The PeerRank method identifies the good and bad agents, and rewards them appropriately.

PROPERTIES
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The PeerRank rule has a number of desirable (axiomatic) properties. Several of these properties (e.g. no dummy and no discrimination) are properties that have been studied by in peer selection of a prize [@hmeconometrica2013]. First, we argue that the PeerRank rule returns a normalised grade.

The PeerRank rule returns grades in $[0,1]$.

[[**Proof:  **]{}\
]{}Clearly $X^n_i \geq 0$ for all $n$ as it is the sum of two terms which are never negative. We prove that $X^n_i \leq 1$ by induction on $n$ In the base case, $X^0_i \leq 1$ as it is the average of terms which are themselves less than or equal to 1. In the step case, suppose $0 \leq X^n_i \leq 1$ for all $i$. Let $X^n_i = 1 - \epsilon$ where $0 \leq \epsilon \leq 1$. Then $$\begin{aligned}
X^{n+1}_i & = & (1-\alpha)(1-\epsilon) + \frac{\alpha}{\sum_j X^n_j}
\sum_j X^n_j . A_{i,j}  \\
& \leq & 1 -\alpha - \epsilon (1-\alpha) +
\frac{\alpha}{\sum_j X^n_j} \sum_j X^n_j  \\
& = & 1 -\alpha - \epsilon (1-\alpha) + \alpha \\
& = & 1 - \epsilon (1 - \alpha) \ \leq 1
$$ Note 