V. Reinhart wrote on Sun, Jul 23, 2017 03:16 PM UTC:
I agree that the concept of power density involves some assumptions that might cause the value to be an approximation. As you mentioned, it does assume that pieces have fixed values, even with a different mix of pieces, and different board sizes.
I do believe that if every game has a mix of pieces (as they do), such errors would tend to cancel out. For example, as board size changes, some pieces might gain slightly in value, while others lose value.
The only way to overcome such possible errors is if there was an accurate way to identify a piece's value based on the specific board size. I'm not aware of any work that has been completed to do this for a range of board sizes. At best, maybe we know the rough difference in value of a few pieces when they go from an 8x8 to 10x8 board. To my knowledge, there is no piece which has its value altered by such a large amount that it would render power density as grossly innacurate.
I believe the biggest error currently found in the power density table is the data for Chess on an Infinite Plane . Here a board size of 18 x 20 was assumed because it's the approximate span of pieces in the starting position. But the bulk of the dynamics in actual play is usually found in a much smaller area.
In fact, the tendency of pieces to try to "fight for the center" might be a phenomenon seen in all games, so the stated "board sizes" themselves might be an opportunity for refinement. But I'm reluctant to complicate the formula based only on conjecture. As we learn more about piece valuations for variant chess, I certainly can plan to refine the formula when there is merit to do so. For now, it's based on the theory that "Simple and approximate" is better than "Complex with speculation".
I agree that the concept of power density involves some assumptions that might cause the value to be an approximation. As you mentioned, it does assume that pieces have fixed values, even with a different mix of pieces, and different board sizes.
I do believe that if every game has a mix of pieces (as they do), such errors would tend to cancel out. For example, as board size changes, some pieces might gain slightly in value, while others lose value.
The only way to overcome such possible errors is if there was an accurate way to identify a piece's value based on the specific board size. I'm not aware of any work that has been completed to do this for a range of board sizes. At best, maybe we know the rough difference in value of a few pieces when they go from an 8x8 to 10x8 board. To my knowledge, there is no piece which has its value altered by such a large amount that it would render power density as grossly innacurate.
I believe the biggest error currently found in the power density table is the data for Chess on an Infinite Plane . Here a board size of 18 x 20 was assumed because it's the approximate span of pieces in the starting position. But the bulk of the dynamics in actual play is usually found in a much smaller area.
In fact, the tendency of pieces to try to "fight for the center" might be a phenomenon seen in all games, so the stated "board sizes" themselves might be an opportunity for refinement. But I'm reluctant to complicate the formula based only on conjecture. As we learn more about piece valuations for variant chess, I certainly can plan to refine the formula when there is merit to do so. For now, it's based on the theory that "Simple and approximate" is better than "Complex with speculation".