Reciprocity phase in various 2×2 games by agents equipped with two-memory length strategy encouraged by grouping for interaction and adaptation

This paper numerically investigates 2 × 2 games involving the Prisoner's Dilemma, Chicken, Hero, Leader, Stag Hunt, and Trivial Games in which agents have a strategy expressed by five-bit, two-memory length. Our motivation is to explore how grouping for game interaction and strategy adaptation influence ST reciprocity and R reciprocity (Tanimoto and Sagara, 2007a [Tanimoto, J., Sagara, H., 2007a. A study on emergence of coordinated alternating reciprocity in a 2 × 2 game with 2-memory length strategy. Biosystems 90(3), 728-737]. Enhanced R reciprocity is observed with the stronger grouping for game interaction when a relatively stronger grouping for strategy adaptation is assumed. On the other hand, enhanced ST reciprocity emerged with the stronger grouping for strategy adaptation when the relatively weaker grouping for game interaction is imposed. Our numerical experiment deals with those two groupings independently and dependently.     

An inclusive fitness analysis of synergistic interactions in structured populations

We study the evolution of a pair of competing behavioural alleles in a structured population when there are non-additive or ‘synergistic’ fitness effects. Under a form of weak selection and with a simple symmetry condition between a pair of competing alleles, Tarnita et al. provide a surprisingly simple condition for one allele to dominate the other. Their condition can be obtained from an analysis of a corresponding simpler model in which fitness effects are additive. Their result uses an average measure of selective advantage where the average is taken over the long-term—that is, over all possible allele frequencies—and this precludes consideration of any frequency dependence the allelic fitness might exhibit. However, in a considerable body of work with non-additive fitness effects—for example, hawk–dove and prisoner''s dilemma games—frequency dependence plays an essential role in the establishment of conditions for a stable allele-frequency equilibrium. Here, we present a frequency-dependent generalization of their result that provides an expression for allelic fitness at any given allele frequency p. We use an inclusive fitness approach and provide two examples for an infinite structured population. We illustrate our results with an analysis of the hawk–dove game.     

Gradients for the evolution of bimatrix games

The evolutionary dynamics of bimatrix games is studied for rescaled partnership games and zero sum games. The former case leads to gradient systems. The selection equations for sexual and asexual reproduction of genotypes corresponding to mixed strategies are analysed. As examples, the origin of anisogamy and cyclic chases for predator-prey coevolution are studied.     

Inclusive fitness in finite populations—effects of heterogeneity and synergy

I review recent results concerning the relationship between the inclusive fitness (IF) effect and standard measures of allele fitness in a finite‐population, with attention to the effect of heterogeneity in population structure and nonadditive fitness effects. In both cases, existing theoretical work is somewhat technical and I try to provide a more transparent account. In a heterogeneous population it is known that inclusive fitness will generally fail to incorporate the effects of selection on the distribution of alleles among states unless a reproductive‐value weighting is used. But even given that, recent work shows that under certain updating rules, the IF effect can fail to be equivalent to standard measures such as fixation probability. In terms of synergistic fitness effects, I review the result that in the finite population model, the IF effect can be calculated using only “additive” relatedness coefficients so that computational difficulties found in the infinite‐population model do not arise. In my own work, there is an interaction here in that my 2012 work on synergy with Maciejewski made an assumption about inclusive fitness that my 2014 work on heterogeneity with Tarnita showed to be wrong. I include (Appendix C) a corrected argument for the 2012 result.     

One-shot reciprocity under error management is unbiased and fragile

The error management model of altruism in one-shot interactions provides an influential explanation for one of the most controversial behaviors in evolutionary social science. The model posits that one-shot altruism arises from a domain-specific cognitive bias that avoids the error of mistaking a long-term relationship for a one-shot interaction. One-shot altruism is thus, in an intriguingly paradoxical way, a form of reciprocity. We examine the logic behind this idea in detail. In its most general form the error management model is exceedingly flexible, and restrictions about the psychology of agents are necessary for selection to be well-defined. Once these restrictions are in place, selection is well defined, but it leads to behavior that is perfectly consistent with an unbiased rational benchmark. Thus, the evolution of one-shot reciprocity does not require an evoked cognitive bias based on repeated interactions and reputation. Moreover, in spite of its flexibility in terms of psychology, the error management model assumes that behavior is exceedingly rigid when individuals face a new interaction partner. Reciprocity can only take the form of tit-for-tat, and individuals cannot adjust their behavior in response to new information about the duration of a relationship. Zefferman (2014) showed that one-shot reciprocity does not reliably evolve if one relaxes the first restriction, and we show that the behavior does not reliably evolve if one relaxes the second restriction. Altogether, these theoretical results on one-shot reciprocity do not square well with experiments showing increased altruism in the presence of payoff-irrelevant stimuli that suggest others are watching.     

More 'altruistic' punishment in larger societies

If individuals will cooperate with cooperators, and punish non-cooperators even at a cost to themselves, then this strong reciprocity could minimize the cheating that undermines cooperation. Based upon numerous economic experiments, some have proposed that human cooperation is explained by strong reciprocity and norm enforcement. Second-party punishment is when you punish someone who defected on you; third-party punishment is when you punish someone who defected on someone else. Third-party punishment is an effective way to enforce the norms of strong reciprocity and promote cooperation. Here we present new results that expand on a previous report from a large cross-cultural project. This project has already shown that there is considerable cross-cultural variation in punishment and cooperation. Here we test the hypothesis that population size (and complexity) predicts the level of third-party punishment. Our results show that people in larger, more complex societies engage in significantly more third-party punishment than people in small-scale societies.     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

Dynamics of Vaccination Strategies via Projected Dynamical Systems

Previous game theoretical analyses of vaccinating behaviour have underscored the strategic interaction between individuals attempting to maximise their health states, in situations where an individual's health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here, we extend such analyses by applying the theories of variational inequalities (VI) and projected dynamical systems (PDS) to vaccination games. A PDS provides a dynamics that gives the conditions for existence, uniqueness and stability properties of Nash equilibria. In this paper, it is used to analyse the dynamics of vaccinating behaviour in a population consisting of distinct social groups, where each group has different perceptions of vaccine and disease risks. In particular, we study populations with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). We find that a population with a vaccine-inclined majority group and a vaccine-averse minority group exhibits higher average vaccine coverage than the corresponding homogeneous population, when the vaccine is perceived as being risky relative to the disease. Our model also reproduces a feature of real populations: In certain parameter regimes, it is possible to have a majority group adopting high vaccination rates and simultaneously a vaccine-averse minority group adopting low vaccination rates. Moreover, we find that minority groups will tend to exhibit more extreme changes in vaccinating behaviour for a given change in risk perception, in comparison to majority groups. These results emphasise the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by PDS and VI in mathematical epidemiology.     

Optimal Timing of Disease Transmission in an Age-Structured Population

It is a common medical folk-practice for parents to encourage their children to contract certain infectious diseases while they are young. This folk-practice is controversial, in part, because it contradicts the long-term public health goal of minimizing disease incidence. We study an epidemiological model of infectious disease in an age-structured population where virulence is age-dependent and show that, in some cases, the optimal behavior will increase disease transmission. This provides a rigorous justification of the concept of “endemic stability,” and demonstrates that folk-practices may have been historically justified.     

Population dynamics with a stable efficient equilibrium

We propose a game-theoretic dynamics of a population of replicating individuals. It consists of two parts: the standard replicator one and a migration between two different habitats. We consider symmetric two-player games with two evolutionarily stable strategies: the efficient one in which the population is in a state with a maximal payoff and the risk-dominant one where players are averse to risk. We show that for a large range of parameters of our dynamics, even if the initial conditions in both habitats are in the basin of attraction of the risk-dominant equilibrium (with respect to the standard replication dynamics without migration), in the long run most individuals play the efficient strategy.