After a game, the new rating for a player is updated according to the formula:
r' = r + K · (s - E[S])
...where r is the elo rating, K is a "constant" factor, s is the score (i.e. 1 for win, 0 for loss, 0.5 for draw), and E[S] is the expected score (e.g. Magnus Carlsen has E[S] = 0.9999999 against Average Joe).
Player A and player B have a total score of s_A + s_B = 1
, since either exactly one player wins, or both players draw. By the proof given on Wikipedia, player A's expected score E[S_A]
and player B's expected score E[S_B]
also sum to 1, i.e., E[S_A] + E[S_B] = 1
.
Thus, if K = K_A = K_B
, the total new rating of the players after the game is:
r_A' + r_B' = r_A + r_B + K · ((s_A + s_B) - (E[S_A] + E[S_B]))
= r_A + r_B + K · ((1) - (1))
= r_A + r_B
That is, the total rating is conserved after a game iff K = K_A = K_B
.
In Elo's original work, K = K_A = K_B = 10
is symmetric, but more modern rules may sometimes use asymmetric factors for K_A, K_B
depending on the situation.