Evolution of altruistic punishments among heterogeneous conditional cooperators

The above developed intuition is converted into an agent-based evolutionary model in the context of public goods provision. In the proposed evolutionary agent-based model⁴², all the agents play a linear public goods game by using conditional cooperative strategies, enforcing altruistic punishments based on relative differences in their cooperation tendencies, and imitating successful role models’ social behavior with certain errors. The process is iterated several thousands of generations.

Population type

In the proposed model, the individuals or agents in the population behave like conditional cooperators and the population is heterogeneous in its conditional nature. The agents who are more willing to cooperate are also more willing punish to potential free riders. Each individual is born with an arbitrary conditional cooperative criterion (CCC) and a propensity, β. Both are positive values. The agents with higher CCC donate less frequently than the agents with a higher CCC for the given same amount of past cooperation levels. The same agents can cooperate or enforce altruistic punishments or free ride given the past cooperation levels in the population. β indicates the propensity to implement a conditional cooperative decision and imitate the successful role model’s social behavior. Each agent’s CCC value is drawn from a uniform distribution (0, N), where N is the population size, and β is drawn from a uniform distribution (0, 3). With β = 0 the actions of the individuals are random and with β = 3 the individuals behave like ideal conditional cooperators. With intermediate values the individuals behave like non-ideal conditional cooperators. The consideration is equal to the natural selection designing the conditional cooperative strategies. The combinations of CCC and β create heterogeneous populations with varieties of propensities. The consideration is close to the conditional nature of the population observed in experimental settings^12,36.

Conditional cooperative decision

The conditional cooperative decision of the agent is operationalized in the following way^43,44. For instance, in the rth round, an agent i (with CCC = CCC_i value) donates to the public good with probability, q_d,

n_C indicates the number of donations in the (r − 1)th round. The parameter β_i controls the steepness of the probability function. For the higher β_i, the agent is highly sensitive to the (n_C-CCC_i). For instance, as β_i → ∞, the q_d is sensitive to the sign of the (n_C –CCC_i), i.e., if (n_C –CCC_i) > 0 then q_d = 1 and if (n_C -CCC_i) < 0 then q_d = 0. Either with (n_C -CCC_i) = 0 or with β_i = 0 and both are zero, the agent donates to PGG with 50% time. In the model 0 < β_i < 3. With β_i > 2, the agent is more sensitive to the conditional rule. For β_i = 0, the agent ignores the rule and behaves randomly. In β_i < 2 and (n_C -CCC_i) < 2 the individual does not follow the conditional rule, occasionally. In this construction both agents with the same CCC value may act differently with different β_i values. For example, an agent with n_C = 25, CCC_i = 24, and β_i > 2 donates with probability close to one and β_i < 2, donates with the probability less than one. With a non-zero amount of cooperation in the zeroth round, very low CCC value agents potentially behave like altruists, middle CCC value agents behave like conditional cooperators and very high CCC value agents behave like free riders. In the model the population and the conditional cooperators are heterogeneous. The assumption is similar to the experimental and field observations in repeated public goods^9,12,36,37.

Public goods game

All the individuals in the population play a linear public goods game³⁷ three times in each generation. In each round, each agent is given an initial endowment E and individuals potentially donate an amount u_i using Eq. (1). After all the individuals make their decisions, the collective amount is enhanced by a factor (α > 1) and the resulting public goods are distributed equally among all the agents, irrespective of their contributions towards public accounts. An agent’s total payoff from each linear public goods game is calculated using the following equation.

({pi}_{Gi}) is payoff of agent i from the game. The first term (left(E-{u}_{i}right)) indicates the payoff from what was not contributed to the public goods (the private payoff). The second term indicates payoff from the public goods. Each unit donated becomes worthα > 1 unit. Due to the ‘non-excludable’ nature of public good, all the agents gain equal payoff from the public goods game. Clearly, all the individuals by donating can create an efficient PGG and by free riding increase his/her private payoff but reduce the group’s payoffs. Overall, the free riders gain relatively higher payoffs than the donors if the game starts with few initial donations. Given the same amount of cooperation level, the higher CCC agents donate less frequently and gain relatively higher payoffs than the payoffs of the lower CCC agents who donate more frequently.

Altruistic punishments

The assumptions of the model allow the following two rules: (i) the agents who are more willing to donate are also more willing to punish^9,12,37. (ii) the agents punish the free riders more frequently than the occasional free riders^12,37. The CCC value of an agent acts as a proxy measure of an individual’s donation tendency and punishment tendency. The individuals enforce altruistic punishment based on the relative CCC difference of randomly matched pairs. An agent i (with CCC_i) has the potential to punish another agent j (with CCC_j) with probability ({q}_{p}).

With (CCC_j-CCC_i) > 1 and β_i > 2 or with (CCC_j-CCC_i) > 2 and β_i > 1, the agent i punishes the agent j with high probability. With (CCC_j-CCC_i) × β_i < 1, the agent i punishes the agent j occasionally. In this construction, a lower CCC agent with β > 2 punishes a higher CCC agent more accurately than a slightly lower CCC agent. A lower CCC agent with β < 2 punishes a higher CCC agent less accurately and may not punish a slightly lower CCC agent. For example, in a random pair, with (CCC_j-CCC_i) = 1 with β_i > 2 agent i punishes the agent j with a high probability close to one and β_i < 2 punishes rarely. Depending on the differences in CCC values and β many possibilities exist. The above considerations are different from the existing models of altruistic punishments^31,34 and close to the experimental observations in public goods provision^12,37. In the population of heterogeneous conditional cooperators, the lower CCC agents donate and punish frequently, the moderate CCC agents occasionally free ride and punish, and very high CCC agents mostly free ride and do not punish.

Reproduction

After altruistic punishments, each agent’s total payoff equals to the sum of the payoff from the PGG and the potential costs incurred in imposing altruistic punishments (π_alt) and the cost paid if the punishment is received (π_in). For instance, ith agent’s payoff will be π_{i =} π_Gi + π_alt + π_in. All the agents occasionally update their strategies by pairing another randomly matched agent and adapting the role model’s strategy with a probability proportional to the payoff difference^45,46. In terms of cultural evolution each individual imitates the successful agent’s social behavior; all the agents update their CCC and β_i values simultaneously with certain mutations. The updating is done by the following procedure^4,44. An agent i potentially imitates successful individual j’s social behavior (CCC_j) and β_i with probability, q_r,

where (π_j-π_i) is the cumulative payoff difference of agents j and i respectively. β_i is ith agent’s propensity, which controls the steepness of Eq. (4). An agent with a higher β_i more accurately imitates the social behavior of the role model. In each generation the population undergoes 10% mutations, i.e., each individual miscopies successful role models’ properties with probability 0.1. From Eqs. (1) and (4) it suggests that the agents who are more willing to donate are also more willing to punish free riders.

Simulations

In the simulations, the initial propensity (β) values are drawn from uniform distribution [1, 3] and CCC values are drawn from [1, N], where N is the population size = 100 similar population size is chosen elsewhere^45,47. Agents enter into the PGG with the initial payoff (u) = 50 units and number of donations in the zeroth round = 10. Each agent decides their donation by using a stochastic conditional decision rule, Eq. (1). The donation cost is 1 unit and the enhancing factor of the collected donation is α = 3 units. The total payoff of an agent is given by Eq. (2). A generation consists of three rounds of PGG. After each generation, altruistic punishments were implemented and, subsequently, the population updates or reproduces by using Eq. (3) with 10% mutations.

Mutations are created by adding a random value, drawn from a Gaussian distribution with mean zero and s.d. = 5 (max = 50 and min =  − 50). Each agent in the population miscopies the role model properties with a probability 0.1. After updating by using Eq. (4), a randomly drawn CCC value is added to the updated CCC value and a β value is replaced by a randomly drawn value from uniform distribution of (0, 3). If the updated CCC value is greater than N, it is rounded off to N; if the resultant CCC value is negative, it is rounded off to zero. This allows the population to have unconditional free riders (CCC = N) and unconditional cooperators (CCC = 0).

The number of donations in the zeroth round is 10. A generation consists of three rounds of repeated PGGs without enforcing altruistic punishments. After each generation altruistic punishments were implemented and subsequently all the individuals in the population simultaneously update their strategies with 10% mutations. To reduce individual trial variations, each experimental condition (a fixed set of parameters) is run over 20,000 and iterated 20 times and the average data is used to plot the results. We observed that there is no difference in the results after the first few thousand generations; therefore, we have plotted donation fractions for the first 10,000 generations. We plotted the distribution of CCC and β values for different experimental conditions in the last 20,000th generation. We measured donation fractions, i.e., the fraction of donations in a specified number of generations. Asymptotic donation fractions are computed by taking donations in the last 20,000 generations of the 20,000th generation. We computed the distribution of CCC of the population and β values, which indicates the composition of population (or strategies). In simulations, we kept the following parameters constant: the donation cost (u) = 1, enhancing factor of collected donations (α) = 3. The probability of altruistic punishments after each generation is designated by w. Simulations are performed with varies w(w < 1) and inflected costs. In the model, if the punishing cost is x, the inflicted punishment cost will be 3x.

Source: Ecology - nature.com