Super-additive cooperation | Nature

A sequential social dilemma with a continuous action space

In the models, pairs of players play a social dilemma. Players choose how much to cooperate from a continuous action space, and moves are sequential. The game is thus an apt description of many social dilemmas past and present. Food sharing45,46 and alloparental care47, for example, must be sequential social dilemmas with continuous action spaces. They are not simple prisoner’s dilemma games in which players simultaneously decide to defect fully or cooperate fully. The emphasis on continuous action spaces is not trivial. As results from the repeated interactions scenario show, intuitions honed on the analysis of reciprocal altruism in repeated prisoner’s dilemma games48 do not extend to settings where cooperation can vary continuously. Moreover, by developing models and experiment (see below) with parallel designs, we recruit the complementary strengths of both methods in a way that renders the link between theory and empiricism transparent49. We do not need a vague intermediate step where we extrapolate from models based on one type of social interaction to experiments involving another type of social interaction, with misleading predictions as a result44.

With respect to the stage game in the models (supporting information section 126), each player has an endowment normalized to one. The first mover can transfer any amount up to and including her full endowment to the second mover, and the transfer is doubled. Then, the second mover can transfer any amount up to and including her full endowment to the first mover, and this transfer is also doubled. Because transfers are doubled, expected relatedness cannot explain cooperation. Given what we know about average relatedness within groups in small-scale societies22, efficiency gains would have to be much higher than this for relatedness alone to be adequate.

A one-shot interaction is one stage game. Repeated interactions consist of repeated stage games, where each repetition involves new endowments. An individual’s strategy has two parts, an initial transfer and a response function. The initial transfer specifies how much the individual transfers, if first mover, for the first interaction only. For all choices after the initial transfer, the response function specifies an individual’s current transfer as a function of her partner’s most recent transfer. Specifically, the second mover always responds to the first mover’s transfer in the same interaction. If interactions are repeated, from the second interaction onward, the first mover responds to the second mover’s transfer from the preceding interaction (supporting information sections 2.1.3, 2.2 and 2.326).

The three scenarios

The repeated interactions scenario consists of models of populations subdivided into 40 groups of 24 individuals each without any competition between groups. Individuals within groups pair off randomly to play the game. Individuals only play the social dilemma with ingroup partners, and we consider both one-shot games and repeated interactions (supporting information section 2.1.426). Because individuals only play with ingroup partners, the repeated interactions scenario isolates the effects of repeated interactions and the reputational concerns they create from the effects of intergroup competition and more generally outgroup interactions of all sorts. We ignore uncertainty about whether a game is one-shot or repeated2,3, which maximizes the scope for repeated interactions to support cooperation when relationships actually do last a long time.

The group competition scenario also consists of models in subdivided populations. In this scenario, however, groups compete, and games are always one-shot. Groups are paired within a generation (supporting information section 2.1.526). Each individual plays both a one-shot social dilemma with a randomly selected ingroup partner and a one-shot social dilemma with a randomly selected outgroup partner from the paired group. The individual has separate strategies for ingroup versus outgroup interactions. The opportunity to cooperate with outgroup partners in our models is different from most evolutionary models of parochialism because most models limit attention to outgroup strategies that range from defection to outright aggression34. Defection in these models is the most generous feasible option for an outgroup interaction.

After game play, we model the occurrence of group competitions by assuming that paired groups compete against each other with relatively low probabilities (supporting information section 2.1.726) that decrease as the groups become more similar (supporting information section 2.1.526). This approach reflects the idea that paired groups assess each other and avoid competing when they have trouble identifying the probable winner, which is consistent with both past modelling work and ethnographic evidence4,50. We can think of a competition as a violent conflict, a competition for some limited resource, or a process where the culture of one group displaces the culture of another group31. In general, the group competition scenario isolates the effects of intergroup competition from the effects of repeated interactions and associated reputational concerns within groups. The joint scenario combines both repeated interactions within groups and competition between groups (supporting information section 2.1.626). It is identical to the group competition scenario with one exception; ingroup interactions are always repeated.

A framework for comprehensive variation in model structure

To develop a set of models that examine a wide range of potential ancestral conditions, we cross the six model characteristics below in all possible combinations.

The dimensionality of strategy space (all scenarios)

We vary the dimensionality of the strategy space as a way of manipulating the set of possible strategies. When a strategy is two-dimensional, it consists of an initial transfer and a second quantity controlling the slope and location of a linear response function (supporting information sections 1.1 and 2.226). Possible response functions include perfect reciprocity, escalating reciprocity, and de-escalating reciprocity. A perfectly reciprocal response function means a focal individual’s transfer is exactly the same as her partner’s most recent transfer (Fig. 1). When two perfect reciprocators interact, all transfers are identical to the initial transfer of the first mover. Escalating reciprocity means the focal player increases the degree of cooperation when possible (Fig. 1a), and unconditional full cooperation is an extreme case. When two escalators interact repeatedly, they converge on full cooperation, and in this sense escalation is a cooperative form of reciprocity. De-escalating reciprocity means the focal player decreases the degree of cooperation when possible (Fig. 1b), and unconditional full defection is an extreme case. When two de-escalators interact repeatedly, they converge on full defection, and thus de-escalation is an uncooperative form of reciprocity.

In a three-dimensional strategy space, a strategy consists of an initial transfer, as well as left and right intercepts for a linear response function (supporting information sections 1.2 and 2.126). Three dimensions allow for all the strategies feasible in two dimensions, but with a number of additional possibilities. For example, three dimensions allow for ambiguous reciprocity. Ambiguous reciprocity means the focal player has a non-negatively sloped response that escalates low transfers and de-escalates high transfers (Fig. 1c). If an ambiguous reciprocator interacts repeatedly with a partner having any positively sloped response function, the players converge on intermediate levels of cooperation (supporting information section 1.2.826). A four-dimensional strategy space adds strategies involving a wide range of non-linear response functions (supporting information section 2.326). Some of the new possibilities include non-linear analogues of ambiguous reciprocity (Fig. 1d). New possibilities also include non-linear forms of reciprocity that do the opposite of ambiguous strategies by de-escalating low transfers and escalating high transfers (Fig. 1d). Such strategies punish low transfers with even lower transfers and reward high transfers with even higher transfers.

Cancellation effects at the individual level (all scenarios)

When a population is subdivided into groups and some individuals remain in the groups where they were born, relatedness within groups is present. When individuals play the social dilemma with ingroup partners, this relatedness allows cooperators to channel the benefits of cooperation towards other cooperators. Relatedness within groups can support the evolution of ingroup cooperation as a result, but it does not necessarily do so. Life history details, demography, and local ecological conditions can offset the effects of related individuals playing the game together51. Offsetting effects of this sort are cancellation effects at the individual level. Our models vary these cancellation effects by relying on two different life cycles (supporting information section 2.1.226). In one case, the order of events in the life cycle is birth, game play, migration, group competition when relevant, and finally individual selection within groups. Game play and individual selection are decoupled. Individuals play the ingroup social dilemma with partners who are on average related to some extent. Relatedness increases the probability that cooperators end up playing together, which supports mutual cooperation. However, when individuals later compete within the group to reproduce, they compete against a different set of individuals precisely because migration occurs after game play but before individual selection. The timing of migration decouples the patterns of relatedness that hold when individuals play the social dilemma from the patterns of relatedness that hold when individuals compete to reproduce. As a result, related cooperators impose the gains from mutual cooperation as a relative advantage on others who are unrelated.

In the other case, the life cycle is birth, migration, game play, group competition when relevant, and individual selection within groups. Under this life cycle, game play and individual selection are coupled. Relatedness within groups ensures that cooperators are relatively likely to play with other cooperators. However, because migration occurs before game play, not after, cooperators who play together also end up competing against each other to reproduce. This cancels, to some extent, the degree to which relatedness supports the evolution of cooperation24,25. In our case, this cancellation effect at the individual level does not completely offset the value of playing the social dilemma with relatives. Under both life cycles, the evolution of cooperation increases with relatedness, though the effect is weak. Playing the game with related partners thus provides some limited support for the evolution of cooperation (supporting figures 15 and 1626). That said, cancellation effects at the individual level also play a role in the following precise sense. In models without group competition, the decoupled life cycle supports more cooperation than the coupled life cycle (supporting figures 15 and 1626).

Importantly, in terms of the link between game play and individual selection, decoupling is a relative concept. Under the decoupled life cycle, related cooperators who play the social dilemma together might still end up competing against each other at the selection stage. This outcome is possible simply because, even when migration rates are high, some individuals remain in the natal group. Thus, two individuals who play the game together may both stay in the same group and end up competing to reproduce later. The timing of migration does not completely eliminate this possibility because not everyone migrates. Instead, the decoupled life cycle ensures that individuals who play the social dilemma together are less likely to compete against each other than they would be under the coupled life cycle.

Cancellation effects at the group level (group competition and joint scenarios)

Cancellation effects can also operate at the group level8, and the intuition parallels that at the individual level precisely. Imagine a competition between two groups, one group composed of cooperative individuals and the other of uncooperative individuals. The cooperative group wins and replaces the losing group with a descendant group that is also relatively cooperative. If the parent and descendant groups go on to compete with two entirely different groups in the subsequent generation, both groups are relatively likely to compete against less cooperative groups and thus win their respective competitions. This maximizes the extent to which the group-level benefits of cooperation support the evolution of cooperation via group selection. In contrast, if the parent and descendant groups go on to compete against each other, then two cooperative groups compete against each other, with neither enjoying a relative advantage. This cancels the effects of the group-level benefits that result from both groups having many cooperative individuals.

Apart from a recent and important exception8, multi-level selection models are like the former example. However, if ancestral human groups did not rove freely across the landscape in search of new competitions, which seems entirely plausible, ancestral conditions were at least somewhat like the latter example. To examine this distinction, we use a novel approach to manipulate cancellation effects at the group level (supporting information section 2.1.226). The 40 groups in a population constitute a population of groups. In each generation groups are paired and have a competition with positive probability. We can interpret this setting as one in which paired groups occupy adjacent territories that place the two groups in close contact. At the beginning of each generation, Ξ  {0, 20, 40} groups are randomly selected to enter a pool of migrating groups that move around in space. These migrating groups are randomly redistributed to the open territories. The population of groups is well mixed when Ξ = 40. Groups move around a lot, and groups that win intergroup competitions are relatively unlikely to compete against their descendant groups in the subsequent generation. This minimizes cancellation effects at the group level. Anchoring the opposite extreme, Ξ = 0, which means groups never move. This maximizes group-level cancellation effects.

The importance of differences in aggregate resources between groups (group competition and joint scenarios)

If paired groups engage in a group competition, as explained above, the group with more resources may or may not win the competition. Specifically, the probability of winning can be more or less sensitive to the difference in total resources between the two groups. We consider four levels of sensitivity (supporting information section 2.1.526) controlled by the parameter λ {0, 10, 25, 100}. If λ = 0, which group wins is unrelated to the difference in total resources. Groups compete in this case, but outcomes are unsystematic. Therefore, group selection cannot occur, and in this sense λ = 0 is effectively like the repeated interactions scenario. As λ values increase, the group with more resources is increasingly likely to win, and the group competition and joint scenarios are increasingly different from the repeated interactions scenario.

Migration rates (all scenarios)

We vary the migration rate and by extension the relatedness within groups by allowing either 8 or 16 out of 24 individuals to migrate (mj) per group per generation (supporting information sections 2.1.4−2.1.6 and 2.1.1926).

Initial conditions (all scenarios)

In the initial generation, we seed the population with either (1) perfect reciprocators who initially transfer the full endowment, (2) unconditionally selfish individuals, or (3) individuals having random strategies drawn from a uniform distribution over the strategy space (supporting information section 2.1.826). Perfect reciprocators start by transferring the maximum possible amount, if first mover, in the first interaction. For all subsequent choices, perfect reciprocators do exactly what their partners just did. In other words, they match the most recent transfers of their partners measure for measure. Seeding the population with perfect reciprocators represents initial conditions that are favourable for the evolution of cooperation, while seeding the population with unconditionally selfish individuals represents initial conditions that are unfavourable.

Altogether, the three scenarios and six model characteristics yield 936 combinations. For each combination, we simulated 50 independent populations. In the main paper we focus on simulation results based on three-dimensional strategies. We occasionally discuss analytical results and simulation results based on two- and four-dimensional strategies. We especially do so for the repeated interactions scenario, where the dimensionality of the strategy space is decisive (supporting information section 1.2.1226). The supporting information26 includes additional results and analyses, including those that go beyond the core project outlined here, and we also mention these results in the main paper as appropriate.

Adding mistakes

The main paper presents results based on models that assume individuals never make mistakes. Theory based on repeated play of the standard prisoner’s dilemma suggests this may not be an innocent assumption. Without mistakes, different cooperative strategies can drift in and out of the population because the strategies in question lead to identical choices36,52. The population eventually drifts towards some mix of cooperative strategies that is vulnerable to invasion by an uncooperative strategy, and cooperation collapses. With mistakes, however, these same cooperative strategies no longer generate identical choices. Drift accordingly plays a reduced role, and mistakes can stabilize a specific cooperative strategy from among a glut of cooperative strategies52.

Because of the potential importance of mistakes, we added mistakes and repeated our entire simulation study. A mistake occurs when an actual transfer deviates from the transfer stipulated by an individual’s strategy. We implemented mistakes by distributing actual transfers around the stipulated transfer (supporting information section 526). Mistakes are thus common, but they vary in magnitude. For three-dimensional and four-dimensional strategies, results remain, in effect, exactly the same. In the two-dimensional case, under repeated interactions as a stand-alone mechanism, mistakes dramatically slow down the invasion of cooperative strategies compared to otherwise identical situations without mistakes. As a result, over long but finite time scales, repeated interactions cannot support the evolution of cooperative strategies even when strategies are two-dimensional. This limitation opens the door for group competitions to interact positively with repeated interactions, which is exactly what happens (supporting information section 5.326). Mistakes thus expand the range of conditions that lead to the evolution of super-additive cooperation. Future research could vary the structure of mistakes when actions are continuous to see how robust this conclusion is.

Sequential social dilemma in Papua New Guinea

We conducted our experiment with members of Perepka and Ngenika groups, two horticultural groups in the Western Highlands of Papua New Guinea (supporting information section 626). The Western Highlands are an ideal place to evaluate evolutionary theories of human cooperation because the people who live there, relatively speaking, are beyond the reach of state institutions. Social preferences, local norms, reciprocity and group affiliation are the main forces that govern social life. These forces were probably pervasive for much of the human evolutionary past, and so they are the primary points of contention with respect to the evolution of human cooperation. By contrast, the enforceable contracts and legal institutions of contemporary large-scale societies introduce additional forces that are recent in evolutionary terms. This can confuse the interpretation of empirical findings by confounding ancestral psychologies with incentives, norms, and expectations tied to contemporary institutions.

At the time of the experiment, the Ngenika and Perepka groups inhabited territories separated by about 30 km in the Western Highlands. Although each group was aware of the other’s existence, no one had any memory of hostilities between the two groups. With adult participants, we implemented a sequential social dilemma that included both ingroup and outgroup pairings (supporting information sections 7 and 826). One author (H.B.) grew up and lived in the local area for 15 years, speaks the local language (Tok Pisin) fluently, and has a detailed knowledge of the values and cultural practices of local populations. This knowledge ensured that the experiments could be conducted in the local language and in a manner respectful of local cultures. Participants provided informed consent verbally. The Internal Review Board of the Faculty of Business, Economics and Informatics at the University of Zurich approved the study.

The players in a pair were each provided with an endowment of five Papua New Guinean Kina. This endowment was roughly half of a high daily wage for a labourer in the informal sector of the local workforce. Most participants earned less than this daily wage on average because they were not working for money on a regular basis. After receiving the endowment, the first mover in a pair transferred some amount between zero and five Kina, in increments of one Kina, to the second mover. The experimenter doubled this transfer. Before learning the amount actually transferred, the second mover specified an amount she wished to back transfer to the first mover for each of the first mover’s possible transfer levels, yielding six observations per second mover. This is the strategy method of eliciting second mover responses, and previous research has shown it to be a reliable method for measuring behavioural strategies53. After eliciting the second mover’s strategy, the experimenter revealed the amount actually transferred by the first mover and implemented the appropriate back transfer. The experimenter also doubled the back transfer.

Using a between-subjects design, we implemented four treatments that differed in terms of the group affiliations of the two players. We varied affiliations in all combinations, which yielded two ingroup treatments (Ngenika/Ngenika and Perepka/Perepka) and two outgroup treatments (Ngenika/Perepka and Perepka/Ngenika). We used no statistical methods to pre-determine sample size (see Reporting Summary). All players knew the rules of the game. Each player also knew the group affiliation of her partner. The experimenter mediated all interactions in private, and so interactions were anonymous apart from information about group affiliations.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

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