The goal of paired comparison statistics is to deduce a ranking from an uneven matrix of observed results, from which the contestants can be sorted from best to worst. In the knowledge that crushing all the complexities of the situation into just one number is a large simplification, one wishes to have the best one-dimensional explanation of the data.

Each game between two players (*P _{i}, P_{j}*) can be thought
of as a random experiment where there is a probability

where *w _{ij}* is the number of wins by player

We wish to assign a *relative strength* (RS) parameter
to each of the players involved in a tournament, where
implies that player *P _{i}* is better than player

A probability function *F* such that *F*(0)=0.5 and
*F*(*x*)=1-*F*(-*x*)(for all )
is chosen arbitrarily; following [73] we
use the logistic function

The model describes the probabilities *p _{ij}* as a function of
the RS parameter
for each player:

so the outcome of a game is a probabilistic function of the difference between both opponent's strengths. The conditions imposed on

The observed data is a long sequence of games between opponent pairs, each one
a either a win or a loss. According to eq. 3.7, the probability of
that particular sequence was

for any choice of _{i}'s.The set of _{i}'s
that best explains the observations is thus the one that maximizes this probability.
The well known method of maximum likelihood can be applied to find the maximum
for eq. 3.8, generating a large set of implicit simultaneous equations
on
that are solved by the Newton-Raphson
algorithm.

An important consideration is, the _{i}'s are not the
true indeterminates, for the equations involve only paired differences,
.
One point has to be chosen arbitrarily to be the zero of the RS scale.

A similar method permits assigning a rating to the performance of any smaller
sample of observations (one player for example): fixing all the _{i}'s
on equation (3.8), except one, we obtain

where is the only unknown -- all the other values are known. The single indeterminate can be found with identical procedure.

A player's history of games is a vector
of win/loss
results, obtained against opponents with known RS's
,
respectively. Eq. (3.9) can be solved iteratively, using a ``sliding
window'' of size * n<N,* to obtain strength estimates for
,
then for
,
and so on. Each successive value
of
estimates the strength with respect to the games contained
in the window only.

With this window method we can do two important things: analyze the changing performance of a single player over time, and, putting the games of a group of players together into a single indeterminate, observe their combined ranking as it changes over time.

Altogether, the paired comparisons model yields:

- A performance scale that we have called Relative Strength (
**RS**). The zero of the scale is set arbitrarily (to the one of a fixed sample player: agent 460003). - An ordering of the entire set of players in terms of proficiency at the game, as given by the RS's.
- An estimation, for each possible game between two arbitrary players, of the win-lose probability (eq. 3.7). With it, an estimation of exactly how much better or worse one is, as compared to the other.
- A way to measure performance of individuals or groups over time.
- A possible fitness measure: the better ranked players can be chosen to survive.