that these bounds are reachable. In particular, if all peer grades are 0 (1) then the PeerRank rule gives every agent this grade.

Next we argue that if all agents give an agent the same grade then this is their final grade.

If all agents give an agent the grade $k$ then the PeerRank rule gives this grade $k$ to the agent.

[[**Proof:  **]{}\
]{}Suppose all agents give agent $i$ the grade $k$. Consider the $i$th component of the fix point equation: $$X_i = (1-\alpha).X_i + \frac{\alpha}{\sum_j X_j} \sum_j X_j . A_{i,j}$$ Rearranging gives: $$\alpha X_i = \frac{\alpha}{\sum_j X_j} \sum_j X_j . A_{i,j}$$ Dividing by $\alpha$ and multiplying up the fraction gives: $$\begin{aligned}
(\sum_j X_j) . X_i & = & \sum_j X_j . A_{i,j} \\
& = & (\sum_j X_j) . k\end{aligned}$$ Dividing by the common term, $\sum_j X_j$, we get: $X_i = k$.

The PeerRank rules also satisfies a no discrimination axiom. Every vector of grades is possible.

Given any vector of grades, there exists a grade matrix with which the PeerRank rule returns this vector.

[[**Proof:  **]{}\
]{}Suppose we want agent $i$ to get the grade $k_i$. Then we construct the grade matrix with $A_{i,j} = k_i$ and 