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.
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.