Kevin Pacey wrote on Tue, Nov 7, 2017 10:13 PM UTC:
World Chess Champion Emmanual Lasker (and some other chess writers) put the fighting value of a king at 4 pawns. I have been wondering how he arrived at that estimate, and today came up with a theory, however unlikely that it's true. If we take his (also commonly used) values of P=1;N=B=3;R=5 and Q=9, we can reach an average fighting value for the king if we look at 4 basic endgame situations:
1) K normally stops P+N, so in this case let's say the K's stopping power (or fighting value)=P+N=4;
2) K normally stops P+B, so in this case the K's fighting value =4 again;
3) Let's say K+P draws vs. R considerably more often than not, so in this case the K's fighting value =4 again;
4) Let's say K at least restrains 3 passed pawns more often than not, so in this case the K's fighting value =3.
The average fighting value of a king based on the above would be (4+4+4+3)/4 =3.75, or rounded to the nearest 0.5 pawns would be =4 pawns, Lasker's given figure.
Another matter I've wondered about is, if we assume a Guard's value is at least that of a king's fighting value, in chess variants that include Guard(s), what values should a King and a Guard have respectively in such variants (assuming for a start that the values will be close)?
Well, if a chess variant was 8x8 and had all the piece types in chess, plus one or more Guards, then the above 4 endgame situations would still be relevant to a Guard or King's value. In fact, the above four would lead to the final value for a Guard in such variants, the way they did for a king in chess. A king's value must be recomputed for such variants, however, by averaging in a fifth endgame situation:
5) Let's say a K+P and Guard+P(on the same file) situation favours the Guard about 1 time out of 4, in that the Guard drives away the K and wins the enemy P - in this case the fighting value of a K is only worth about Guard-P=3. All guesswork, I suppose, but I'm only demonstrating the sort of calculation I'd use if I knew more about this case.
Thus, the average fighting value for a king in such 8x8 variants as I described earlier would be (4+4+4+3+3)/5=3.6 pawns, or 3.5 if rounding to the nearest 0.5 pawns.
For the record I would use P=1; N=3.49 (or 3.5 when rounded); B=3.5; R=5.5; Q=10 if doing my own calculations for the above values of the K and Guard. Also, computer studies put the value of a Guard at 3.2 pawns, I recall.
World Chess Champion Emmanual Lasker (and some other chess writers) put the fighting value of a king at 4 pawns. I have been wondering how he arrived at that estimate, and today came up with a theory, however unlikely that it's true. If we take his (also commonly used) values of P=1;N=B=3;R=5 and Q=9, we can reach an average fighting value for the king if we look at 4 basic endgame situations:
1) K normally stops P+N, so in this case let's say the K's stopping power (or fighting value)=P+N=4;
2) K normally stops P+B, so in this case the K's fighting value =4 again;
3) Let's say K+P draws vs. R considerably more often than not, so in this case the K's fighting value =4 again;
4) Let's say K at least restrains 3 passed pawns more often than not, so in this case the K's fighting value =3.
The average fighting value of a king based on the above would be (4+4+4+3)/4 =3.75, or rounded to the nearest 0.5 pawns would be =4 pawns, Lasker's given figure.
Another matter I've wondered about is, if we assume a Guard's value is at least that of a king's fighting value, in chess variants that include Guard(s), what values should a King and a Guard have respectively in such variants (assuming for a start that the values will be close)?
Well, if a chess variant was 8x8 and had all the piece types in chess, plus one or more Guards, then the above 4 endgame situations would still be relevant to a Guard or King's value. In fact, the above four would lead to the final value for a Guard in such variants, the way they did for a king in chess. A king's value must be recomputed for such variants, however, by averaging in a fifth endgame situation:
5) Let's say a K+P and Guard+P(on the same file) situation favours the Guard about 1 time out of 4, in that the Guard drives away the K and wins the enemy P - in this case the fighting value of a K is only worth about Guard-P=3. All guesswork, I suppose, but I'm only demonstrating the sort of calculation I'd use if I knew more about this case.
Thus, the average fighting value for a king in such 8x8 variants as I described earlier would be (4+4+4+3+3)/5=3.6 pawns, or 3.5 if rounding to the nearest 0.5 pawns.
For the record I would use P=1; N=3.49 (or 3.5 when rounded); B=3.5; R=5.5; Q=10 if doing my own calculations for the above values of the K and Guard. Also, computer studies put the value of a Guard at 3.2 pawns, I recall.