Diversity and persistence of group tags under replicator dynamics

Everyday I walk to the Stabile Research Building to drink espresso and sit in my cozy — although oversaturated with screens — office. Oh, and to chat about research with great people like Arturo Araujo, David Basanta, Jill Gallaher, Jacob Scott, Robert Vander Velde and other Moffitters. This walk to the office takes about 30 minutes each way, so I spend it listening to podcasts. For the past few weeks, upon recommendation from a friend, I’ve started listing to the archive of the Very Bad Wizards. This is a casual — although oversaturated with rude jokes — conversation between David Pizarro and Tamler Sommers on various aspects of the psychology and philosophy of morality. They aim at an atmosphere of two researchers chatting at the bar; although their conversation is over Skype and drinks. It is similar to the atmosphere that I want to promote here at TheEGG. Except they are funny.

While walking this Wednesday, I listed to episode 39 of Very Bad Wizards. Here the duo opens with a Wilson & Haidt’s TIME quiz meant to quantify to what extent you are liberal or conservative.[1] They are 63% liberal.[2]

To do the quiz, you are asked to rate 12 statements (well, 11 and one question about browsers) on a six point Likert scale from strongly disagree to strongly agree. Here are the three that caught my attention:

  1. If I heard that a new restaurant in my neighborhood blended the cuisines of two very different cultures, that would make me want to try it.
  2. My government should treat lives of its citizens as being much more valuable than lives in other countries.[3]
  3. I wish the world did not have nations or borders and we were all part of one big group.[4]

Do you strongly agree? Strongly disagree? What was your overall place on the liberal-conservative scale?


Regardless of your answers, the statements probably remind you of an important aspect of your daily experience. The world is divided into a diversity of groups, and they coexist in a tension between their arbitrary, often artificial, nature and the important meaning that they hold to both their own members and others. Often this division is accompanied by ethnocentrism — a favoring of the in-group at the expensive of, or sometimes with direct hostility toward, the out-group — that seems difficult to circumvent through simply expanding our moral in-group. These statements also confront you with the image of what a world without group lines might look like; would it be more cooperative or would it succumb to the egalitarian dilemma?[5]

As you know, dear reader, here at TheEGG we’ve grappled with some of these questions. Mostly by playing with the Hammond & Axelrod model of ethnocentrism (2006; also see: Hartshorn, Kaznatcheev & Shultz, 2012). Recently, Jansson’s (2015) extension of my early work on the robustness of ethnocentrism (Kaznatcheev, 2010) has motivated me to continue this thread. A couple of weeks ago I sketched how to reduce the dimensionality of the replicator equations governing tag-based games. Today, I will use this representation to look at how properties of the game affect the persistence and diversity of tags.

General model

As before, I will consider the simplest tag-based models. Each agent has an arbitrary tag, unrelated to their strategy, and can condition their decision to cooperate or defect on if the tags of their interaction partner are the same or different from their own. Evolution then follows replicator dynamics. A couple of weeks ago, I showed how to use the symmetries of the model with n tags to reduce the dimensionality to 3n – 1 equations that fall into three families:

\begin{aligned}  \frac{ds_k}{dt}& = s_k(p^\text{in}_k f_k(C_\text{in}) + (1 - p^\text{in}_k) f_k(D_\text{in}) + p^\text{out}_k f_k(C_\text{out}) + (1 - p^\text{out}_k) f_k(D_\text{out}) - \langle f \rangle), \\  \frac{dp^\text{in}_k}{dt} & = p^\text{in}_k(1 - p^\text{in}_k)(f_k(C_\text{in}) - f_k(D_\text{in})) + \Delta_k(f_k(C_\text{out}) - f_k(D_\text{out})), \\  \frac{dp^\text{out}_k}{dt} & = p^\text{out}_k(1 - p^\text{out}_k)(f_k(C_\text{out}) - f_k(D_\text{out})) + \Delta_k(f_k(C_\text{in}) - f_k(D_\text{in})).  \end{aligned}

Where s_k is the proportion of agents with tag k, among them p^\text{in}_k is the fraction of in-group cooperators and p^\text{out}_k is the fraction of out-group cooperators. These are the quantities that describe our population. f_k(\cdot) is the payoff function for behavior \cdot for an agent with tag k; this is a function of the population parameters and defines our game. \langle f \rangle is the average fitness over the whole population. Finally, \Delta_k is a constant the depends on initial conditions. For more clarification and explanation, return to the previous post.

In the rest of the post, I will assume that all \Delta_k = 0, and that we are considering a model with just two tags. This simplifies the notation by letting me writing p_\text{in} for p^\text{in}_1 and q_\text{in} for p^\text{in}_2, and similarly for sup/sub-script out. Since there is only 2 strategies, we will not need subscripts on our tag proportions, and s will be the proportion of tag-1 strategies, with (1 – s) being the proportion of tag-2.

Instability of tag diversity

As a starting point, let us consider what happens in the long term when there is no direct out-group interaction. Alternatively, there is a constant fitness effect from meeting out-group members, and since global additive factors don’t matter, we can subtract that effect from all fitnesses. Thus, the only effect of meeting an out-group agent is opportunity cost: you will have one less in-group interaction. This means that the dynamics for \{p,q\}_\text{out} will be static, and those two equations can be ignored.

Let \delta_1(p_\text{in}) be the difference in utility between cooperating with a tag-1 agent and defecting against them, given that the proportion of tag-1 cooperators is p_\text{in}; and similarly for tag-2 and \delta_2(q_\text{in}). Then our dynamics for in-group cooperation in the two groups become:

\begin{aligned}  \dot{p}_\text{in} & = p_\text{in}(1 - p_\text{in})s\delta_1(p_\text{in}), \\  \dot{q}_\text{in} & = q_\text{in}(1 - q_\text{in})(1 - s)\delta_2(q_\text{in}).  \end{aligned}

Note that these equations are completely decoupled from each other. Their only dependence on s is a global rate term of s or (1 – s), respectively. Thus, in the long term, these dynamics will converge towards their equilibrium points p_\text{in}^* and q_\text{in}^* where the agents are getting an average payoff of \langle f_1(\cdot_\text{in}) \rangle_{p^*_\text{in}} =: sg^\text{in}_1(p_\text{in}^*,p_\text{in}^*) =: sg^\text{in*}_1 and similar for tag-2 with 1 replaced by 2, p by q, and s by (1 – s).

Plugging these equilibrium solutions into the equations for the proportion of tag-1, we get:

\dot{s} = s(1 - s)(sg^\text{in*}_1 - (1 - s)g^\text{in*}_2)

which has an internal fixed point at s^* := \frac{g_2^\text{in*}}{g_1^\text{in*} + g_2^\text{in*}} if g_1^\text{in*} and g_2^\text{in*} have the same sign. If they have different signs then there is a flow towards all agents having the tag with the positive sign. An obvious case of destruction of diversity, but not the interesting one. The interesting case is when both gs are positive then the internal fixed point is unstable (otherwise if both are negative then it is stable): instead of moving towards the fixed point, the population flows away from the fixed point towards the strategy that was over-represented (where equal representation is given by s*).

In words, this means that if it is better to interact with an in-group agent than interact with nobody (or with an out-group agent that provides constant fitness effect) then tags will tend to decrease in diversity. Unless there is another mechanism in place to stabilize the population. Since we see a diversity of tags in most systems that we want to model, that means that either we are observing a very long transient or such a mechanism exists.

Ethnocentrism by itself is not a mechanism that would facilitate diversity of tags. If we look at the stable strategy profile for a prototypical model of ethnocentrism then the whole point is that the there is cooperation with the in-group and defection from the out-group. That means that the in-group payoffs are higher than the out-group payoffs. If we subtract the out-group payoffs from all fitness effects then we end up with g^\text{in*} > 0. Thus, the diversity is not stable and in the long-term a single tag would come to dominate the population.[6]

But, a population of selfish or ethnocentric agents isn’t the only possible stable configuration that we encounter in tag-based models. Laird (2011) observed cases where we find a stable coexistence between two populations of traitorous agents with differing tags. In that case, there is cooperation with the out-group and defection with the in-group: g^\text{in*} < 0. My inviscid analysis would then suggest that this would help stabilize diversity in the population. Unfortunately, it isn’t clear how often such configurations are found in biology.

In this model, a static and stable population with high rates of in-group cooperation and persistent tag diversity seems unlikely. The next step is to consider dynamic equilibria or to go back to a large number of tags in a finite population. Since this post, especially with the footnotes, has run extremely long, I will save that analysis for next week.

Notes and References

  1. For some of the research that this is based on, see Alexander Yartsev’s post on ethics. Alex focuses heavily on physicalist explanations, and if that interests you then also see my answer to this CogSci question. Later in this post, I focus on the wording of the statements, and for background on that I recommend looking at this CogSci question.
  2. Since there are two of them, David and Tamler averaged their answers. I would have expected that the final result would then be in-between their individuals results. With one being slightly more liberal and the other being more conservative. So on the questions they disagreed, I entered their differing answers instead of the average answers they gave. I wanted to see who would be more liberal, I suspect it would be David. Turns out that they are both more conservative than their average:


    In the figure above, at top you see the results for the show as a whole. At the bottom, however, you see their individual results. With David on the left and Tamler on the right. I was amused by this, even if it is completely irrelevant to this post.

  3. This questions contains a nice subtlety that Tamler pointed out on the podcast. It’s isn’t saying that the “lives of citizens of my country are much more valuable than lives in other countries,” but that the government should treat them as such. That this is the duty of the government towards its citizens, even if thus duty disagrees with the objectively equality of human lives. At first, it might seem that it is preposterous to have such an irrational government that doesn’t observe basic facts. However, this might seem less noxious in the interpersonal domain. Consider a mother’s attention towards her children. Objectively, the lives of all children are equal. Yet the mother is willing to pay $800 to send her child to a better school, instead of spending that money to get twenty kids mosquito nets and potentially save them from dying of malaria. But we don’t view her action with disdain, because she stands in a relation of care to her child that does not extend to other children. A similar argument could be made for the role of government.

    The analogy to the mother relies on an ethics of care (or something similar; maybe virtue ethics) as our meta-ethical position. I think that is a great basis for interpersonal morality, but it is not the appropriate position for treating the role of the institution of government. In After Virtue, MacIntyre makes a convincing — for me, at least — argument that modern bureaucratic institutions, like the State, are inherently committed to a consequentialist meta-ethic. For a strict consequentialist, however, it is much more difficult to argue that a mother should treat her own child as exceptional to other children. The difficulty remains — or is made worse — if the mother is replaced by government.

    An anarcho-communist like Kropotkin might then conclude that the idea of the State is incoherent. Although it isn’t clear how that would help you decide if you agree or disagree with the original statement.

  4. For anybody who travels a lot, had to apply for visas, or interact with an obnoxious border patrol officer, this might seem like an obvious “strongly agree”. A commitment to your government treating everybody, citizen or not, equally also makes borders and other nation-divisions inconsequential. However, since the question closes with the vague “group” instead of the more specific “state”, it might be worth trying to imagine to what extent a group-less world is coherent or desirable. For many on the left, such as myself, it is easy to imagine a world where nation-borders don’t divide groups. This is because nationality is not how we have create our mental borders, but that doesn’t mean we don’t have any: we can always find a blue tribe and a red tribe.

    Note, for example, the not-so-subtle group membership messages that I littered throughout the post. How do they affect when you give me the benefit of the doubt and when your reject my statements out of hand? How do they affect your judgements of my expertise? What if I was affiliated with a different tribe from yours?

  5. It is easy to imagine group boundaries as always obstructive. But this might be an unwarranted universal. For example, Blau & Schwartz (1984) argued for this universal in the spread of ideas: if group boundaries were eliminated then there would be greater social integration and easier spread of ideas. But Centola (2015) suggests that this is true only up to a point (thank you to Helga Vierich for pointing me to this work in an earlier comment). Of course, that doesn’t mean that we should swing to the other extreme, and create more group boundaries. It seems wiser to approach each division on a case by case basis.
  6. This decrease of tag diversity can be turned into oscillations by considering finite populations or introducing a non-negligible mutation rate. In that case, a new tag could come to replace an old tag after being brought near extinction by a mechanism like the one observed by Riolo et al. (2001) in one of the first models of ethnocentrism. After a tag approaches dominance, the selection strength on the out-group strategy is relaxed due to such infrequent out-group interactions. In the dominated tags, however, the selection for out-group strategy is very strong and so they can evolve to exploit the accidental generosity of the dominant group, and replace them.

    An alternative possibility in finite populations, closer to the mechanism behind cooperation found by Hauert et al. (2006) in the ecological public goods game, is that as the group grows the selection strength for the in-group strategy increases and the proportion of in-group cooperators decreases due to being out-competed by defectors. A less populous tag, due to stochastic fluctuations of the finite population and the lesser selection strength, might have more in-group cooperators and thus bid to replace the dominant tag. It would be interesting to analyze the tag diversity in my previous observations on inviscid ethnocentrism, to see if either (or both) of these mechanisms are in play.

Blau, P.M. & Schwartz, J.E. (1984). Crosscutting Social Circles: Testing a Macrostructural Theory of
Intergroup Relations
. Academic Press, New York.

Centola, D. (2015). The Social Origins of Networks and Diffusion1. American Journal of Sociology, 120(5), 1295-1338.

Jansson, F. (2015). What games support the evolution of an ingroup bias? Journal of theoretical biology, 373, 100-10 PMID: 25794651

Hammond, R., & Axelrod, R. (2006). The Evolution of Ethnocentrism. Journal of Conflict Resolution, 50(6): 926-936

Hartshorn, M., Kaznatcheev, A., & Shultz, T. (2013). The evolutionary dominance of ethnocentric cooperation. Journal of Artificial Societies and Social Simulation, 16(3): 7.

Hauert, C., Holmes, M., & Doebeli, M. (2006). Evolutionary games and population dynamics: maintenance of cooperation in public goods games. Proceedings of the Royal Society B: Biological Sciences, 273(1600), 2565-2571

Kaznatcheev, A. (2010). Robustness of Ethnocentrism to Changes in Interpersonal Interactions. In AAAI Fall Symposium: Complex Adaptive Systems.

Laird, R.A. (2011) Green-beard effect predicts the evolution of traitorousness in the two-tag Prisoner’s dilemma. Journal of Theoretical Biology 288(7): 84-91.

Riolo, R., Cohen, M., & Axelrod, R. (2001). Evolution of cooperation without reciprocity. Nature, 414 (6862), 441-443

About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

3 Responses to Diversity and persistence of group tags under replicator dynamics

  1. Pingback: Cataloging a year of blogging | Theory, Evolution, and Games Group

  2. Soham says:

    Shameless self-promotion here, but I follow this blog regularly and think you might be interested in our recent work on the evolution of ethnocentrism published in the Nature journal Scientific Reports titled: “The Inevitability of Ethnocentrism Revisited: Ethnocentrism Diminishes As Mobility Increases” (here’s the link: http://www.nature.com/articles/srep17963). In the paper we present a model that builds on Hammond and Axelrod’s model to show that ethnocentrism is not inevitable under all situations, and it critically depends on the amount of mobility of the individuals in the society. In societies with low mobility, higher rates of ethnocentric behavior emerges, while in societies with high mobility, lower rates of ethnocentric behavior emerges.

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