The cultural Red King effect

ABSTRACT

Why do minority groups tend to be discriminated against when it comes to situations of bargaining and resource division? In this article, I explore an explanation for this disadvantage that appeals solely to the dynamics of social interaction between minority and majority groups—the cultural Red King effect (Bruner, 2017). As I show, in agent-based models of bargaining between groups, the minority group will tend to get less as a direct result of the fact that they frequently interact with majority group members, while majority group members meet them only rarely. This effect is strengthened by certain psychological phenomenon—risk aversion and in-group preference—is robust on network models, and is strengthened in cases where preexisting norms are discriminatory. I will also discuss how this effect unifies previous results on the impacts of institutional memory on bargaining between groups.

KEYWORDS:

• Bargaining
• discrimination
• evolutionary
• game theory
• inequity
• networks
• Red King effect
• social dynamics

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Acknowledgments

Many thanks to Simon Huttegger, Edouardo Gallo, Brian Skyrms, James Weatherall, Peyton Young, and Justin Bruner for comments and suggestions. I would also like to thank the UCI Social Dynamics seminar for wonderful feedback.

Notes

¹ The term “Red Queen” comes from Lewis Carroll’s Through the Looking Glass where the Red Queen tells Alice, “Now, here, you see, it takes all the running you can do, to keep in the same place” (Carroll, 1871).

² Thanks to Jean-Paul Carvalho for this analogy.

³ Bergstrom and Lachmann (2003), Bruner (2017), O’Connor and Bruner (2017) use the mini-game approach (see Sigmund, Hauert, & Nowak, 2001) of investigating small, tractable games that capture the strategic scenario of interest for similar purposes. Throughout this article, I will do the same.

⁴ The version I present is actually often termed an “anti-coordination” game since actors must take opposite strategies to succeed. However, they still succeed by coordinating their action, so I find this terminology unhelpful. O’Connor (2017) distinguishes between correlative coordination games, where actors must correlate actions to succeed, and complementary coordination games, where actors need to take complementary actions. I use a complementary coordination game here, rather than a correlative one as in Bergstrom and Lachmann (2003), since this makes clearer the connection to the work of Axtell et al. (2000) on bargaining, which I will later elaborate on.

⁵ These are the most commonly used model of selection in evolutionary game theory. They assume that strategies that beat the population average payoff will become more prevalent, while those that are less successful will contract. The two-population replicator equations specify change in proportional representation, (), for each strategy, , in a population with proportions (). They are . This equation can be read as stating that the rate of change of a particular strategy () is equal to the current proportion of that strategy () multiplied by the difference between the payoff to that strategy given the state of the population and the average payoff for the entire population given the state of the . Strategies for population update according to analogous dynamics.

⁶ These figures, and others like them in the article, were generated using the program Dynamo (Sandholm, Dokumaci, & Franchetti, 2012).

⁷ These simulation results were generated using the discrete time replicator dynamics, where population proportions update at discrete time steps rather than continuously. The multiplier then represents the number of replications that the faster population undergoes at each step.

⁸ This is not necessary in the Bergstrom and Lachmann (2003) models because they consider only interactions between species, so actors only interact with one (out-group) type.

⁹ These three dynamics, as well as the replicator and best response dynamics, are described by Sandholm (2010) as some of the most important in evolutionary game theory. As Bruner (2017) outlines, the cultural Red King effect does not arise under myopic best response dynamics because the rate of change of a strategy is not directly impacted by its payoff, meaning that both groups adapt at the same rate.

¹⁰ This interpretation comes from the observation that the replicator dynamics are the mean field dynamics of explicit models of cultural imitation (Björnerstedt, Weibull, et al., 1994; Schlag, 1998; Weibull, 1997). It has been observed in anthropology, for example, that humans do seem to imitate successful behaviors of group members (Richerson & Boyd, 2008).

¹¹ Wood and Eagly (2012), for example, extensively outline how children are explicitly directed to correctly gendered behavior, including own-gender imitation.

¹² I make one small change from the implementation in Axtell et al. (2000). They randomly initialize the memories of the agents at the start of simulation. For low values of , this will mean that on average more agents will start by demanding High than Low, whereas for higher values of , this will mean that more agents will start by demanding Low than High. These initial demands matter because we are looking at a case where one type learns more quickly on average as a result of its size. If the initial demands are skewed in one direction, the minority type will adapt to this, meaning they will be likely to ultimately end up complementing whatever the majority type is doing at the beginning of simulation. To avoid this, I start the agents with no memories and determine their first strategies using random coin flips. Afterwards, they best respond to whatever memories they have.

¹³ These are conventions in the sense outlined by Young (1993a).

¹⁴ I ran models with error rate , memory , population size , low demand , and proportion of larger type .

¹⁵ I incorporate this into the model by using a utility function . This is a somewhat arbitrary function chosen because it respects the 0 payoff point, is concave, and is monotonically increasing. Other risk averse utility curves will have a similar effect.

¹⁶ In particular, Young shows that as the partition of demands for the actors gets finer and finer, the SSE approachs a version of the Nash bargaining solution weighted by memory length. He also considers heterogenous groups and finds that the actor with the shortest memory, i.e., the “least steady” actor, in each group determines the SSE. Whichever group has the least steady actor of all is expected to be disadvantaged as a result.

¹⁷ In this setup, as in the Young models, the least networked agent is the one that matters in determining the expected split between the two sides.