Convergence of Recursively Calculated Ratings

from a Single Round Robin (Table 1)

               Iteration                 A                                  B                                 C                           D

Linear Formula [4.1],  K = 1,  values initialized at .5

                       5                0.751029                      0.751029                     -0.00205761             0.5
                       6                0.749657                      0.749657                      0.000685871            0.5
                       7                0.750114                      0.750114                     -0.000228624           0.5
                       8                0.749962                      0.749962                      7.62079e-005           0.5
                       9                0.750013                      0.750013                     -2.54026e-005          0.5
                     10                0.749996                      0.749996                      8.46754e-006           0.5
                     11                0.750001                      0.750001                     -2.82251e-006           0.5
                     12                0.75                              0.75                              9.40838e-007           0.5
                     13                0.75                              0.75                             -3.13613e-007          0.5
                     14                0.75                              0.75                              1.04538e-007           0.5
                     15                0.75                              0.75                             -3.48459e-008          0.5

Berkin Formula [5.6],  values initialized at .5

                     50                 0.913513                       0.783039                     0.130499                 0.651468
                     60                 0.913294                       0.782406                     0.130387                 0.652222
                     70                 0.912974                       0.782585                     0.130433                 0.652239
                     80                 0.913025                       0.782631                     0.13044                   0.652163
                     90                 0.913052                       0.782609                     0.130435                 0.652168
                   100                 0.913045                       0.782606                     0.130434                 0.652175
                   110                 0.913043                       0.782609                     0.130435                 0.652174
                   120                 0.913043                       0.782609                     0.130435                 0.652174
                   130                 0.913044                       0.782609                     0.130435                 0.652174
                   140                 0.913043                       0.782609                     0.130435                 0.652174
                   150                 0.913043                       0.782609                     0.130435                 0.652174

Elo Rating Formula [5.4],  values initialized at 2200

                    100               1328.48                         1328.48                        1028.48                   1238.17
                    110               1231.57                         1231.57                          931.567                 1141.26
                    120               1134.66                         1134.66                          834.657                 1044.35
                    130               1037.75                         1037.75                          737.747                   947.438
                    140                 940.837                         940.837                        640.837                   850.528
                    150                 843.927                         843.927                        543.927                   753.618
                    160                 747.017                         747.017                        447.017                   656.708
                    170                 650.107                         650.107                        350.107                   559.798
                    180                 553.197                         553.197                        253.197                   462.888
                    190                 456.287                         456.287                        156.287                   365.978
                    200                 359.377                         359.377                          59.377                   269.068

Ratio Rating Formula [5.2], values initialized at 1

                    10                   7.80455                          7.80455                        1.15829                  4.76605

                    20                   38.3958                          38.3958                        5.69836                  23.4469

                    30                   188.893                        188.893                        28.0339                  115.35

                    40                   929.287                        929.287                      137.917                    567.483

                    50                 4571.75                        4571.75                        678.499                  2791.81

                    60               22491.4                        22491.4                        3337.97                  13734.7

                    70             110649                         110649                         16421.6                    67569.7

                    80             544355                         544355                         80788.4                  332418