I forgot to provide the formula that gives the probability that a person won on merit assumed by Quadratic Weighting. I'm rectifying that in this post.

First of all, let p_win be the probability that a person wins against the opponent according to their ratings and deviations. This is provided by Glickman. For posterity, it is repeated here. Given Player 1's Rating R_1 and Deviation RD_1 and Player 2's Rating R_2 and Deviation R_2, the expected probability that Player 1 beats Player 2 is:

Code:

P_Win = 1 / (1 + 10^(((R_2 - R_1) / (400 * sqrt(1 + C * (RD_1^2 + RD_2^2))))))
where C = 3 * ln(10)^2 / (400 * pi)^2 (approximately 0.0000100724)

Now let W be equal to 1 if Player 1 won against Player 2, and 0 if Player 1 lost against Player 2, and let X be the difference between W and P_Win. That is:

The nearer X is to 0, the more accurate was Glickman's prediction of Player 1's result against Player 2. On the other hand, the nearer X is to 1, the less accurate was Glickman's prediction. Hence, the nearer X is to 0, the less there is a chance that Player 1 won due to luck. Quadratic Weighting assumes that the chance that Player 1 won on merit (i.e. not due to luck)is:

Code:

P_Merit = 1 - (1-p) * X * (1+2*X)

Note that p in the above equation is the constant we're looking for. It is actually the probability that a player wins on merit **against another equally skilled player**. In this particular scenario, i.e. when two equally-skilled players play each other, X would be equal to 0.5 whether or not the players win or lose. In fact, when X = 0.5, P_Merit = p.

By comparison, Linear Weighting's formula was

Code:

P_Merit = 1 - (1-p) * X * 2

i.e. it replaces the (1+2*X) for Quadratic Weighting with 2. Here, it is also the case that when X = 0.5, P_Merit = p.

Constant Weighting's formula was simply P_Merit = p. Here it is assumed that the probability that Player 1 won on merit against Player 2 is constant, no matter what the relative skill of both players is.

As a last note, **the important thing, for those that want to find a value of p using empirical means, is that you play against people of roughly your same skill, otherwise the value of p that you find won't be good.**