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Check out Janggi (Korean Chess), our featured variant for December, 2024.
Check out Janggi (Korean Chess), our featured variant for December, 2024.
Jörg, I'm not sure you've given due consideration to board geometry. Betza's mobility calculations attempt to account for both the probability that a move will be blocked by another piece AND the probability that a move will be blocked by the edge of the board. If you assume that pieces are distributed randomly, then the odds of being blocked by a piece are p^(distance-1), as your formula suggests, but the odds of being blocked by the edge of the board follow a completely different pattern (e.g. for a rook, it's 1 - (distance/8)). That means it can't be accurately represented simply by plugging a different value for p into the polynomial shown in your post.
You also seem to be assuming that piece value is directly proportional to mobility. Most people believe that value has a super-linear relationship to mobility; the evidence being that compound pieces tend to be worth more than their component parts. There are also many things other than mobility that might affect a piece's value; Betza attempted to compile a list here.
Finally, you might want to consider that Betza believed a full Rook had a value closer to 3/2 of a Knight than to 5/3 of a Knight. If that's the case, then it's unclear whether a piece priced at 4 points should aim to be worth 4/5 of a Rook or 4/3 of a Knight--a difference of perhaps 10%, or roughly the difference in your calculated mobility between an R4 and a R5.