2. Nonstrategic author model

We will begin with a simple model of the journal-selection process. There is a universe of papers that will submit to a single journal. We’ll assume that each paper has a quality, $q$ , that is represented by a real number in $\left[ {0,1} \right]$ . We remain agnostic about what this quality represents. It could represent something intrinsic to the paper, such as the epistemic quality of the work. It might also be a judgment about something extrinsic to the paper, such as the number of citations the paper will receive. In an empirical field, it might be the probability that the paper replicates. In a theoretical field, it might represent the importance of the theoretical advance. The only significant assumption is that quality can be represented on a single dimension.

Although it won’t matter until section 3, we also assume that the author is aware of the quality of her paper: she knows how good her paper is. This is an idealizing assumption for the purposes of this article, but we expect our results could be generalized to any setting where the author knows substantially more about their paper’s quality than the journal. ^{Footnote 3}

For simplicity, we will assume that there is a paper for every real number in $\left[ {0,1} \right]$ . This represents a setting where the papers are uniformly distributed over that range. In our first model, we will assume that every paper is submitted to the journal regardless of any decision made by the journal. Hence, the authors are behaving nonstrategically.

Journals make two decisions. First, they decide on a quality threshold, denoted by ${Q_T}$ , which is the minimally acceptable paper. A journal can say, “We’re only going to accept papers that are in the best 10% of the field” ( ${Q_T} = 0.9$ ), or “in the best 1%” ( ${Q_T} = 0.99$ ), or “only the very best paper” ( $Q_T = 1$ ). At the other extreme, they might say, “We’ll publish anything” ( ${Q_T} = 0$ ). ^{Footnote 4} The second decision is the quality of peer review, which determines the probability that any particular quality paper is accepted or rejected, given the journal’s quality threshold.

The peer-review process is modeled as a noisy quality-estimation process. That is, the peer-review process returns a number that represents the estimated quality of the paper. The estimated quality is $q + e$ , where $e$ is normally distributed with mean 0 and variance $\varepsilon $ . We model the decision to adopt a given quality of peer review as the decision to set $\varepsilon $ to a particular value: high $\varepsilon $ entails large errors and thus poor-quality review, and vice versa. ^{Footnote 5}

If the paper’s apparent quality $q + e$ is higher than or equal to the journal’s threshold, ${Q_T}$ , the journal publishes the paper. If the paper’s apparent quality is below the threshold, then the journal rejects. In this idealized scientific world, there is no process to revise and resubmit. ^{Footnote 6}

To sum up, the journal makes two decisions: First, what’s the threshold? And second, how much effort will we put into peer review?

We compare two different ways that one can incentivize a journal. We might judge a journal by its quality. In this case, we will consider a journal that wants to maximize the average quality of papers published in the journal (call this the quality-incentivized journal). We will represent the average quality of papers published in the journal as $\bar q$ . The other way journals are incentivized is by selectivity. We represent a journal’s rejection rate as $r$ , and in our second type of model, we assume journals strive to maximize this (call this the selectivity-incentivized journal). ^{Footnote 7}

Figure 1 shows two panels representing equivalent choices of ${Q_T}$ and $\varepsilon $ from the journal’s perspective. The horizontal axis is the announced quality threshold, ${Q_T}$ . The vertical axis here relates to the reviewing quality, $\varepsilon $ , which is the variance of the journal’s error distribution. At $\varepsilon = 0$ , peer review is perfect. The lines in this space represent lines of equivalent goodness from the perspective of the journal. The right-hand plot is for a journal that cares about its selectivity. From the journal’s perspective, for any target rejection rate, it is indifferent between any points along the relevant line.

Figure 1.

Indifference curves for the journal when choice of ${Q_T}$ and $\varepsilon $ is unconstrained. Within each panel, each line represents choices that yield an equivalent payoff for the journal.

The plot on the left-hand side represents the journal incentivized by the average quality of papers published in the journal. This looks a little bit different, of course, because now it’s not concerned about how many papers it’s rejecting. It’s concerned about the quality of the papers it’s accepting. Both panels illustrate one important point: that journals can achieve equivalent results, from either perspective, with several different pairs of choices. They might achieve the same payoff with very good peer review and a slightly lower ${Q_T}$ or by having bad peer review and a different ${Q_T}$ . This basic point will underwrite much of what is to come.

So far, there is no cost to any choice the journal makes, so in both cases, the overall optimal behavior for the journal is to choose ${Q_T} = 1$ (only accept the very best paper) and $\varepsilon = 0$ (have perfect peer review). This will result in a journal that has an average quality of $1$ and a rejection rate of $1$ , both the highest values possible.

In reality, however, there is a cost for high-quality peer review. Identifying appropriate reviewers, soliciting reviews for a paper, and reading and evaluating the quality of the reviews take time and effort. Some journals even pay their reviewers. All of these things are costly. In order to account for this, we must include a cost for the quality of peer review. If we introduce this cost, we get an interestingly different result.

Our two utility functions are as follows:

(1) $$\matrix{ {{u_R}\left( {{Q_T},\varepsilon } \right)} \hfill\ { = r - {c_J}\left( \varepsilon \right)} \hfill \cr {{u_Q}\left( {{Q_T},\varepsilon } \right)} \hfill\ { = \bar q - {c_J}\left( \varepsilon \right),} \hfill \cr } $$

where ${u_R}$ represents the utility for the selectivity-incentivized journal, and ${u_Q}$ represents the utility for the quality-incentivized journal. The function ${c_J}\left( \cdot \right)$ is arbitrary and represents the cost of improving peer review. We will assume that ${c_J}$ is decreasing in $\varepsilon $ , meaning that the cost increases as peer review gets better.

With that relatively weak assumption, we can prove that a journal incentivized by its rejection rate will want to set ${Q_T} = 1$ (this is proven in the appendix). The journal will set the threshold for publication at the highest value it can. It does this because changing ${Q_T}$ is free, whereas improving peer review is not.

Once the journal sets ${Q_T}$ at $1$ , it will then optimize $\varepsilon $ to balance the costs and benefits from improving peer review. This will depend on the functional form of ${c_J}\left( \cdot \right)$ , but in many plausible cases, it will result in an intermediate value of $\varepsilon $ that represents neither perfect nor completely unreliable peer review. ^{Footnote 8}

Interpreting these results, we get the following conclusions: the journal wants to claim to be the perfect journal. It claims that it will only accept the very best papers, but it won’t go all the way to perfect peer review. It will choose an intermediate quality of peer review that represents the trade-off between the benefits and the cost of its peer-review process.

To say more beyond this observation, we assume a particular functional form for ${c_J}\left( \cdot \right)$ . For the remainder of this article, we assume

$${c_J}\left( \varepsilon \right) = {1 \over {{{(1 + \varepsilon )}^k}}},$$

where $k$ is a parameter that allows us to vary the steepness of the cost function. For “low” values of $k$ , the cost increases steeply as peer review becomes better. For “high” values of $k$ , cost increases less steeply, so good peer review is relatively cheaper.

Figure 2 is the optimal error in peer review, from the journal’s perspective, for various values of $k$ . Very low $k$ values are not plausible because they lead to uninteresting results. ^{Footnote 9} For what we think of as plausible values of $k$ , which are around 6 and higher, a journal incentivized by its quality will do better at peer review than a journal incentivized by its rejection rate. In all subsequent plots, we set $k = 8$ .

Figure 2.

Optimal $\varepsilon $ value, under two incentive regimes, as a function of $k$ . Note: In these calculations, we cap $\varepsilon $ at 1 because that is already a very low reliability of peer review, whereby even the worst paper has approximately a one-in-six chance of being accepted in a journal with the maximum quality threshold.

When a journal is judged by its rejection rate, it is indifferent about which papers it rejects. If it rejects good papers, the journal is rewarded just the same as it is if it rejects bad papers. However, if we judge a journal by its quality, this is not the case: a journal is punished for rejecting good papers and rewarded for rejecting bad ones. As a result, when there is a cost to peer review, journals judged by their quality will dedicate more effort to improving peer review. A quality-incentivized journal better serves the epistemically valuable sorting function than a selectivity-incentivized journal.

However, there is still a significant concern about both the quality-incentivized and selectivity-incentivized journals. Both journals are setting ${Q_T} = 1$ . In effect, the journals are saying, “We are the very best journal. Our quality standards are the highest they could possibly be.” But they choose some intermediate value for the quality of peer review, which creates a gap between the quality of papers that are published in the journal and the announced quality of the journal. The journal, in some sense, claims to be enforcing the highest standards—but the claim is somewhat hollow, given the low investment in peer review. In terms of serving a certification function for the public, it might be more beneficial for a journal to adopt a lower quality threshold with more accurate enforcement.

3. Strategic authors

In the first version of the model, we assumed that all authors submitted their papers regardless of the probability of acceptance. In reality, an author who knows the quality of her own paper might choose not to submit to a journal where she thinks her chances of being accepted are low. In order to accommodate this possibility, we now allow authors to decide whether they will submit to the journal.

In reality, authors may not know ahead of time the precise threshold or quality of peer review of the journal; however, for simplicity, we assume this knowledge is public. Given widespread discussion among academics about journal reputation, as well as citation metrics and published rankings in some fields, authors clearly have some knowledge of journal quality, and that knowledge should be widely shared. We will assume an order of operations as follows: The journal announces its peer-review policy ( $\varepsilon $ ) and its threshold ( ${Q_T}$ ). This is known by the authors. The authors produce a paper and observe its quality, $q$ . For the moment, we will treat a successful publication as worth utility $1$ for all authors. This will be revisited later. Because better papers are more likely to pass peer review, authors of better-quality papers stand to gain more from submitting to a journal, so the expected benefit of submission is increasing with paper quality.

We also assume that rejection comes with a cost (denoted “ ${c_A}$ ” for author cost). That should be no surprise to an academic. Rejection comes with a psychological cost, at least, but also comes with other forms of cost, too. For example, one must write a bespoke cover letter or conform to an arbitrary journal style. There is also opportunity cost. Submitting to one journal forecloses the option to submit to another journal for some period. ^{Footnote 10} If the paper is rejected and then takes longer to come out in another venue, the paper might be scooped by someone else, or the topic may no longer be relevant or exciting. Some of these costs, such as the burden of complying with formatting rules, are present whenever an author submits, whereas the psychological cost of rejection and the opportunity cost of having missed out on time to submit elsewhere arise only when the paper is rejected. We ignore this difference as inconsequential for modeling purposes and treat all costs as arising only when the paper is rejected—but for convenience, we will switch between the terminology of rejection cost and submission cost, depending on the context.

We will assume that the cost depends on (at most) the quality of the underlying paper, and we therefore represent it as a function ${c_A}:\left[ {0,1} \right] \to \mathbb{R}$ . In this article, we will model cost in two ways. One is a constant cost: regardless of the quality of the paper, the cost of rejection is the same. This cost will be represented by ${c_A}\left( q \right) = c$ . Alternatively, the cost might vary with the rejected paper quality; it’s worse to have a paper rejected when the paper is relatively high quality. This cost will be ${c_A}\left( q \right) = c \times q$ .

A constant-cost model reflects many of the frustrating difficulties of the submission process. One must format a paper for the journal, write a cover letter, suggest suitable reviewers and/or editors, and so forth. The variable model of costs, on the other hand, reflects opportunity costs. If an author has a low-quality paper, the opportunity cost of having it rejected is relatively low because if they had taken it to a different journal, it probably would have been rejected there, too. Further, having a bad paper rejected is not as bad as having a good paper rejected because having a good paper rejected will mean that citations and other impacts of the paper will be delayed, whereas a sufficiently bad paper will have a negligible impact whenever it is ultimately published. In reality, publishing features some combination of both kinds of costs. For analytic purposes, we treat them separately.

Given the announced strategy of the journal and the quality of their own papers, the authors know the probability that their papers will be accepted. The authors are also aware of the cost of submission and will choose to submit their papers if the expected utility of doing so is positive. (We normalize not submitting a paper as utility 0.)

In order to understand behavior in this model, we must consider an equilibrium between the authors and the journal. In the previous section, a journal could count on all authors to submit and would simply maximize its expected quality or rejection rate against that. In the strategic-authors model discussed in this section, the journal must anticipate that some authors may not submit. ^{Footnote 11}

This consequence is most obvious in the context of the rejection-rate model. If the selectivity-incentivized journal sets the threshold very high with very high-quality peer review, authors of bad papers may choose not to submit. In such a case, the rejection rate will go down because the journal does not have the opportunity to reject the bad papers. So, the journal might do better by increasing the chance that bad papers are accepted in order to entice the authors to submit. So far, this is just a claim, but when we analyze the model, this is what we find.

3.1. Constant cost

Suppose the setting where ${c_A}\left( q \right) = c$ for all qualities $q$ , so all authors face the same rejection costs. Many of the same basic facts remain from the nonstrategic model. Journals remain incentivized to set a quality threshold of 1, and then they tweak the peer-review quality to alter the results of the process of submission and review. In the nonstrategic model, this was a simple one-party optimization problem: the journal wanted to set the error in peer review to balance the benefits of a more reliable review process, in terms of either quality or rejection rate, against the attendant costs. In the strategic model, the journal must now anticipate the prospect that some authors will not submit.

To understand the behavior of the authors and the journal in equilibrium, we first observe that if any author does not submit, then all authors of worse papers will also not submit because authors of worse papers have even less chance of being accepted and the cost of rejection is constant. This means that self-selection works from the “bottom up” (see the appendix for a proof of this claim; it is illustrated in fig. 3).

Figure 3.

Submission set for a version of the constant-cost/constant-benefit model.

Bottom-up selection has very different effects on the two differently incentivized journals. From the perspective of the quality-incentivized journal, this self-selection is always good. Unless peer review is perfect, there is always a small probability that a bad paper passes peer review. If the bad paper never submits, the probability of that paper being published drops to zero—thus increasing the expected quality of published papers at no cost to the journal. ^{Footnote 12}

Although self-selection is good from the journal’s perspective, it does come with a consequence for the authors. When authors of bad papers are self-selecting out, the journal has less of an incentive to maintain high-quality peer review. The journal no longer has to worry about ensuring that bad papers are not published because the bad papers are not being submitted. As a result, in equilibrium, a journal will have worse peer review than in the situation of the previous section where all authors are submitting. This is illustrated in figure 4 (left plot) by noting that as $c$ increases, the journal chooses a higher $\varepsilon $ , thus resulting in lower-quality peer review.

Figure 4.

Journal strategy and outcomes in equilibrium for a version of the constant-cost/constant-benefit model. (Left) Error rate adopted by the journal as a function of the author’s rejection cost. Higher rejection costs lead to worse peer review (higher error rates). The small dips that occur near $\varepsilon $ =1 are the result of small rounding errors and should not be taken to be meaningful. (Right) Average quality of papers published and journal payoffs as a function of the cost to authors of rejection. For the quality-incentivized journal, published quality and journal payoff are both monotonically increasing with the author’s submission cost. For the selectivity-incentivized journal, quality of published papers exhibits a nonmonotonic relationship with the author’s costs, whereas the journal’s utility is always decreasing in the author’s costs. Observe that the quality of papers published in the selectivity-incentivized journal is never higher than that of the quality-incentivized journal. ( $k = 8$ ).

Although we do not model this choice, if the journals were capable of influencing the cost of submission ( $c$ in our model), they might prefer to increase it in order to encourage self-selection. This illustrates a point made by Tiokhin et al. (Reference Tiokhin, Panchanathan, Lakens, Vazire, Morgan and Zollman2021): that high costs for submissions might be used as a filtering device. Unlike the conclusion discussed in that article, and as we show in figure 4, this then leads to lower-quality reviewing. In turn, the lower-quality review has a negative epistemic consequence, in that the chance of a better paper being rejected while a worse one is accepted increases. Difficult submission processes come to substitute for reviewing quality. However, evaluated altogether, the quality of published papers goes up, indicating that the epistemic benefit of self-selection outweighs the loss from lower-quality peer review (see the right-hand plot in fig. 4, where $\bar q$ increases as $c$ increases for a journal incentivized by quality). ^{Footnote 13}

All this is different for the journal incentivized by rejection rates. For a journal incentivized by selectivity, self-selection is a problem. If a paper is not submitted, it cannot be rejected, and if it cannot be rejected, it doesn’t count toward the journal’s rejection rate. Thus, there is no reason for such journals to prefer a higher submission cost. Indeed, in this variant of our model, where authors self-select from the bottom up, a journal motivated by selectivity would ideally prefer to set the quality of peer review low enough that even authors of the worst papers think it is worth it to submit. Even with very poor reviewing, however, as the cost of submission gets sufficiently high, authors of the worst papers calculate that the chances of acceptance are too low and won’t submit. Hence, the journal’s utility declines with increasing $c$ .

This illustrates two important incentives that lead journals to choose low-quality reviewing. Journals incentivized by quality can use self-selection to take the place of high-quality peer review. Journals incentivized by rejection rate want to keep reviewing sufficiently poor to combat self-selection. Both are epistemically undesirable, but as we show, the latter leads to worse reviewing than the former.

These conclusions can all be seen in figure 4: see the dotted lines (where journals are incentivized by rejection rate). In the left-hand plot, the journal judged by rejection rate maintains a lower quality of review (higher $\varepsilon $ ) than one incentivized by quality. In the right-hand plot, the payoff of the journal incentivized by its rejection rate goes down as the submission cost parameter, $c$ , goes up. This occurs because as cost increases, fewer bad papers are submitted (see fig. 3). and the rejection rate is lower.

What are the epistemic consequences of this state of affairs? In particular, what might the effect be on people who are neither authors nor journal editors and rely on journals as a source of information? If a journal is incentivized by rejection rate and could eliminate the cost to authors, it would lead to a journal that was (a) inferior in quality to a quality-incentivized journal with the same costs (compare the solid and the dashed red lines at $c = 0$ ) and (b) inferior in quality to a journal incentivized by rejection rate but where the cost to authors is significantly higher. This is illustrated by the dashed red line in figure 4, which represents the average quality of papers published in a journal that is incentivized by its rejection rate. ^{Footnote 14}

3.2. Variable costs

The previous model reflects the fixed costs that come with journal rejection. There is a cost to filling out forms and formatting the paper to follow submission guidelines. Beyond the time, there are psychological costs that come with receiving a rejection. It is not a bad first approximation to treat these as constant across paper quality.

Some costs, however, vary with respect to the quality of the paper. A low-quality paper may make little impact even when published, so the cost of a delay is small. A high-quality paper may have a significant risk of being scooped or might lose out on early uptake if publication is delayed. In order to model this, we will consider a second model where ${c_A}\left( q \right) = cq$ for some $c$ .

This immediately changes the dynamics of author behavior. Authors of the lowest-quality papers have no reason not to submit because the cost of rejection is zero. ^{Footnote 15} Consequently, in this model, self-selection proceeds either from the middle out or from the top down (as illustrated in fig. 5).

Figure 5.

Submission set for a version of the variable-cost/constant-benefit model.

It is no longer obvious whether the quality-incentivized journal does better by discouraging submissions. It does eliminate some papers the journal would want to reject, but it also does so by increasing the proportion of submitted papers that are quite bad. The journal now faces a more complicated trade-off.

It remains the case that the quality-incentivized journal will rely on some self-selection. As $c$ increases, all journals will, for the most part, reduce their quality of review (increase their $\varepsilon $ ), relying on self-selection to take the place of quality reviewing. This is illustrated in figure 6. As was the case in the constant-cost/constantbenefit model, the rejection-incentivized journal will choose to have much worse peer review because it wants to incentivize as many paper submissions as possible.

Figure 6.

Journal strategy and outcomes in equilibrium for a version of the variable-cost/constant-benefit model. ( $k = 8$ ).

There is one important difference between this model and the previous one that requires discussion. In the previous model, we showed that the selectivity-incentivized journal always preferred lower submission costs to encourage more submissions. In this version of the model, things are somewhat more complicated. In this version of the model, self-selection does not proceed “from the bottom up,” so a journal incentivized by its rejection rate might not always want to encourage submissions.

If a journal can discourage the best papers from submitting while holding its quality threshold for acceptance constant, it improves the journal’s rejection rate. Consider a journal where only the very worst papers are being submitted: in this case, even with relatively poor peer review, the journal can have a rejection rate near 100%. ^{Footnote 16} If the best paper were then submitted, the journal’s rejection rate might actually go down. So, unlike in the previous version of the model, the journal does not always benefit from encouraging more submissions.

One can see this complicated relationship in the right panel of figure 6, where the cost is related to the journal’s payoff. A journal incentivized by its rejection rate may want to set $c$ either very low or very high. In this particular case, choosing a low $c$ is epistemically superior to choosing a high one because the average quality of published papers is higher in the former case than the latter.

What should we make of these divergent results? As mentioned earlier, we think the real situation features costs of both types: those that are insensitive to the quality of the paper and those that vary with the quality of the underlying paper. However, we think that the journal would, for the most part, have control over those costs that are constant—things like submission charges and formatting requirements.

4. Discussion of assumptions

We have developed a series of models to analyze how well a laissez-faire system will incentivize high-quality peer review. We have identified several impediments that will be summarized in the next section. Before we do that, however, it is important to discuss the ways our model is limited by its assumptions.

First, we have chosen two extreme ways to model the cost of submission to authors: constant across paper quality or variable as an increasing function of paper quality. In reality, there is a complex web of costs in between those two extremes. We believe that our results would be robust to more complicated hybrid cost functions, but this was not tested.

On the journal end, we have treated the journal as incentivized by either the quality of its papers or its rejection rate. It is unlikely that any journal is exclusively incentivized by its rejection rate. However, as noted earlier, there is considerable anecdotal evidence that a high rejection rate serves as a proxy of quality and may sometimes be adopted as desirable in itself.

We have also assumed away any hard constraints, such as page constraints, that some journals face. Many journals could not reject all submitted papers, even if they wanted to, because publishers expect them to publish a certain number of issues each year. Journals also face constraints regarding the maximum number of issues they can publish. We see no reason to believe that our results would be qualitatively different if we introduced such constraints, but this remains untested.

Regarding the process of peer review, our model treats this as a black-box process. We do not model the peer reviewers as influenced by the population of papers that are submitted. Were those agents more sophisticated, their judgments might be more informed, and the model might yield very different conclusions. We have made this assumption quite intentionally: we think the black-box model is a more accurate model of how reviewers typically work rather than modeling them as ideal Bayesian agents. Reviewers, of course, are aware of the general quality of the journal, and they are attempting to make a judgment as to whether a particular paper is good enough for that particular journal—our model is consistent with this. But reviewers generally know very little about the overall population of papers submitted to the journal and very little about other inputs to the editorial decision process.

Perhaps most critically, we assume that there is a single journal that exercises monopoly power. We think this model is appropriate for circumstances where there is a single “top” journal that is regarded as a critical journal for promotion and tenure. Our results might change in a setting where journals compete with one another in order to attract the best papers. ^{Footnote 17}