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Cooperation, correlation and the evolutionary dominance of tag-based strategies

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Abstract

Cooperation in the prisoner’s dilemma is possible if interactions are sufficiently correlated. We show that when conditions favorable to the evolution of cooperation hold (rb > c) tag-based strategies dominate. Thus, well-meaning interventions aimed at promoting cooperation may succeed but will often lead to in-group favoritism and ethnocentric behavior. Exploring ways that promote cooperation but do not usher in tag-based strategies should be a focal point of future work on the evolution of cooperation.

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Notes

  1. For more on Hamilton’s rule see (Birch 2017; Skyrms 2002; Sober 1992) and Bergstrom (2003).

  2. More generally, if there are N strategies in the population, then the probability an agent interacts with an agent using strategy i conditional on them using strategy i is \(x_{i} + r\left( {1 - x_{i} } \right)\). The probability an agent interacts with someone using strategy j conditional on them using strategy i is \(x_{j} - rx_{j}\).

  3. “It makes no difference if altruists settle with altruists because they are related... or because they recognize fellow altruists as such, or settle together because of the pleiotropic effect of some gene on habitat preference.” (Hamilton 1975).

  4. See (Hales 2000, 2005; Riolo et al.2001).

  5. As this passage suggests, tags often refer to permanent or semi-permanent visible features of the individual. In this sense they are analogous to what is often referred to as a ‘cue’ in the animal communications literature.

  6. For a very accessible introduction to the biological basis of in-group favoritism see (Sapolsky 2017). See also (Greene 2014) for a discussion of cooperation, in-group favoritism and morality.

  7. Keller and Ross (1998) were the first to document the existence of a greenbeard gene in red fire ants. Soon after this a green beard gene was identified in slime mold (Queller et al. 2003).

  8. There has been some recent discussion of green-beards in the philosophy of biology literature. See Heath and Rioux (2018) and Stanford (2018). Handfield et al. (2018) provides an interesting response to Stanford (2018).

  9. See (Robson 1990). In particular, the strategy Green (Purple) Unconditional Defect is not an evolutionarily stable strategy. More generally, any population consisting entirely of unconditional defectors can be invaded by conditional cooperators using a novel tag.

  10. From here on we refer to ‘\(r\)’ as the level of assortment due to the external correlation mechanism. This is worth pointing out because, as we noted in Sect. 3, tags are themselves a way of ensuring some level of assortment. Thus, the actual level of correlation can be above \(r\) because we are exploring a scenario where two correlation mechanisms are at work.

  11. See García et al. (2014) for more on these cycles.

  12. We assume that mutants are so rare they have no impact on the fitness of the natives (that is, \(x_{m} \approx 0\), where \(x_{m}\) is the proportion of mutants). Thus, our native Green universalists and conditional cooperators secure a payoff of b. Mutants, on the other hand, are likely to interact with both types of natives and, due to correlation mechanism, fellow mutants. The probability a mutant meets another mutant is just \(x_{m} + r(1 - x_{m} )\). Since we assume \(x_{m}\) is extraordinarily small, this is essentially equal to \(r\). The probability of meeting a Green conditional cooperator is thus \(\left( {1 - r} \right)x_{gc}\) while the probability of interacting with a Green universalists as a mutant is \(\left( {1 - r} \right)(1 - x_{gc} ).\) The expected fitness of a mutant (Purple conditional cooperator) is thus \(rb + \left( {1 - r} \right)cx_{gc} + \left( {1 - r} \right)\left( {b + c} \right)\left( {1 - x_{gc} } \right)\), which simplifies to \(c + (1 - x_{gc} )b\) and is greater than \(b\) (fitness of natives) when \(x_{gc} > \frac{c}{b}.\)

  13. In particular, we observe the proportion of conditional cooperators after 10,000 generations, meaning the population has settled at the polymorphic equilibrium involving conditional cooperators and universal cooperators of the same tag. In all 10,000 simulation runs the population settles at the cooperative equilibrium.

  14. In particular, a Green universalist is exploited by Green traitors, Green and Purple unconditional defectors, Purple conditional cooperators.

  15. A Green conditional cooperator is exploited by Green unconditional defectors but take advantage of Purple unconditional cooperators and Purple traitors.

  16. For instance, if strategic type i is 7/8 of the population they lose 7/8 \(\mu\) to mutations but only gain \(\mu\)/8 due to mutations.

  17. What occurs if not all mutations occur with the same probability? Consider the case where it is easier to mutate one’s strategy than one’s tag. In this setting universalists are on slightly better footing—they are not exploited as frequently since there are fewer members of the out-group. Thus we’d expect the balance between universalists and conditional cooperators at the cooperative equilibrium to be closer to a fifty-fifty split.

  18. See (Smead and Forber (2013)) as well as Forber and Smead (2014) for more on negative assortment and the evolution of social behavior.

  19. Recall that tags-based strategies are one source of correlation, so the actual level of positive correlation in the model may be lightly above r.

  20. Whether conditional cooperators or unconditional cooperators constitute a majority at the cooperative equilibrium hinges on whether the hare hunting equilibrium is risk-dominant.

  21. Bruner (2019) finds that splits that disadvantage minority groups are overwhelmingly likely in many cases. Following up on this, O’Connor notes that inequitable divisions are more likely if the agents are also risk-averse. Finally, Mohseni et al. observe that discriminatory unequal splits are likely to emerge in the economics laboratory.

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  1. Department of Political Economy and Moral Science, University of Arizona, Tucson, USA

    Justin P. Bruner

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Appendix

Appendix

In this short appendix we introduce the model of negative correlation discussed in section "Negative assortment" and derive stability conditions for the polymorphic equilibrium. Negative correlation occurs when individuals are less likely than random chance to interact with another of the same strategic type. Formally, this means we modify the replicator dynamic by once again altering the probabilities with which two agents meet. Assume focal agent is of type T1. With probability \(\left( {1 - e} \right)\Pr \left( {T1} \right)\) this individual interacts with a fellow T1 individual. The focal agent interacts with an individual of type T2 with probability \(\left( {1 - e} \right)\Pr \left( {T2} \right) + e\Pr \left( {T2} \right)/\left( {1 - \Pr \left( {T1} \right)} \right)\).

As mentioned, we uncover a polymorphic equilibrium at which the population is evenly split between Green and Purple ‘traitors’. We consider the expected payoff associated with various mutants at the polymorphic equilibrium. A mutant unconditional cooperator is exploited by in-group members and cooperates with out-group members, resulting in an expected payoff of b/2. Conditional cooperators, on the other hand, are exploited by in-group members and exploit out-group members, resulting in an expected payoff of \(\left( {b + c} \right)/2\). Finally, unconditional defectors secure the highest payoff of \(\frac{b}{2} + c\). We determine when natives can secure a higher payoff than a mutant unconditional defector. Without loss of generality, a Purple traitor at the polymorphism interacts with in-group members with probability \(\left( {1 - e} \right)/2\) and with out-group members with probability \(\frac{1 - e}{2} + e\), meaning her expected payoff at the polymorphic equilibrium is \(\frac{{\left( {1 - e} \right)c}}{2} + \left[ {\frac{1 - e}{2} + e} \right]b\). This expectation is larger than b/2 + c whenever \(e > c/\left( {b - c} \right)\).

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Bruner, J.P. Cooperation, correlation and the evolutionary dominance of tag-based strategies. Biol Philos 36, 24 (2021). https://doi.org/10.1007/s10539-021-09799-x

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