Johnny Luken wrote on Thu, Oct 11, 2012 11:29 PM UTC:
I didn't produce any actual results here, but a certain hierarchy becomes pretty obvious-the remover and withdrawer are extremely weak as they have zero probability of capture beyond adjacent squares, furthermore the withdrawer also relies on the probability of having an empty square in the opposite direction. For infinite range pieces, the displacer (queen) has a clear advantage over the advancer and long leaper, as it only relies on intermediate spaces being empty to make a capture. The advancer on the other hand has zero probability of capturing adjacent pieces. The long leaper does, but has to factor in the probability of an empty square behind the target piece.
On an infinitely large board, in starting conditions (all pieces on the board), to use a simple example, the displacer has capture probability in a starting direction of simply 1 in 4. The advancer approaches this value for infinitely large board. The long leapers is less-approaching a value of(1/4)*(3/4)=3/16, so 3/4 that of the other 2. The queen>advancer>long leaper hierarchy looks to be preserved for all conditions (all combinations of f, e, and A), however we have the additional caveat that the long leaper can capture multiple pieces a turn.
So what do we with that-well renaming the calculated pieces value as "statistical capture probability PER UNIT PIECE", we simply add the additional probabilities for capturing 2nd, 3rd etc pieces onto the long leapers previous value. On an infinitely large, full board the long leapers value approaches a series looking like (3/16)+[(3/16)*(3/16)]+[(3/16...], which converges to a value~0.93 advancer/queen. So while the long leaper has an additional relative increase in its term it never exceeds the queen/advancer in value.
Note that the long leapers value on a rococo board is improved, possibly beyond that of an advancer, as while it otherwise would have no probability of capturing a piece on the edge of a board from a direction approaching the edge, this term now becomes (P encountering enemy piece) times (P capturing=space behind piece being empty=1)=P encountering enemy piece=1/4 in starting conditions. This is now contributed to the long leapers value, as in this case, (1/4)(P of being able to reach that edge square) added to all its terms (all positions, directions, 1st/2nd/3rd capture). The advancer on the other hand gets a minute increase in its value, basically (probability the target piece itself making having recently made a capture and not moved)(probability of reaching square in front of it to make the capture)(probability of not having made a previous capture), averaged across the board...
I didn't produce any actual results here, but a certain hierarchy becomes pretty obvious-the remover and withdrawer are extremely weak as they have zero probability of capture beyond adjacent squares, furthermore the withdrawer also relies on the probability of having an empty square in the opposite direction. For infinite range pieces, the displacer (queen) has a clear advantage over the advancer and long leaper, as it only relies on intermediate spaces being empty to make a capture. The advancer on the other hand has zero probability of capturing adjacent pieces. The long leaper does, but has to factor in the probability of an empty square behind the target piece.
On an infinitely large board, in starting conditions (all pieces on the board), to use a simple example, the displacer has capture probability in a starting direction of simply 1 in 4. The advancer approaches this value for infinitely large board. The long leapers is less-approaching a value of(1/4)*(3/4)=3/16, so 3/4 that of the other 2. The queen>advancer>long leaper hierarchy looks to be preserved for all conditions (all combinations of f, e, and A), however we have the additional caveat that the long leaper can capture multiple pieces a turn.
So what do we with that-well renaming the calculated pieces value as "statistical capture probability PER UNIT PIECE", we simply add the additional probabilities for capturing 2nd, 3rd etc pieces onto the long leapers previous value. On an infinitely large, full board the long leapers value approaches a series looking like (3/16)+[(3/16)*(3/16)]+[(3/16...], which converges to a value~0.93 advancer/queen. So while the long leaper has an additional relative increase in its term it never exceeds the queen/advancer in value.
Note that the long leapers value on a rococo board is improved, possibly beyond that of an advancer, as while it otherwise would have no probability of capturing a piece on the edge of a board from a direction approaching the edge, this term now becomes (P encountering enemy piece) times (P capturing=space behind piece being empty=1)=P encountering enemy piece=1/4 in starting conditions. This is now contributed to the long leapers value, as in this case, (1/4)(P of being able to reach that edge square) added to all its terms (all positions, directions, 1st/2nd/3rd capture). The advancer on the other hand gets a minute increase in its value, basically (probability the target piece itself making having recently made a capture and not moved)(probability of reaching square in front of it to make the capture)(probability of not having made a previous capture), averaged across the board...