Exact relative value of chess pieces and fairy chess pieces

With the advent of chess programs and engines, especially self-learning artificial intelligence engines such as AlphaZero and Leela, the search for the exact relative value of chess pieces has become a thing of the past, as artificial intelligence is demonstrating the true dynamic essence of chess, not the static relative values.

However, for programmers of traditional chess programs and engines, it still has some relevance.

Since about 2015 I started to find out about the relative value of chess pieces, partly because I wanted to create my own chess variant, which is on hiatus, but it will most likely be paralyzed since it would be a cross between chess traditional and the Congo variant, not very popular.

In 2016, a year later, I discovered what I think are the exact values, partly thanks to the help of Microsoft Excel, since there I was able to enter the data and graphs for the exact calculations. In 2017 I posted a very general entry (in Spanish) on the subject on my blog, and due to my disappointment in creating a new chess variant, I posted the exact values ​​in an entry (in Spanish) this year.

I consider that it was an excellent discovery since in my research, many experts on the subject give imprecise and changing values, even grandmasters disagree a lot with each other. In Wikipedia there is an article about the values ​​of the pieces and the different values ​​given throughout history.

To calculate the relative values ​​of the chess pieces, it must be taken into account that they are relative values, worth the redundancy, because everything will ultimately depend on the position.

There is something very important to take into account, something that many ignore or overlook and is the biggest and most common error when discovering the relative value of the pieces, is that the relative value of two simple pieces, for example Rook and Bishop is never the sum of both, since in terms of coordination, time and mobility they interfere with each other (something similar to Moskalenko's T3 and T5 factors).

Instead, in a composite piece, like the Queen, a unique synergy converges. It is for this reason that if the Rook is worth 5 and the Bishop is worth 3.375, both pieces can never be worth the sum of both, that is, 8.375, while the Queen is worth 8.375. To alleviate this situation, historically and pragmatically, the Queen has been given a value of 9, but this is not exact because, as I mentioned, they ignore the factors of time, coordination and mobility mentioned above.

To get an idea of ​​the above, see the following image, which shows the power of the Queen in front of the Rook and the Bishop working together on a 4x4 board, the number of squares that attack in each case, both in the optimal position as in the worst. It is observed how the pieces are hindered and that having a Rook and a Bishop is generally much inferior to having a Queen.

Queen vs. Rook and Bishop

Now, we see that even if the pieces were alone or transposed, the value would be less than that of a Queen, the potential value, or rather, the number of squares that attack both pieces is equal to that of the Queen, but in this, for example, as in the previous example, the time factor should also be taken into account since you can only move one piece per time in chess:

Queen vs. Rook and Bishop alone

Regarding the names of the proposed fairy pieces, I have to say that it has also been thought and I have focused on chess as a war game and on politics and culture, especially Western.

Chess lovers of fairy pieces may not be familiar with the names "Vizier", "General" and "Cardinal". Below I explain the suggested names for each piece:

Vizier: A vizier is a prime minister of the king, especially in the Middle East. It is known that in practice, being head of government, he holds even more powers than the king himself, the plots and the political relationship between the king and the vizier is often stormy. Hence, I preferred this name to the more fanciful "Amazon" as it is commonly known.

General: The idea is to be a general of the army or supreme commander of the same. If there is a chess piece that takes us back to the past and the origin of the game, it is precisely the Knight, and together with the Rook, the Knight gives an idea of ​​the cavalry, and the Rook, of a heavy or siege unit. The combination of both gives the idea of ​​a piece that leads the army in general.

Cardinal: The idea is a piece superior to the Bishop but with the same religious connotation, a high priest. The cultural origin of the Bishop has been lost, initially it represented a war elephant, from the Arabic al-Fil (الفيل), so it was the Alfil which today is another fairy piece that moves leaping a space in diagonal. If we talk about other fairy pieces, in fact, the King is a piece composed of two fairy pieces: the Wazir (1 orthogonal movement) and the Ferz (1 diagonal movement). Returning to the theme of the Cardinal, the name seemed appropriate to represent this piece, also in Spanish it is very similar: Cardenal. Ironically, the church and the state, present in chess.

How did I get the exact relative values ​​of the parts?

In view of the above and working on the spreadsheets to calculate both values: the relative value of the simple pieces individually and as a group, and of the combined pieces. The base of the calculation is first obtained, the value of the pawn, then it is equal to 1 (1.000) and the values ​​of the other pieces are obtained. The key is to use two boards and place the piece there, on one board as if it were free and on the other as if it were surrounded by enemy pieces. All the values ​​are added and it is divided to the value of the pawn that turns out to be 224, then for example the Bishop gets 756 squares on both boards, this is divided by 224 and the result is 3.375 (or 3 plus three eighths), the following graph shows it:

Calculating the values

From the value of the pawn you can calculate the values ​​of the other pieces, including the most popular fairy pieces.

Anyway, the exact relative value (I think so) of the chess pieces that I have discovered is as follows:

King = 3.750 or infinite, in practice 268.000 or more.
Queen + Knight (Vizier) = 11.375
Queen = 8.375
Rook + Knight (General) = 8.000
Bishop + Knight (Cardinal) = 6.375
Rook = 5.000
Bishop = 3.375
Knight = 3.000
Pawn = 1.000

A very interesting piece that I think has a chance to enter to the club of "celebrities" of the fairy pieces is the so called Centurion or the fancy name "squirrel". It is basically a super Knight or a super leaper. The value, I have calculated it of course, is 6.000 in a 8x8 board. It moves like the image below:

From Chessvariants.com

Later I realized that these values ​​were discovered a year after my discovery, but using another method. Jeff Coakley and François Labelle published an article with the same values ​​that I discovered, but only of the traditional pieces.

For my part, for four years I consider that the issue of the relative value of chess pieces is closed because the exact values ​​have been discovered, but it just coincides with the onslaught of artificial intelligence, and this issue remains for history, as very referential data, and for programmers of traditional chess engines.

Have a great year and great games!

Hasan