The Blessing of Sexuality : Evolution of Altruismwith Mating Preference

Current simulation practices in artificial societies typically ignore the contribution of sexuality as a driving force for the evolution of prosocial behaviours. As recent researches in biology and genetics argued, sexual attractiveness, via the method of sexual selection, can explain many aspects of the second-order social dilemma. The basic hypothesis is that altruism is a sexually attractive virtue. To introduce the hypothesis into the analysis of human altruism, we employ the concepts of altruistic punishment and the behaviour-based sexual attractiveness to develop a gender-based evolutionarymodel wheremating preference acts as the compensation to themale punishers from females in the given public goods game. In themodel, the force of sexual selection is expressed as the e ect of mating preference on altruism. The computer simulation indicates that social cohesion can be achieved by the existence of sexuality in an artificial society where the co-evolution of mating preference, altruistic punishment and cooperation exist. We then extend the model in two ways: (1) we employ the variable size population assumption to test the invasion capacity of cooperators, and (2) individual variation in altruistic investment is introduced to replace the average population payo function in the baseline model. The variable size population and individual variation in investment are found to have amplifying e ects on the evolution of altruism from di erent perspectives. Finally, we discuss the definition of altruism in dynamic evolutionary games, as well as the gender di erences in the formation of altruism in primitive tribes.


Introduction
. Sexuality has been viewed as the nemesis of morality in nearly every traditional culture, yet in previous literatures in biology and genetics, sexuality is considered as one of the driving forces for the evolution of prosocial behaviours (Miller ).Selection in relation to sex provides a unique perspective to solve the persistent puzzle of the existence of social cooperation and altruistic behaviour (the strategy that benefits others and costs the player) that is referred to as the social dilemma (Dawes & Messick ).In other fields of virtues, biological evolutionary models showed that inter-sexual selection can explain the evolution of male-only care even if sexual selection may disfavour parental care when care comes at a net loss to fertilization success (Alonzo ), and provided templates for the virtue of altruism.

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To solve the puzzle of cooperation, researchers proposed that altruism can benefit the fitness of the cooperator's genes (Preston ).Based on this proposal, biologists have investigated in ways of inclusive fitness (Hamilton ; Dugatkin ) and various selections at di erent levels (Alexander ; Boyd & Richerson ; Sober & Wilson ; McAndrew ).Particularly, pioneering works have attempted to explain the maintenance and evolution of kin altruism (Hamilton ) and reciprocal altruism (Trivers ; Ridley ; Axelrod & Hamilton ; Sugiyama et al. ) via various forms of game theory, social selection models and empirical studies.In the field of pure altruism, the introduction of a costly altruistic punishment found in economic experiments (Fehr & Gächter ; Gintis ) paves the way for the emergence of altruism involving no genetic or direct rewards.

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A costly punishment from a cooperator is not an advantageous strategy in the game of life, given other cooperators' non-punishment strategy (Ye et al. ).This violation of causality, which is also referred to as the second-order social dilemma (Hauert et al. ), was partially solved by Hauert et al. ( ) via a dynamic evolutionary model where players could opt out of the collective action.The possibility of playing the nonparticipant strategy requires crucial assumptions concerning human society.Otherwise, the model loses its power to provide a cogent mechanism for the establishment of punishment (Boyd & Mathew ).To expand the breadth of the 'narrow way to cooperation' proposed by Hauert et al. ( ), various adjusted model and simulations have been conducted without the assumption of nonparticipation.Ja e & Zaballa ( ) proposed a new form of punishment called cooperative punishment that was contingent upon the concurrent participation of other players.With the existence of cooperative punishment, social cohesion might be achieved in computer simulations.Ye et al. ( ) introduced the moral concept of sympathy as compensation for the punisher.In Ye's model of punishment and sympathy, individuals in the society are classified into three types: cooperator, defector and punisher.They share the benefits from public goods equally while only defectors do not contribute.Punishers impose an altruistic punishment on defectors at a cost to themselves, and cooperators who do not punish the defectors directly sympathize with the punisher and bear a cost due to the sympathy.The existence of sympathy in the model operates as a stabilizer for the social cohesion, where the punishers will persistently dominate the population.

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All the adjusted models based on altruistic punishment are contributive and salient.However, taken alone they do not elucidate an understanding of how sexuality shapes the evolutionary equilibrium of altruism in human society.Besides, theoretical biologists and economists have broadly studied the benefit-cost analysis of altruism (Axelrod & Hamilton ; Axelrod ) without exploring the human motivation for selfless help (Preston ), while sexuality possibly plays the role of motivation to invest in altruism according to sexual selection theory and costly signalling theory (Gintis et al. ; McAndrew ).The ignorance of sexuality in models involving alternations of generations is usually associated with the implicit assumption of asexual reproduction, which is a tradition from the Moran process (Moran ).As has been mentioned before and noted by many, sexual selection is a key factor in the evolution of human altruism.Without the analysis of the e ect of sexual selection in altruism, the models are not complete for the simulations of human societies.

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In this paper, we propose a dynamic model of punishment and sexual selection, where the social cohesion can be achieved by the behaviour-based sexual attractiveness from males that will increase the fitness of male punishers.Sexual attractiveness via sexual selection is an analogous to the sympathy in Ye's model ( ).The fundamental di erence between Ye's sympathy and sexual attractiveness in this paper is the division between asexual and sexual reproduction.The concept of sexual attractiveness in the model opens up the novel sources to study the game of public goods, altruism and other prosocial behaviours from the perspective of sexuality.

Baseline Model
Model setup .We model the dynamic evolution of an isolated population of a constant scale N (we will modify this constant population assumption in next sections).The population is composed of four types of people including: • MP: Male Punisher • MD: Male Defector • FC: Female Cooperator • FD: Female Defector .
The strategies and behaviours of each type are described as follows: MP is altruistic; that is, he will invest I into the altruistic behaviour.The investment incurs a cost c, which is assumed to be proportional to the investment (i.e., c = αI ).However, MP also punishes defectors regardless of their genders.The strength of the punishment is δ≥c, which will incur a cost of γ < δ to MP.The population of MP is assumed to be N MP .
MD is egoistic, contributes nothing to the altruistic behaviour and hence bears no altruistic cost.However, as a member of the society, he benefits from MP and FC's contribution just.MD su ers punishments from MP.The population of MD is N MD .
FC is altruistic.She also invests I into the altruistic behaviour (we assume that I is identical in every MP and FC, and this assumption will be generalised to individual variation in the later section).FC also admires MP with the strength of s (i.e., the strength of sexual attractiveness or MP's sexual capital), while the cost incurred by the s is set to .The population of FC is N FC FD is the same as MD except for their genders.For simplicity, we assume that FD will not admire any MP.The population of FD is N FD .
. Every individual equally shares the benefit from the altruistic contribution from MP and FC.
. It should be noted that the names of 'defector' or 'cooperator' do not necessarily imply that the individuals are involved in a typical public goods game defined in the field of economics.It would be more appropriate to rename the individuals in the society as 'altruist' and 'egoist'.However, to pay tribute to the previous contribution by pioneers, we utilize their system of names in this paper.
. By the definition of birth rate by Koella ( ), the expected payo s for each individual in each of the four types in a single period are defined as where P MP , P MD , P FC ∧ P FD are the payo s of MP, MD, FC and FD, respectively.The benefit from the entire society is calculated by the mean altruistic investment where N is the total population of the society, . We adopt the exponential form for the function of fitness (i.e., f = e ωP ), where f denotes the fitness of an individual, P denotes the payo and ω denotes the selection strength ( 0 < ω < 1).The form of fitness comes from Traulsen et al. ( ).
. We adopt a generalised Moran Process adjusted for sexuality and sexual reproduction to model the evolution of the society given the assumptions above.The probability of death d is identical to each individual in the society and remains constant during each period.To keep the population unchanged, the number of newborn o springs is set to be equal to the number of the dead in the period.The process of determining the type of the o spring is completed through two steps: ( ) nature randomly decides whether the o spring is male/female with a probability of % respectively; ( ) if the o spring is a male, then his probability of being an MP (MD) is proportional to the fitness of an MP (MD), and if the o spring is a female, then her probability of being an FC (FD) is proportional to the fitness of an FC (FD).This process permits the balance of both genders.
. Additionally, we assume that any individual of the population could randomly switch to another type within the same gender with a small probability µ, which is referred to as the mutation rate.

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The outcome of the simulation is notable.Figure shows that when punishment and sexual attractiveness exist (Figure C), being altruistic is advantageous, which means that MP and FC are the dominant types in the society, whereas MD and FD are at the limit of near-extinction a er some initial periods.However, in Figure A (no punishment and no attractiveness) and B (punishment without attractiveness), the status where the egoists become dominant over the altruists (especially in the male society) is a superlative example of the second-order social dilemma in a society with no sexual attractiveness.The di erence in results between the simulations in
We perform robustness checks on various parameters and initial settings to test the power of our model.We select several important parameters to report their robust test results.The conclusion withstands the robust test on the initial proportion of the four types in the society.Figure illustrates the results of the robust test with no MP and FC at period with three di erent treatments as in Figure .
. It is necessary to clarify that the exact timing of the turning point where altruists take power in the society from the egoists is inconsistent in repeated computer simulations due to the randomness of the model dynamics.
Figure indicates the various timings of the turning points from three repeated computer simulations that report di erent evolutionary paths and routines.
. We now conduct the robust test on the strength of sexual attractiveness, s.
. In simulations where s > 0.2, MP and FC take dominance over MD and FD; in the simulation where s = 0.1, MD takes dominance over MP while FC and FD coexist.The parameters in these simulations are the same as those in Figure C, with the exception of the value of s.It should be noted that the results for s = 1 and s = 0 have been reported in previous simulations. .

Figure tests
the value of mutation rate µ, which is .in the baseline model.
. Figure shows that mutation plays the role of 'revolution', trying to redistribute the population.A smaller mutation rate can widen the gap between cooperators and defectors.
.  moral psychologists cannot compensate the punisher directly by increasing the punisher's fitness.Mate preference (or equivalently sexual attractiveness from the opposite perspective) serves as a particular form of sympathy that can increase the fitness of the individual to whom the preference is given.

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The simulation results describing the evolutionary path of each type of individual provide critical evidence for sexual attractiveness as a strong stabilizer to generate a social equilibrium where prosocial strategy (i.e., being altruistic) is advantageous in an artificial human society.The introduction of sexuality to the traditional  altruistic punishment game is beyond the scope of the previous literatures on social evolution via methods of computer simulation.We found that the introduction of sexuality and sexual attractiveness changed the social equilibrium from the sole dominance of punishers (Ye et al. ) to the coexistence of both (male) punishers and (female) cooperators, which fits reality better.

Extension to Variable Size Population
. A strict constraint in Moran process is the constant population size, which greatly simplifies the calculations for theoretical modelling and serves as a common default in the field of evolutionary simulations.However, the assumption of fixed population will be fulfilled only in models with hard resource limitations (Claussen & Traulsen ), e.g.lattice models with viscous populations (Koella ).By assuming constant population, the interplay between reproductive strategies and population dynamics is neglected and only the e ect of altruism on each individual's fitness is taken into account.Even at capacity, the behaviour of altruism can increase the population size due to cooperators' contribution to the public goods.The dependency of population size on cooperators happens when individuals in an isolated society compete for limited resources: egoism leads to the so-called common tragedy and altruism leads to abundant public goods due to the scale of economics (Houchmandzadeh ). Simulations have led to the outcome that an increasing population has a diminishing e ect to produce a higher mean altruistic level in structured societies with a spatial distribution of agents (Shutters & Hales ).In Houchmandzadeh's asexual model ( ), where population size is designed to be a function of the proportion of cooperators, clarified that a deleterious mutant (i.e., altruism) may possibly invade the community than the wild type (i.e., defectors).Based on this approach, it is reasonable to suppose that the variable size population (as a function of altruism) is an amplification factor for the social cohesion made by altruistic punishment and sexual selection as we have demonstrated in the preceding sections.

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In this section, we apply the variable size population condition instead of the original constant population constraint to test the contribution of both sexual selection and population dynamics.

Variable size population assumption
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Consider an artificial society with four types of individuals as described before.Population size N is now an increasing function of the proportion of cooperators in the population (denoted as x), following Houchmandzadeh ( ) assumption.In each generation, the population reaches its capacity for the particular population structure (i.e. the distribution of each type), that is, no more individuals can survive under the resource limitations.To simplify the model, we adopt the linear form for N (x) (Houchmandzadeh ): where N C and N D (N C < N D ) are the carrying capacities of the society when composed of cooperators and defectors, respectively.x is the proportion of cooperators as defined in ( ).The dynamics process follows the formula in the Baseline Model section except for the variable size population assumption.

Simulation results
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No remarkable di erences in x from the baseline model can be detected when the initial population distribution was set to be evenly distributed between four types.To extract the invasion capacity of cooperators under   variable size population assumption, we plot a comparison of simulation results for di erent treatments (Figure ).
. The results of A , A and A indicate that the di erences in x are not significant, and the di erences in x in C , C and C are not significant as well.Now we compare the plots for FC and FD in B , B and B to see the role of variable size population treatment in the change of invasion capacity.The value of x increases from B to B .In B , the plots of FC overlap with FD, that is, FC and FD share nearly the same population frequency (about %).When the variable size population treatment is employed (B and B ), the plots of FC and FD separate and FC gains the majority of females.The advantage of FC is enhanced in B compared to B due to a larger di erence between N C and N D .Thus, in the society with altruistic punishment but without sexual attractiveness, variable size population treatment o ers a more cooperative female society (i.e., FC dominates FD).In other words, FC in variable size population model has a higher invasion capacity than in constant population model with this particular setting.The result coincides partially with Houchmandzadeh's theoretical outcomes in an asexual society. .Overall, these simulation results imply that the treatment of variable size population has an amplifying e ect on the invasion of FC in the case of altruistic punishment model.Varying population size serves as a partial substitute for attractiveness in the altruistic evolution of females.The mechanism of variable size population in altruistic evolution is not clear, but it is clear that varying population size benefits the cooperative society as a whole: a defector-dominated society is designed to have less population size in the next period as a punishment.The increase in population induced by cooperators can overcome the altruistic cost (Houchmandzadeh ) and promote the rational consumption of limited resources.

Extension to the Individual Variation
. Variations in an individual's social environment and state variables, such as colour, size and age, can influence the evolution of altruism, but the interaction between individual variation and cooperation have rarely been studied in an evolutionary way (Barta ).Therefore, an important problem in the evolutionary models based on the average population payo functions, as the simulations we have done in the previous sections, is how to model the individual variation in the society.The study of penduline tits has been done based on the individual variation to resolve the conflict over parental care (Van Dijk et al. ).In the baseline simulation models, we classify the society into four types (i.e., MP, MD, FC, FD) for the sake of simplicity, but it is reasonable to question the accuracy of the model since a well-fitted simulation requires more than four types of individuals.We believe that in the real world, if altruistic investment can be scaled, the value of I follows a certain distribution and thus there are no strict stereotypes as punishers, cooperators or defectors.

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In the extended models in this section, the society is composed of males and females investing I into altruism.I takes value from to , with a random mutation changing the I in steps of ±0.1 (the settings are adopted from Koella's ( )) with a constant and identical mutation rate µ.Initially, altruistic investments in males and females both follow an identical normal distribution with a mean of .and standard deviation of . .

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The strategies and behaviours of each individual are assumed to satisfy: . For male investing I * into altruism, he will punish individuals whose I < I * regardless of his/her gender.
In turn, he will su er from males whose I > I * .
. For females investing I * * into altruism, she will su er from males whose I > I * * .She will favour males whose I > I * * .
. To summary, a male will punish individuals with a lower level of altruism than him, and a female will favour males with more altruistic than her (i.e., a male favoured by a female will obtain a unit of sexual attractiveness s).Note that only males punish, only females show favour to males and only males receive female's favour and thus increase their sexual attractiveness.
. The dynamic process is the same as in previous sections except the second step of birth.In this section, if the o spring is a male, his probability of having where f (I * |male) represents the fitness of a male with I = I * .
. For the simulation results, we report the line graphs of the average value of I of males and females respectively.    .Finally, we report the final section (at period , ) of the investment distribution in Figure .
. Given the normal distribution initially, individual's investments are decentralised from the midpoint I = 0.5, the mean value of the normal distribution) in A. In B, most males invest about ., and most females invest between .and , with a mean value larger than . .In C and D, the plots of males completely overlap with females, and more than males (females) invest , making them saints of their generation.
. In all, the results in this section are quite similar to the baseline model.The treatment of individual variation appears to be a significant amplifier for the evolution of altruism.Moreover, a superlative cohesion of society will be established where everyone invests when both punishment and sexual attractiveness are employed.Surprisingly, the societies in C and D are even more cooperative than the cohesion achieved in the baseline model.
. The above simulation results suggest that the evolution of altruism with individual variation assumption is a self-reinforcing process.The altruism in the society may be reinforced because the punishment mechanism runs synchronously with the evolution of altruism.Punishment and sexual attractiveness permit the advantage of cooperators, and the society structure may, in turn, enhance the punishment and attractiveness.For example, in simulation in Figure B, a male with I = 0.8 su ers nearly no punishment and receives the highest honour of attractiveness at early period (i.e., period less than ), but at later period (say , or more) he will su er punishment from nearly every males and receive nearly no admiration if his investment remains .because the majority invest now.The case for females is somehow similar.Our conclusion is quite optimistic in the sense that our society is likely to coordinate on high-level altruism equilibrium.

Discussion
. In the baseline model and the extended models, the role of sexual attractiveness, variable size population and individual variation in the evolution of altruism are present in simulation results.In this paper, we provide a di erent and sexual approach to the puzzle of altruism, extending the existing evolutionary models in asexual evolutions.Based on the hypothesis that cooperation might be a costly commitment to public interest (Bird et al. ), we speculate that altruistic punishment is highly related to the sense of justice and law enforcement, and sexual attractiveness can be one of the predecessors of morality in civilisations.

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Feminists may argue that the model setups in this paper suggest inequality between genders because only males are set to be qualified as punishers.Actually, females in this model can punish MD by not selecting them as sexual partners, which will lead to the corresponding e ects of altruistic punishment on MD.Thus, in a generalised perspective, altruistic punishment is conducted by both MP and FC in the proposed model.Additionally, altruistic punishment can be performed in various forms: even uninvolved bystanders can pay to punish defectors to benefit the group (Raihani et al. ).Our simple assumptions on the gender-based behaviours are based on the historical observations, and for simplicity we do not model the e ects of female punishers and male cooperators (who admire female cooperators/punishers).In modern civilisations, altruistic punishment is generally conducted by social norms and authorities, while mate preference to males/females (especially to males), such as superheroes and super heroines, with a strong sense of justice still exists.

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Yet we have not talked about the incentives for human beings to value altruism as a sexually attractive trait, but it is hard to deny the fact that altruism, as well as other moral virtues, can inspire long-term love and admiration (Miller ).Researches in biology suggest that moral traits may serve as indicators of genetic quality and health condition in the process of mate choice (Ne & Pitcher ; Miller ).The hypothesis is that altruism evolves as a costly but honest signal to advertise the altruist in competition with others (Gintis et al. ).Religions and social norms also reinforce the mate preference to altruism by virtue of education and social scrutiny, and in turn provide infrastructures for the evolution of altruism through the mate preference.Religions also take part in creating of social cohesion directly via inhibition against norm violation (Potgieter ).The mate preference we illustrate in this paper is not the only origin of the social cohesion because religions, social structures and politics also shape the society. .
Finally, we have to note that the concept of altruism is understood di erently in various disciplines (Clavien & Chapuisat ).In biology, altruism is defined by the behaviour that increases the fitness of another individual while decreasing the actor's own fitness (Bell ).In our model, MP and FC are considered altruistic because at a particular instance (e.g., period ) they sacrifice their own fitness for the sake of the whole society, although MP receives admiration from FC, which may lead to a net increase in MP's fitness.The possible growth in MP's fitness does not suggest that MP is biologically egoistic because the agent in the simulated model is treated as an artificial life or artificial molecular (Langton ) with no self-consciousness (i.e., MP will never realise that it is in his own interest to o er public goods).In this sense, the admiration from FC to MP is a life-long hidden blessing of sexuality.

Figure :
Figure : Simulation Results of the Punishment-Sexuality Game with % Defectors: (A) The timing does not appear before periods.(B) The timing appears between and periods.(C) The timing appears approximately at periods.The parameters in all three simulations are identical to the simulation in Figure C except for the numbers of periods.

Figure :
Figure : Simulation Results of the Punishment-Sexuality Game with Di erent Strengths of Sexual Attractiveness.The parameters in these simulations are the same as those in Figure C, with the exception of the value of s.

Figure :
Figure : Simulation Results of the Punishment-Sexuality Game with Di erent mutation rate.The parameters in these simulations are the same as those in Figure C, with the exception of the value of µ.The mutation rate µ has been labeled on each figure.

Figure :
Figure : Simulation Results of the Punishment-Sexuality Game with Di erent Periods.The parameters in these simulations are the same as those in Figure C, with the exception of the value of period.

Figure :
Figure : Simulation Results for Variable Size Population Assumptions.(A -C ) Simulation results of constant population models.(A -C ) Simulation results of variable size population models where N C = 1, 000 and N D = 200.(A -C ) Simulation results of variable size population models when N C = 2, 000 and N D = 200.Treatments and numerical simulation results are shown in Table .
Figure examines the e ect of punishment and attractiveness on the mean value ofI.Model settings in Figure are listed in Table .

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The plots in Figure B show that altruistic punishment favours the evolution of high altruistic level females ( I > 0.7 and disfavours the evolution of high altruistic level males ( I < 0.4.Nearly no fluctuations are observed in the plot of males in B. In Figure C and D, we employ both punishment and attractiveness (s= and respectively).Plots in both C and D reach the level where I = 1 regardless of genders.

Figure.
Figure : Mean Value of I in , Figure A-D once again show the role of punishment and attractiveness in the altruistic evolution.

Figure :
Figure : Investment Distribution at Period ,

Table :
Treatments and numerical simulation results of Figure .