True value of the king?

If the king wasn't the heart of chess then what would its value be, as a slow poorman's queen?

16 cents, three mints, a dazzling fingernail design, or my opinion.

I'll say 1. I don't think it's more valuable than a Pawn since at least a Pawn has potential to be more.

a king is actually quite a powerful piece, johnkorean.

Obviously the king's material value is infinity, but what is it's positional value? It depends on the position.

I think about 4 points. Maybe 3.

2 pesos and a coffee bean

the way you're saying it, you're applying to all the other pieces. 

                         PRICELE$$

I've always heard that the king has an attacking power equivalent to a knight = about 3 

The King equals "roughly" 4 points.

It's typically stronger than either a Bishop or Knight in the endgame.  But only if you can reach a (fairly level) and simple endgame.  Then the King "comes alive" and can be quite powerful.

This is true for many Rook and Pawn, or Minor Piece endings.

Naturally, it all "depends on the postion."

1/32 of the price of the set.

If a game of fairy chess is played on an 8x8 board with Princes (moves like king, but not royal) then a Prince is considered to be approximately equal in value to a Knight.

I have done piece-value measurements using the Fairy-Max engine, and it turned out the opening value of such a non-royal King (usually referred to as 'Commoner' or simply 'Man') is slightly less (~ 0.3 Pawn) than that of a Knight. I.e. give one side two Commoners in stead of the usual Knights, and he will lose roughly 60-40 from an opponent that starts with orthodox Knights.

In the end-game a Commoner can be quite strong, though. It is very effective against Pawns at close range, but its main weakness is that it moves quite slowly, and cannot catch up with them. So its position on the board has a much larger effect on its value than in the case of the Knight (and even there the effect is much stronger than for the Bishop).

Also, a Commoner can be a very strong defender. KQKR is lost to the Rook, but generally K+Q cannot beat K + Commoner, provided the latter two can connect. (Otherwise the Queen wins quite easily by forking them.) This because the attacking King cannot approach a Commoner, so it is not possible to attack it twice, and all the defending side has to do is keep its Comonner protected by the King while sheltering behind it.

Of course another important aspect is that K + Commoner have mating potential. So where KNPKN can be easily drawn by giving the Knight for the last remaining Pawn, you could not do that in KCPKN. Similarly, being a  commoner up is winning where being a minor up is draw, e.g. KCNKN as opposed to KBNKN. An N-trade is no longer a viable defence in that case, and that makes all the difference.

Have you tried replacing the Bishops with Commoners? That at least equal for the side with the commoners because they are closer to the centre.

When I do such measurements I usually average over many different shuffles of the back-rank pieces. I never found much effect of the initial setup. A B-Pair is about 0.5 Pawn stronger than two Knigts (or B+N), so just replacing both B for C would make the Commoners lose even more badly (something like 70-30). But moving B to the b- and g-file, and putting commoners on c- and f-file, did not make the Commoners fare much better. (Despite what Betza claims...)

But I cannot exclude that developing some good opening theory would bring out the difference. Fairy-Max plays he opening pretty stupidly, almost random, (as far as it doesn't blunder away material), because I add a quite heavy random score during the first few moves, to prevent duplicate games. (So 1. h4 is not uncommon...) That could make it difficult to cash in on a subtle positional advantage.

King        -              -                 4 

Another thing is that the smaller the board range is the stronger the King (and the N) are, but I don't know how this can be quantified

True, the numbers I gave are for 8x8 board. The Knight loses about 0.25 Pawn on 10x8 boards, while the Bishop gains 0.25 (compared to Rook). So there is a B-N gap of 0.5 Pawn between lone B and N in Capablanca Chess. To that, the pair bonus has still to be added. So trading your first Bishop for N+P is an even trade there!

 Could you calculate the values  of the pieces from 8 x 3 boards to 8x8 boards?

For example this:

has a practical board range of 8x4 if you give one free column to the right for outflanking

it could be interesting to calculate the variations of the values of the pieces with the different boards

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