promote, they could still guard that square. And assuming the double-move can't go through pieces, a bishop could still block a pawn. But yeah, minor pieces would be seriously weakened in the end-game. Two passed pawns would be worth more than a bishop.
    – Acccumulation
    Commented Mar 29, 2018 at 17:07

7 Answers 7

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It's possible to use logistic regression (a statistical method) to estimate the predictive values. This way, you wouldn't need anyone to try the game at all.

http://www.sumsar.net/blog/2015/06/big-data-and-chess has the details. I personally tried the method, and it was a good start.

The method estimates the value of each piece by predicting how they relates to the log-odd of winning.

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    Fascinating and cool... but doesn't this type of analysis require a dataset of expert games from which to extract these piece data? So while you don't need experts to convene on the specific issue of piece valuation, you do need experts (or at least proficient players) to come up with the games in the first place.
    – Daniel
    Commented May 23, 2016 at 23:07
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    @Daniel it's possible to code an engine for it. It's also possible to use Monte Carlo to simulate games.
    – SmallChess
    Commented May 23, 2016 at 23:10
  • I doubt random games would give good values.
    – hkBst
    Commented May 26, 2016 at 6:32
  • @hkBst It's fine to have random games, but the number must be huge and random. That's how Monte-Carlo works. That's also how AlphaGo works.
    – SmallChess
    Commented May 26, 2016 at 6:34
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    Alphago only does that for the final evaluation step and it was fed a lot of expert games to study to build up its knowledge/intuition.
    – hkBst
    Commented May 26, 2016 at 6:37
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Ralph Betza tried to do this and he wrote a series of six articles about this, starting with this one: http://www.chessvariants.com/piececlopedia.dir/ideal-and-practical-values.html

Ideas to determine the piece values include the following factors

  • average mobility (clearly the dominant factor, but hard to bring it down to numbers)
  • colourboundness
  • type of movement (jump vs. ride)
  • levelling effect (Scharnagl calls it "elephantiasis correction")

Practical experience with chess variants shows that an empirical determination through playtesting cannot be replaced completely by a determination from first principles. For instance, the compound piece formed from Bishop and Knight (known under many names, including Archbishop, Princess, Janus, Cardinal, Paladin, Equerry, and Minister) is much stronger than an a priori analysis suggests.

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The values of the pieces derive from which piece exchanges are considered desirable and which are not. Knowledge of desirability of piece exchanges usually comes from having played many games, but it is probably also possible to mechanically extract this knowledge from a large collection of games played by skilled players.

Another option is to use an evolutionary process to determine piece values. You start with a large collection of random piece values and hold one-on-one elimination matches (or maybe tournaments is better?) to determine some best fraction (half, top ten percent) of random piece values. Then you create a new generation of random piece values through some method of combining values from that best fraction together with small random perturbations. Repeat until the values stabilize. The values you get will probably depend on the specific chess engine (and time controls) you use, but I don't know how strong that effect is.

Once you have a reasonably good sense of where the values are, you may want to use the scientific method to answer specific questions, such as whether the value of your new pawn is more or less than half a knight. You could have your chess engine play many games at different strengths (time controls or ply depth) and use statistical analysis to determine an answer up to a certain confidence level.

You may also be interested in deriving piece values in a more analytical way; many people have thought that there must be a relation between piece mobility and piece values. Relevant factors may include: board-average mobility, board-maximum mobility, reachable fraction of the board, triangulation ability, mating ability, and (most confoundingly) the other pieces on the board. Nothing very general seems to have been discovered though.

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We could start guessing the approximate value of this hypothetical "superpawn" or "enhanced pawn" in terms of "mobility", in the order of E~2P because of the definition (move up to 2 squares instead of only 1 square).

Next we adjust this initial guess by forming an 8x8 matrix, where each square has a number indicating how "mobile" is the analyzed piece (P=pawn, E="enhanced pawn") when placed at that square:

Pawn    xxxxxxxx<--last rank    Enhanced pawn   xxxxxxxx
        11111111                                22222222
        11111111                                22222222
        11111111                                22222222
        11111111                                22222222
        11111111                                22222222
        22222222<--first rank                   22222222
Pawn    xxxxxxxx               Enhanced pawn    xxxxxxxx

Here we have an average mobility of 2 squares for the enhanced pawn vs 7/6 for the normal pawn (who can only jump 2 squares when located at the initial rank). The relative power E/P appears to be 2/(7/6)=12/7~1.7 slightly below E=2P.

But there are normally other pieces that populate the board and limit the mobility. In a real game, we will find that at some locations our new "superpawn" is completely surrounded by other pieces and does not differ from a "normal pawn". So the tentative number E=1.7P should be pushed somewhat lower.

In order for these numbers to be of any value, we should imagine certain tasks or situations and see how a particular piece or group of pieces performs. A similar analysis has been made for the standard chess pieces. Some examples:

  • 1 Queen cannot corner and checkmate a lonely rival King, while 2 Rooks can. That suggests 2R>Q which is in accordance with the normally accepted values Q~9P, R~5P. (Or Q~10P R~5.5P).
  • King+Rook can checkmate an enemy King, while kNight+Rook cannot (they need the aid of the King). So in this case K+R>N+R, K>N.
  • But a kNight can cross a barrier formed by a Rook, while a King cannot. So there are opposite situations where N>K.

  • For some tasks K>N, for other tasks N>K. This behaviour is supported by the official point scales, which evaluate the difference of King vs kNight to be in the order of a pawn or fraction of pawn.

  • And where does our new enhanced pawn fit? He can cross the barrier of a rook, while a King cannot. That means that in some situations, he can outperform a King, E>K (being K between ~3P and ~4P)

  • But he cannot cross a barrier formed by 2 Rooks, while a Bishop can. So here is B>E.
  • And he cannot cross a barrier formed by 2 Bishops, while a kNight can. So here is N>E.
  • If we build a big table with lots of tasks, we can count how many "E>K" and how many "K>E", "E>B", "B>E"...etc we have, and calculate an average.

A more powerful approach would be to access a big database of complete games, not just individual "tasks". As has been already mentioned in this site, with the aid of a game database it is possible to analyze the result of trading pieces. Applying this idea to our "superpawns", with thousands of games we could answer questions like "Is a superpawn really worth 2 pawns? Or is 2P>E? The player who loses 1E while taking 2P from the rival, does he normally lose? Or does he retain a reasonable expectation of winning? What about 2E vs 3P? E vs B? 2E vs B? 2E vs N?

It is often said that everything depends on the position, but with big (very big!) sets of data we could think that the variations of particular positions tend to cancel out and what remains after averaging is what we call "piece value".

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The value for pieces in terms of pawn units was originally determined by collecting experience while actually playing the game. The same can be applied to the modified game.

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In a different reality, I would do this by creating an pool of experts, then asking them.

1) Ensure an educated set of experts.

Hold a chess tournament (or perhaps several) with attractive first, second, and third prizes. This will entice the best players to attend. They'll play and become educated.

2) Have the experts tell you the value of the pawn

As part of the tournament, on the last day perhaps, have the top X players estimate the new value of the pawn. The GM who most closely guesses the value you feel is accurate wins another cash prize. From the estimates, calculate an average (or whatever) and pay the person making the closest guess.

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There are many factors that affect the value of a piece, three of which are most influential:

  1. Piece mobility in situation - e.g., A knight in an open position is less valuable, but in a closed situation is more valuable.
  2. Piece qualities(can it cause zugzwang, can it be easily trapped, etc.)
  3. Piece's positioning - e.g., if a queen was on the other side of the board unable to save the king from an attack, it is significantly weaker than a queen on the 'correct' side of the board.
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