Value of Pieces for "Infinite Chess"

"If a Queen goes up in strength on a larger board, why does it take longer to issue the checkmate of K + Q vs. K on a larger board vs. a smaller one?"

Because the power of a piece is not determined by how long it takes to deliver checkmate, it is determined by its ability (relative to other pieces) to domiante the tactical side of things so one gains an advantage that ultimately allows checkmate to become inevitable.

The queen can't even deliver checkmate alone in this format. It really doesn't matter how long it takes a piece to checkmate the enemy king on a clear board, that isn't how most chess games are won, and certainly none will be won like that on an infinite plane.

We're not really playing on an infinite board. It's potentially infinite, but it's always limited to the most remote piece. You simply make the board bigger by moving away from the other pieces, but the board will never be infinite in any practical sense.

To illustrate that point, there comes a point where two moves are identical in strategic value... for example, R(1,999) or R(1,9999999). The rook controls the 1st rank either way, and the file it controls is irrelevant in both cases. Furthermore, from both squares it has zero risk of being hit by the queen with tempo, unless there happens to be another piece near it anyway. So for all practical purposes, the two moves are identical.

This idea that it's infinite isn't quite true.

Not sure I agree with this. A rook can still get from any square to any other square in two moves unless it's blocked or attacking a piece. And if it's blocked or attacking that gives it value too.😐

KR vs. K is a draw on an unbounded board. The board is unbounded by the game definition, not by what the players do. FWIW, I believe (but can't prove) that very long moves are usually poor play. Pieces which are too far lose their ability to fork. One interesting thing is that since pawns can't move backwards, they probably have the effect of pulling the action into a zone between the two armies. But other pieces are pretty much free to move where they want. Whether moves are accurate or not is an entirely different question.

Give sliding pieces more points. The bishop in 4 player is even worth more in infinte chess.

Better make it infinte.

You obviously can't use a King and another piece (exception B+K) to mate on an edgeless board because the King can only move one space every other turn and thus all the other King must do is continue to move away. The King can't coordinate the mate if there is any flight direction.

at least 2 pieces is possible

This is a quirk of the board geometry. The bishops attack the players to the left and right, while the rooks only attack the player opposite.

The bishops are of course strong here, but it's got little to do with why they're strong in 4p.

The problem is that you can't use board size for valuation in this variant.

Yes, when the board actually inhibits movement a larger board will decrease the value of non-sliders, but the reality is that the value is based on playable area, that is the squares where pieces will tend to stay within to effectively attack the King.

It's possible to put check on a King from 1000 spaces away but this has almost no bearing on the game since it is almost exactly like putting check from 200 spaces away. It's only when you are close enough to be attacked by a different kind of piece (from a different direction) that you can justifiably call that an estimated board size.

If you have a system for estimating piece values which use board size, those simply aren't going to work in this variant.

Also, attempting to work out values for pieces without [edited:typo] using the pawn as a base value of 1 is, IMHO, stupid.

"Also, attempting to work out values for pieces with using the pawn as a base value of 1 is, IMHO, stupid."

I don't agree here, it's no more stupid than using the pawn as a base unit in regular chess.

Anyone who understands chess knows that a pawn does not have a fixed value. Pawns become stronger when they are advanced, stronger yet when they are passed. An unstoppable pawn is worth at least a bishop/knight, possibly a rook if that is all the enemy has to stop it queening. A pawn that is blocking a bishop from becoming active is also worth the same as the bishop, at least if the blockage is permanent.

But still we assign it a value of 1. The only reason to do this is to give an approximate value of the other pieces,

If we can't assign the pawn a value, then there's no point in valuing any other piece.

Any other system of valuation would be more complicated. The pawn has a value of 1 at the start of the game. It fluctuates, as do the values of the other pieces. This fluctuation will be more profound in infinite chess compared to classical chess. That doesn't mean we can't at least try to assign values to pieces. We need to, otherwise how can we have any idea which pieces to trade?