---
abstract: 'We propose the PeerRank method for peer assessment. This constructs a grade for an agent based on the grades proposed by the agents evaluating the agent. Since the grade of an agent is a measure of their ability to grade correctly, the PeerRank method weights grades by the grades of the grading agent. The PeerRank method also provides an incentive for agents to grade correctly. As the grades of an agent depend on the grades of the grading agents, and as these grades themselves depend on the grades of other agents, we define the PeerRank method by a fixed point equation similar to the PageRank method for ranking web-pages. We identify some formal properties of the PeerRank method (for example, it satisfies axioms of unanimity, no dummy, no discrimination and symmetry), discuss some examples, compare with related work and evaluate the performance on some synthetic data. Our results show considerable promise, reducing the error in grade predictions by a factor of 2 or more in many cases over the natural baseline of averaging peer grades.'
author:
- Toby Walsh
bibliography:
- '/Users/twalsh/Documents/biblio/a-z.bib'
- '/Users/twalsh/Documents/biblio/a-z2.bib'
- '/Users/twalsh/Documents/biblio/pub.bib'
- '/Users/twalsh/Documents/biblio/pub2.bib'
title: 'The [PeerRank]{} Method for Peer Assessment'
---

INTRODUCTION
============

We consider how to combine together peer assessments of some work to construct an overall evaluation of this work. An important application of our proposed framework is to evaluation in massive open online courses (MOOCs). In such a setting, it may be impractical to offer anything but automated marking (where this is possible) or peer assessment (e.g. for essays where this might not be possible). Another application of the proposed framework is to peer assessment of grant applications. Often there is only a small pool of experts who are capable of reviewing grant applications in a particular sub-area. In many cases, these people have also submitted grant applications themselves. It is natural therefore to consider designing a mechanism in which those people submitting proposals also review them.

Unfortunately, peer assessment suffers from several fundamental problems. First, how can we provide an incentive to agents to assess their peers well? Second, as peers may have different expertise, how do we compensate for any unintentional biases that peer assessment may introduce? Third, as peers may not be disinterested in the outcome, how do we compensate for any intentional biases that peer assessment may introduce? In this paper, we view this as a mechanism design problem in which we look to provide incentives for peers to assess well, as well as a means to try to compensate for any biases.

Our main contribution is to propose the PeerRank method for peer assessment. This constructs a grade for an agent based on the grades proposed by the agents evaluating the agent. The PeerRank method makes two basic assumptions about how peer grades should be combined. First, it supposes that the grade of an agent is a measure of their ability to grade correctly. Hence, grades are weighted by the grades of grading agents. Second, agents should be rewarded for grading correctly. This gives agents an incentive to provide accurate peer assessments. We identify some formal properties of the PeerRank method. We also evaluate the performance on some synthetic data. As our method favours consensus, it is most suited to domains where there are objective answers but the number of agents is too large for anything but peer grading.

We hope that this work will encourage others to consider peer assessment from a similar (social choice) perspective. There are other axiomatic properties we could formalise and study. For instance, the PeerRank rule is not monotonic. Increasing the grade for an agent can hurt an agent if they thereby receive a bigger proportion of their support from agents that grade poorly. On the other hand, the PeerRank rule likely satisfies a more complex form of monotonicity, in which reducing the error in the grade of an agent only ever helps that agent. We expect there are important axiomatic results to be obtained about peer assessment. Finally, an interesting extension would be to return a distribution or interval of grades, reflecting the uncertainty in the estimate.

PEER RANK RULE
==============

We suppose there are $m$ agents, and agent $j$ provides a grade $A_{i,j}$ for the exam of agent $i$. Grades are normalised into the interval $[0,1]$. We suppose agents grade their own work but this can be relaxed. In addition, as we show in the experimental section, the proposed PeerRank rule is relatively insensitive to any bias that an agent might have towards grading their own work or that of other agents. The grade of each agent is constructed from the grades of the agents evaluating the agent. Since the grade is a measure of their ability to grade correctly, we weight the grade an agent gives 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 grade, this choice is not very critical and we will typically reach the same fixed point with other initial seeds. Similarly, the choice of the exact value of $\alpha$ is not critical and largely affects the speed of convergence. This is because the fixed point is an eigenvector of the grade matrix $A$.

The PeerRank rule returns grades that are an eigenvector of the grade matrix $A$.

[[**Proof:  **]{}\
]{}In matrix notation, at the fixed point, we have: $$X = (1-\alpha)X + \frac{\alpha}{|X|} A.X$$ That is, $$X = X - \alpha X + \frac{\alpha}{|X|} A.X$$ Rearranging and cancelling gives: $$\frac{\alpha}{|X|} A.X = \alpha X$$ Dividing by $\alpha$ and letting $\lambda = |X|$, we get: $$A.X = \lambda X$$

SOME EXAMPLES
=============

To illustrate how the PeerRank rule works on some simple cases, we consider a few examples.

Unanimous grade matrix {#unanimous-grade-matrix .unnumbered}
----------------------

Suppose that every entry in the grade matrix $A$ is the grade $k$ with $0 \leq k \leq 1$. Now an eigenvector of $A$, and the PeerRank solution assigns each agent with this grade $k$. The weighted average of identical grades is always the same whatever the weights. This is what we might expect. The grade matrix tells us nothing more than this.

Identity grade matrix {#identity-grade-matrix .unnumbered}
---------------------

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 an eigenvector of $A$, and the PeerRank solution assigns each agent with the average grade $\frac{1}{m}$. Again, this is what we might expect. The grade matrix tells us nothing more than all agents are symmetric, and so dividing the mark between them might seem reasonable.

Bivalent grade matrices {#bivalent-grade-matrices .unnumbered}
-----------------------

Suppose that agents partition into two types: good and bad. The good agents give a grade of 1 to other good agents, and 0 to bad agents. The bad agents gives a grade of 1 to every agent. In each iteration of the PeerRank method, the grades of the good agents remain unchanged at 1. On the other hand, the grades of the bad agents monotonically decrease towards their fixed point at 0. We also get the same fixed point if the bad agents give a grade of 0 to every agent besides themselves (irrespective of the grade that they give themselves). Again, this is what we might expect. The PeerRank method identifies the good and bad agents, and rewards them appropriately.

PROPERTIES
==========

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 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 appeal to unanimity.

The PeerRank rules also satisfies a no dummy axiom since every agent has some influence over the final grade.

There exist two grade matrices which differ in just the grades assigned by one agent for which PeerRank returns different final grades.

[[**Proof:  **]{}\
]{}Consider the grade matrix in which every agent gives the maximum grade of 1 to every other agent, and the grade matrix which is identical except agent $i$ gives every agent the minimum grade of 0. Then PeerRank gives a grade of 1 to agent $i$ in the first case and 0 in the second. Hence $i$ is not a dummy.

The PeerRank rules also satisfies a simple symmetry axiom.

If we swap the grades of two agents and the grades that the two agents are given then the PeerRank rule swaps the grades assigned to the two agents.

It is also interesting to identify properties that the PeerRank rule does not have. For example, it is not impartial. Your grades of others do affect your own final grade. As a second example, it is not anonymous. It does matter who gives you a grade. It is better to get a good grade from an agent who themself receives good grades than from an agent who themself receives poor grades.

GENERALIZED PEERRANK
====================

The PeerRank rule proposed so far does not incentivize agents to evaluate other agents or even themselves accurately. We therefore add an additional term to provide such an incentive. Suppose $\alpha$ and $\beta$ are parameters with $\alpha + \beta \leq 1$. Then we define the generalised PeerRank rule recursively by the following equation: $$\begin{aligned}
X^{n+1}_i & = & (1-\alpha-\beta) . X^n_i + \frac{\alpha}{\sum_j X^n_j} . \sum_j X^n_j . A_{i,j} + \\
 &  &  \ \ \frac{\beta}{m} . \sum_j 1-|A_{j,i}-X^n_j| \end{aligned}$$ This degenerates to the earlier form of the rule when $\beta=0$. The new term measures the normalised absolute error in the grades given by an agent. This is similar to the reward given in the recent mechanism for reviewing NSF proposals in the SSS program [@nsf1]. The agent “receives” a credit towards their grade of $\beta$ times this normalised error.

If $A_{j,i} = X^n_j$ for all $j$ then the grades assigned by an agent are exact and we add $\beta$ to their score. If $|A_{j,i} - X^n_j|=1$ for all $j$ then the grades assigned by an agent are completely wrong (either the agent gives a grade of 1 when it should be 0 or vice versa). In this case, their grade is reduced by a factor $\beta$ for evaluating incorrectly.

The generalised PeerRank rule continues to satisfy the domain, no discrimination, no dummy, and symmetry properties. For no discrimination, and no dummy, we can just set $\beta=0$ and appeal to the previous results. For the domain property, we need to prove afresh that the additional term cannot take us outside the interval $[0,1]$.

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

[[**Proof:  **]{}\
]{}Clearly $X^n_i \geq 0$ for all $n$ as it is the sum of terms which are not negative. We prove that $X^n_i \leq 1$ by induction on $n$ 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 $$\begin{aligned}
X^{n+1}_i & = & (1-\alpha-\beta)(1-\epsilon) + \frac{\alpha}{\sum_j X^n_j}
\sum_j X^n_j . A_{i,j} + \\
& & \ \ \ \ \frac{\beta}{m}. \sum_j 1 - |A_{j,i} -X^n_j| \\
& \leq & 1 -\alpha - \beta - \epsilon (1-\alpha - \beta) +
\frac{\alpha}{\sum_j X^n_j} \sum_j X^n_j  +  \\
& & \ \ \ \ \ \frac{\beta}{m} \sum_j 1 \\
& \leq & 1 -\alpha - \beta - \epsilon (1-\alpha - \beta) + \alpha + \beta \\
& \leq & 1 - \epsilon (1 - (\alpha + \beta))\end{aligned}$$ 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}
----------------------

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}
---------------------

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.

Bivalent grade matrices {#bivalent-grade-matrices-1 .unnumbered}
-----------------------

Suppose that agents partition into two types: good and bad. The good agents give a grade of 1 to the good agents, and 0 to the bad agents. The bad agents gives a grade of 1 to every agent. Now the generalised PeerRank method give the good agents a grade less than or equal to 1, and the bad agents a grade more than or equal to 0. The bad agents get some credit for grading the good agents (semi-)accurately. This means that the grade of the bad agents by the good agents was a little too harsh, and their own grade suffers.

EXPERIMENTAL EVALUATION
=======================

We tested the performance of the generalised PeerRank rule on some synthetic data. In all experiments, we set $\alpha=\beta=0.1$. Results are, however, relatively insensitive to the actual choice of $\alpha$ or $\beta$. Based on the promise shown in these experiments, we are currently preparing a real world test with undergraduate students. Our typical experimental setup is 10 agents who give an integer mark to each other of between 0 and 10, and an actual mark of between 0 and 100. Therefore a simple baseline against which we compare is the sum of these peer graded marks (or equivalently the average of the normalised peer grade). We denote this as the AVERAGE rule.

We studied a number of different distributions of marks amongst the agents (e.g. binomial, normal, uniform). These are discussed in more detail in the next section. We also need a marking model to determine how well the grading agents grade. We used a simple model based on each mark being awarded independently with a probability given by the grade of the grading agent. In our experiments, this means that the agents are effectively answering 10 questions, that the probability of each of these questions being answered correctly is their actual grade, and that the probability of each of these questions being graded correctly is the grade of the grading agent. This gives a distribution of marks that is the sum of two binomials.

For instance, if the actual mark of an agent is 62 out of 100, then we expect their peer grade to be (on average) 6 out of 10. Suppose their work is marked by an agent whose actual mark is 72 out of 100. On the 6 questions that the agent is expected to get right, we suppose that each is marked correctly by this peer with probability 0.72. This gives a binomial distribution of 6 marks with a probability of 0.72. On the 4 questions that they got wrong, we suppose also that they are marked correctly by this peer (as false) with probability 0.72, and incorrectly (as true) with probability 1-0.72. This gives again a binomial distribution of 4 marks with a probability of 1-0.72. Hence, the final mark given to the agent by their peer is the sum of these two binomial distributions: $bin(6,0.72)+bin(4,1-0.72)$ where $bin(m,p)$ is a binomial distribution of $m$ trials with probability $p$.

We tried other marking models including normally distributed peer grades with a standard deviation that is inversely proportional to the grade of the marking agent, and uniform distributed peer grades with a range that is also inversely proportional to the grade of the marking agent. As we obtained similar results with these other marking models, we focus here on our simple sum of binomials model.

Mark distributions
------------------

We begin with a simple binomial distribution of marks. We let the actual mark of the agents be a binomial distribution of 100 trials with a given probability $p$. In Figure 1, we plot the RMSE (root mean square error) of the predicted mark as a percentage of the 100 marks for varying $p$. Thus a RMSE of 5% means that the PeerRank grade is off with a root mean square error of 5 marks (out of the 100 possible marks). If we map back onto grades out of 10, this means we are off by less than half a grade. For $p>0.6$ (in other words, for where the marks are typically above 60 out of 100), the generalised PeerRank method outperforms simply averaging the peer grades. For $p>0.65$, the error is 4% or less. This compares well with the error returned by simply averaging the peer grades (which is mostly above 10% in this region). Note that for PeerRank to get any useful signal out of the data, we need $p>0.5$. At $p=0.5$, we will often answer (or mark) an exam just as well by tossing a coin. With the PeerRank method, we need the exam to be informative (that is, to have $p > 0.6$), to be able to extract much information from the grade matrix.

![Performance of the generalised PeerRank method on marks coming from a binomial distribution with parameter $p$.](varpn1000m10)

We next turned to a normal distribution of marks. This permits us to study the impact of increasing the standard deviation in marks. With the previous binomial distribution of marks, the standard deviation is $\sqrt{100p(1-p)}$ which is fixed by $p$. In Figure 2, we plot the error in the predicted mark for varying standard deviations. The mean grade is fixed at 70 marks out of 100. We again see that the generalised PeerRank method outperforms simply averaging peer grades except when there is a very large standard deviation in marks.

![Performance of the generalised PeerRank method on marks from a normal distribution with a mean of 70 as we vary the standard deviation.](varnormn1000m10mean70)

Finally, we consider a simple uniform distribution of marks. We suppose that every mark from $lo$ to 100 is equally likely.

![Performance of the generalised PeerRank method on marks coming from the uniform distribution $[lo,100]$ for varying lower bound $lo$. ](varunilon1000m10)

In Figure 3, we plot the error in the predicted grade whilst we vary $lo$, the lowest possible mark. From $lo>20$, the generalised PeerRank method outperforms simply averaging the peer grades. For $lo \geq 50$, the error is less than 10%. As with binomially distributed marks, the exam needs to be informative (that is, for marks to be above 50), to be able to extract information from the grade matrix.

Group size
----------

So far, we have supposed that there are 10 agents who grade each other. We next consider the impact that the size of this group has on the accuracy of the peer grades. We therefore ran an experiment in which we varied the number of agents peer marking. We again use a binomial distribution of marks with a mean of 70. With 5 or more agents, the error of the generalised PeerRank method was less than 5% and was half or less of that of simply averaging the peer grades. With 10 to 20 agents, the error of the generalised PeerRank method was less than a third of that of simply averaging the peer grades. These results suggest that the PeerRank method does not need many peer grades in order to obtain an accurate result. Ideally, we need around 10 grades for each agent, but even with just 5 grades, we are often able to obtain acceptable results.

Biased marks
------------

Peer grades may be systematically biased. For instance, students may collude and agree to grade each other generously. Even if there is no explicit collusion, there are studies which suggest that students grade each other generously (e.g. [@kpdjeb95]). To study this, we inflate or deflate the mean of the peer grades by a factor $r$. For instance, if $r=1.1$ then the mean peer mark is increased by 10%. On the other hand, if $r=0.9$ then the mean peer mark is decreased by 10%. We again use a binomial distribution of actual marks with a mean of 70.

![Performance of the generalised PeerRank method on marks coming from a binomial distribution with a mean of 70 (i.e. $p=0.7$) as we vary the bias in the peer marking. For $r>1$, peers return inflated marks. For $r<1$, peers return deflated marks. ](varerrorn1000p07)

In Figure 4, we plot the RMSE of the predicted grade again as a percentage of the 100 marks whilst we vary the bias in peer grades. For $0.75 \leq r \leq 1.25$, the error of the generalised PeerRank method is 5% or less of the total marks. That is, we are able to tolerate a bias of 25% in peer grades without significantly increasing the error. These results suggest that the generalised PeerRank method has some robustness against bias. The minimum in errors for averaging peer grades at around $r=1.5$ is likely an artefact of the model. Averaging peer grades tend to under-estimate the actual grade. Therefore a positive bias on the peer grades tends to reduce this.

RELATED WORK
============

There is a large literature on peer assessment but the focus is mostly on pedagogical aspects of peer assessment (for example, how peer assessment itself contributes to the learning experience). There is less literature on how best to combine peer assessments together. Many of the peer assessments systems being used in practice today often have simple and rather ad hoc mechanisms for combining together peer assessments. In addition to multiple choice questions and computer grading, peer assessment has been used on a number of Coursera courses. Students first train with a grading rubric. To get feedback on their own work, a student has to grade five essays. The student then receives peer grades from five other students. Using machine learning algorithms, Piech [*et al.*]{} have estimated and corrected for grader biases and reliabilities [@edm2013]. They demonstrate significant improvement in grading accuracy on real world data of over 63,000 peer grades. Their models are probabilistic so give a belief distribution over grades (as opposed to a single score) for each student.

PEAS is a new peer evaluation extension for the EdX open source MOOC platform [@peas]. Students are incentivized to grade accurately by a calibration method that constructs an incentive score based on the accuracy of their grading. To improve review quality, students are divided into groups based on this incentive score, and each assignment is peer graded by one student from each group. A simple normalisation of grades is also performed to reduce bias in peer grading. Expert grading can be used to resolve discrepancies in peer grades, and to provide training data for Machine Learning algorithms to grade automatically. An important difference with our work is that the calibration in PEAS is just once, whilst PeerRank potentially uses multiple rounds of adjustment of grades.

One of the closest works to ours is a peer reviewing mechanism being piloted by the National Science Foundation (NSF) for the Signal and Sensing Systems (SSS) program. The mechanism is designed to help deal with an increase in proposals which is putting an increasing stress on the grant reviewing process. This increase in proposals has led to a degradation in the quality of reviews, as well as a shrinking pool of qualified but non-conflicted reviewers. The NSF has therefore decided to pilot a mechanism for peer review that is adapted from one first proposed by Merrifield and Saari [@nsf1]. To incentivize applicants to review well, and to deter strategic ranking, reviewers receive additional score for reviewing well which can increase them a maximum of 2 places in the final ranked list. This mechanism is somewhat different to ours as the NSF mechanism ranks proposals, whilst our mechanism returns a grade. In our mechanism, final grades returned may not totally rank the proposals. Another work which is close to ours is the CrowdGrader mechanism for peer evaluation from UC Santa Cruz [@crowdgrader]. CrowdGrader lets students submit and collaboratively grade solutions to homework assignments. The tool permitted both ranking and grading of homework. However, de Alfaro and Shavlovsky found that students much preferred to grade than to rank. They expressed uneasiness in ranking their peers, perceiving ranking as a blunt tool compared to grading. At the heart of CrowdGrader is the [*Vancouver*]{} algorithm for combining peer grades. There are two significant differences between the Vancouver algorithm and the PeerRank rule. First, the Vancouver algorithm measures variance in grades whilst PeerRank (like the NSF rule) measures the absolute deviation. A rule based on variance will tend to penalise inexperienced agents more greatly. Second, the reward term in the Vancouver algorithm is added after a fixed point is reached, whilst in the PeerRank rule, it is part of the fixed point calculation. We conjecture that it is more robust to include it in the fixed point calculation when there is significant variation in the accuracy of grades assigned by a single agent. There are a number of closely related problems to ours in the social choice literature. For example, Holzman and Moulin have studied a related problem in which a set of agents wish to select one amongst them to receive a prize [@hmeconometrica2013]. A fundamental assumption of this work is that nominations are impartial: your message never influences whether you win the prize. In our setting, such an assumption is less appropriate. We want your evaluation of the work of another agent to influence your evaluation. There are several reasons behind this change. First, your ability to evaluate the work of other agents measures in part your command of the subject being examined. Second, you will be incentivized to grade accurately by a better final grade. If your message cannot influence your evaluation, then you have no incentive to provide good evaluations. For this reason, even if we extend the sort of methods proposed by Holzman and Moulin to the task of ranking, the starting assumptions are very different. Another related problem is “selection from the selectors” [@selectingselectors]. The goal here is to select a subset of $k$ agents from a group (e.g. to select a subcommittee). The problem of awarding a prize from a group of peers can be seen as the special case of $k=1$. As approval voting is not impartial, Alon, Fischer, Procaccia and Tennenholtz look for impartial rules that approximate approval voting (that is, guarantee that the total approval scores of the $k$ winners are within a fixed fraction of the optimal answer). Again, a difference with this work is that we are not trying to achieve or approximate impartiality.

A closely related problem is the division of cash between a group of partners [@dividedollar]. Each partner cares selfishly about their share but is supposed to be disinterested about the distribution of the money that he or she does not get. Partners rate the relative contributions of the other partners. With four or more partners, there exist symmetric and impartial division rules. By comparison, whilst our PeerRank rule is symmetric, it is not designed to be impartial. The grades assigned by an agent can definitely influence their final grade.

Conclusions
===========

We have proposed the PeerRank method for peer assessment. The PeerRank method weights grades by the grades of the grading agents. In addition, it rewards agents for grading well and penalises those that grade poorly. As the grades of an agent depend on the grades of the grading agents, and as these grades themselves depend on the grades of other agents, the PeerRank method is defined by a fixed point equation similar to the PageRank method. We have identified some formal properties of the PeerRank method, discussed some examples, and evaluated the performance on some synthetic data. The method reduces the error in grade predictions by a factor of 2 or more in many cases over the natural baseline of simply averaging peer grades. As the method favours consensus, it is most suited to domains where there are objective answers but the number of agents is too large for anything but peer grading.

There are many possible extensions. For example, we might consider peer assessments where agents only grade a subset of each other. The PeerRank rule lifts easily to this case. As a second example, we might permit external calibration by having some agents graded externally. As a third example, we might consider peer assessment when agents order rather than grade. They might be willing to say “agent $a$ should be graded higher than agent $b$”, or “agent $a$ should receive a similar grade as agent $b$” but the grading agents might be less willing to give an absolute grade without seeing the work of all other agents. Another interesting direction would be to return a distribution or interval of grades, reflecting the uncertainty in the estimate. This could be calculated based on the intermediate grades seen before the fixed point is reached.
