A problem in the mathematical biophysics of interaction of two or more individuals which may be of interest in mathematical sociology

The behavior of an individual may be discussed in terms of a “satisfaction function”. An individual may be considered to always behave in such a way as to make his satisfaction function a maximum. The interaction of two individuals may consist in a cooperation in the production of any kind of objects of satisfaction. Those objects may be either material goods or anything else. The satisfaction of each individual is determined by his share in the total output as well as by the effort he makes. It is shown that for a prescribed method of sharing a behavior in which each individual attempts to maximize the total satisfaction of both individuals results in a greater output than a behavior in which each individual attempts to maximize his own satisfaction.     

Outline of a mathematical biology of leadership

Leadership, whether executive, political or any other type, is connected with the achievement of some goal by the social group through an appropriate organization of that group. From this point of view different leadership ranks in a group would be assigned to individuals according to their ability to organize the group for the purpose of reaching a specific goal. The situation is actually complicated by the circumstance that an individual may have the necessary ability but may not like the responsibility connected with the leadership, or vice versa. Also, he may not be interested in the goal. The suggested mathematical approach is to consider that the satisfaction of an individual is a function of his leadership rank, of the goal, and of several other parameters. If each individual tends to adjust his position in society so as to maximize his own satisfaction, this condition gives us the equations which determine the leadership rank of each individual. It is found that, in general, the rank of an individual depends not only on his ability, but on the abilities of all other individuals. The method enables us to calculate the distribution function of abilities among individuals of a given rank, and leads to results which allow, in principle, experimental verification.     

Mathematical theory of motivation interactions of two individuals: II

The behavior of two individuals, consisting of effort which results in output, is considered to be determined by a satisfaction function which depends on remuneration (receiving part of the output) and on the effort expended. The total output of the two individuals is not additive, that is, together they produce in general more than separately. Each individual behaves in a way which he considers will maximize his satisfaction function. Conditions are deduced for a certain relative equilibrium and for the stability of this equilibrium, i.e., conditions under which it will not “pay” the individual to decrease his efforts. In the absence of such conditions “exploitation” occurs which may or may not lead to total parasitism. Some forms of the inverse problem are considered, where the form of behavior is given and forms of the satisfaction function are deduced which lead to it.     

The recognition of social alliances by vervet monkeys

Vervet monkeys (Cercopithecus aethiops) are significantly more likely to threaten a particular individual if they have previously been involved in a fight with that individual's close kin. They thus show evidence of recognizing that certain individuals associate regularly with each other. Vervets over the age of 3 years are also significantly more likely to threaten a particular individual if that individual's close kin and their own close kin have previously been involved in a fight. It therefore seems possible that adult vervets can recognize, from experience, that certain sorts of relationships share similar characteristics, regardless of the individuals involved.     

Forms of output distribution between two individuals motivated by a satisfaction function

Motivations of two individuals governed by a satisfaction function are assumed to determine their respective “efforts”, which result in the production of “output”, i.e., objects of satisfaction. In previous papers the sharing of output was prescribed in advance. In the present article, however, the sharing formula itself is determined to a certain extent by the satisfaction function. The rate of remuneration per unit of output for each individual is taken to be proportional to the derivative of the satisfaction of the other individual with respect to the effort of the first. The formulation of this condition leads to a partial differential equation whose solutions determine the sharing formula. Sharing determined in this way is referred to as sharing according to the Condition of Mutual Need (C.M.N.). Satisfaction resulting from five different situations are the computed and compared: (1) an individual producing and consuming alone; (2) two individuals sharing equally and neither taking the “initiative” to determine the optimum output; (3) sharing determined by C.M.N. with optimum output determined as in (2); (4) equal sharing but with one individual taking “initiative” in determining optimal output; and (5) sharing determined by C.M.N. and optiml output by the “initiative” of one individual. further considerations concern conditions imposed on the arbitrary function occurring in the solution of the above-mentioned partial differential equation.     

A note on a possible mathematical approach to the theory of individual freedom

As suggested in previous publications, freedom may be defined quantitatively as a restriction upon the choice of a number of activities. If the choice is determined by maximizing the satisfaction function, it is suggested that freedom may be defined in terms of the satisfaction function. If an individual is isolated and no physical restrictions limit his choice of activities, he is free to choose any activity in an amount which maximizes his satisfaction. This isolated state may be considered therefore as that of maximum freedom. If the individual interacts with another, he will choose different amounts of his object of satisfaction depending on whether he behaves egoistically or altruistically. But in either case the value chosen will not maximize his satisfaction function considered alone. A simple analytical expression is suggested as a measure of freedom in this case, and some problems which arise from this suggestion are mentioned.     

Mathematical theory of motivation interactions of two individuals: I

The behavior of two individuals is considered as consisting of an increase or a decrease of productive output. Motivation for increase is the derivative of a “satisfaction function”. This is an algebraic sum of the well-known Thurstone satisfaction curve and another essentially negative quantity, which is a product of a “reluctance parameter” and the “effort”. Each individual attempts to maximize his own total satisfaction. The resulting behavior is examined under a variety of conditions; namely, 1) equal sharing of produced without prescribed sharing of effort; 2) various contracts prescribing the sharing of effort; 3) situations in which one individual is more aware of the underlying motivations than the other. It is these latter situations which under the simplest assumptions of equal sharing, without prescribed sharing of effort, lead to parasitism, i.e. total cessation of effort on the part of one individual. This happens when one individual becomes aware of the other'sautomatic adjustment of his effort so as to bring about a total optimum output, which is a constant. Parasitism is prevented by various forms of contracts in which either the effort necessary for the total optimum output is shared according to a prescribed ratio or the effort of one individual is fixeda priori as a function of the effort of the other. In the latter case the respective efforts become a function of a single variable, and each of the satisfaction functions is maximized by a particular value or values of this variable. In general, these critical values do not coincide for the two satisfaction functions. The problem of finding forms of contract which will result in identical critical (maximizing) values of the variable for both satisfaction functions leads to a functional equation.     

Mathematical biology of division of labor between two individuals or two social groups

The concept of the satisfaction function is applied to the situation in which two individuals may each produce two needed objects of satisfaction, or may each produce only one of the objects and then make a partial exchange. It is shown that with a logarithmic satisfaction function there is no advantage in a division of labor, unless such a division materially increases the purely physical efficiency of production. This result appears to be connected with the particular choice of the form of satisfaction function (logarithmic). While the problem has not been solved for other forms, it is made plausible that satisfaction functions which have an asymptote will lead to a different result. Next the case is studied in which division of labor occurs between two groups of individuals. It is shown that in this case the relative sizes of the two groups are determined from considerations of maximum satisfaction. Possible applications to problems of urbanization are suggested.     

Does evolution lead to maximizing behavior?

A long‐standing question in biology and economics is whether individual organisms evolve to behave as if they were striving to maximize some goal function. We here formalize this “as if” question in a patch‐structured population in which individuals obtain material payoffs from (perhaps very complex multimove) social interactions. These material payoffs determine personal fitness and, ultimately, invasion fitness. We ask whether individuals in uninvadable population states will appear to be maximizing conventional goal functions (with population‐structure coefficients exogenous to the individual's behavior), when what is really being maximized is invasion fitness at the genetic level. We reach two broad conclusions. First, no simple and general individual‐centered goal function emerges from the analysis. This stems from the fact that invasion fitness is a gene‐centered multigenerational measure of evolutionary success. Second, when selection is weak, all multigenerational effects of selection can be summarized in a neutral type‐distribution quantifying identity‐by‐descent between individuals within patches. Individuals then behave as if they were striving to maximize a weighted sum of material payoffs (own and others). At an uninvadable state it is as if individuals would freely choose their actions and play a Nash equilibrium of a game with a goal function that combines self‐interest (own material payoff), group interest (group material payoff if everyone does the same), and local rivalry (material payoff differences).     

A suggestion for a new approach to the mathematical theory of imitative behavior

The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.     

The problem of exchange between two or more individuals,motivated by hedonistic considerations

Let two or more individuals each possess different quantities of two objects of satisfaction. Under certain conditions they may agree to exchange part of the objects if this leads to an increase of each one's satisfaction. The equations which govern this process have been derived by G. E. Evans (1930) for the case of two individuals. A different proof of these equations is given here and the equations are generalized to the case of more than two individuals.     

Historical and ecological controls on phylogenetic diversity in Californian plant communities

An unresolved controversy regarding social behaviors is exemplified when natural selection might lead to behaviors that maximize fitness at the social-group level but are costly at the individual level. Except for the special case of groups of clones, we do not have a general understanding of how and when group-optimal behaviors evolve, especially when the behaviors in question are flexible. To address this question, we develop a general model that integrates behavioral plasticity in social interactions with the action of natural selection in structured populations. We find that group-optimal behaviors can evolve, even without clonal groups, if individuals exhibit appropriate behavioral responses to each other's actions. The evolution of such behavioral responses, in turn, is predicated on the nature of the proximate behavioral mechanisms. We model a particular class of proximate mechanisms, prosocial preferences, and find that such preferences evolve to sustain maximum group benefit under certain levels of relatedness and certain ecological conditions. Thus, our model demonstrates the fundamental interplay between behavioral responses and relatedness in determining the course of social evolution. We also highlight the crucial role of proximate mechanisms such as prosocial preferences in the evolution of behavioral responses and in facilitating evolutionary transitions in individuality.     

The effect of trauma on Neanderthal culture: A mathematical analysis

Traumatic lesions are often observed in ancient skeletal remains. Since ancient medical technology was immature, severely traumatized individuals may have frequently lost the physical ability for cultural skills that demand complex body movements. I develop a mathematical model to analyze the effect of trauma on cultural transmission and apply it to Neanderthal culture using Neanderthal fossil data. I estimate from the data that the proportion of adult individuals who suffered traumatic injuries before death was approximately 0.79–0.94, in which 0.37–0.52 were injured severely and 0.13–0.19 were injured before adulthood. Assuming that every severely traumatized individual and a quarter to a half of the other traumatized individuals lost the capacity for a cultural skill that demands complex control of the traumatized body part, I estimate that if an upper limb is associated with a cultural skill, each individual had to communicate closely with at least 1.5–2.6 individuals during adulthood to maintain the skill in Neanderthal society, and if a whole body is associated, at least 3.1–11.5 individuals were necessary. If cultural transmissions between experts and novices were inaccurate, or if low frequency skills easily disappeared from the population due to random drift, more communicable individuals were necessary. Since the community size of Neanderthals was very small, their high risk of injury may have inhibited the spread of technically difficult cultural skills in their society. It may be important to take this inhibition into consideration when we study Neanderthal culture and the replacement of Neanderthals by modern humans.     

Bell miner provisioning calls are more similar among relatives and are used by helpers at the nest to bias their effort towards kin

Kin selection predicts that helpers in cooperative systems should preferentially aid relatives to maximize fitness. In family-based groups, this can be accomplished simply by assisting all group members. In more complex societies, where large numbers of kin and non-kin regularly interact, more sophisticated kin-recognition mechanisms are needed. Bell miners (Manorina melanophrys) are just such a system where individuals regularly interact with both kin and non-kin within large colonies. Despite this complexity, individual helpers of both sexes facultatively work harder when provisioning the young of closer genetic relatedness. We investigated the mechanism by which such adaptive discrimination occurs by assessing genetic kinship influences on the structure of more than 1900 provisioning vocalizations of 185 miners. These 'mew' calls showed a significant, positive linear increase in call similarity with increasing genetic relatedness, most especially in comparisons between male helpers and the breeding male. Furthermore, individual helping effort was more heavily influenced by call similarity to breeding males than to genetic relatedness, as predicted if call similarity is indeed the rule-of-thumb used to discriminate kin in this system. Individual mew call structure appeared to be inflexible and innate, providing an effective mechanism by which helpers can assess their relatedness to any individual. This provides, to our knowledge, the first example of a mechanism for fine-scale kin discrimination in a complex avian society.     

The effects of pathogen infection and mutation on life-history characters in Arabidopsis thaliana

The nature of the interaction among deleterious mutations is important to models in many areas of evolutionary biology. In addition, interactions between genetic and environmental factors may affect the predictions of such models. Individuals of unknown genotypes of Arabidopsis thaliana, ecotype Marburg, were exposed to five levels of chemical (EMS) mutagenesis and three levels of Pseudomonas syringae infection. Survival, growth and flowering characteristics of each individual were measured. The logarithm of fitness is expected to be a linear function of mutation number if mutations act independently. Furthermore, the expected number of mutations should be approximately a linear function of time of exposure to mutagen. Therefore, nonlinear effects of mutagen exposure on the logarithm of fitness characters would suggest epistasis between mutations. Similarly, if pathogen infection and mutation act independently of each other, their effects should be additive on a log scale. Statistical interactions between these factors would suggest they do not act independently; particularly, if highly mutated individuals suffer more when infected than do less mutated individuals, this suggests that pathogens and mutations act synergistically. Pseudomonas-infected individuals were shown to have an increased probability of flowering under conditions of short day length, but to ultimately produce fewer flowers than uninfected individuals. This suggests a plastic response to stress and, despite that response, an ultimately deleterious effect of infection on fitness. Leaf rosette growth was negatively and linearly related to the expected number of mutations, and the effects of mutation on different life-cycle stages appeared to be uncorrelated. No significant interactions between pathogen and mutation main effects were found. These results suggest that mutations act multiplicatively with each other and with pathogen infection in determining individual fitness.     

The moral importance of selecting people randomly

This article discusses some ethical principles for distributing pandemic influenza vaccine and other indivisible goods. I argue that a number of principles for distributing pandemic influenza vaccine recently adopted by several national governments are morally unacceptable because they put too much emphasis on utilitarian considerations, such as the ability of the individual to contribute to society. Instead, it would be better to distribute vaccine by setting up a lottery. The argument for this view is based on a purely consequentialist account of morality; i.e. an action is right if and only if its outcome is optimal. However, unlike utilitarians I do not believe that alternatives should be ranked strictly according to the amount of happiness or preference satisfaction they bring about. Even a mere chance to get some vaccine matters morally, even if it is never realized.     

Mutual trust and cooperation in the evolutionary hawks–doves game

Using a new dynamical network model of society in which pairwise interactions are weighted according to mutual satisfaction, we show that cooperation is the norm in the hawks–doves game when individuals are allowed to break ties with undesirable neighbors and to make new acquaintances in their extended neighborhood. Moreover, cooperation is robust with respect to rather strong strategy perturbations. We also discuss the empirical structure of the emerging networks, and the reasons that allow cooperators to thrive in the population. Given the metaphorical importance of this game for social interaction, this is an encouraging positive result as standard theory for large mixing populations prescribes that a certain fraction of defectors must always exist at equilibrium.     

Implicit frequency dependence and kin selection in fluctuating environments

Summary I consider a general model of a fluctuating environment in which the environmental state each year is drawn at random from some given distribution. Each year organisms must choose what action to perform before the environmental state for that year is known. There is no interaction with kin. In this scenario, natural selection will tend to produce organisms which maximize their geometric mean fitness. In this paper I introduce the idea of the profile of a strategy. This function quantifies how the strategy peforms for each environmental state. I show that there is a unique profile such that a strategy is optimal if and only if it has this profile. I then give a characterization of the optimal profile which generalizes previous work by others in this area. The characterization of the optimal profile has a game theoretical interpretation. Motivated by this I introduce a game in which individuals play the field in a constant environment. This game may be interpreted as a cooperative game between kin. The key result of this paper shows that a strategy maximizes geometric mean fitness in the original fluctuating environment problem if and only if it is an evolutionarily stable strategy of the deterministic environment game. It is well known that an optimal strategy in a fluctuating environment may be mixed, involving adaptive coin-flipping. Others have previously noted that this may result in some individuals sacrificing individual reproductive success for the good of the genotype. My analysis shows that one may regain the concept of individual optimization if the quantity maximized is suitably defined. Under an optimal strategy every action taken maximizes the expected number of offspring produced, where this expectation is not calculated using the true distribution of environmental states, but a distribution modified to take account of the actions of kin.     

Development of structure in a society with a dominance relation when new members are added successively

A society with a dominance relation is considered to be built up by starting with a small society and adding new members in succession. As each member is added he engages in contests with each of the older members to determine the dominance relation between them. The probability that the older member dominates is considered to depend on the size of the society and linearly on the older members score. A recurrence relation for the hierarchy index is derived. The approach of the society to a hierarchical structure is considered for various special cases of this probability. Reasonable assumptions concerning this dominance probability are shown to lead to structures close to the hierarchy. If the new member dominates all the older ones below a certain rank, and is dominated by all those above this rank, then the hierarchy will persist if it is the initial structure, or the structure will tend to hierarchy as the size increases, if it is not the initial structure.