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Minority (dis)advantage in population games


We identify a novel ‘cultural red king effect’ that, in many cases, results in stable arrangements which are to the detriment of minority groups. In particular, we show inequalities disadvantaging minority groups can naturally arise under an adaptive process when minority and majority members must routinely determine how to divide resources amongst themselves. We contend that these results show how inequalities disadvantaging minorities can likely arise by dint of their relative size and need not be a result of either explicit nor implicit prejudices, nor due to intrinsic differences between minority and majority members.

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  1. While the replicator dynamic was initially constructed to model evolution by natural selection, it is formally equivalent to models of social learning via imitation, as well as models of individual learning such as the Herrnstein reinforcement model used in psychology.

  2. See Ernst (2001) for alterations to the replicator dynamic that lead to other evolutionary possibilities.

  3. Skyrms and Zollman (2010), p. 271.

  4. See, for instance, Bruner (2015), Skyrms (2004), Alexander (2007) and Smead (2008).

  5. In this case, the average fitness of Blue individuals utilizing strategy i is \(2\sum _{l=1} \pi (i,l)x_l(t) + \sum _{k=1}\pi (i,k)y_k(t)\).

  6. Once again, the initial composition of both populations was determined by randomly sampling the space of all possible population compositions.

  7. As I first noted in my dissertation (Bruner 2014), an asymmetry of response can be the result of differential population size, differential network connectivity (Gallo 2015) as well as the result of differences in institutional memory (Young 1993). Both Gallo and Young investigate the evolution of bargaining in these different contexts and arrive at results which are similar to those uncovered in this paper. Namely, in many bargaining contexts, asymmetries between parties results in one party responding at a slower rate than their counterpart, which in turn, yield benefits.

  8. See, for instance, McPherson et al. (2001).

  9. For more on how norms disadvantaging minorities could develop in the workplace or in academia, see Bruner and O’Connor (forthcoming) and O’Connor and Bruner (forthcoming).

  10. Likewise, the chance of interacting with a Green out-group member is \(p(G)(1-e)\) instead of p(G).

  11. This was determined by computer simulation.

  12. That said, the average payoff of the minority is higher when interactions are correlated since, as mentioned earlier, minorities are more likely to interact with in-group members.

  13. Of course, since interactions are correlated, both minority and majority are interacting with out-group members less frequently, which means both are less responsive to the behavior of out-group members. Yet the minority is still significantly more responsive to the majority than the majority is to the minority. To see this, consider the situation in which Greens and Blues constitute 20 and 80% of the total population, respectively. In the absence of correlation, 80% of the minority’s interactions are with out-group members while only 20% of the majority’s interactions are with out-group members. Thus the minority is four times as likely as the majority to interact with an out-group member (80/20). Now consider the situation involving positive correlation. In particular, set e to 0.2. Recall that the likelihood of interacting with an out-group member is \((1-e)pr(Out)\), where pr(Out) is the proportion of out-group members in the population. Thus members of the Green minority will interact with out-group members with probability \((1-.2)0.8\) or 64% of the time. Blue majority members will interact with members of the minority with probability \((1-.2).2\) or 16% of the time. Note that while both groups are now less likely to interact with out-group members it still remains the case that members of the minority are four times more likely than members of the majority to interact with out-group members (64 / 16). Thus the relative responsiveness of the two populations remains the same despite the increased level of correlation. Since this asymmetry in responsiveness is what determines whether the minority or majority disadvantage equilibrium is reached, the cultural red king effect uncovered in Sect. 3 still holds.

  14. This was established on the basis of computer simulations.

  15. However, this would not be the case if the best response dynamic was modified so the same number of individuals from each group was selected to best respond. In this case, a larger proportion of the minority group would update their strategic behavior than the majority group, and as a result the minority would quickly learn to accommodate the majority.

  16. See Sandholm (2010) for a discussion of these various adaptive dynamics and their properties. It is worth noting that these three dynamics along with the best response dynamic and replicator dynamic are among the most popular deterministic dynamics used in evolutionary game theory.

  17. To be clear here, what we mean is that for \(r>0.95\) the minority disadvantage equilibrium is more likely to emerge than compared to the ‘baseline’ model in which \(r=1\) and both groups are of equal size. When r dips below this 0.95 threshold the equal split becomes overwhelmingly likely.

  18. This threshold, of course, depends on parameters in the model (such as the size of the minority group). In general, however, a relatively high level of signal noise still allows for the red king effect.


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Thanks to Brian Skyrms, Simon Huttegger, Jean-Paul Carvalo, Peter Vanderschraaf, Cailin O’Connor, Kim Sterelny and Bob Goodin for comments and suggestions. Additional thanks to audience members of the Social Dynamics Seminar at UC Irvine, Moral, Social and Political Theory Workshop at ANU, Formal Ethics 2015 at Bayreuth, the 2014 meeting of the Association for the Study of Religion, Economics and Culture at Chapman University and the Philosophy Seminar Series at Monash University.

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Correspondence to Justin P. Bruner.

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Bruner, J.P. Minority (dis)advantage in population games. Synthese 196, 413–427 (2019).

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  • Social philosophy
  • Social norms
  • Game theory
  • Evolutionary game theory
  • Bargaining
  • Distributive justice