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On the emergence of minority disadvantage: testing the cultural Red King hypothesis


The study of social justice asks: what sorts of social arrangements are equitable ones? But also: how do we derive the inequitable arrangements we often observe in human societies? In particular, in spite of explicitly stated equity norms, categorical inequity tends to be the rule rather than the exception. The cultural Red King hypothesis predicts that differentials in group size may lead to inequitable outcomes for minority groups even in the absence of explicit or implicit bias. We test this prediction in an experimental context where subjects divided into groups engage in repeated play of a bargaining game. We ran 14 trials involving a total of 112 participants. The results of the experiments are statistically significant and suggestive: individuals in minority groups in these experiments end up receiving fewer resources than those in majority groups. Combined with previous theoretical findings, these results give some reason to think that the cultural Red King may occur in real human groups.

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  1. 1.

    Adams and Freedman (1976), Freedman and Montanari (1980) discuss the prevalence of equity norms. Tilly et al. (1998) describes research on the prevalence of categorical inequity.

  2. 2.

    There is some relation here to inversion thesis from standpoint epistemology. The idea there is that sometimes those in oppressed groups have special access to knowledge or epistemic privilege (Wylie 2013). In Bruner’s models, minority groups learn more quickly about their social situation.

  3. 3.

    Nozick (1974) in particular requires that such distributions derive from just initial acquisitions of goods and just transfers of holdings, though he does not do much fill in the details of what justice here entails.

  4. 4.

    Many thanks to Nick Adams and Mike Ashfield for this point.

  5. 5.

    This hypothesis is named after the Red Queen in Lewis Carrol’s Through the Looking Glass. She tells Alice, “Now here, you see, it takes all the running you can do to keep in the same place”.

  6. 6.

    This game was first introduced by Nash (1950), and has subsequently been used to represent countless scenarios of resource division—salary decisions, international trade agreements, bargaining over household labor, etc. It has also been called divide the dollar, divide the pie, divide the cake, the Nash bargaining game, or just the bargaining game.

  7. 7.

    Sigmund et al. (2001) introduce the term ‘mini-game’ and defend the usefulness of small, simple game theoretic models. In the case of evolutionary bargaining models there is a long history of using mini-games to capture bargaining scenarios as in Skyrms (1994), Axtell et al. (2000).

  8. 8.

    Pure strategies are those where the actors always take the same action rather than probabilistically mixing. There is also a mixed strategy equilibrium of this game, but it does not emerge in the evolutionary models we will describe.

  9. 9.

    Notice that these three outcomes are the population level equilibria between groups in the model described. I.e. in a population playing these strategies, no individuals may change their between-group strategy and improve their payoff. In these models, conventional behavior also emerges within each of the two in-groups. Usually this involves everyone treating each other fairly. Sometimes ‘fractious’ patterns emerge where some individuals demand High and others Low (Skyrms 1994).

  10. 10.

    Taylor and Jonker (1978) introduce the replicator dynamics to model evolution by natural selection, but they have also often been used as a good representation of cultural change. In particular, they are equivalent to the average expected population change of explicitly cultural evolutionary and learning models (Weibull 1997; Börgers and Sarin 1997; Hopkins 2002). Since this is a simulation, we employ the discrete time version of the dynamics adapted to a two population model where all individuals interact and the populations are of different sizes.

  11. 11.

    For more on this particular model see O’Connor (2018a).

  12. 12.

    See Bruner (2017), O’Connor (2017) for a more in-depth explanation.

  13. 13.

    In this particular model, we give each agent a limited memory of past interaction. Each round we randomly choose agents for interaction. They select whatever strategy would perform best against their memories. For example, an agent with many memories of their out-group playing High will do best to respond by playing Low. This agent-based model is based off an influential analytic modeling paradigm used by Young (1993) to investigate bargaining between groups. Axtell et al. (2000) develop an agent-based version of the model, and O’Connor (2017) demonstrates the cultural Red King effect in this agent-based version. The choice of dynamics here is somewhat arbitrary. We used this learning rule since it has been previously used to explore the cultural Red King, but results will be qualitatively robust across many rules.

  14. 14.

    See, for example, Blume et al. (1998), Bruner et al. (2018) who find that subjects in the lab conform to evolutionary predictions in the signaling game, but by no means accord to the perfect conventions predicted.

  15. 15.

    Though, see Sect. 6 for a discussion of how subjects’ decision making might still have been affected by inequity aversion.

  16. 16.

    Random selection of rounds for payment after an experiment is used to prevent known distortions in subject behavior that occur as participants perceive their level of wealth to change (Davis and Holt 1993).

  17. 17.

    All of the data from our experiments is registered at the Open Science Framework ( The URL to the data repository is located in “Appendix A”.

  18. 18.

    This said, there are a few points that strengthen the worry. First, Nozick adheres to a ‘Lockean proviso’ that individuals cannot leave interactive partners worse off than they started. In these models, interpreted as involving joint action, individuals always benefit from interaction. It is just that some may benefit more than others. Second, one might worry that equitable distributions are unstable, because the very talented will be able to (fairly) garner high payments for their services. This may generally be true, but in the world of the models we consider here fairness is a stable possibility. Unfairnesses emerges nonetheless.


  1. Adams, J. S., & Freedman, S. (1976). Equity theory revisited: Comments and annotated bibliography. Advances in Experimental Social Psychology, 9, 43–90.

    Article  Google Scholar 

  2. Arrow, K. J. (1978). Nozick’s entitlement theory of justice. Philosophia, 7(2), 265–279.

    Article  Google Scholar 

  3. Axtell, R., Epstein, J. M., & Young, H. P. (2000). The emergence of classes in a multi-agent bargaining model. Papers 9, Brookings Institution-Working Papers.

  4. Bergstrom, C. T., & Lachmann, M. (2003). The red king effect: When the slowest runner wins the coevolutionary race. Proceedings of the National Academy of Sciences, 100(2), 593–598.

    Article  Google Scholar 

  5. Binmore, K., & Shaked, A. (2010). Experimental economics: Where next? rejoinder. Journal of Economic Behavior and Organization, 73(1), 120–121.

    Article  Google Scholar 

  6. Blume, A., DeJong, D. V., Kim, Y.-G., & Sprinkle, G. B. (1998). Experimental evidence on the evolution of meaning of messages in sender–receiver games. The American Economic Review, 88(5), 1323–1340.

    Google Scholar 

  7. Börgers, T., & Sarin, R. (1997). Learning through reinforcement and replicator dynamics. Journal of Economic Theory, 77(1), 1–14.

    Article  Google Scholar 

  8. Bruner, J., O’Connor, C., Rubin, H., & Huttegger, S. M. (2018). David lewis in the lab: Experimental results on the emergence of meaning. Synthese, 195(2), 603–621.

    Article  Google Scholar 

  9. Bruner, J. P. (2017). Minority (dis)advantage in population games. Synthese, 13, 1–15.

    Google Scholar 

  10. Buhrmester, M., Kwang, T., & Gosling, S. D. (2011). Amazon’s mechanical turk: A new source of inexpensive, yet high-quality, data? Perspectives on Psychological Science, 6(1), 3–5. PMID: 26162106.

    Article  Google Scholar 

  11. Chen, D. L., Schonger, M., & Wickens, C. (2016). oTree-An open-source platform for laboratory, online, and field experiments. Journal of Behavioral and Experimental Finance, 9, 88–97.

    Article  Google Scholar 

  12. Cooper, D. J. (2014). A note on deception in economic experiments. Journal of Wine Economics, 9(2), 111–114.

    Article  Google Scholar 

  13. Croson, R. (2005). The method of experimental economics. International Negotiation, 10(1), 131–148.

    Article  Google Scholar 

  14. Davis, D., & Holt, C. (1993). Experimental economics: Methods, problems, and promise. Estudios Económicos, 8, 179–212.

    Google Scholar 

  15. Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. The Quarterly Journal of Economics, 114(3), 817–868.

    Article  Google Scholar 

  16. Fehr, E., & Schmidt, K. M. (2010). On inequity aversion: A reply to binmore and shaked. Journal of Economic Behavior and Organization, 73(1), 101–108.

    Article  Google Scholar 

  17. Freedman, S. M., & Montanari, J. R. (1980). An integrative model of managerial reward allocation. Academy of Management Review, 5(3), 381–390.

    Article  Google Scholar 

  18. Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., Gintis, H., et al. (2001). In search of Homo economicus: Behavioral experiments in 15 small-scale societies. American Economic Review, 91(2), 73–84.

    Article  Google Scholar 

  19. Hopkins, E. (2002). Two competing models of how people learn in games. Econometrica, 70(6), 2141–2166.

    Article  Google Scholar 

  20. Kees, J., Berry, C., Burton, S., & Sheehan, K. (2017). An analysis of data quality: Professional panels, student subject pools, and amazon’s mechanical turk. Journal of Advertising, 46(1), 141–155.

    Article  Google Scholar 

  21. Nash, J. (1950). The bargaining problem. Econometrica: Journal of the Econometric Society, 18, 155–162.

    Article  Google Scholar 

  22. Nozick, R. (1974). Anarchy, state, and utopia. New York basic book. New York: Basic Books.

  23. O’Connor, C. (2017). The cultural red king effect. The Journal of Mathematical Sociology, 41(3), 155–171.

    Article  Google Scholar 

  24. O’Connor, C. (2018a). The evolution of inequity. Oxford: Oxford University Press. (forthcoming).

    Google Scholar 

  25. O’Connor, C. (2018b). Modeling minimal conditions for inequity (pp. unpublished).

  26. O’Connor, C., Bright, L. K. & Bruner, J. P. (2017). The emergence of intersectional disadvantage. Working Paper.

  27. O’Connor, C., & Bruner, J. (2017). Dynamics and diversity in epistemic communities. Erkenntnis, 84, 1–19.

    Google Scholar 

  28. Ogbu, J. U. (1978). Minority education and caste: The American system in cross-cultural perspective. New York: Academic Press.

    Google Scholar 

  29. Paolacci, G., & Chandler, J. (2014). Inside the turk: Understanding mechanical turk as a participant pool. Current Directions in Psychological Science, 23(3), 184–188.

    Article  Google Scholar 

  30. Rubin, H., & O’Connor, C. (2018). Discrimination and collaboration in science. Philosophy of Science, 85(3), 380–402.

    Article  Google Scholar 

  31. Schelling, T. C. (1980). The strategy of conflict. Cambridge: Harvard University Press.

    Google Scholar 

  32. Sigmund, K., Hauert, C., & Nowak, M. A. (2001). Reward and punishment. Proceedings of the National Academy of Sciences, 98(19), 10757–10762.

    Article  Google Scholar 

  33. Skyrms, B. (1994). Sex and justice. The Journal of philosophy, 91, 305–320.

    Article  Google Scholar 

  34. Smith, V. L. (1976). Experimental economics: Induced value theory. The American Economic Review, 66(2), 274–279.

    Google Scholar 

  35. Stewart, Q. T. (2010). Big bad racists, subtle prejudice and minority victims: An agent-based analysis of the dynamics of racial inequality. Annual Meeting of the Population Association of America.

  36. Tajfel, H. (1970). Experiments in intergroup discrimination. Scientific American, 223(5), 96–102.

    Article  Google Scholar 

  37. Taylor, P. D., & Jonker, L. B. (1978). Evolutionary stable strategies and game dynamics. Mathematical Biosciences, 40(1–2), 145–156.

    Article  Google Scholar 

  38. Tilly, C., et al. (1998). Durable inequality. Berkeley: University of California Press.

    Book  Google Scholar 

  39. Weibull, J. W. (1997). Evolutionary game theory. Cambridge: MIT Press.

    Google Scholar 

  40. Wylie, A. (2013). Why standpoint matters. Science and other cultures (pp. 34–56). Abingdon: Routledge.

    Google Scholar 

  41. Young, H. P. (1993). An evolutionary model of bargaining. Journal of Economic Theory, 59(1), 145–168.

    Article  Google Scholar 

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This research was supported by NSF Grant 1535139. Thanks to the UC Irvine Social Dynamics seminar for feedback, to the ESSL lab at UC Irvine, to Nicholas Smith, Mike Ashfield, and to Michael McBride. Many thanks to participants at the California Philosophy Workshop for comments, especially Wendy Salkin, Joshua Armstrong, David Plunkett, Gabbrielle Johnson, and Karl Schafer. Thanks also to audiences at the Philosophy, Politics, and Economics Society meeting in New Orleans, the Agent-Based Models in Philosophy conference in Bochum, the Complex Systems seminar in Moscow, Idaho, and the APA Pacific Division meeting in Vancouver. Special thanks to Justin Bruner for inspiration, and for feedback.

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Correspondence to Aydin Mohseni.

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Appendix A: Experiment data

All of the data from our experiments can be found, registered at the Open Science Framework, at the following URL:

Appendix B: Plots of individual and group behavior


Appendix C: Screenshots of the experiment interface


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Mohseni, A., O’Connor, C. & Rubin, H. On the emergence of minority disadvantage: testing the cultural Red King hypothesis. Synthese 198, 5599–5621 (2021).

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  • Social epistemology
  • Social justice
  • Evolutionary game theory
  • Experimental economics
  • Red King hypothesis