Adam Hair wrote:pichy wrote:The relative value of pieces changes as a game progresses to the endgame. The relative value of pawns and rooks may increase, and the value of bishops may increase also, though usually to a lesser extent. The knight tends to lose some power, and the strength of the queen may be slightly lessened, as well. Some examples follow.
• A queen versus two rooks
• In the middlegame they are equal
• In the endgame, the two rooks are somewhat more powerful. With no other pieces on the board, two rooks are equal to a queen and a pawn
According to long time control engine matches between engines greater than 2700 Elo (CCRL scale) and with an Elo difference of 50 or less, with no minor pieces the 2 rooks are worth about 0.3 pawns more than the queen. If you add a pawn to the queen, then the queen + pawn is worth 0.7 pawns more than the 2 rooks.
Of course, the above situation (Q vs 2 R with no minors) depends on how many pawns are on the board. The rooks have their greatest advantage when there are 2 or 3 pawns on the board for each side (+0.5 pawns for the rooks vs a queen).
If you let this endgame to be played out to the end what do you think the outcome will be without using tablebase, Komodo is evaluating the position as favorable to White, but I gave Black to Houdini and White to Komodo and it ended in a draw search depth set to 34:
[D]4k3/5r2/8/8/8/8/3P2Q1/1r2K3 w - - 0 1
FEN: 4k3/5r2/8/8/8/8/3P2Q1/1r2K3 w - - 0 1
Komodo3-32:
1 00:00 4 39.999 -0.21 Ke1e2
2 00:00 858 8.579.999 -0.07 Ke1e2 Rf7e7+ Ke2d3 Re7d7+ Kd3c2
3 00:00 1.366 13.659.999 +0.02 Ke1e2 Rf7e7+ Ke2d3 Re7d7+ Kd3c2 Rb1b6
4 00:00 3.114 31.139.999 +0.02 Ke1e2 Rf7e7+ Ke2d3 Re7d7+ Kd3c2 Rb1b6
5 00:00 5.054 0 -0.01 Ke1e2 Ke8f8 Qg2a8+ Kf8g7 Qa8e4 Rb1c1 Ke2d3
5 00:00 5.054 50.539.999 -0.01 Ke1e2 Ke8f8 Qg2a8+ Kf8g7 Qa8e4 Rb1c1 Ke2d3
6 00:00 6.268 0 0.00 Ke1e2 Ke8f8 Qg2a8+ Kf8g7 Qa8e4 Rb1f1 Ke2d3 Rf7d7+ Kd3c2
6 00:00 6.268 62.679.999 0.00 Ke1e2 Ke8f8 Qg2a8+ Kf8g7 Qa8e4 Rb1f1 Ke2d3 Rf7d7+ Kd3c2
7 00:00 13.956 894.611 +0.04 Ke1e2 Ke8f8 Qg2d5 Kf8g8 Ke2d3 Rb1b4 Qd5c5
7 00:00 13.956 894.611 +0.04 Ke1e2 Ke8f8 Qg2d5 Kf8g8 Ke2d3 Rb1b4 Qd5c5
8 00:00 29.926 383.665 +0.08 Ke1e2 Ke8f8 Qg2d5 Kf8g8 Qd5g5+ Kg8h7 Qg5e5 Rb1f1 Ke2d3 Rf1f4
8 00:00 29.926 383.665 +0.08 Ke1e2 Ke8f8 Qg2d5 Kf8g8 Qd5g5+ Kg8h7 Qg5e5 Rb1f1 Ke2d3 Rf1f4
9 00:00 50.043 458.268 +0.07 Ke1e2 Ke8f8 Qg2d5 Kf8g8 Qd5g5+ Kg8h7 Qg5h4+ Kh7g7 Qh4g4+ Kg7f8 Qg4d4 Kf8g8 Ke2d3
9 00:00 50.043 458.268 +0.07 Ke1e2 Ke8f8 Qg2d5 Kf8g8 Qd5g5+ Kg8h7 Qg5h4+ Kh7g7 Qh4g4+ Kg7f8 Qg4d4 Kf8g8 Ke2d3
10 00:00 123.799 466.812 +0.07 Ke1e2 Ke8f8 Qg2d5 Rb1f1 Ke2d3 Rf1f5 Qd5d4 Kf8g8 Kd3c2 Rf5b5 Qd4e4
10 00:00 123.799 466.812 +0.07 Ke1e2 Ke8f8 Qg2d5 Rb1f1 Ke2d3 Rf1f5 Qd5d4 Kf8g8 Kd3c2 Rf5b5 Qd4e4
11 00:00 244.797 506.196 +0.06 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8d4 Rb6d6 Qd4c5 Rf7f6 Ke2d1 Ke7f8 Kd1c2
11 00:00 244.797 506.196 +0.06 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8d4 Rb6d6 Qd4c5 Rf7f6 Ke2d1 Ke7f8 Kd1c2
12 00:00 372.430 518.992 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Qc3c5+ Rf7e7+ Ke2d3 Rb6f6 Kd3c2 Kf8f7 Qc5d5+ Kf7g7 Qd5c4
12 00:00 372.430 518.992 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Qc3c5+ Rf7e7+ Ke2d3 Rb6f6 Kd3c2 Kf8f7 Qc5d5+ Kf7g7 Qd5c4
13 00:01 504.340 521.442 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Kd1c2 Kf8g7 Qc3g3+ Rf6g6 Qg3e5+ Rf7f6 Qe5d4
13 00:01 504.340 521.442 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Kd1c2 Kf8g7 Qc3g3+ Rf6g6 Qg3e5+ Rf7f6 Qe5d4
14 00:01 758.570 534.353 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3a3+ Kf8g7 Qa3g3+ Rf6g6 Qg3e5+ Rg6f6 d2d4 Kg7f8 Kd1c2 Rf6f1
14 00:01 758.570 534.353 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3a3+ Kf8g7 Qa3g3+ Rf6g6 Qg3e5+ Rg6f6 d2d4 Kg7f8 Kd1c2 Rf6f1
15 00:02 1.309.882 541.720 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g1+ Kg7h7 Qg1d4
15 00:02 1.309.882 538.247 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g1+ Kg7h7 Qg1d4
16 00:04 2.199.003 518.240 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g1+ Kg7h7 Qg1d4 Kh7g7 Qd4g4+ Kg7f8
16 00:04 2.199.003 518.240 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g1+ Kg7h7 Qg1d4 Kh7g7 Qd4g4+ Kg7f8
17 00:06 3.161.019 530.442 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g1+ Kg7h7 Qg1d4 Kh7g7 Qd4g4+ Kg7f8
17 00:06 3.161.019 530.442 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g1+ Kg7h7 Qg1d4 Kh7g7 Qd4g4+ Kg7f8
18 00:10 5.458.408 541.636 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g5+ Kg7f8 Qg5d5 Kf8g7 Qd5e5 Rf7b7 Qe5c5
18 00:10 5.458.408 541.636 +0.05 Ke1e2 Ke8f8 Qg2h2 Rb1b6 Qh2h8+ Kf8e7 Qh8c3 Ke7f8 Ke2d1 Rb6f6 Qc3c8+ Kf8g7 Qc8c5 Rf7d7 Kd1c2 Rd7f7 Qc5g5+ Kg7f8 Qg5d5 Kf8g7 Qd5e5 Rf7b7 Qe5c5
19 00:24 13.576.984 549.443 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1f1 Qh2d6 Kf8f7 Qd6d5+ Re7e6 Kd3c2 Rf1f6 Qd5h5+ Kf7g7 Qh5g4+ Kg7f8 Qg4b4+ Re6e7 Qb4c5 Kf8f7 Qc5c4+ Kf7g7 Qc4c8 Re7f7 Qc8g4+ Kg7f8
19 00:24 13.576.984 549.443 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1f1 Qh2d6 Kf8f7 Qd6d5+ Re7e6 Kd3c2 Rf1f6 Qd5h5+ Kf7g7 Qh5g4+ Kg7f8 Qg4b4+ Re6e7 Qb4c5 Kf8f7 Qc5c4+ Kf7g7 Qc4c8 Re7f7 Qc8g4+ Kg7f8
20 00:35 19.625.224 558.873 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8b2 Rc1c6 Qb2b7+ Kd7d6 Qb7b8+ Kd6d7 Qb8a7+ Rc6c7 Qa7d4+ Kd7c8 Qd4h8+ Kc8b7 Qh8b2+ Kb7c8 Qb2f6 Re7d7+ Kd3e2 Rc7c2 Qf6a6+ Kc8c7 Qa6a5+ Kc7c8 Qa5a8+ Kc8c7
20 00:35 19.625.224 558.873 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8b2 Rc1c6 Qb2b7+ Kd7d6 Qb7b8+ Kd6d7 Qb8a7+ Rc6c7 Qa7d4+ Kd7c8 Qd4h8+ Kc8b7 Qh8b2+ Kb7c8 Qb2f6 Re7d7+ Kd3e2 Rc7c2 Qf6a6+ Kc8c7 Qa6a5+ Kc7c8 Qa5a8+ Kc8c7
21 00:54 30.537.120 566.079 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8b7+ Rc1c7 Qb7d5+ Kd7c8 Qd5f5+ Kc8b7 Qf5b5+ Kb7c8 Qb5b6 Rc7d7+ Kd3c2 Rd7b7 Qb6g6 Re7d7 Kc2d1 Rb7c7 Qg6f6 Rd7e7 Qf6a6+ Kc8b8 Qa6b5+ Kb8c8
21 00:54 30.537.120 565.916 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8b7+ Rc1c7 Qb7d5+ Kd7c8 Qd5f5+ Kc8b7 Qf5b5+ Kb7c8 Qb5b6 Rc7d7+ Kd3c2 Rd7b7 Qb6g6 Re7d7 Kc2d1 Rb7c7 Qg6f6 Rd7e7 Qf6a6+ Kc8b8 Qa6b5+ Kb8c8
22 01:23 47.270.442 565.245 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8a7+ Kd7d8 Qa7a8+ Rc1c8 Qa8a5+ Rc8c7 Qa5b6 Re7d7+ Kd3e2 Rd7e7+ Ke2d1 Kd8c8 Qb6f6 Kc8b7 Qf6b2+ Kb7a7 Qb2a3+ Ka7b8 Qa3b3+ Rc7b7 Qb3d5 Rb7d7 Qd5a5
22 01:23 47.270.442 565.245 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8a7+ Kd7d8 Qa7a8+ Rc1c8 Qa8a5+ Rc8c7 Qa5b6 Re7d7+ Kd3e2 Rd7e7+ Ke2d1 Kd8c8 Qb6f6 Kc8b7 Qf6b2+ Kb7a7 Qb2a3+ Ka7b8 Qa3b3+ Rc7b7 Qb3d5 Rb7d7 Qd5a5
23 02:25 83.027.682 570.498 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8a7+ Kd7d8 Qa7a8+ Rc1c8 Qa8a5+ Rc8c7 Qa5b6 Re7d7+ Kd3e2 Rd7e7+ Ke2d1 Kd8c8 Qb6f6 Kc8b7 Qf6b2+ Kb7a7 Qb2a3+ Ka7b8 Qa3b3+ Rc7b7 Qb3g8+ Kb8a7 Qg8d8 Re7c7 Kd1e2
23 02:25 83.027.682 570.494 +0.05 Ke1e2 Ke8f8 Qg2h2 Rf7e7+ Ke2d3 Rb1c1 Qh2f4+ Kf8e8 Qf4b8+ Ke8d7 Qb8a7+ Kd7d8 Qa7a8+ Rc1c8 Qa8a5+ Rc8c7 Qa5b6 Re7d7+ Kd3e2 Rd7e7+ Ke2d1 Kd8c8 Qb6f6 Kc8b7 Qf6b2+ Kb7a7 Qb2a3+ Ka7b8 Qa3b3+ Rc7b7 Qb3g8+ Kb8a7 Qg8d8 Re7c7 Kd1e2
Here is another interesting position where the Queen is not as strong as two rook
[D]3k4/2rr4/8/8/8/8/8/3KQ3 w - - 0 1