THE TRADE-DOWN BONUS

 

In the Camelot middlegame, the trading down of material to reach a winning endgame is of prime importance.  That trade-down action can be quantified in the following manner.

 

The Knight-Man formula (discussed here) calculates the value of a Knight (N) as:

 

Let: Man Value = 100

Let: User-defined value of N at start of game = 2.25

P = Total uncastled friendly Pieces

 

N = 100 + 125 ln(P) / ln(14)

 

The Trade-Down Bonus formula calculates the value of the bonus (B) as:

 

Let: D = Difference in Material (White minus Black)

Let: T = Total Material (White plus Black)

 

B = 1.125 D - .00025 DT

 

This Bonus, expressed in numerical terms, can then be directly compared to material value.  Following are the Bonus values for all 1,430 combinations of Knights and Men.

 

(WN = Number of White Knights)

(WM = Number of White Men)

(BN = Number of Black Knights)

(BM = Number of Black Men)

(B = Trade-Down Bonus)

 

WN WM BN BM B
3 7 2 9 0.01571095
4 6 2 10 0.34775532
4 7 3 9 0.42240121
1 9 2 7 0.52391611
1 10 3 6 0.70523783
0 9 1 7 1.01740377
3 8 4 6 1.81792284
4 8 3 10 1.84168878
0 10 2 6 1.88529977
3 8 2 10 2.16567816
2 8 1 10 2.35188853
0 6 3 1 2.48786951
1 8 0 10 2.54111384
2 8 3 6 3.05712636
4 1 0 8 3.56576575
2 1 0 4 3.76283863
1 8 2 6 4.42641360
2 9 4 5 5.04280598
3 7 4 5 5.05851692
0 8 1 6 5.69294600
2 7 3 5 7.27038177
1 9 3 5 7.79429788
4 0 2 3 8.11047545
1 3 3 0 8.54638508
4 1 1 6 9.25871175
3 6 4 4 9.54930828
1 7 2 5 9.61603283
1 10 4 4 10.25454611
0 9 2 5 10.63343660
0 7 1 5 11.78501400
2 8 4 4 12.60643464
2 6 3 4 12.93617203
0 10 3 4 14.82147180
3 5 4 3 15.65336361
1 6 2 4 16.47831978
3 1 1 4 17.26223878
1 8 3 4 17.36258563
1 5 4 0 17.51036065
0 6 1 4 19.75010829
2 5 3 3 20.52531874
3 2 0 7 22.03277458
0 8 2 4 22.17126579
2 7 4 3 22.92374538
4 10 4 9 23.20842293
1 9 4 3 23.44766149
3 4 4 2 23.92601089
1 5 2 3 25.62083610
4 1 2 4 25.73703153
4 2 0 9 26.25251731
2 2 0 5 27.16386131
4 2 1 7 27.26992108
0 7 4 0 29.29537465
4 9 4 8 30.03636840
1 7 3 3 30.14135158
0 5 1 3 30.34018218
2 4 3 2 30.79595963
0 9 3 3 31.15875535
4 9 3 10 31.87805718
2 0 1 1 32.84355631
1 1 0 2 33.38249989
0 3 2 0 33.77394381
3 2 1 5 33.81778858
3 3 4 1 35.27547890
3 10 4 7 35.28499330
3 10 3 9 35.70739452
2 6 4 2 36.86218292
4 2 2 5 36.88595391
4 8 4 7 37.12668208
0 7 2 3 37.40585010
4 8 3 9 37.54908329
1 4 2 2 38.08603039
0 10 4 2 38.74748269
3 3 0 8 38.84124465
0 5 3 0 38.88656727
4 3 1 8 39.20183458
1 8 4 2 41.28859653
4 3 0 10 41.74294841
3 9 3 8 42.30988438
2 3 0 6 42.59414990
4 7 3 8 42.73228560
4 3 2 6 43.62824818
3 9 4 6 44.12780723
3 9 2 10 44.47556254
3 3 1 6 44.53419066
4 7 4 6 44.55020844
4 7 2 10 44.89796376
0 4 1 2 44.94163895
2 3 3 1 45.08201941
4 4 1 9 46.63682169
2 10 3 7 47.00079386
2 10 2 9 47.01650481
4 4 2 7 47.16073779
1 6 3 2 47.27427942
4 6 3 7 47.34854918
3 0 2 1 47.35059411
4 6 2 9 47.36426012
4 5 2 8 48.23738583
2 1 1 2 48.70447758
3 8 3 7 49.16647202
3 8 2 9 49.18218296
1 2 0 3 50.05836105
3 4 0 9 50.17852820
4 5 1 10 50.58927435
4 0 0 6 50.70462535
3 0 0 4 51.11343273
3 4 1 7 51.19593197
4 5 3 6 51.29451219
3 2 4 0 51.32814923
2 10 4 5 52.05931078
4 6 4 5 52.40706610
2 4 0 7 52.82873421
0 8 3 2 52.96722542
4 0 3 1 53.19249486
4 10 4 8 53.24479133
2 9 2 8 53.28019180
3 7 2 8 53.29590275
3 8 4 5 54.22498894
4 4 3 5 54.43111956
3 5 1 8 54.85519818
4 10 3 10 55.08648011
3 1 2 2 55.34826918
2 9 1 10 55.63208033
3 7 1 10 55.64779128
2 5 4 1 55.80079765
1 3 2 1 55.89697919
3 6 1 9 56.18612996
2 9 3 6 56.33731816
3 7 3 6 56.35302911
4 1 3 2 56.53299117
4 3 3 4 56.56442021
3 6 2 7 56.71004607
1 10 1 9 56.89136780
3 5 0 10 57.39631202
4 2 3 3 57.41127266
1 10 2 7 57.41528391
2 2 1 3 57.50404350
0 6 2 2 57.83613869
2 8 1 9 59.24325632
3 5 2 6 59.28161179
2 5 0 8 59.36656340
3 2 2 3 59.43862468
1 3 0 4 59.65981782
2 8 2 7 59.76717243
3 4 2 5 60.81196481
4 5 4 4 60.84382046
3 3 2 4 61.01251044
2 7 1 8 62.12557995
2 3 1 4 62.34425819
1 9 1 8 62.64949606
2 6 0 9 63.11470023
3 6 3 5 63.98042784
2 6 1 7 64.13210401
2 4 1 5 64.61374822
2 7 0 10 64.66669379
1 10 3 5 64.68566568
0 10 0 9 65.00000000
2 5 1 6 65.05950940
1 9 0 10 65.19060990
1 4 0 5 65.24989171
1 7 4 1 65.41683048
2 9 4 4 65.88662644
3 7 4 4 65.90233739
0 10 1 7 66.01740377
2 2 3 0 66.05042858
2 0 0 2 66.22605619
0 9 4 1 66.43423425
2 7 2 6 66.55199356
0 3 1 1 66.61750011
2 8 3 5 67.03755420
1 9 2 6 67.07590966
4 9 4 7 67.16305048
1 8 0 9 67.54111384
4 9 3 9 67.58545170
1 5 0 6 68.21498600
1 8 1 7 68.55851761
1 7 0 8 68.98259623
1 6 0 7 69.30705400
0 9 0 8 70.00000000
4 4 4 3 70.08448317
4 0 1 4 70.45473364
1 5 3 1 70.70285551
3 5 3 4 72.21778382
2 6 2 5 73.74813684
1 7 1 6 74.67554223
0 8 0 7 75.00000000
0 10 2 5 75.63343660
0 9 1 6 75.69294600
3 10 3 8 78.01727890
1 8 2 5 78.17455044
4 1 0 7 78.56576575
2 7 3 4 79.48816559
3 6 4 3 79.63379145
3 10 4 6 79.83520174
4 8 3 8 79.85896768
0 7 0 6 80.00000000
1 9 3 4 80.01208170
3 10 2 10 80.18295706
1 10 4 3 80.33902928
4 3 4 2 80.49043110
1 6 1 5 81.09206800
3 4 3 3 81.33728355
2 5 2 4 81.53782918
4 8 4 6 81.67689052
4 8 2 10 82.02464584
2 4 4 0 82.12410887
0 7 3 1 82.48786951
3 1 0 5 82.51213049
2 8 4 3 82.69091781
1 2 2 0 83.83230486
0 6 0 5 85.00000000
0 5 2 1 86.23716137
0 8 1 5 86.78501400
1 5 1 4 87.96509429
0 5 0 4 90.00000000
2 4 2 3 90.23458432
4 1 1 5 90.35077975
4 10 4 7 90.37147341
4 10 3 9 90.79387463
1 7 2 4 91.15386201
3 9 3 7 91.47635640
3 9 2 9 91.49206735
3 3 3 2 91.80847007
4 7 3 7 91.89875762
4 7 2 9 91.91446856
0 9 2 4 92.17126579
4 2 4 1 92.68675156
2 6 3 3 94.27345558
0 4 0 3 95.00000000
1 4 1 3 95.59007389
3 0 1 2 96.05507168
3 5 4 2 96.14379471
0 10 3 3 96.15875535
4 2 0 8 96.25251731
3 9 4 5 96.53487332
4 7 4 5 96.95727454
1 6 4 0 98.60242865
1 8 3 3 98.69986918
2 1 0 3 98.76283863
0 7 1 4 99.75010829
0 3 0 2 100.00000000
2 10 2 8 100.29669661
2 3 2 2 100.43028859
4 6 2 8 100.64445193
4 2 1 6 101.94546331
3 2 0 6 102.03277458
3 8 2 8 102.46237477
2 10 1 10 102.64858514
4 6 1 10 102.99634045
2 10 3 6 103.35382297
2 7 4 2 103.41417648
4 6 3 6 103.70157829
1 9 4 2 103.93809259
1 4 3 0 104.13645897
0 8 4 0 104.29537465
3 2 3 1 104.52064409
1 3 1 2 104.60145677
3 8 1 10 104.81426329
0 2 1 0 105.00000000
0 2 0 1 105.00000000
3 8 3 6 105.51950113
1 6 2 3 106.71290410
4 3 0 9 106.74294841
4 5 1 9 107.48064215
4 3 1 7 107.76035218
4 1 4 0 107.86114040
4 5 2 7 108.00455826
4 0 2 2 108.54076403
4 4 1 8 109.28631775
4 9 3 8 109.89533608
4 9 4 6 111.71325892
4 4 0 10 111.82743158
4 9 2 10 112.06101424
2 5 3 2 112.33378882
0 8 2 3 112.40585010
2 9 1 9 112.52344813
3 7 1 9 112.53915907
3 1 1 3 112.85231267
2 10 4 4 112.90313125
2 9 2 7 113.04736424
3 7 2 7 113.06307518
4 6 4 4 113.25088656
2 2 2 1 113.40102269
4 4 2 6 113.71273135
3 3 0 7 113.84124465
3 8 4 4 115.06880940
4 5 3 5 115.27494003
0 6 1 3 115.34018218
4 1 2 3 115.97161585
3 4 4 1 116.61276245
1 2 1 1 116.67586116
2 2 0 4 117.16386131
4 3 2 5 117.37638502
4 2 2 4 118.42378310
3 6 1 8 118.83562602
1 10 1 8 119.54086386
3 4 0 8 120.17852820
2 9 3 5 120.31774601
3 7 3 5 120.33345695
3 6 0 10 121.37673986
3 1 3 0 121.39869776
3 2 1 4 121.78288287
2 8 1 8 121.89275238
1 7 3 2 121.94982165
1 10 0 10 122.08197770
3 5 0 9 122.39631202
0 9 3 2 122.96722542
3 6 2 6 123.26203963
3 5 1 7 123.41371579
0 6 3 0 123.88656727
1 10 2 6 123.96727746
2 8 0 10 124.43386622
3 3 1 5 125.62625865
3 4 1 6 125.87147421
1 5 2 2 126.05112468
2 8 2 6 126.31916599
4 4 3 4 126.64890338
3 10 3 7 127.18375092
3 10 2 9 127.19946187
2 3 0 5 127.59414990
0 4 2 0 128.77394381
4 8 3 7 129.02543970
4 8 2 9 129.04115064
2 6 4 1 129.54893449
2 7 0 9 129.66669379
1 9 0 9 130.19060990
2 7 1 7 130.68409756
4 5 4 3 130.92830364
1 9 1 7 131.20801367
0 10 4 1 131.43423425
3 10 4 5 132.24226784
2 1 2 0 132.53678243
2 4 0 6 132.82873421
3 5 2 5 133.02974862
4 10 3 8 133.10375901
2 6 0 8 133.11470023
1 8 4 1 133.97534809
4 8 4 5 134.08395662
2 5 0 7 134.36656340
4 10 4 6 134.92168185
0 5 1 2 134.94163895
0 10 0 8 135.00000000
4 10 2 10 135.26943717
2 4 3 1 135.31660372
4 0 0 5 135.70462535
2 9 4 3 135.97110961
3 7 4 3 135.98682056
3 6 3 4 136.19821166
1 10 3 4 136.90344949
1 8 0 8 137.54111384
0 7 2 2 137.83613869
4 3 3 3 137.90170376
1 1 1 0 138.38249989
1 1 0 1 138.38249989
2 6 1 6 138.80764624
2 8 3 4 139.25533802
2 7 2 5 140.30013039
0 10 1 6 140.69294600
1 9 2 5 140.82404650
3 4 2 4 142.34979399
3 3 4 0 143.13661931
1 8 1 6 143.23405984
1 7 0 7 143.98259623
3 9 2 8 144.77225915
0 9 0 7 145.00000000
4 7 2 8 145.19466037
3 0 0 3 146.11343273
2 5 1 5 146.15157740
3 9 1 10 147.12414768
4 7 1 10 147.54654889
3 9 3 6 147.82938551
4 7 3 6 148.25178673
4 2 3 2 149.21974273
1 6 0 6 149.30705400
1 2 0 2 150.05836105
4 4 4 2 150.57491427
3 3 2 3 151.24709475
1 4 2 1 151.48705308
1 6 3 1 151.79492351
2 4 1 4 152.57884251
1 5 0 5 153.21498600
3 5 3 3 153.55506737
1 3 0 3 154.65981782
0 8 0 6 155.00000000
1 4 0 4 155.24989171
2 6 2 4 155.28596602
1 7 1 5 155.76761023
0 9 1 5 156.78501400
0 10 2 4 157.17126579
3 9 4 4 157.37869379
0 8 3 1 157.48786951
4 7 4 4 157.80109500
2 3 1 3 157.93433208
4 1 0 6 158.56576575
4 9 3 7 159.06180810
4 9 2 9 159.07751905
2 10 1 9 159.53995293
1 8 2 4 159.71237962
3 2 2 2 159.86891327
4 6 1 9 159.88770825
2 10 2 7 160.06386904
3 6 4 2 160.12422255
4 6 2 7 160.41162436
2 7 3 3 160.82544914
1 10 4 2 160.82946039
4 1 3 1 161.05363526
1 9 3 3 161.34936525
0 4 1 1 161.61750011
3 8 1 9 161.70563109
2 2 1 2 162.10550027
3 8 2 7 162.22954720
2 8 4 2 163.18134891
2 5 4 0 163.66193805
4 9 4 5 164.12032502
0 7 0 5 165.00000000
2 1 1 1 165.38033874
4 0 1 3 166.04480753
2 3 3 0 166.48071717
2 10 3 5 167.33425081
4 6 3 5 167.68200613
3 1 2 1 168.74929186
1 6 1 4 169.05716229
3 8 3 5 169.49992897
4 5 1 8 170.13013821
2 0 1 0 171.22605619
2 0 0 1 171.22605619
0 6 2 1 171.23716137
4 2 0 7 171.25251731
2 5 2 3 171.77241350
3 1 0 4 172.51213049
4 5 0 10 172.67125205
3 4 3 2 173.14575362
4 3 4 1 173.17718266
1 7 4 0 173.27797088
0 9 4 0 174.29537465
4 5 2 6 174.55655181
4 0 3 0 174.59119261
0 8 1 4 174.75010829
0 6 0 4 175.00000000
2 9 1 8 175.17294419
3 7 1 8 175.18865514
4 3 0 8 176.74294841
4 4 0 9 176.82743158
2 9 0 10 177.71405802
3 7 0 10 177.72976897
4 4 1 7 177.84483536
4 1 1 4 178.31587404
2 9 2 6 179.59935779
3 7 2 6 179.61506874
3 0 2 0 179.88737654
3 10 2 8 180.47965367
1 7 2 3 181.38844633
4 10 3 7 182.27023103
4 10 2 9 182.28594198
4 8 2 8 182.32134245
0 9 2 3 182.40585010
4 3 1 6 182.43589442
3 10 1 10 182.83154220
2 10 4 3 182.98761442
4 2 1 5 183.03753131
4 6 4 3 183.33536974
3 10 3 6 183.53678003
1 5 1 3 183.55516818
4 8 1 10 184.67323097
0 5 0 3 185.00000000
3 8 4 3 185.15329258
4 8 3 6 185.37846881
2 6 3 2 186.08192565
3 6 0 9 186.37673986
3 2 0 5 187.03277458
1 10 0 9 187.08197770
4 10 4 5 187.32874795
3 6 1 7 187.39414363
4 4 2 5 187.46086819
4 5 3 4 187.49272385
0 10 3 2 187.96722542
1 10 1 7 188.09938147
1 3 2 0 188.43376162
3 5 4 1 188.83054627
2 8 0 9 189.43386622
2 8 1 7 190.45126999
1 8 3 2 190.50833926
2 4 2 2 190.66487290
1 5 3 0 192.10155326
3 5 0 8 192.39631202
2 9 3 4 192.53552982
3 7 3 4 192.55124077
3 10 4 4 193.08608831
3 3 0 6 193.84124465
4 8 4 4 194.92777708
0 4 0 2 195.00000000
3 4 0 7 195.17852820
0 7 1 3 195.34018218
2 7 4 1 196.10092804
3 3 3 1 196.32911416
1 9 4 1 196.62484415
3 6 2 5 197.01017646
1 10 2 5 197.71541430
3 5 1 6 198.08925802
2 1 0 2 198.76283863
4 3 2 4 198.91421420
2 7 0 8 199.66669379
2 8 2 5 200.06730282
1 9 0 8 200.19060990
1 4 1 2 200.19153066
4 2 4 0 200.54789196
0 7 3 0 203.88656727
3 9 1 9 204.01551548
4 7 1 9 204.43791669
3 9 2 7 204.53943158
4 7 2 7 204.96183280
0 3 1 0 205.00000000
0 3 0 1 205.00000000
2 7 1 6 205.35963979
1 9 1 6 205.88355590
3 4 1 5 206.96354220
1 6 2 2 207.14319268
4 4 3 3 207.98618693
2 6 0 7 208.11470023
4 2 2 3 208.65836741
0 10 0 7 210.00000000
4 5 4 2 211.41873474
3 9 3 5 211.80981335
2 2 0 3 212.16386131
4 7 3 5 212.23221457
4 9 2 8 212.35771085
1 8 0 7 212.54111384
3 0 1 1 212.73093285
0 8 2 2 212.83613869
3 3 1 4 213.59135295
2 3 2 1 213.83131127
2 5 0 6 214.36656340
3 5 2 4 214.56757781
4 9 1 10 214.70959938
4 9 3 6 215.41483721
4 1 2 2 216.40190443
2 9 4 2 216.46154072
3 7 4 2 216.47725166
2 5 3 1 216.85443291
3 2 1 3 217.37295676
3 1 1 2 217.45376944
3 6 3 3 217.53549521
2 3 0 4 217.59414990
2 4 0 5 217.82873421
1 10 3 3 218.24073304
0 5 2 0 218.77394381
2 6 1 5 219.89971424
0 6 1 2 219.94163895
2 8 3 3 220.59262157
1 3 1 1 221.27731793
0 10 1 5 221.78501400
2 7 2 4 221.83795958
4 0 2 1 221.94178672
2 10 1 8 222.18944900
1 9 2 4 222.36187568
4 6 1 8 222.53720431
1 7 0 6 223.98259623
1 8 1 5 224.32612784
3 8 1 8 224.35512715
3 4 4 0 224.47390286
2 10 0 10 224.73056283
4 9 4 4 224.96414549
0 9 0 6 225.00000000
4 6 0 10 225.07831815
4 0 0 4 225.70462535
3 2 3 0 225.91934185
1 7 3 1 226.47046574
2 10 2 6 226.61586260
3 8 0 10 226.89624099
4 6 2 6 226.96361791
3 9 4 3 227.46317696
0 9 3 1 227.48786951
4 7 4 3 227.88557818
3 8 2 6 228.78154075
4 3 3 2 229.71017383
3 4 2 3 232.58437830
2 5 1 4 234.11667169
1 6 0 5 234.30705400
4 10 2 8 235.56613378
2 6 4 0 237.41007489
4 5 0 9 237.67125205
4 10 1 10 237.91802230
4 10 3 6 238.62326014
4 5 1 7 238.68865582
0 10 4 0 239.29537465
1 5 2 1 239.45214737
2 10 3 4 239.55203463
3 10 1 9 239.72290999
4 6 3 4 239.89978995
0 8 0 5 240.00000000
3 10 2 7 240.24682610
4 8 1 9 241.56459877
3 8 3 4 241.71771279
1 8 4 0 241.83648849
4 8 2 7 242.08851488
2 9 0 9 242.71405802
3 7 0 9 242.72976897
1 5 0 4 243.21498600
4 4 4 1 243.26166584
4 1 0 5 243.56576575
2 9 1 7 243.73146180
1 7 1 4 243.73270452
3 7 1 7 243.74717274
0 9 1 4 244.75010829
3 5 3 2 245.36353744
2 6 2 3 245.52055034
2 2 2 0 245.93780512
3 0 0 2 246.11343273
4 4 0 8 246.82743158
0 10 2 3 247.40585010
3 10 3 5 247.51720787
2 4 1 3 248.16891640
4 10 4 4 248.17256842
4 5 2 5 248.30468865
4 8 3 5 249.35889665
1 8 2 3 249.94696394
1 4 0 3 250.24989171
0 7 2 1 251.23716137
4 2 0 6 251.25251731
0 5 1 1 251.61750011
3 3 2 2 251.67738334
4 3 0 7 251.74294841
4 4 1 6 252.52037759
2 7 3 2 252.63391921
3 6 4 1 252.81097411
1 9 3 2 253.15783532
2 9 2 5 253.34749463