3.1.1. Model

We use multitype random graphs, networks that are used to model populations with multiple social identity groups (Golub and Jackson Reference Golub and Jackson2012). In this setup, agents are classified according to which type they are (in this case, whether they are a minority or majority group member). Each agent has some probability of forming a link with an agent of the same type, p _in, and some probability of forming a link with an agent of a different type, p _out. If $p_{\mathrm{in}}>p_{\mathrm{out}}$, we say that the agents exhibit homophily.

Once the network is formed, agents update their strategies based on the payoffs they receive by interacting with their collaborators—those they are connected to on the collaboration network. Each agent’s strategy consists of two parts: a demand when interacting with an in-group member and a demand when interacting with an out-group member. These strategies are initially randomly assigned. Each round, agents interact with all of their collaborators and, with a small probability, will decide to update their strategy. Strategies are updated using myopic best response: in the next round, the strategy agents will use is the one that would have gotten them the best payoff in the current round, given the strategies of their collaborators. This captures the fact that agents are trying to choose a strategy that is likely to result in their getting the most out of a successful collaboration while avoiding the poor payoff from a failed collaboration.

3.1.2. Results

We look at the frequencies with which populations converged to different bargaining norms. Cases in which two or fewer agents were playing strategies outside the equilibrium expectation were counted as converged since, on the basis of the probabilistic nature of the model, these agents may not have had a chance to update their strategies. The probability of updating a strategy in any given round was .1. Simulations were run for 1,000 rounds over networks ranging from 20 to 100 agents (in intervals of 20), where the high demand (H) ranged from 6 to 9 and the minority group constituted 10%–50% of the population. While for all simulations the probability of a within-group link was held fixed at $p_{\mathrm{in}}=.4$, the probability of an out-group link p _out ranged from .2 to .8. (That is, we look at a range of cases, from cases in which the minority was twice as likely to collaborate with in-group members to cases in which the minority is twice as likely to collaborate with out-group members.) Each combination of parameter values was run 100 times.

Within each group, populations nearly always evolved to the norm of equal division. Between groups, populations most often evolved to the norm of fair division, but a significant amount of the time they also evolved to a discriminatory norm, as in Poza et al. (Reference Poza, Galán, Santos and López-Paredes2010).

We now look at whether minority status has any effect on the likelihood of being discriminated against. Figure 2 shows results for high demand, H = 6. For a small minority group, it is more likely that the majority group will end up demanding High against the minority. As the size of the minority group increases, the fair division becomes more likely and both groups become equally likely to discriminate.

Figure 2.

Convergence to different norms over size of minority group for $H=6$.

This effect results from an asymmetry in the average number of links for each group. Consider a simple demonstrative example. If there are 10 majority group members, five minority group members, and 10 total between-group links, on average, majority group members will have one between-group link and minority group members will have two. If a minority group member has two links, both of these collaborators would have to be demanding 4 in order for their best response to be 6. At least initially, this would happen with only probability 1/9. On the other hand, only one of these collaborators would have to be demanding 6 in order for the best response to be demand 4, and, at least initially, this happens with probability 5/9.Footnote ¹¹ In contrast, a majority group member having one link to the minority would have a 1/3 chance of each of the strategies being a best response (she demands the complement of whatever her collaborator demands). So, at the start, a minority group member is much more likely than a majority group member to demand 4 and much less likely to demand 6. This asymmetry drives populations toward the outcome in which the majority demands 6.

Our findings are somewhat similar to previous results on the cultural Red King/Queen effect described above. However, minority disadvantage does not arise here due to an increased speed of learning in the minority group, but rather due to differences in the best responses of the group. Furthermore, while we observe minority disadvantage, which is analogous to the cultural Red King effect when H = 6, we do not observe an analogue of the Red Queen for higher levels of H.

To understand why nothing similar to the Red Queen occurs, consider the same simplified example as above, but now with H = 9. Again, majority group members having one link to the minority would have a 1/3 chance of each of the strategies being a best response, as they still care only about demanding the complement of their collaborator. Now, though, the minority group members are very unlikely to make the Low demand of 1; it is a best response only with probability 1/9 when both of their collaborators demand 9. They are just as likely as the majority to demand 9 (doing so with probability 1/3, whenever both their collaborators demand 1 or one collaborator demands 1 while the other demands 9). More importantly, though, it is most likely that the minority group member’s best response will be to demand 5. This will be the best response whenever at least one of their collaborators demands 5, which happens with probability 5/9. So the evolution of strategies is again affected by the minority group’s initial best responses; but in this case it is likely that the frequency of the Med demand in the minority will increase, followed by the majority learning to also make Med demands.

We also found that varying levels of homophily did not influence whether the majority or minority group is more likely to discriminate.Footnote ¹² Since agents are updating their in-group and out-group strategies separately, how the within-group linking probability compares to the out-group linking probability does not have an effect on how often the network converges to a discriminatory norm. So the existence of majority group advantage does not depend on the existence of homophily.

3.2.1. Model

We employ a model similar to that of Watts (Reference Watts, Dutta and Jackson2003) in which agents can choose to form or break links with other agents in the community based on their payoffs from bargaining with those other agents. Each link represents a collaboration and therefore a payoff from the Nash demand game. Since we are investigating the effect of a preexisting discriminatory norm, majority group members receive a payoff of 6 from a between-group link while minority members receive a payoff of 4.Footnote ¹³ Agents receive a payoff of 5 from within-group links, where there is a norm of fair division. Agents have a maximum number of links, capturing the fact that there are a limited number of projects academics can work on. A player can unilaterally sever a link, but both players must consent to a new link being formed. This represents the fact that all the researchers involved in a collaboration must consent to be part of the collaboration. If one person no longer wishes to collaborate, the collaboration fails and the link is broken.Footnote ¹⁴

The evolution of the collaboration network proceeds as follows. We begin with an empty network (there are no links between any nodes). At each time step, two nodes are chosen at random. One of these is an agent who will update her links and the other is a potential or current collaborator of the agent.

If we have chosen a potential collaborator, we determine whether both researchers will consent to form a new link. If neither player has reached the maximum number of links, they will both consent. (Getting some payoff is better than none.) If either, or both, of them already have the maximum number of links, we check whether they will break any of their existing links to form the new link. If players can increase their payoff by breaking their link with the lowest payoff in order to form this new link, they will consent to the new link. If both players consent to the link forming, this new link will form, and agents will break links to the collaborator with which they receive their lowest payoff.Footnote ¹⁵ If at least one of them does not consent, no links are formed or broken.

By contrast, if we have chosen a current collaborator, the agent has an option to break the link and form a new one. A potential collaborator is chosen at random from the community. Then, if the agent would get a higher payoff from linking to this new potential collaborator, she tries to form a link. If the potential collaborator would also like to form a link with the agent, the link is formed. The agent breaks the link with the old collaborator (and the agent’s new collaborator breaks her link with the lowest payoff collaborator if she already had the maximum number of links).

3.2.2. Results

Across a wide range of parameters, the network reliably converges to the point where researchers collaborate only within their own group.Footnote ¹⁶ Figure 3 provides an example of how this occurs. Initially, links form steadily within each group and also between groups, as researchers have not yet formed the maximum number of links. Once minority group members have reached the maximum number of links, they begin to break their links with majority group members whenever they have the opportunity to form a new link with another minority group member, which yields them a payoff of 5 rather than a payoff of 4. So between-group links decline and links within the minority group increase.

Figure 3.

Example of the evolution of a network with discriminatory norms, with 100 researchers, the minority group making up 40% of the population, $H=6$, and the maximum number of links set to 3.

Simultaneously, links within the majority group increase. The reason is that as minority group members break their links with majority group members, the majority group members look to form new links. Minority group members will refuse to form new links with them, but other majority group members (if they also have less than the maximum number of links) will agree to form the link. Note that in figure 3, fewer links exist within the minority than within the majority simply because there are fewer minority group members and so fewer possible links. The important trend is that links between groups decrease over time until they are essentially nonexistent.Footnote ¹⁷

To this point, we have interpreted credit in the models here as resulting from purposeful division among collaborators, but it is worthwhile pointing out that academics do not fully control how credit is divvied out. There is evidence, in particular, that members of traditionally disadvantaged groups do not receive as much credit for collaborative work, even when author order is determined fairly. Sarsons (Reference Sarsons2017) finds that in economics, where author order is alphabetical, coauthoring does not decrease men’s chances of getting tenure, but women who coauthor are less likely to be tenured. This effect is not as strong when women coauthor with other women. The implication is that the effects we have been describing might occur even when collaborators attempt a fair division. If the payoff to minority types for between-group collaboration is, in fact, lower nonetheless, they still might learn to avoid such collaborations.

3.2.3. Bigger Academics, Bigger Pies

In the model we have just presented, the communal norm determines credit share for all agents, and this predictably affects the network structure. As we have pointed out, those who are being discriminated against will want to avoid this discrimination. One assumption of this model, though, is that agents of both sorts generate the same amount of credit, which may not apply in certain cases.

There are (at least) two plausible scenarios in which members of one social group will produce more credit than those of another. The first is an interpretation of these models in which the groups are stages of the academic hierarchy—graduate students, postdocs, early professors, established professors, and so forth. In such a case, established academics have the experience and social standing to publish better papers in better journals and to expect that their results will be more widely read and cited. The second plausible scenario of this sort is one in which members from a traditionally excluded social group are first moving into a discipline. We can expect members from such small minority groups to have less access to the sorts of experience and prestige just described and so to produce less credit on average.Footnote ¹⁸

Importantly, such considerations can change the outcome of the models just discussed. Assume that the different types of collaborative pairings—in-group collaborations among established academics, between-group collaborations, and in-group collaborations between less established academics—generate descending amounts of credit for the reasons just described. Perhaps the total credit generated in the three cases is 15, 12, and 10. Demands in the game are then for proportions of these resources, say .4, .5, and .6.

Varying these parameters, maintaining our assumption that discriminatory norms hold, and always assuming that those who produce more credit are the discriminators, we can yield three different sorts of outcomes. The first are those we have described in section 3.2.2. Second, there are parameters in which members of both types are incentivized to engage in only out-group collaboration. Suppose the three pie sizes are 15, 14, and 10. The established group will get 7.5 for within-group collaboration, and the less established will get 5. For between-group collaboration, the payoffs will be 8.4 and 5.6, respectively. In other words, the less established group is willing to do more work and get a smaller proportion of the credit in order to share in the larger pie. The more established group is willing to produce a smaller pie as long as the members receive a larger portion of it. Such a situation seems to correspond well with what we see when professors and graduate students collaborate, for example.

There is a third outcome in which we again see complete segregation, but for a different reason. In this case, the established group produces such an excess of credit that its members are unwilling to collaborate with the out-group even when they can take advantage of out-group members. For instance, if our pie sizes are 18, 13, and 10, the in-group payoffs are then 9 and 5, while the between-group payoffs are 7.8 and 5.2. Here the members of the less established group would like to collaborate, even if they do more work and receive a smaller proportion of credit. But the established group is unwilling to do so.

3.2.4. Diversity and Credit

In the introduction we discussed the idea that diversity might be important to scientific progress. If so, we might expect that the credit generated by diverse collaborative groups will sometimes be greater than that generated by homogeneous ones. Some empirical findings back this up. Freeman and Huang (Reference Freeman and Huang2015), for example, show that cross-cultural collaborations generate more credit in the form of citations. Campbell et al. (Reference Campbell, Mehtani, Dozier and Rinehart2013) find the same thing for gender diverse collaborations in ecology. And Barjak and Robinson (Reference Barjak and Robinson2008) find that culturally diverse academic teams are more productive.

To capture such a possibility, we can alter the model so that while in-group collaborations generate 10 units of credit, between-group collaborations generate more. Whether or not homophily emerges will then depend on the level of discrimination and the size of the credit boost from diversity. For instance, if between-group collaborations generate a pie of size 12 and the minority group gets 40% of the credit generated, their expected payoffs are 5 with their in-group and 4.8 with the out-group, meaning we should still expect segregation despite the extra payoff diversity generates. If we increase this number to 14, the minority group now gets 5.6, and we expect individuals to form only between-group collaborations. If we suppose more serious discrimination has emerged, so that minority types get only 30% of the credit, say, it will take a between-group pie of 17 to prevent segregation. Still, for any level of discrimination, if there is enough of a credit boost for diverse collaborations, we should not expect homophily to emerge.

Schneider et al. (Reference Schneider, Rubin and O’Connor2017) use models like those presented here to discuss possibilities for promoting diverse collaborations, despite natural tendencies toward homophily. One such possibility is for funding bodies to give more grants to diverse collaborative teams, increasing their average credit generated. This yields a model like the one just described: between-group collaborations split a bigger pie than within-group collaborations. As they point out, this does decrease homophily in network formation models. There is a downside, though. In the models we describe in section 3.2.2, while there is complete homophily, there is no active discrimination occurring because no one maintains a collaborative bond where they receive the Low demand. In models in which between-group collaboration is incentivized, by either funding agencies or the natural benefits of diversity, minority groups will be willing to accept inequitable arrangements with out-group members. These arrangements provide a net benefit over in-group collaboration but provide a much more significant benefit for those in the majority. What we see in such a case is a tension between a social good (diverse collaborations that improve the progress of science) and equity between individuals.Footnote ¹⁹

The general takeaway here is that the collaboration network we expect to evolve will vary on the basis of both the structural situation for academics (i.e., who is poised to create credit) and the existing norms of bargaining. In particular, there are multiple, realistic configurations of these features that yield segregated coauthorship networks.

3.3.1. Model

We start with an empty network, with each agent’s strategy randomly determined. In each round, there is some small probability that each agent will take an action. There are two types of possible actions: updating your bargaining strategy and updating your set of collaborators. So if agents take an action in the round, there is a chance they will update their set of collaborators and a chance they will update their strategy (agents do not update both at once). Updating sets of collaborators is done via breaking and reforming links, as described in section 3.2. Strategies are updated via best response, as described in section 3.1.

3.3.2. Results

We look at results for a network of 100 researchers where H = 6. We ran simulations for 20,000 rounds, varying the size of the minority group from 10% of the population to 50%, in intervals of 5%, and varying the maximum number of links (either 3 or 9). Each combination of parameter values was run 100 times. In each round, agents take actions with probability .1. If they act, there is a 20% chance they will update their set of collaborators and an 80% chance they will update their strategy. (The particular probabilities are not important; similar results can be obtained for a variety of values.)

As in section 3.1, within groups, the likely outcome was fair division, so we will focus on between-group strategies. We are particularly interested in the likelihood of majority discrimination and the effect it has on the collaboration network. As figure 4 shows, the larger the minority group is, the less likely it is that majority group members will discriminate against them, just as in section 3.1.

Figure 4.

Level of majority discrimination, given by the percentage of the majority group whose out-group strategy is to demand High, with different sizes of minority groups.

However, the patterns of discrimination in these models are different from those in previous models. Groups do not end up at consistent norms where all members make the same demand of the out-group. Instead we see outcomes in which some discrimination and some fair treatment emerge between groups. Figure 5 shows the evolution of between-group strategies for two different runs of the same simulation.Footnote ²⁰ In panel a we see an outcome in which more majority group members demand High and minority members Low. Panel b shows a run in which both sides tend to demand Med. Note that in both cases, as mentioned, the network does not fully settle into either norm. To understand why this is the case, let us look at the evolution of the collaboration network.

Figure 5.

Evolution of between-group strategies over time. Data for the majority demanding Low and the minority demanding High are omitted for simplicity. Panel a shows an outcome in which the high demand quickly spreads in the majority population. Panel b shows an outcome in which the medium demand quickly spreads in the majority population.

As in section 3.2, both within- and between-group links increase initially, and then between-group links decrease. In particular, discriminatory between-group links tend to be dropped, while fair ones remain. Once an agent has no more links to the other group, she stops updating her between-group strategy. This explains why the network never fully settles on one norm or another. Discriminators who have lost their minority group links retain latent discriminatory strategies that are never updated.

Figure 6 shows the network structure of the two runs shown in figure 5. As is easy to see, this process leads to collaboration networks that are homophilic but not fully segregated. In fact, there is a continuum of possible outcomes, ranging from everyone in the network reaching the norm of fair division (with no homophily in the network) to all majority members discriminating (with a totally segregated network having no links between social identity groups). Nearly all outcomes are somewhere between these two extremes: partially segregated networks with some members of the majority group upholding the discriminatory norm.

Figure 6.

Possible collaboration network outcomes. Pink nodes represent minority group members and blue nodes represent majority group members. Panel a shows an outcome in which demanding High is common in the majority population and homophily is .4921. Panel b shows an outcome in which demanding Med is common in the majority population and homophily is .1005.

In order to quantify how the level of discrimination in a network affects homophily, we use the following measure of inbreeding homophily (which we will denote I_i) from Currarini, Jackson, and Pin (Reference Currarini, Jackson and Pin2009):

$I_i=\frac{H_i-w_i}{1-w_i},$

where H_i is the proportion of a group i’s links that are within-group links and w_i is the fraction of the population that group makes up. Thus, inbreeding homophily measures how homophilic groups are (by calculating the proportion of their links that are within-group) compared to how homophilic they could be (the 1 in the denominator represents 100% of links being within-group), while controlling for the possibility that groups could make up different proportions of the total population (by subtracting w_i in both the numerator and denominator).

As figure 7 shows, more discrimination means more homophily. This is a very stochastic process, as one can see from the wide spread of possible levels of homophily for each level of majority discrimination. However, if we look at the regression line, we see an overall trend where as more of the majority group discriminates, homophily is stronger.Footnote ²¹

Figure 7.

Level of homophily, measured using inbreeding homophily, with changes in majority discrimination, given by the percentage of the majority group whose out-group strategy is to demand High.