Networks

The Networks Dynamics of “Mean Girls”: A Study on the Collapse of Cooperation

The question of who around us is trustworthy has plagued human social groups for millennia. It has occasionally led us to some dark places– the formation of in- and out- groups is often detrimental to the well-being of society as a whole, and especially its most vulnerable members. But if this is the case, why do we keep seeing these groups appear?

A plausible reason is that mirroring the behavior of well-established group members when you first encounter a new population can be beneficial. Consider, for example, Cady Heron’s behavior in the movie “Mean Girls”– when she joins the school, she initially mirrors Janice and Damian’s behavior in distrusting the Plastics: a decision that, as the movie progresses, we find was probably wise. But sociologically, why did Cady have the instinct to mirror their behavior? She might intuitively argue that people who have been in a group for a while, like Janice and Damian, know who to trust and who to distrust. However, if Cady had initially become friends with Gretchen (another well-established node in the North Shore High School friendship network), she might have never known to be wary of the toxic Regina George. As we think about dramatized High School social dynamics, this brings us to a few more general questions: on the aggregate level, how likely is it that a new student will copy the behavior of someone like Gretchen with questionable taste in friends? If it is very likely, the school environment will likely become toxic, and lead to a Burn Book incident and the collapse of the social system (as we see in the movie). Is there any behavior we can encourage among the new students as they join the network to maximize their chances of meeting collaborative, supportive individuals?

In their article “Information Cascades and the Collapse of Cooperation”, Csikász-Nagy et. al. attempt to answer these questions using game theory and a network model of evolutionary social dynamics in a community. They define two types of nodes: cooperators and cheaters. Cooperators pay a small price to benefit others, trusting that the others will pay the same price to benefit them. Cheaters simply act in their own pure best interest, receiving the benefits from being connected to cooperators without offering anything in return. Through a game theory perspective, we can draw an analogy to the Prisoner’s Dilemma: Cheaters get the highest payoff for accusing their co-conspirator, while cooperative nodes never name their co-conspirators. A representative payoff matrix for cheaters and collaborators is below, where Node 1 chooses columns and Node 2 chooses rows:

 

                                                                              Node 1

Cheater          Cooperator

Node 2           Cheater	-5, -5	0, -10
Cooperator	-10, 0	-1, -1

 

Despite the incentive to always act selfishly, we see that sociologically, cooperative groups do develop. This is likely because cooperation maximizes social welfare when you know you can trust the people around you. For example, if  the two prisoners cared about each others’ well-being, they might choose the welfare-maximizing situation in the Prisoner’s Dilemma. Over a long period of time, experiencing multiple Prisoner’s Dilemma scenarios with many people, a population more likely to act selflessly will have higher payoffs than a population more inclined to act selfishly.

Given the networks of trust we see in the world around us, Csikász-Nagy then asks the question: what happens when nodes connected by trust already exist in a network, and new nodes try to join? To model this situation, the following rules are established:

1. Newcoming nodes select a node whose behavior to copy within the network, then imitate its strategy in connecting to other nodes (essentially, taking the more experienced node’s input into consideration). They select this node taking its fitness into consideration, which is some function of how likely the newcomer thinks it is that they make good decisions. With a probability p, the node will make its own choices about with whom to connect. With a probability 1-p, the node will copy its role model’s strategy and consider connecting with the same nodes as the role model. (note that p is usually small.)
2. When connecting, the newcoming node considers both its private information for who is trustworthy, and the public information from which nodes have a high in-degree: it seems plausible that someone popular would be good to connect with, if other nodes have logically made the decision to connect with them.
3. To keep the number of nodes constant, a node is randomly selected to be deleted when a new node is added.

It is useful to note that the consideration of public information can lead to information cascades– large numbers of nodes following the crowd based on little real information. (In Mean Girls, this might explain why Regina is so popular, even though she is cruel to most of her fellow students– since she is already popular, others assume that she must be worthy of popularity.)

To see how large networks develop with these rules, the authors run several computer models with differing initial conditions. They vary the number of cheaters and cooperators, as well as all of the probability parameters that we have defined above. Additionally, they create contrived situations in which a cheater is introduced into a tightly-knit networks of cooperators, or a sequence of cooperators is trying to connect amidst a crowd of cheaters. These situations can mirror what we see in real life, or in the movies. It is arguable whether Cady is a cooperator or a cheater (one might cite Janice’s line: “You’re plastic. Cold, shiny, hard Plastic!”) but either way, she is navigating a network with popular and successful cheaters like Regina George, and some moderately successful, smaller connected components of cooperators, like Janice and Damian. These contrived situations help us understand how Cady should act, depending on if she wants to join a group of cooperators or become successful but run the risk of social exile like Regina.

The authors of this paper find that nodes essentially need to have a specific mix of suspicion and willingness to collaborate with others for the network to be successful, and that this mix highly depends on the “selection strength” of the network– essentially, how likely a node is to emulate the behavior of another successful node of its type, rather than a random node. To numerically analyze the results further, the authors define several quantities as follows:

“If a newcomer decides to establish a connection with a cooperative node x then this constitutes a TP [true positive]; if a newcomer decides to not establish a connection with a cooperative node, then it is a FN [false negative]. If a newcomer decides to establish a connection with a cheater then it is a FP [false positive], while if a newcomer decides to not establish a connection with a cheater, then it is a TN [true negative]. The corresponding specificity and sensitivity are defined as specificity=TN/(TN+FP) and sensitivity=TP/(TP+FN).”

Additionally, the strategic introduction of a cooperator or cheater into a group can disrupt the network. If a cheater is introduced into a well-connected group of cooperators, newcoming cheaters can see that they are doing well and emulate their behavior, destroying the cooperative network. This generally happens when specificity is low (number of false positives increases) and lots of connections are established between newcoming and existing nodes. On the other hand, if the whole population is made of cheaters before a series of cooperative nodes enters the network, the cooperators are able to build a cooperative network in the cooperate almost solely with each other. They generally are more likely to succeed if the sensitivity is low (i.e. the number of false negatives increases).

The balance of specificity and sensitivity to produce optimal outcomes for the collaborators largely depends on other characteristics of the network. How do nodes weigh their public information against their private information? How likely are they to emulate someone who seemed to have a high payoff (i.e., how high is their selection strength)?

We find that when nodes strongly emulate someone else of their type with a high payoff, it pays better to be suspicious of those around you and avoid connections; additionally, the network is more fragile altogether, because the introduction of one cheater into a group of collaborators could trigger an invasion of cheaters. However, when nodes emulate behavior more randomly, this is not as much of an issue– connecting with one cheater will not attract that many others. Thus, it becomes more beneficial to make a larger number of connections, in the hopes that you will find other cooperators.

This analysis implies something we, and Cady, know to be true– social dynamics can be quite complicated. However, those of us for whom navigating the high school social scene proved difficult might be pleased to know: a rudimentary model for good interpersonal decision making does, in fact, exist! Maybe instead of modeling the social dynamics she saw in school with animal behavior in the ecosystem, Cady should have observed her peers from afar, calculated the necessary network characteristics, run a computational model to understand the network dynamics, and thus become the true queen of High School.

Sources:

Yang, G., Csikász-Nagy, A., Waites, W. et al. Information Cascades and the Collapse of Cooperation. Sci Rep 10, 8004 (2020). https://doi.org/10.1038/s41598-020-64800-z

Information Cascades and the Collapse of Cooperation | Scientific Reports (nature.com)

 

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