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.     

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.     

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.     

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.     

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.     

Mathematical biological of social behavior: II

A group of individuals is considered in which each individual has tendencies to exhibit one or another of two mutually exclusive behaviors. Neurobiophysically this may be described in terms of Landahl's reciprocally inhibited parallel reaction chains. The spontaneous excitations ε₁ and ε₂ at the central connections of each chain are a measure of the “natural” tendency of the individual toward one or the other of the two behaviors. According to equations derived by H. D. Landahl, the probability of one or the other behavior is determined by the difference ε₁ − ε₂. A population of individuals is considered in which ε₁ − ε₂ is distributed in some continuous way, and therefore in which the probability of a given behavior is distributed continuously between 0 and 1. The effect of other individuals exhibiting a given behavior is to increase the corresponding ε of the individual. Thus behavior of others affects the probability for a given behavior of each individual. It is shown that the equations describing the behavior of the population on the basis of this neurobiophysical picture reduce in the first approximation to the differential equations which were postulated by the author in his previous work on social behavior.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

Mathematical biology of social behavior

When an individual grows up in a society, he learns certain behavior patterns which are “accepted” by that society. He may in general have a tendency toward behavior patterns other than those which are “accepted” by the society. This tendency toward such unaccepted behavior may be due to a process of cerebration which results in doubt as to the “correctness” of the accepted behavior. Thus, on the one hand, the individual learns to follow the accepted rules almost automatically; on the other hand, he may tend to consciously break those rules. Using a neural circuit, suggested by H. D. Landahl in his theory of learning, a neurobiophysical interpretation of the above situation is outlined. Mathematical expressions are derived which describe the social behavior of an individual as a function of his age, social status, and some neurobiophysical parameters.     

Suggestions for a mathematical biophysics of some psychoses

We may consider that most of the human behavior is a set of learned responses to certain patterns which recur frequently in the course of human life. Some “abnormal” events or experiences may result in the learning of abnormal responses, and thus in abnormal behavior. The “abnormal” responses may begin to be learned after some of the normal response patterns have been fairly well established. The development of both normal and abnormal behavior may thus be represented by learning curves of the type studied by H. D. Landahl. Applying some of the results of the theory of learning curves and considering that the normal and abnormal reactions may reciprocally inhibit each other, a quantitative theory of some psychoses may be developed. In particular, the effects of shock may be deduced from the assumption that they cause the more recently learned abnormal reactions to be “unlearned” more readily, than the earlier learned “normal” reactions. The effectiveness of shock treatments as a function of the duraction of psychosis is discussed from this point of view.     

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.     

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.     

Studies in imitative behavior: A generalization of the rashevsky model; Its mathematical properties

This paper is based on N. Rashevsky's theory of imitative behavior, the underlying idea being that performance of one reaction by a given individual produces an increased stimulation (or tendency) toward the same reaction in other individuals. For simplicity, consideration is limited to cases in which each individual may choose only between two (or two main categories of) reactions, denoted byA andB in the following. However, upon suggestion from Dr. Rashevsky and certainly in better agreement with actual facts, the strength of imitative interaction is assumed to vary from individual to individual. More precisely, if Ψ_i denotes the additional excitation caused by imitation in theith individual,PA_i the probability for performance of reactionA, andPB_i the probability for performance of reactionB by theith individual, we postulate that where the constants α _ik ^′ and β _ik ^′ (coefficients of imitative interaction) measure the amount of imitative influence exerted by thekth individual upon theith,N being the total number of individuals in the population. The term — α_i Ψ_i accounts for the spontaneous decay of excitation, and the quantities α _ik ^′ and α _ik ^′ are assumed to benon-negative. The expressions forPA_i andPB_i are obtained from H. D. Landahl's theory of conflicting stimuli; they depend non-linearly on the values Ψ_i. It is implicit in this formulation that the theory can only be applied if the frequency of contacts between individuals is not too small. Some further shortcomings and limitations of the model are outlined, and the discussion includes suggestions for reinterpretation and improvement of the theory. If all the quantities α _ik ^′ and α _ik ^′ have the same value, sayA, we return to the case treated by Rashevsky (and Landau, 1950); these authors, however, replace the sums in the equation above by integrals, which automatically restricts the validity of their results to very large values ofN. Their work may therefore be characterized by the assumption of uniform interaction in large populations. Our equations, on the other hand, are applicable even to very small groups, and therein lies one of their main advantages. In this paper the mathematical properties of the non-linear system of equations above are studied with particular reference to the existence and stability of steady states [dΨ_i/dt ≡ 0;, i = 1 , 2, . . . N]. A sufficient condition for the existence of only one stable steady state is derived. It may be formulated roughly by stating that all the coefficients of interaction should be sufficiently small. It that is not the case, there may exist a greater number of stationary states. In particular, two of them (called “extremal”) have the following properties: they arestable and such that the average number of individuals in the group performing one or the other reaction is the largest (or smallest) possible as compared with the other steady states. Hence the situation is qualitatively similar to that found by Rashevsky and Landau.Quantitatively, however, important differences may arise, depending on the nature of the matrix specifying the interaction. A stable state may be approached through damped oscillations, but this effect is important only if the damping is sufficiently small for the oscillations to become practically observable. Little information could be obtained on this point, due to mathematical difficulties. As mentioned above, the most interesting applications of this theory will be with respect to small populations or to populations partitioned into subgroups with varying amounts of imitative interaction within as well as between groups.     

Does kin-biased territorial behavior increase kin-biased foraging in juvenile salmonids?

We examined the effects of kin-biased territorial defense behavioron the distribution of foraging attempts and percent weightchanges (fitness benefits) in juvenile Atlantic salmon (Salmosalar) and rainbow trout (Oncorhynchus mykiss) in an artificialstream channel. The individual percent weight changes and frequencyof aggressive interactions and foraging attempts were quantifiedin kin (full sibling) and non-kin groups of salmon and troutWe observed kin groups of both species to obtain significantlygreater mean and less variable percent weight gains that non-kingroups. In addition, faster-growing (dominant)individuals ofboth species within kin groups exhibited significantly feweraggressive interactions than did faster-growing nonkin individuals,while we observed no difference between kin and non-kin slower-growing(subordinate) individuals. Slowergrowing kin individuals ofboth species obtained significantly more foraging opportunitiesthan slower-growing non-kin individuals while there was no differencebetween faster-growing kin and non-kin individuals. These datasuggest that reduced aggression by faster-growing individualstowards slower-growing kin enables slower-growing kin to obtainmore foraging opportunities, resulting in higher and less variablepercent weight changes. These data also suggest that as a resultof kin-biased territorial defense and foraging behavior, juvenileAtlantic salmon and rainbow trout may be able to maximize inclusivefitness potential by defending territories near related conspecifics.     

Mathematical biology of social behavior: IV. Imitation effects as a function of distance

In previous publications, social groups have been studied in which each individual has a preference for one of two possible mutually exclusive activities. This preference is measured by a quantity ϕ. The value ϕ=0 corresponds to no preference; a preference for one activity is measured by a positive ϕ, the preference for the other by a negative ϕ. The quantity ϕ varies from individual to individual. It has been shown previously that, owing to effects of imitation, even when the average ϕ for the group is zero, one of the two behaviours will be chosen by the majority of the group. Whereas in previous studies the imitation effect was considered as independent of the distance between the imitating and imitated individuals, in the present study the case is considered in which the effect of imitation decreases with the distance between the individuals. It is found that under certain conditions a greater percentage near the center of the area occupied by the group, rather than near the periphery, exhibits the chosen behavior. The possible sociological meaning of this gradient of behavior is discussed.     

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.     

Mathematical biosociology of beliefs and prejudices

Following a suggestion made previously (Bull. Math. Biophysics,13, 61, 1951), it is assumed that every individual has both a tendency to behavearationally, by accepting everything on faith, and rationally, by subjecting everything to rational analysis. Arational behavior is characterized by various beliefs, prejudices, etc., which are considered to be conditioned reactions, learned by the individual before he completely develops his faculties for rational thinking. The two tendencies are assumed to be due to excitations of two different regions of the central nervous system, and are measured by the intensities ɛ _f and ɛ _r of those excitations. Those intensities are further assumed to increase linearly with time, the increases of the two beginning, in general, at different ages. The rates of increase are considered as normally distributed in the population. The relative frequency of arational and rational behavior is determined by the difference φ=ɛ _f =ɛ _r according to equations 0 developed previously (Bull. Math. Biophysics,11, 255, 1949). It is shown that with the above assumptions the majority of the population, which starts with arational behavior, will, within two or three of generations, either change to rational behavior or continue indefinitely to behave arationally. This will hold as long as imitative factors are present. Expressions for the numbers of individuals who behave rationally and arationally are derived. If the intensity of conditioning toward an arational behavior decreases with increasing size of the rationally behaving minority, or, if the rationally behaving individuals are not influenced by imitation, then a slow secular trend toward rational behavior may be present. An expression is also derived for the fraction of individuals who behave rationally as a function of age. This fraction increases with increase of the age at which the beginning conditioning toward any beliefs or prejudices begins.     

Contribution to the mathematical theory of mass behavior: I. The propagation of single acts

The propagation of a single act in a large population is supposed to depend on some external circumstance and on an “imitation component”, where encounters with individuals who are performing or have already performed the act contribute to the tendency of an individual to perform it. The “tendency” to perform is supposed to be measured by the average frequency of stimuli, randomly distributed in time, impinging on the individual. The deduced equation is a relation between the fraction of the population who have performed the act and time, provided the time course of the “external circumstance” and the way in which the imitation component contributes are known. Several special cases are studied, in particular, cases without the imitation component, cases with imitation only, and various mixed cases. Examples are given of social situations in which such factors may operate and general suggestions are made for the systematization of observations and/or experiments to test the assumptions of the theory.