Abstract
Reciprocal altruism is a fact. There are many within-species and between-species examples of such strategic interaction. At the same time, reciprocal altruism would appear to be impossible. In deep evolutionary time, exceedingly small differences in costs will suffice to make the difference between life and death. Genotypes carrying extra costs will perish. How could individuals carrying extra altruistic costs survive? This study searches for the answer by looking at the mathematical features of the centipede game with what we call Trivers-payoffs. The notion refers to the reciprocally asymmetrical structure of costs and benefits between two players. There is an altruist individual X who gives significant benefits to another individual Z at a negligible cost to herself, getting similarly weighted favors back from Z. (1) When relative payoffs of the centipede game are defined as Trivers-payoffs, when the game is modeled in a finite population of players, and when strategies are depicted as genetically hard-wired operational instructions, the originally lone altruist mutation is capable of surviving and reproducing for quite some time, and (2) when the lone altruist mutation gets her first partner, changes in payoffs are dramatic. The altruist mutation begins to gain in payoffs and hence also in frequency.
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21 August 2022
Correction in Footnote 2 to bring the two universal quantifiers back to the parentheses.
Notes
Gamba’s (2013) model of the evolution of altruistic preferences, and Rand and Nowak’s (2012) model of evolutionary dynamics in finite populations, come probably closest to our own (much simpler) model. Gamba analyzes a theoretical centipede game in which players featured by altruist preferences always opt for pass (here: the a-strategy), whereas the behavior of selfish types depends on their beliefs concerning the actual game situation. The share of altruists in the community pool (q)–denoting, by the same token, the probability of coming across an altruist in a given game (as the other player)–is one of the key variables in the model. The success of the two types depends on this number as well as on the beliefs held by the selfish types about the type of the other player in the game, which (the beliefs) need not be correct. It is possible for the altruists to perform better than the selfish types, in Gamba’s model. It is also possible for the equilibrium dominated by altruists to be an evolutionarily stable self-confirming equilibrium (SCE, a weaker equilibrium concept than the Bayesian Nash equilibrium, BNE). The material payoffs of Gamba’s model that actually determine the objective success of the two different types, over time, do not, however, reflect what we call Trivers-payoffs in the current paper, so her results differ markedly from the results of our study. The same applies to Rand and Nowak’s (2012) model, where the cost of altruism is very high (50.0%, in the sense that the scale is defined in the current paper, whereas in our model the cost of altruism is 0.5%). Rand and Nowak find that weak selection pressure (w) makes it possible for altruists to succeed over the long term (w = 0), whereas strong selection pressure (w = 1) makes life difficult for the altruists. Specifically, the selection pressure parameter w in their model determines how much the payoff from the game actually contributes to evolutionary fitness. As stated, Rand and Nowak’s results also differ markedly from the results in our study. In our model, high selection pressure does not prevent the evolutionary success of altruists. The survival threshold we find in our model is much lower than the ones discovered by Gamba and Rand and Nowak in their models.
In modern game theory, ‘strategy’ is defined as a complete set of instructions for how to play the game (Dixit et al., 2015, pp. 27-28). The model of this study differs from the definition in that here the strategies are pre-programmed, or fixed, for the entire game. Reference is made to a- and b-genotype players, both equipped with hard-wired operational instructions for how to play the game. Note that the interpretation is open in the sense that, while the pre-programmed strategies are here understood as genetically coded instructions for how to play the game, they may also be understood–still in the context of our mathematical model–as pre-programmed behavioral or cultural patterns of action (e.g., by mimic or memetic adoption of patterns of action) working as instructions for playing the game (see the way Dawkins, 1976, p. 69, defines ‘strategy,’ following Maynard-Smith; on memetic replicators, see Dawkins, ibidem: 189–201; on the effects of cultural accumulation of knowledge on the emergence of human cooperation, see Henrich & Muthukrishna, 2021).
It may seem odd to define and depict strategies as hard-wired operational instructions. Game theory, after all, deals with responsive, subgame-perfect, rational decision-making. Evolutionarily, however, going far enough back in time, rational analysis in the form of backward induction and common knowledge rationality had only a limited effect, or no effect at all, on strategic interaction between individual players.
However, there is more to the significance of fixed strategies than this. They are important in iterated games, especially when considering how originally unlikely equilibrium states of affairs can be maintained in strategic interaction, once they have been reached. The Folk Theorem defined for infinitely iterated prisoner’s dilemma game, originally discovered by Aumann (1959; see also, e.g., Binmore, 2004), is a good example of this. The cooperative Nash equilibrium solution characteristic of the Folk Theorem can be maintained by the players, once such a state of affairs has been reached–and they have an incentive to do so, for the equilibrium state of affairs is backed up by rational analysis concerned with the chosen (cooperative) outcome versus non-cooperative outcomes of the game (the threat points)–but not without fixed commitment by both players to the established pair of equilibrium outcomes, and corresponding fixed strategies. If this commitment breaks down, the equilibrium state of affairs breaks down and cannot be expected to be regained. Cooperation comes to an end.
There is generality to this idea: once a collective action dilemma has been solved in an iterated game, say in an iterated prisoner’s dilemma game, in an iterated tragedy of commons game, or in an iterated public goods game, it is rational for the players to fix their strategies, in order to keep to the cooperative solution. This will never solve the original collective action dilemma, but it makes it possible to defend the cooperative solution. It also shows that assuming fixed strategies with sudden, idiosyncratic shifts from one strategy to another (here from ∀b to a to ∀a), may be a technically good and empirically valid idea in modeling.
Golden jackals, for example, have better hunting success per jackal when hunting in pairs than when hunting alone (Bshary, 2010).
In a sense, the cost of altruism is still rather high. We could have realistically gone to 0.01% or to 0.001% cost structure, but settle for the 0.5% cost structure here, for the purposes of graphical representation of the results of the study.
On the assumption of fixed strategies, see endnote 2.
The exact number of individuals is not decisive from the point of view of our analysis. The number of 15 individuals, however, has come up regularly in analyses of group sizes of primates, early hominins, and modern human social networks. It appears to be one of the numbers around which group sizes actually cluster (Dunbar et al., 2018).
We note in passing that, according to Richard Wrangham (2019, pp. 103–104, more comprehensively, pp. 84–111 and passim), the key to the self-domestication of bonobos in the wild (reduction in reactive aggression) was interaction and cooperation in-between the female members of bonobo communities. In bonobo communities, females stay close to each other. Contrary to this, in chimpanzee communities, where no self-domestication has been detected, females follow the small parties that roam the territory, often as a minority within the party, and are separated from other females.
This is not an implausible proposition. The emergence of a genotype taking a new, alternative decision (a) from a binary choice (a, b) may be closer to a necessity (p = 1.0), at some point in time over several millennia, rather than an impossibility (p = 0.0). The number of modifications that evolution produces, on a continuous basis (by mutations, by sexual selection, by natural selection), is astronomical. Due to the immense range of differences in individuals, the same probably applies to behavioral, cultural, and even to rational modifications (the varying individual preferences, combined with logical inference), if you choose to think about the model in these terms.
The 2.004 payoff here is actually about 2.00357–a fact that one needs to know in wanting to compute the cumulative payoffs.
There were three players who cooperated all along, and opted for the a-strategy also in the last node of the centipede game, thus giving away some of their own payoff for the benefit of the other player. When asked for the reason for this last choice, they all answered in essentially the same manner. They said that they thought it was the other player’s cooperation that enabled them to reach the last node, so ‘(…) they felt obligated (…),’ in words of Bornstein et al. (2004, p. 602), ‘(…) to reciprocate (…).’.
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Vuorensyrjä, M. The Rise of Reciprocal Altruism–a Theory Based on the Centipede Game with Trivers-Payoffs. Evolutionary Psychological Science 9, 13–25 (2023). https://doi.org/10.1007/s40806-022-00326-z
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DOI: https://doi.org/10.1007/s40806-022-00326-z