The Chess Variant Forum

Some time ago I did a quick calculation of "Material Power Density" of some games that I play, and a few others for comparison. The calculation is simply a ratio of the value of all pieces on the board, divided by the board area (number of playing squares).

Although this indicator cannot precisely give a complete measure of the style or quality of any chess game, it is a partial indicator of the overall attacking power on the board.

Assuming play is good quality, and the opponents are approximately equally matched, a higher power density often leads to games with faster exchanges, and the attacking patterns can be more complex and dynamic. Each side can quickly create threats, and the other side will need to react quickly to defend and create counter-attacks. On average, games will often progress into an endgame quicker, often without pawn advances playing a dominant role.

In games with a lower power density, the opening game and development will usually last longer, permitting both sides to have more moves to strategically create defensive formations, while initiating threats to disturb the opponent. Accurate play and strategy is required in advancing pawns, as this is both necessary to create defensive formations, and also to create opportunities to promote one or more pawns (or at least create the threat of doing so). In these games it is much more likely the endgame will feature one or more queens earned by a pawn promotion.

A brief description of the games included is below. (I plan to add some other games to this calculation, and possibly refine some calculations when I'm able to find some time).

1) Classical Chess: (the "baseline")

2) Classical Chess, Endgame only: Chess endgames can be challenging puzzles in their own right. They represent a condition where there is much less piece value on the board, but tactical strategy can still be complex and interesting. I included a chess ending with KQRR vs. KQR, which is the highest value of pieces from an endgame of seven pieces (assuming no promotions). This represents a "lower limit" game condition, where if there was any further reduction of power (simplification), the game could start to become less interesting.

3) Janus Chess: Invented about 40 years ago. Has two januses (= bishop + knight) for each player in addition to other normal chess pieces.

4) Capablanca Chess: Invented in the 1920s. Uses a chancellor (= rook + knight), archbishop (= bishop + knight), and other normal other chess pieces.

5) Seirawan Chess: Invented about 10 years ago. Uses a hawk (= bishop + knight), elephant (= rook + knight), and other normal chess pieces.

6) Musketeer Chess: A more modern chess variant, which allows the players to choose from ten special pieces to be added with other normal pieces. The ten pieces include archbishop (= bishop + knight), chancellor (= rook + knight), dragon (= queen + knight) and seven other powerful pieces. In this analysis I use only a sample game using archbishop and chancellor.

7) Bulldog Chess: A variant with two guards and two bulldogs with other normal chess pieces. The bulldogs are a pawn-type piece, and the guards move with king-like ability.

8] Bulldog Chess with Witch: A Bulldog variation where each player has one witch, one guard, and other normal chess pieces. The witch does not capture, but pieces adjacent to her become transparent to pieces of her color.

9) Bulldog Legacy Chess: Another Bulldog variation where each side starts with 18 pieces rather than 20. The game uses a guard along with other normal chess pieces. The two outside files have only a single pawn for each player and no other pieces.

10) Waterloo Chess (5th Edition): This edition of Waterloo was released early in 2017, and features seven variant pieces in addition to other normal chess pieces, played on a 10x10 board.

11) Chess on an Infinite Plane: Play for this variant started early in 2017. Each side has two chancellors, two guards, two hawks, and other normal chess pieces. The playing area is unbounded.

12) Amsterdam Medieval Chess: A chess-variant designed as an intermediate form between classical (FIDE) chess and the more complicated Waterloo chess.

13) Chu Shogi: A chess-like game inspired by Japanese Shogi. To calculate the power density, piece exchange values were used from (http://www.chushogi.de/strategy/chu_strategy_exchange_values.htm) and all values normalized so that a piece equivalent to the rook has a value of 5.

Thanks, I believe Gothic Chess has the same material density as Capablanca Chess because it has the same pieces and board size. But I know there are game differences which will of course affect the style and quality of games.

I added Gothic Chess to one of the headings (bar for Capablanca and now Gothic Chess). As you might presume this is only a somewhat inexact measurement of a game, but I often use it when thinking about adding new pieces to a game, or whether I'm interested in trying a new game. Updated chart with Gothic Chess is here:

How would you expand the formula to factor drops? Crazyhouse I think would be easy, but shogi has a couple of restrictions on where you can drop pawns.

You have not given a formal definition, but intuitively I would say that 'power' is something different then 'value'. Chess pieces can be valuabable because of some latent trait not expressed now, and 'power' seems more a measure of the current situation. E.g. in Shogi most pieces promote (e.g. a Rook would get the moves of a King added), and that makes them more valuable than an unpromotable piece that has the same move (a Rook obtained by promotion of a Gold General in Chu Shogi). But their move is exactly the same, so I would say their power is the same, and the promotion is just something you can hope for in a distant future.

Shogi- / Crazyhouse-style drops are diffferent, however. They are an immediate tactical concern when you use a piece. E.g. a Crazyhouse Knight in an otherwise normal Chess games would be a much less valuable piece than a normal Knight. As sacrificing it immediately loses you two Knights. So it would be unwise to have it venture onto squares attacked by, say, a Rook, even when it is protected. I am not sure if this should be included in the measure of 'power', but it surely suppresses the effective instant value of the piece. Of course all pieces with a high value (e.g. Queen) suffer similar restrictions, and indeed their effetive value drops when they face more low-valued pieces. (Which is the reason 3 Queens lose to 7 Knights on 8x8.) But this interdiction effect (against the normal average mix of opponent pieces) is already included in their classical values, so if we equate power to value for those, it would be reasonable to also hold it against the power of a Crazyhouse Knight that it is droppable.

A piece already in hand are quite different beasts from on the board; they don't have their normal moves, but can be dropped everywhere. (But capture nowhere!) So it seems easonable to say a piece in hand has an entirely different 'power' from the same piece on the board. Of course the effective power of a normal piece also depends on where it is on the board; Knights in corners are kind of powerless. But the point there is hat this can often be changed with a single move, putting it in a better place. But you cannot simply take a piece in hand if it happens to be more powerful there (in Shogi/Crazyhouse, at least). You can get it in the opponent's hand quite easily, but that doesn't help. So that piece in hand usually have more power than those on the board doesn't affect the power of pieces on the board.