another agent by their own grade. The grade of an agent is thus the weighted average of the grades of the agents evaluating the agent. Now the grades of the agents evaluating an agent are themselves the weighted averages of the grades of the agents evaluating the agents. Hence we set up a set of equations and look for the fixed point of the system. This is reminiscent of the problem faced by the PageRank algorithm [@pagerank]. In PageRank, web-pages are ranked according to the ranks of the web-pages that link to them, these ranks depend on the ranks of the web-pages that like to them, and so on.

Let $X^n_{i}$ be the grade of agent $i$ in the $n$th iteration of the PeerRank rule and $0 < \alpha < 1$. We define the grades at each iteration as follows: $$\begin{aligned}
X^0_i & = & \frac{1}{m} \sum_j A_{i,j} \\
X^{n+1}_i & = & (1-\alpha) . X^n_i + \frac{\alpha}{\sum_j X^n_j} \sum_j  X^n_j . A_{i,j}\end{aligned}$$ The last term is merely the average grade of an agent weighted by the current grades. The PeerRank grades are the fixed point of these set of equations. Note that whilst we start with the (unweighted) average 