\frac{\beta}{m} \sum_j 1 \\
& \leq & 1 -\alpha - \beta - \epsilon (1-\alpha - \beta) + \alpha + \beta \\
& \leq & 1 - \epsilon (1 - (\alpha + \beta))\end{aligned}$$ Recall that $\alpha + \beta \leq 1$ and $\epsilon \geq 0$. Thus, $\epsilon (1 - (\alpha + \beta)) \geq 0$. Hence $X^{n+1}_i \leq 1$.

To demonstrate the impact of the new term that encourages accurate peer grading, we consider again the simple grade matrices considered previously.

Unanimous grade matrix {#unanimous-grade-matrix-1 .unnumbered}
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Suppose every entry in the grade matrix $A$ is the grade $k$. Now the generalised PeerRank solution assigns each agent with a grade greater than or equal to $k$ (with equality when $m=1$, $k=1$ or $\beta=0$). Grades increase above $k$ as agents receive some credit for grading accurately.

Identity grade matrix {#identity-grade-matrix-1 .unnumbered}
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Suppose the grade matrix $A$ is the identity matrix. That is, each agent gives themselves the maximum grade 1, and every other agent the minimum grade 0. Now the generalised PeerRank solution assigns each agent a grade greater than or equal to $\frac{1}{m}$ (with equality when $m=1$ or $\beta=0$). Grades are larger than $\frac{1}{m}$ as agents receive credit for grading themselves semi-accurately.

