Social climbing and Amoroso distribution
SSOCIAL CLIMBING AND AMOROSO DISTRIBUTION
GIACOMO DIMARCO AND GIUSEPPE TOSCANI
Abstract.
We introduce a class of one-dimensional linear kinetic equations of Boltz-mann and Fokker–Planck type, describing the dynamics of individuals of a multi-agentsociety questing for high status in the social hierarchy. At the Boltzmann level, themicroscopic variation of the status of agents around a universal desired target, is builtup introducing as main criterion for the change of status a suitable value function inthe spirit of the prospect theory of Kahneman and Twersky. In the asymptotics ofgrazing interactions, the solution density of the Boltzmann type kinetic equation isshown to converge towards the solution of a Fokker–Planck type equation with vari-able coefficients of diffusion and drift, characterized by the mathematical properties ofthe value function. The steady states of the statistical distribution of the social statuspredicted by the Fokker–Planck equations belong to the class of Amoroso distributionswith Pareto tails, which correspond to the emergence of a social elite . The details ofthe microscopic kinetic interaction allow to clarify the meaning of the various param-eters characterizing the resulting equilibrium. Numerical results then show that thesteady state of the underlying kinetic equation is close to Amoroso distribution evenin an intermediate regime in which interactions are not grazing.
Keywords.
Amoroso distribution; Generalized Gamma distribution; Log-Normal dis-tribution; Kinetic models; Fokker–Planck equations.
AMS Subject Classification : 35Q84; 82B21; 91D10, 94A17 Introduction
More than one century ago, the economist Vilfredo Pareto [67, 68] observed that humansocieties tend to organize in a hierarchical manner, with the emergence of social elites . Hefurther noticed that social mobility in this hierarchical state appears to be higher in the middleclasses than in the upper and lower part of the hierarchy. The correctness of Pareto’s socialanalysis has been recently investigated in a number of studies dealing with the modeling of thedynamics of social networks in which individuals quest for high status in the social hierarchy[2, 53, 54, 80]. There, the social status of individuals is usually ranked according to a certainmeasure strongly related to centrality in the society. Since individuals in top ranked positionsenjoy social advantages compared to those with low rankings, individuals are motivated toclimb the social ladder to reach a better rank [27].These models captured at a numerical level some key ingredients enough to reproduce themain characteristics predicted by the sociological analysis of Pareto [67, 68]. In particularthe numerical simulations of the dynamics driven by the models introduced in [2, 27] ledto conclude that, in accord with Pareto’s observations, the hierarchical state reached byindividuals in the social network looking for high status is very stable.
Date : June 5, 2020.Department of Mathematics and Informatics of the University of Ferrara e.mail: [email protected]. Department of Mathematics of the University of Pavia, and IMATI CNR, Italy. e.mail:[email protected]. a r X i v : . [ phy s i c s . s o c - ph ] J un GIACOMO DIMARCO AND GIUSEPPE TOSCANI
By contrast with the numerical simulations that form the core of most previous studies, weshow that the analytic techniques of statistical mechanics are ideally suited to the study ofthe phenomenon of social climbing, and can shed much light on its structure and behavior. Inparticular, by resorting to the well-established mathematical tools of kinetic theory of multi-agent systems [65], we pursue an almost entirely analytic approach. Starting from a suitabledescription of the microscopic behavior of agents, we build a class of one-dimensional linearkinetic equations of Boltzmann and Fokker–Planck type, suitable to describe the dynamics ofindividuals of a multi-agent society tending for high ranking in the social hierarchy, in whichboth the properties of the transient and of the steady state are explicitly computable. Thisapproach is coherent with the classical kinetic theory of rarefied gases, where the formationof a Maxwellian equilibrium in the spatially uniform Boltzmann equation is closely related tothe microscopic details of the binary collisions between molecules [24, 44].The methods and results of this paper are part of the numerous studies dealing withthe modeling of social and economic phenomena in a multi-agent system. In economics,this modeling attempted to justify the genesis of the formation of Pareto curves in wealthdistribution of western countries [26, 28, 29, 32, 34, 43, 73], and to share light on the reasonsbehind opinion formation [11, 12, 13, 14, 16, 17, 18, 30, 35, 39, 40, 41, 42, 74, 76].Due to the human nature of social phenomena, these investigations naturally includedspecific behavioral aspects of agents in the modeling. In the field of collisional kinetic theorya pioneering approach has been proposed in [61] to model price formation of a good in amulti-agent market. The kinetic model of Boltzmann type describes the dynamics of twodifferent trader populations, playing different rules of trading, including the possibility foragents to move from one population to other. The kinetic description, inspired by the well-known Lux–Marchesi model [59, 60] (cf. also [55, 56]) included in the mechanism of tradingthe opinion of traders [76], and behavioral components of the agents, like risk’s perception.This last component has been done, by resorting to the prospect theory by Kahneman andTwersky [50, 51], in terms of a suitable value function .The analysis of [61] enlightened the importance of modeling kinetic interactions by takinginto account aspects of human behavior [7, 8, 9, 10], as pioneered by Zipf in [81] (a recentcollection of contributions on this topic can be found in two special Issues of this journal[5, 6]). The possible connections between the kinetic modeling of human phenomena andtheir description in terms of value functions, has been recently developed in [45], where themathematical translation of this relationship justified at a microscopic level first the mecha-nism of formation of the service time distribution in a call center, and subsequently a numberof other social phenomena which lead to a stationary state in the form of a lognormal distri-bution [46].The leading idea in [45] is based on a general principle which can be easily verified in anumber of social activities of agents in which one identifies the possibility of a certain addiction[33, 77].The forthcoming kinetic modeling follows along the lines of the recent approaches of [33, 45,46], where a relevant number of phenomena involving measurable quantities of a populationand fitting suitable probability distributions (in particular log-normal distribution) was shownto be consequence of a certain human behavior. The list of social phenomena to which thekinetic description furnishes a convincing explanation of the reasons behind the formationof a certain statistical profile is remarkable. Furthermore, the kinetic approach of [45, 46],recently extended to addiction phenomena [33, 77], gives a unified view to various socialaspects, previously treated resorting to different methods. The list include the distribution ofbody weight [22], women’s age at first marriage [69], drivers behavior [47], the level of alcohol
OCIAL CLIMBING AND AMOROSO DISTRIBUTION 3 consumption [52, 70], or, from the economic world, the level of consumption in a westernsociety [3], the size of cities [4], the length of call-center service times [21].In agreement with the modeling assumption of [46], the microscopic agent’s behavior isexpressed by assuming that the agents change their state aiming to approach an optimaltarget, and that this change requires an asymmetric effort, depending on whether the currentstate is above or below the optimal one. On the basis of the prospect theory by Kahnemanand Tversky [50, 51], the elementary changes of state of agents depend on a suitable valuefunction expressing this asymmetry in the whole range of possible values. In the situationleading to the log-normal or other rapidly decaying probability densities, the asymmetry ofthe value function translates at the mathematical level a fundamental property: it is easierto reach the optimal target starting from below, than to approach it from above.To clarify this idea through a simple example, an asymmetry of this type is present whenlooking at body weight of individuals characterized by common age and sex, where the desiredtarget can be identified in the mean ideal weight . It is clearly easier to reach the ideal weightstarting from lower values, since this implies no restrictions on eating, while it is hard toreduce the weight when above the ideal target, since this requires to respect a certain diet.In analogy with the previous phenomena, the analysis of social climbing leads to concludethat the microscopic dynamics still contains a marked asymmetry, which, however, for lowvalues of the social rank, works in the opposite direction. Indeed, while it is clear that agentstend as before to improve their social status, trying to approach an optimal target that canbe identified in a certain level of well-being, it is usually hard to exit from the very low zone,since this implies to possess, in addition to personal skills and competences, a certain amountof financial resources, possibility of access to good schools, and other facilities.With respect to the choice made in [46, 77], this asymmetry will be expressed through anew class of value functions, still obtained from the classical ones proposed by Kahneman andTversky [50, 51], which now express Pareto’s analysis about social mobility: the microscopicvariations are higher for agents in the middle class than for agents in the upper and lowerpart of the hierarchy. The new value functions are used to built the microscopic interactionsuitable to describe the elementary variation of the social rank, and to study in this way thephenomenon through the variation in time of the density f = f ( w, t ) of agents in the systemwith social rank measured by w > , at time t ≥ .As usual in the kinetic setting [37], the change of f due microscopic interactions generates alinear kinetic equation of Boltzmann type for the density of agents, that will be subsequentlystudied in the asymptotic regime of grazing effects [79]. The grazing regime describes thesituation in which a single interaction produces only a very small change of the variable w .In this regime, the variation of density of the social rank of the agent’s system is driven by apartial differential equation of Fokker–Planck type. If we denote by g = g ( w, t ) the density ofagents which have a social rank equal to w at time t ≥ , this density is solution of a linearFokker–Planck equation with variable coefficients of diffusion and drift, given by(1.1) ∂g ( w, t ) ∂t = (cid:26) σ ∂ ∂w (cid:16) w δ g ( w, t ) (cid:17) + µ ∂∂w (cid:20) δ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) w δ g ( w, t ) (cid:21)(cid:27) . In (1.1) ¯ w L represents the target value of social rank that agents tend to reach, while σ, µ and δ are positive constants closely related to the typical quantities of the phenomenon understudy, satisfying further bounds imposed by the physical conditions on the value functionin the microscopic interaction. In particular < δ ≤ . The equilibrium density of the GIACOMO DIMARCO AND GIUSEPPE TOSCANI
Fokker–Planck equation (1.1) is given by the Amoroso-type distribution [1](1.2) g ∞ ( w ) = g ∞ ( ¯ w L ) (cid:16) ¯ w L w (cid:17) δ + γ/δ exp (cid:26) − γδ (cid:18)(cid:16) ¯ w L w (cid:17) δ − (cid:19)(cid:27) . In (1.2) γ = µ/σ , where σ and µ are the coefficients of the diffusion (respectively of the drift)terms of the Fokker–Planck equation (1.1). This asymptotic procedure was used in [31, 36]for a kinetic model for the distribution of wealth in a simple market economy subject tomicroscopic binary trades in presence of risk, showing formation of steady states with Paretotails, in [78] on kinetic equations for price formation, and in [76] in the context of opinionformation in presence of self-thinking. A general view about this asymptotic passage fromkinetic equations based on general interactions towards Fokker–Planck type equations canbe found in [37], where also an exhaustive discussion about the large-time behavior of thesolution to equations (1.1) is presented. Other relationships of this asymptotic procedurewith the classical problem of the grazing collision limit of the Boltzmann equation in kinetictheory of rarefied gases have been recently enlightened in [45].Assuming that the steady state distribution (1.2) is taken of unit mass, the equilibriumstate is a probability density belonging to the class(1.3) f ∞ ( w ; θ, α, β ) = 1Γ( α ) (cid:12)(cid:12)(cid:12)(cid:12) βθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) wθ (cid:17) αβ − exp (cid:110) − ( w/θ ) β (cid:111) , for non-negative values of w and positive values of the parameters α, θ and β (cid:54) = 0 . When β > ,the function (1.3) was considered as a generalization of the Gamma distribution by Stacy [75],and includes the familiar Gamma, Chi, Chi-squared, exponential and Weibull densities asspecial cases. Generalized Gamma distributions, also known as Amoroso and Stacy-Mihramdistributions [1, 48], are widespread in physical and biological sciences [49, 57, 58], as well asin the field of social sciences [20, 33].Note that the density (1.2) corresponds to the choice of the negative value β = − δ in (1.3).Negative values of β lead to distributions with only a limited number of moments bounded,and represent, among others, a careful approximation of the statistical distribution of wealth,in accord with Pareto’s discovery of fat tails in this case [66]. In particular, if δ = 1 thefunction (1.2) coincides with an inverse Gamma distribution.The choice of β = − establishes a strong connection between the steady distributions ofsocial rank, and, respectively, of wealth. As a matter of fact, this connection is evident ifwe consider that the social status of individuals is closely related to their wealths. However,the mechanism of wealth formation in a multi-agent system has been modeled at a kineticlevel in terms of binary trades. Beside the kinetic models of Boltzmann type introduced inrecent years to enlighten the formation of an unequal distribution of wealth among tradingagents [25, 65], a Fokker–Planck type equation assumed a leading role. This equation, whichdescribes the time-evolution of the density f ( t, w ) of a system of agents with personal wealth w ≥ at time t ≥ reads(1.4) ∂f ( w, t ) ∂t = σ ∂ ∂w (cid:0) w f ( w, t ) (cid:1) + λ ∂∂w (( w − m ) f ( w, t )) . In (1.4), σ, λ , and m denote positive constants related to essential properties of the trade rulesof the agents.The Fokker–Planck equation (1.4) has been first obtained by Bouchaud and Mézard [19]through a mean field limit procedure applied to a stochastic dynamical equation for the wealthdensity. Then, the same equation was derived in [32] by resorting to an asymptotic procedure OCIAL CLIMBING AND AMOROSO DISTRIBUTION 5 applied to a Boltzmann-type kinetic model for binary trading in presence of risk. Note thatthe steady state of equation (1.4) is an inverse Gamma density.The Boltzmann-type equation leading to (1.4), was assumed to satisfy the strong hypothesisof Maxwellian molecules. This choice has been critically revisited in [38], to give a morecoherent (from the economical point of view) interaction kernel. There, the choice of a variablekernel gave in the grazing limit a Fokker–Planck equation different from (1.4), that whilepossessing an inverse Gamma density as steady state, allowed to prove that the solution isconverging to equilibrium at exponential rate, a result that is missing for the solution to (1.4).The new Fokker–Planck equation considered in [38] is(1.5) ∂f ( w, t ) ∂t = σ ∂ ∂w (cid:0) w ν f ( w, t ) (cid:1) + λ ∂∂w (( w − m ) w ν f ( w, t )) , where < ν ≤ is a constant parameter related to the intensity of the frequencies describedby the kernel in the Boltzmann equation. The Fokker–Planck equation (1.5) interpolatesbetween equation (1.4) ( ν = 0 ) and equation (1.1) ( ν = δ = 1 ).The forthcoming kinetic modeling is not restricted to the sociological aspect of the formationof elites, but it can be easily adapted to describe the distribution of knowledge in a society[44, 64], where the elites in this case can be identified for example in the members of thescientific academies, or in the description of social climbing in sports with a large participation,such as the football game, where the elites correspond to the footballers playing in the bestclubs of a country. This last example is particularly interesting since one can be easily accessdata, and furthermore allows to understand in a clear way the main assumptions leading tothe general kinetic description.In details, in the forthcoming Section 2 we will present on the example of social climbing infootball activity, the main universal assumptions modeling the microscopic kinetic interaction,and the consequent interaction kernel. Then, in Section 3, we discuss from a different point ofview the classical description of the trading interaction in the linear kinetic model for wealthdistribution leading to the Fokker–Planck equation (1.4), which is now rewritten in terms ofthe value function formulation adopted in [45]. This preliminary discussion allows to clarify themain differences between the kinetic models for wealth distribution based on linear trades, andthe present one for social climbing, mainly motivated by social reasons. This will be presentedin Section 4, where we describe a multi-agent system, in which agents can be characterizedin terms of their social rank, measured in terms of a certain unit, and subject to microscopicinteractions which contain the microscopic rate of change of the value of their rank, accordingto the previous general principle. The relevant mechanism of the microscopic interaction isindeed based on a suitable value function, in the spirit of the analysis of Kahneman andTwersky [50, 51], which reproduces at best the asymmetries present in the human behaviorin this situation. Then in Section 5 we will show that in a suitable asymptotic procedure(hereafter called grazing limit) the solution to the kinetic model tends towards the solutionof the Fokker-Planck type equation (1.1).Once the Fokker–Planck equation (1.1) has been derived, some numerical examples will becollected in Section 6, together with a detailed explanation of the relevant mechanism whichleads to the typical microscopic interaction in terms of the value function. The numericalexperiments will put in evidence that the time behavior of both the kinetic model and itsFokker–Planck asymptotics is very similar, and that the steady states are very close eachother even in for moderate values of the grazing parameter. GIACOMO DIMARCO AND GIUSEPPE TOSCANI
Amateurs ProfessionalSemi-professional Top playersYouth players
Figure 2.1.
Distribution of football players in Italy from youth schools totop clubs, 2017-2018 census.2.
The example of football activity
The social hierarchy represents a fundamental aspect of life observed in many differentanimal species, including insects, mammals and birds. Depending on the situation, a hierarchymay be established in different manners but it often results in a ranking of the animals ina group and, its importance is such that can influence the quality of life of the entire group[71]. It is clear that human beings are far from being unaffected from this phenomenon. Onthe contrary, in human societies, one can observe in everyday life, an enormous number ofexamples which could be possibly, in last instance, be related to a construction of a hierarchicalstructure. At this subject, Figure 2.1 shows the distribution of football players in Italy startingfrom the young player academies up to professionalism. In this plot, Top players are consideredthose players belonging to the top five Italian clubs. The image is produced using the datareported in Table 2 extrapolated from the Annual Report of the National Football ItalianAssociation (FIGC) .This table shows the number of Italians playing per categories togetherwith the inverse cumulated number of people belonging to each class. The first level regardschildren starting to play football, the second one people that play independently on the age atdifferent amateur levels, while the third level contains the young professionals (referred to assemi-professionals). In the fourth class, we extrapolated the estimated number of professionalplayers coming from the Italian sport Academies. This datum has been obtained eliminatingthe number of players from Foreign countries from the total number of registered professionalsplaying in the Italian leagues at the moment of the survey. Finally, the last level (the footballelite ) gives the number of estimated players coming from the Italian Academy sector arrivingin the top five Italian clubs. This last information has been extrapolated considering theaverage number of players in each club and discarding the number of foreign practitioners.The Figure is obtained by reporting the relative number of players belonging to each categoryversus the inverse cumulative distribution, which is intended as the measure of the socialposition in the football society, in logarithmic scale. One can observe that the empiricalinverse cumulated distribution exhibits tails. A similar Figure, concerning the distributionof school knowledge in Italy, has been obtained in [44] using the data relative to the 2011census. In this case, the knowledge elite was represented by members of the historical scientificacademies.The example of climbing in football activity can serve as a prototype case to understandand justify the main rules of both the personal and social behavior of players characterizing OCIAL CLIMBING AND AMOROSO DISTRIBUTION 7
Level Italians (%) Cumulated values (%)Youth players 673555 63.6% 1054791 100%Amateurs 370540 35.1% 381236 36.14%Young professional 9480 0.9% 10696 1.014%Professional 1136 0.108% 1216 0.1152%Top players 80 0.007% 80 0.007%Totals 1054791 100
Table 2.1.
Distribution of the players for level of activity. Data from FIGC the kinetic model described in the rest of the paper. Inherently to the football activity,progress into the social ladder can be undoubtedly identified in playing for the best clubs ofthe first Italian league. Referring to players at any level to as agents (borrowing this termfrom classical microscopic modeling approaches), it is reasonable to assume that improvingin the practice of football (or any sport in general) can be reached only through a very largenumber of small steps: everyday trainings routine for example. In this context, an agent triesto reach an optimal target identified as the level in which a young football player aims toplay when becoming an adult. In this long process, it is also clear that while it is hard toimprove at the beginning, progresses become faster as the young players grow and learn howto practice. This point of view shares similarities with the common perception that peoplebelonging to low social classes have low chances to improve their status for lack of meansand thus often remain in the so-called bottom of the hierarchy. In the same way, animalsin dominant hierarchies do not consider themselves sufficiently strong to attempt any climband so they simply give up. Coming back to the football example, it is considered usuallyhard to overcome the low skills level zone in football, since this implies to possess, in additionto inbred skills, a certain possibility of access to good instructors and facilities. Trying tomake quantitative the previous observations, we will in the next part of the paper introducea variable w identifying the social status in a society (the football skills in our example).In the same way, as it will be made clear next, we will introduce a value ¯ w identifying thelimit below which an agent does not expect to be able to climb the social ladder. This pointwill characterize a change in the way in which the agent will be able to adjust its socialstatus: passed this value his improvements will be faster. In the same way, many footballplayers consider reaching a certain level in the league enough satisfactory for them. This isalso analogous to what happens to people satisfied of obtaining a certain job not questing forchanges. In order to take into account this second fact, we will define then with ¯ w L > ¯ w thisperceived level which is assumed, at first glance, equal for all agents. The other ingredientswhich completely defines the dynamic under study are the will of football players to improve,as well as, a certain dose of uncertainty depending on external factors which cannot in generalbe controlled. All these aspect will be condensed in a so-called value function Ψ in the sense ofKahneman and Twersky.[50] Finally, regarding players which already reached the professionallevel, we expect for them as natural to be difficult to descend back to lower levels apart ifrare circumstances happen. In the next Section, we will discuss the analogies of the proposedmodel with more classical models describing wealth distribution [65]. GIACOMO DIMARCO AND GIUSEPPE TOSCANI Learning from kinetic theory of wealth distribution
The brief discussion of Section 2 illustrates the main reasons which can justify the profileof Figure 2.1. These reasons need now to be translated at a mathematical level in a model,in order to verify if, in consequence of these rules, this steady behavior appears.Among other approaches, the description of social phenomena in a multi-agent system canbe successfully obtained by resorting to statistical physics, and, in particular, to methodsborrowed from kinetic theory of rarefied gases. The main goal of the mathematical modelingis to construct master equations of Boltzmann type, usually referred to as kinetic equa-tions, describing the time-evolution of some characteristic of the agents, like wealth, opinion,knowledge, or, as in the case treated in this paper, of agent’s ranking in the social ladder[23, 25, 33, 63, 65, 72].The building block of kinetic theory is represented by the details of microscopic interactions,which, similarly to binary interactions between particles velocities in the classical kinetictheory of rarefied gases, describe the variation law of the selected agent’s trait. Then, themicroscopic law of variation of the number density consequent to the (fixed-in-time) way ofinteraction, is able to capture both the time evolution and the steady profile.In economics, the microscopic interactions attempted to simulate the trading activity, aim-ing to justify in this way the formation of Pareto curves in wealth distribution of westerncountries [26, 28, 29, 32, 34, 43, 73]. The qualitative results obtained in [36, 62] then showedthe possibility that Pareto tails could be obtained in consequence of the linear binary tradesbetween agents introduced in [32]. For a better understanding of the reasons leading in gen-eral to the formation of Pareto tails, in the rest of this section we will give a new insight tothis linear trading.We consider a system of agents characterized by their wealths. To fix notations, the amountof wealth will be denoted by v ≥ , and measured with respect to some unit. As usual, thepopulation is considered homogeneous with respect to the way of trading. In addition, it isassumed that agents are indistinguishable [65]. This means that at any instant of time t ≥ agents in the system are completely characterized by the amount of their wealths. Conse-quently, the statistical distribution of wealth of the agents system will be fully characterizedby the unknown density f = f ( v, t ) , of the wealth v ∈ R + and the time t ≥ .The precise meaning of the density f is the following. Given the system of traders, andgiven an interval or a more complex sub-domain D ⊆ R + , the integral (cid:90) D f ( v, t ) dv measures the amount of traders which are characterized by a wealth v ∈ D at time t ≥ . Itis assumed that the density function is normalized to one, that is for all t ≥ (3.6) (cid:90) R + f ( v, t ) dv = 1 . The change in time of the density is due to the fact that agents of the system are subjectto trades, and continuously upgrade their amounts of wealth w at each trade. To maintainthe connection with classical kinetic theory of rarefied gases, it is usual to define elementaryinteraction a single upgrade of the quantity v .Following [38], we will limit ourselves to consider a linear interaction, which while takinginto account all the trading aspects of the original nonlinear model considered in [32], andgiving in the asymptotics of grazing collisions the same Fokker–Planck equation, it is easierto handle. According to the binary trade introduced in [32] the elementary change of wealth OCIAL CLIMBING AND AMOROSO DISTRIBUTION 9 v ∈ R + of an agent of the system trading with the market is the result of three differentcontributes(3.7) v ∗ = (1 − (cid:15)λ ) v + (cid:15)λ ¯ v + η (cid:15) v. In (3.7) λ is a positive constant, while (cid:15) (cid:28) is a small parameter that adjusts the intensityof the exchange, and guarantees that (cid:15)λ (cid:28) . The first term in (3.7) measures the wealththat remains in the hands of the trader who entered into the trading market with a (small)percentage (cid:15)λv of his wealth. The constant (cid:15)λ quantifies the saving propensity of the agent,namely the human perception that it results quite dangerous to trade the whole amount ofwealth in a single interaction. The second term represents the amount of wealth the traderreceives from the market as result of the trading activity. Here ¯ v > is sampled by acertain distribution E which describes the (independent of time) distribution of wealth in themarket. Note that in principle the constant in front of the wealth ¯ v could be different from λ , say ξ . However, the choice λ (cid:54) = ξ will not introduce essential differences in the subsequentdiscussion. Finally, the last term takes into account the risks connected to the trading activity.The uncertainty of the trading result is represented in terms of a random variable η (cid:15) , centeredand with finite variance (cid:15)σ (cid:28) , which in general is assumed such that η (cid:15) ≥ − (cid:15)λ , toensure that even in a risky trading market, the post trading wealth remains non negative. In[62] it is further assumed that the random variable η (cid:15) takes values on a bounded set, thatis − (cid:15)λ ≤ η (cid:15) ≤ (cid:15)λ ∗ < + ∞ . This condition is coherent with the trade modeling, andcorresponds to put a bound from above at the possible random gain that a trader can havein a single interaction.Since ¯ v > , the elementary interaction (3.7) is a particular case of interactions in whichthe value v of the wealth can be modified by two quantities which describe the predictable,and, respectively, the unpredictable behavior of the outcome. The general form of theseinteractions can be written in the form(3.8) v ∗ = v − Ψ (cid:15) (cid:16) v ¯ v (cid:17) v + η (cid:15) v. Using the form (3.8) for the interaction (3.7) we obtain that, for s = v/ ¯ v ≥ (3.9) Ψ (cid:15) ( s ) = (cid:15)λ (cid:18) − s (cid:19) . Note that Ψ (cid:15) ( s ) is a dimensionless increasing concave function, which ranges from −∞ to (cid:15)λ , equal to zero at the point s = 1 . This point is usually referred as reference point . Therole played by the function Ψ (cid:15) ( · ) in the interaction is clear. Since Ψ (cid:15) (¯ v/v ) is negative when v < ¯ v , and positive when v > ¯ v the interaction will increase the value of the wealth in theformer case, while it will decrease the value in the latter. Hence, in absence of randomness,the interaction will move the wealth v towards the value ¯ v .According to the prospect theory of Kahneman and Twersky [50], the function Ψ (cid:15) satisfiesmost of the properties of a value function . The notion of value function was originally relatedto various situations concerned with decision under risk. Kahneman and Twersky in [50]identified the main properties characterizing a value function Φ( s ) , where s ≥ , in a certainbehavior around the reference point s = 1 , expressed by the conditions(3.10) − Φ (1 − ∆ s ) > Φ (1 + ∆ s ) , and(3.11) Φ (cid:48) (1 + ∆ s ) < Φ (cid:48) (1 − ∆ s ) . where ∆ s > is such that − ∆ s ≥ . These properties are well defined for deviationsfrom the reference point s = 1 , and imply that the value function below the reference pointis steeper than the value function above it. In other words, a value function is characterizedby a certain asymmetry with respect to the reference point s = 1 . This asymmetry has aprecise meaning. Given two agents starting at the same distance ∆ s from the reference value s = 1 from below and above, it will be easier for the agent starting below to move closerto the reference value, than for the agent starting above.The function Ψ (cid:15) ( s ) in (3.9) satisfiesproperties (3.10) and (3.11). A third property follows by considering its behavior as (cid:15) → .Since(3.12) lim (cid:15) → Ψ (cid:15) ( s ) (cid:15) = λ (cid:18) − s (cid:19) . the scaled function Ψ (cid:15) ( s ) /(cid:15) is inversely proportional to s .The main result in [32] was to show that, in the limit (cid:15) → , the solution to the kineticequation of Boltzmann type converges towards the solution to the Fokker–Planck equation(1.4), with a steady state in the form of an inverse Gamma density. The inverse Gammadensity is obtained as the distribution of the random variable /X , where X is Gammadistributed. This suggests that a Fokker–Planck equation with a steady state in the form ofa Gamma density is obtained by looking at the interaction (3.8), characterized by a valuefunction that gives the variation of z = 1 /v . To this aim, let us study of the variation of z = 1 /v . In absence of risk, the interaction (3.8), rewritten in terms of z reads(3.13) z ∗ = 1 z − Ψ (cid:15) (cid:16) ¯ zz (cid:17) z . Hence, starting from (3.13), simple computations show that the elementary variation of z isgiven by(3.14) z ∗ = z − Φ (cid:15) (cid:16) z ¯ z (cid:17) z. In (3.14) the function Φ (cid:15) , for s ≥ is given by(3.15) Φ (cid:15) ( s ) = (cid:15)λ s − (cid:15)λ ( s −
1) + 1 . Note that the condition (cid:15)λ < implies the positivity of the denominator. The function Φ (cid:15) ( s ) satisfies the bounds − (cid:15)λ − (cid:15)λ ≤ Φ( s ) ≤ . Hence, at difference with interaction (3.7), the elementary variation of z = 1 /v is driven by afunction Φ (cid:15) , bounded from below and above.It is interesting to remark that Φ (cid:15) still satisfies properties (3.10) and (3.11), typical ofa value function. Moreover, as it happens for Ψ (cid:15) , the quotient between Φ (cid:15) and (cid:15) remainswell-defined in the limit (cid:15) → , and(3.16) lim (cid:15) → Φ (cid:15) ( s ) (cid:15) = λ ( s − , so that the scaled function is proportional to s . Motivated by the study of the statisticaldistribution of alcohol consumption, in [33] a value function suitable to describe the elementaryinteraction was identified in(3.17) Φ (cid:15) ( s ) = µ e (cid:15) ( s − − e (cid:15) ( s − + 1 , s ≥ , OCIAL CLIMBING AND AMOROSO DISTRIBUTION 11 where the constants < µ < and (cid:15) > characterize the intensity of the interaction. Thisfunction satisfies properties (3.10) and (3.11), and(3.18) lim (cid:15) → Φ (cid:15) ( s ) (cid:15) = µ ( s − , namely the same scaling property of the function (3.15). As proven in [33], in presence of aelementary interaction of type (3.8), with value function (3.17), the kinetic model leads indeedto a statistical distribution in the form of a Gamma distribution.This suggests that value functions satisfying the limit property (3.18) generate statisticaldistribution with thin tails, while fat tails are generated by value functions satisfying a scalingproperty like (3.12). We will make use of this observation in the next Section.To conclude this discussion, it is important to remark that interaction (3.7), while leadingto a kinetic equation with a steady state that can have fat tails in presence of high risk [62], itis based on the value function (3.9) which gives a big increment to a small wealth v (cid:28) ¯ v . Thisbehavior is clearly in contrast with the common belief that to increase the personal wealth bytrading, starting from a low value of wealth, is usually much more difficult than to increaseit starting from a high level. As recently discussed in [38], this apparent inconsistency ofthe model can be corrected by introducing a variable frequency of interactions that penalizesinteractions in which the wealth v put into the trade is small. We will be back to this questionlater on. 4. The kinetic description of social climbing
Modeling the elementary interaction.
In this Section, we will deal with the math-ematical modeling of the elementary interaction in the social climbing, treasuring the discus-sions of Section 2 about climbing in football activity, and that of Section 3 about the kineticmodeling of wealth distribution. As before, we assume that the behavior of the populationwith respect to the climbing of social ladder is homogeneous. This homogeneity assumption isclearly quite strong in general, since it requires at least to restrict the population with respectto some characteristics, like age, sex and degree of education.Once the homogeneity assumption is satisfied, the agent’s state at any instant of time t ≥ is completely characterized by the value w ≥ of the social rank occupied in the society. Weassume that this value can be measured in terms of some reasonable unit. If the relationshipbetween social rank and personal wealth is identified as substantial, it is possible to measurethe value w with the unit of the wealth of agents. However, depending on the homogeneousgroup of the society we are considering, other choices can be equally possible. The unknownis the density (or distribution function) f = f ( w, t ) , where w ∈ R + and the time t ≥ , andthe target is to study the statistical features of the subsequent steady state.Since agents in the system tend to improve their social rank, the density f ( w, t ) continuouslychanges in time by elementary interactions. It is clear that these interactions are more generalthan the trading activity described by (3.7), since the social climbing is not only achieved bytrading to increase the wealth, but it involves a variety of different activities finalized to thisgoal. Similarly to the problems treated in [33, 45, 46], the mechanism of social climbingin modern societies can be postulated to depend on some universal features that can besummarized by saying that agents likely tend to increase the value w of their ranking byinteractions, while manifest a certain resistance to decrease it.To obtain a computable and acceptable (from the sociological point of view) expression ofthe elementary variation of the social ranking of agents, we identify the mean values whichcharacterize, at least in a very stylized way, the multi-agent society. A first value, denoted by ¯ w , defines the upper limit of the low social ranking. This value identifies the limit belowwhich agents do not expect to be able to climb the social ladder. The second value, denotedby ¯ w L , with ¯ w L > ¯ w , denotes the value that it is considered as the level of a satisfactory well-being by a large part of the population. Note that both these values are in general differentlyperceived by agents. However, since the variations of these values from agent to agent can beassumed small, we can consider them as mean values universally perceived by agents.The elementary interaction is consequently modeled to describe the behavior of agents interms of these values, and it will express the natural tendency of individuals to reach (atleast) the value ¯ w L .Similarly to the case of football activity introduced in Section 2, and to the case of wealthdistribution treated in the previous Section, there is a strong asymmetry in the realizationof this target. With respect to individuals which enjoy a high level of social ranking, thisasymmetry expresses the objective difficulty of individuals to increase the value w of the rankto reach the desired value ¯ w from below.Proceeding as in Section 3 we will model the elementary interaction by resorting both tovalue functions and to uncertainties. Hence we write the elementary variation of social rankin the form(4.19) w ∗ = w − Ψ (cid:15) (cid:18) w ¯ w L (cid:19) w + η (cid:15) w. In any interaction the value w of the social rank can be modified for two different reasons,both quantified by dimensionless coefficients acting on the actual social rank variable w .The first one is the (value) function Ψ (cid:15) ( w/ ¯ w L ) , which can assume assume both positiveand negative values, that characterizes the asymmetric predictable behavior of agents. Thesecond coefficients characterized a certain amount of unpredictability always present in humanactivities. The uncertainty is contained in the random variable η , and , according to thewealth interactions (3.7), it is designed to be negligible in the mean, and in any case arenot so significant to produce a sensible variation of the value w . Hence, the social rank ofindividuals can be both increasing and decreasing by interactions, and the mean intensity ofthis variation is fully determined by the function Ψ (cid:15) . Last, the positive parameter (cid:15) (cid:28) quantifies the intensity of a single interaction.On the basis of the discussion of Section 3, and resorting to the class of value functionsconsidered in [33], we will describe the microscopic variation of z = 1 /w , in absence ofuncertainty, in the form(4.20) z ∗ = z − Φ (cid:15) (cid:18) z ¯ z L (cid:19) z. In [33], to suitably model the alcohol consumption, the value function was assumed in theform (3.17). Similarly to (3.15) the function (3.17) satisfies conditions (3.10) and (3.11)originally postulated by Kahneman and Twersky in [50], including the property to be positiveand concave above the reference value ( s > ), while negative and convex below ( s < ).[33]This function is bounded from above and below, and satisfies the bounds(4.21) − µ − e − (cid:15) e − (cid:15) ≤ Φ (cid:15) ( s ) ≤ . This property implies that, for small values of the parameter (cid:15) , the maximal amount of increaseof z is of the order of (cid:15) . The function (3.17) belongs to the class of value functions(4.22) Φ (cid:15)δ ( s ) = µ e (cid:15) ( s δ − /δ − e (cid:15) ( s δ − /δ + 1 , s ≥ , OCIAL CLIMBING AND AMOROSO DISTRIBUTION 13
Figure 4.2.
Value function as a function of δ and µ .with < δ ≤ , that interpolate between Φ (cid:15) ( s ) and Φ (cid:15) ( s ) , where(4.23) Φ (cid:15) ( s ) = µ s (cid:15) − s (cid:15) + 1 , s ≥ , The class of value functions (4.22) describe the situation in which it is easier to increase thevalue of z than to decrease it, in the whole range of possible values of the variable z . Then,the Fokker–Planck equations generated in the grazing limit of interactions based on valuefunctions like (4.22) correctly possess a steady state with thin tails.As discussed in Section 3, a correct value function for the problem of the social climbing isthen obtained from (4.20) by writing it with respect to w = 1 /z . If the value function Φ (cid:15) in(4.20) is given by (4.22), one obtains that the elementary variation of the social rank value w is given by (4.19), where, if s = w/ ¯ w L (4.24) Ψ (cid:15) ( s ) = Ψ (cid:15)δ ( s ) = − µ e (cid:15) ( s − δ − /δ − − µ ) e (cid:15) ( s − δ − /δ + 1 + µ , s ≥ . The value function Ψ (cid:15)δ is bounded from below and above, and satisfies the bounds(4.25) − µ − µ ≤ Ψ (cid:15)δ ( s ) ≤ µ − e − (cid:15)/δ (1 − µ ) e − (cid:15)/δ + 1 + µ . Starting from s = 0 , for any fixed value of the positive parameters (cid:15) and δ , the function Ψ (cid:15)δ ( s ) is convex in a small interval contained in the interval (0 , , with an inflection point in ¯ s < ,then concave. A picture of the value function for different values of the parameter is shownin Figure 4.2.We can now relate the inflection point ¯ s < and the reference point s = 1 to the previouslymentioned mean values characterizing the social climbing. Clearly, the value w = ¯ w L , namelythe perceived level of a satisfactory well-being corresponds to the reference point s = 1 . Then, Figure 4.3.
Inflection points for the value function as a function of δ and µ .the upper limit of the low social ranking ¯ w characterizes the inflection point ¯ s < , in view ofthe relationship(4.26) ¯ s = ¯ w ¯ w L < . Indeed, the graph of the function Ψ( s ) can be split in the three regions (0 , ¯ s ) , (¯ s, and (1 , + ∞ ) , such that the graph is steeper in the middle region than in the other two, thuscharacterizing the middle region (the region of the middle social ranks) as the region in whicha higher value of the mobility is present, and it is easier to improve the rank. On the contrary,the mobility is lower for values of the rank below ¯ w and above ¯ w L . In Figure 4.3, we showthe location of the inflection points as a function of δ for different values of µ . This behaviorof the function Ψ (cid:15) is clearly in agreement with the original believe of Pareto.[68]It is interesting to remark that, in the limit δ → the value function (4.24) becomes(4.27) Ψ (cid:15) ( s ) = µ s (cid:15) − ν ) s (cid:15) + 1 − µ , s ≥ , namely a value function of the same type of (4.23). Note however that in this case thelimit value function (4.27), at difference with the value functions (4.24), is concave, andthe inflection point ¯ w is lost. These value functions were originally considered in [45, 46],where it was shown that they characterize the lognormal distribution. Hence, the lognormaldistribution can be defined as a border case that separates fat tailed distributions from thintailed ones.4.2. The kinetic model.
Once the interaction (4.19) has been modeled, for any choice ofthe value function Ψ (cid:15) in the class (4.24) the study of the time-evolution of the statisticaldistribution of the social rank follows by resorting to kinetic collision-like models [24, 65].For any given value of the small parameter (cid:15) , the variation of the density f (cid:15) ( w, t ) is easilyshown to obey to a linear Boltzmann-like equation, fruitfully written in weak form. The weakform corresponds to say that the solution f ( w, t ) satisfies, for all smooth functions ϕ ( w ) (theobservable quantities)(4.28) ddt (cid:90) R + ϕ ( w ) f (cid:15) ( w, t ) dw = (cid:68) (cid:90) R + χ (cid:16) wu (cid:17) (cid:0) ϕ ( w ∗ ) − ϕ ( w ) (cid:1) f (cid:15) ( w, t ) dw (cid:69) . Here expectation (cid:104)·(cid:105) takes into account the presence of the random parameter η (cid:15) in (4.19). In(4.28) the positive dimensionless function χ ( w/u ) measures the interaction frequency of theinteractions with social rank w with respect to the some unit of measure u . OCIAL CLIMBING AND AMOROSO DISTRIBUTION 15
The right-hand side of equation (4.28) quantifies the variation of the observable quantity ϕ of the agents that modify their value from w to w ∗ according to the elementary interaction(4.19).The importance of a variable collision frequency has been outlined in [38] for the kineticdescription of wealth distribution. There, starting from a careful analysis of the microscopiceconomic transactions of the kinetic model, expressed by (3.7), allowed to conclude that thechoice of a constant collision kernel included as possible also interactions which human agentswould exclude a priori . This was evident for example in the case of interactions in which anagent that trades with a certain amount of wealth, does not receive (excluding the risk) somewealth back from the market. In strong analogy with the rarefied gas dynamics [15], wherethe analysis of the Boltzmann equation for Maxwell pseudo-molecules leads to the possibilityto make use of the Fourier transformed version, this makes clear that, in the socio-economicmodeling, the main advantages of the Maxwellian assumption are linked to the possibility toobtain analytical results.In agreement with [38], and treasuring the remarks about the frequency of interactionsmade in Section 2, we express the mathematical form of the kernel χ ( · ) by assuming that, fora given value w of the social rank, the frequency of interactions trying to improve the rank isdirectly proportional to the rank itself. This choice leads to consider collision kernels in theform(4.29) χ (cid:16) wu (cid:17) = (cid:16) wu (cid:17) β · α, for some constants α > and β > . This kernel assigns a low frequency to interactions inwhich individuals have a low rank, and assigns a high frequency to interactions in which thevalue of the rank is greater. This assumption translates in a simple mathematical form thatthe motivations to climb the social ladder are stronger in individuals belonging to the middleand upper classes, which clearly see, at difference with individuals belonging to the low class,the possibility to succeed.The values of the constants α and β need to be suitably chosen to guarantee at best thatthe characteristics of both the value function and the collective phenomenon are maintainedeven for small values of the parameter (cid:15) , and do not disappear in the limit (cid:15) → .To simplify computations, and to retain the essentials of the reasoning, let us set the unitof measure u = ¯ w L . Consequently(4.30) χ (cid:16) wu (cid:17) = α · (cid:18) w ¯ w L (cid:19) β = α s β , For a given value function Ψ (cid:15)δ ( s ) , the individual rate of growth is given by(4.31) ∂ Ψ (cid:15)δ ( s ) ∂s = 2 µ (cid:15) (cid:2) (1 − µ ) e y/ + (1 + µ ) e − y/ (cid:3) s − (1+ δ ) , where y is defined by y = (cid:15) s − δ − δ . Now, consider that, since (cid:15) < , for any given s ≥ , y ≥ − (cid:15)δ ≥ − δ , and the function z ( y ) = (cid:104) (1 − µ ) e y/ + (1 + µ ) e − y/ (cid:105) has a maximum in the point ¯ y = log 1 + µ − µ , where z (¯ y ) = 4(1 − µ ) . Therefore we easily conclude with the bounds(4.32) c δ = 1 (cid:2) (1 + µ ) e /δ + (1 − µ ) e − /δ (cid:3) ≤ (cid:2) (1 − µ ) e y/ + (1 + µ ) e − y/ (cid:3) ≤ C δ = 14(1 − µ ) . This implies that, for a given s > , the individual growth is vanishing as (cid:15) → . To maintaina collective growth different from zero as (cid:15) → , it is enough to fix(4.33) α = 1 τ (cid:15) This corresponds to increase the frequency at the order /(cid:15) , and to introduce at the sametime a relaxation parameter τ . Once the choice of α has been justified to retain a collectivememory of the growth of the value function, one has to face with the collective variation ofthe social rank, now given by τ (cid:15) Ψ (cid:15)δ ( s ) s β . Let us consider first the case s ≤ , namely to the increasing of the social rank. Since Ψ δ ( s ) = 0 , Lagrange theorem implies (cid:15) Ψ (cid:15)δ ( s ) = Ψ (cid:15)δ ( s ) − Ψ δ ( s ) (cid:15) = ∂ Ψ (cid:15)δ ( s ) ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =¯ (cid:15) , ≤ ¯ (cid:15) ≤ (cid:15). Then(4.34) − ∂ Ψ (cid:15)δ ( s ) ∂(cid:15) = 2 µ (cid:2) (1 − µ ) e y/ + (1 + µ ) e − y/ (cid:3) δ (cid:16) s − δ − (cid:17) > . Using the upper bound in (4.32) we can conclude that, when s < , with the choice β = δ thecollective variation of the social rank is uniformly bounded with respect to (cid:15) , and satisfies thebound (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) Ψ (cid:15)δ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s δ ≤ µ δ (1 − µ ) (1 − s δ ) . Note that, if we choose β < δ , the variation of the collective growth of social rank is extremelyhigh for values of s close to zero, namely in the part of population that do not possess manypossibilities to succeed. Hence, this choice is something that we have to exclude a priori.Also, values of β > δ would imply that for values of s close to zero, the collective growth isextremely small, thus practically excluding the possibility to increase the social rank startingfrom very low levels. Hence, the choice β = δ represents a good compromise for most societies.With these assumptions, the weak form of the Boltzmann-type equation (4.28) takes theform(4.35) ddt (cid:90) R + ϕ ( w ) f (cid:15) ( w, t ) dx = 1 (cid:15)τ (cid:68) (cid:90) R + (cid:16) wu (cid:17) δ (cid:0) ϕ ( w ∗ ) − ϕ ( w ) (cid:1) f (cid:15) ( w, t ) dw (cid:69) . OCIAL CLIMBING AND AMOROSO DISTRIBUTION 17
Note that, in consequence of the choice made on the interaction kernel χ , the evolution ofthe density f (cid:15) ( x, t ) is tuned by the parameter (cid:15) , which characterizes both the intensity ofinteractions and the interaction frequency.Due to the presence in the microscopic interaction (4.19) of the nonlinear value function(4.24), it is immediate to show that the only conserved quantity of equation (4.35) is obtainedin correspondence to ϕ = 1 . This conservation law implies that the solution to (4.35) remainsa probability density for all subsequent times t > . The evolution of higher moments is notknown analytically, and it is quite difficult to obtain explicit bounds, which could guaranteethe precise value of the expected tail of the stationary solution.5. Quasi-invariant limit and the Fokker-Planck equation
The grazing limit.
The linear kinetic equation (4.28) describes the evolution of thedensity consequent to interactions of type (4.19), and it is valid for any choice of the parameters δ, µ and (cid:15) . In real situations, however, it is reasonable to assume that in most cases a singleinteraction determines only an extremely small change of the value w . This is certainly truefor the deterministic part of the interaction (4.19), and it is true in the mean for the randompart. This situation is well-known in kinetic theory of rarefied gases, where interactions ofthis type are called grazing collisions [65, 79]. In the value functions (4.22) the smallnessassumption requires to fix (cid:15) (cid:28) . At the same time [37], the balance of this smallness withthe random part is achieved by setting(5.36) η (cid:15) = √ (cid:15)η, where η is a centered random variable of variance equal to σ . In this way the variance of η (cid:15) is of order (cid:15) . Consequently, the scaling assumption (5.36) allows to retain the effect ofall parameters in (4.20) in the limit procedure. An exhaustive discussion on these scalingassumptions can be found in [37] (cf. also [46] for analogous computations in the case of thelognormal distribution, in which the value function is given by (4.23)). For these reasons, weaddress the interested reader to these review papers for exhaustive details.In what follows we only deal with the main differences in the computations, essentially dueto the presence of a non-Maxwellian interaction kernel, and to the presence of a new type ofvalue functions. Without loss of generality, in the rest of the paper we fix in (4.29) the valueof the unit of measure to u = 1 . Note that a different choice will simply correspond to ascaling of time.Let us suppose that the solution to the kinetic equation (4.35) has moments bounded up tothe order two. To have a precise idea of the evolution of the observable in the scaling (cid:15) (cid:28) ,we start by studying the evolution of the mean value m (cid:15) ( t ) = (cid:90) R + w f (cid:15) ( w, t ) dw. In this simple case we have(5.37) ddt m (cid:15) ( t ) = − τ ¯ w δL (cid:90) R + (cid:15) Ψ (cid:15)δ ( w/ ¯ w L ) (cid:18) w ¯ w L (cid:19) δ f (cid:15) ( w, t ) dw, where the integral has been written in terms of the quotient w/ ¯ w L . Using the bounds (4.32)into (5.37) we obtain(5.38) ddt m (cid:15) ( t ) = ≤ C δ τ ¯ w δL (cid:90) w δ (cid:18)(cid:16) ¯ w L w (cid:17) δ − (cid:19) (cid:18) w ¯ w L (cid:19) δ f (cid:15) ( w, t ) dw + c δ τ ¯ w δL (cid:90) ∞ w δ (cid:18)(cid:16) ¯ w L w (cid:17) δ − (cid:19) (cid:18) w ¯ w L (cid:19) δ f (cid:15) ( w, t ) dw ≤ τ ¯ w δL (cid:18) C δ (cid:90) R + w ¯ w L f (cid:15) ( w, t ) dw − c δ (cid:90) R + (cid:16) ¯ w L w (cid:17) δ f (cid:15) ( w, t ) dw (cid:19) . Finally, Jensen’s inequality implies (cid:90) R + (cid:16) ¯ w L w (cid:17) δ f (cid:15) ( w, t ) dw ≥ (cid:18)(cid:90) R + w ¯ w L f (cid:15) ( w, t ) dw (cid:19) δ . Let us set M (cid:15) ( t ) = m (cid:15) ( t )¯ w L . Then M ( t ) satisfies the differential inequality(5.39) ddt M (cid:15) ( t ) ≤ τ ¯ w δL (cid:16) C δ M (cid:15) ( t ) − c δ M (cid:15) ( t ) δ (cid:17) , which shows that, independently of (cid:15) , M (cid:15) ( t ) satisfies the inequality(5.40) M (cid:15) ( t ) ≤ max (cid:40) M (cid:15) ( t = 0) , (cid:18) C δ c δ (cid:19) /δ (cid:41) . Coupling inequality (5.39) with the uniform bound (5.40) shows that the scaling (5.36) issuch that, for any given fixed time t > , the consequent variation of the mean value m (cid:15) ( t ) isbounded from above independently of (cid:15) . Since pointwise(5.41) A δ,(cid:15) (cid:18) w ¯ w L (cid:19) = 1 (cid:15) Ψ (cid:15)δ ( w/ ¯ w L ) → µ δ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) , we have a computable evolution of the mean even in the limit (cid:15) → . As explained in [37],there is a further physical motivation leading to the scaling of the frequency of interactions,as given by (4.33). Since for (cid:15) (cid:28) the interactions produce a very small change of the rankvalue w , in the limit (cid:15) → , a finite variation of the mean density can be observed only if eachagents in the system undergo a great number of interactions in a fixed period of time. Byusing the bounds (4.32) one can observe, at the price of an increasing number of computations,the existence of bounds for the evolution of the second moment of f (cid:15) ( w, t ) , which remainswell-defined also in the limit (cid:15) → (cf. the analysis in [37]).It is now easy to justify the passage from the kinetic model (4.28) to its continuous coun-terpart given by a Fokker–Planck type equation [37]. Given a smooth function ϕ ( w ) , and acollision of type (4.20) that produces a small variation of the difference w ∗ − w , one obtains (cid:104) w ∗ − w (cid:105) = − (cid:15) A δ,(cid:15) (cid:18) w ¯ w L (cid:19) w ; (cid:104) ( w ∗ − w ) (cid:105) = (cid:32) (cid:15) A δ,(cid:15) (cid:18) w ¯ w L (cid:19) + (cid:15)σ (cid:33) w . Therefore, equating powers of (cid:15) , it holds (cid:104) ϕ ( w ∗ ) − ϕ ( w ) (cid:105) = (cid:15) (cid:18) − ϕ (cid:48) ( w ) w µ δ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) + σ ϕ (cid:48)(cid:48) ( w ) w (cid:19) + R (cid:15) ( w ) , OCIAL CLIMBING AND AMOROSO DISTRIBUTION 19 where the remainder term R (cid:15) ( w ) , for a suitable ≤ θ ≤ is such that (cid:15) R (cid:15) ( w ) → as (cid:15) → . Therefore one obtains that the time variation of the (smooth) observable quantity ϕ ( w ) satisfies ddt (cid:90) R + ϕ ( w ) f (cid:15) ( w, t ) dw =1 τ (cid:90) R + w δ (cid:18) − ϕ (cid:48) ( w ) w µ δ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) + σ ϕ (cid:48)(cid:48) ( w ) w (cid:19) f (cid:15) ( w, t ) dw + 1 (cid:15) R (cid:15) ( t ) , where R (cid:15) ( t ) denotes the integral remainder term R (cid:15) ( t ) = (cid:90) R + w δ R (cid:15) ( w ) f (cid:15) ( w, t ) dw. Letting (cid:15) → , shows that in consequence of the scaling (5.36) the weak form of the kineticmodel (4.28) is well approximated by the weak form of a linear Fokker–Planck equation (withvariable coefficients)(5.42) ddt (cid:90) R + ϕ ( w ) g ( w, t ) dw =1 τ (cid:90) R + (cid:90) R + (cid:18) − ϕ (cid:48) ( w ) w δ µ δ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) + σ ϕ (cid:48)(cid:48) ( w ) w δ (cid:19) g ( w, t ) dw. In (5.42) the density function g ( w, t ) coincides with the limit, as (cid:15) → , of the density f (cid:15) ( w, t ) [37]. Provided the boundary terms produced by the integration by parts vanish, equation(5.42) coincides with the weak form of the Fokker–Planck equation(5.43) ∂g ( w, t ) ∂t = ˜ σ ∂ ∂w (cid:16) w δ g ( w, t ) (cid:17) + ˜ µ ∂∂w (cid:18) δ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) w δ g ( w, t ) (cid:19) . In (5.43) we defined ˜ σ = σ/τ and ˜ µ = µ/τ . Equation (5.43) describes the evolution of thedistribution density g ( w, t ) of the social rank w ∈ R + of the agent’s system, in the limit of the grazing interactions. As often happens with Fokker-Planck type equations, the steady statedensity can be explicitly evaluated, and it results to be a generalized Gamma density, withparameters linked to the details of the microscopic interaction (4.20).5.2. Steady states are Amoroso distributions.
Let us set γ = ˜ µ/ ˜ σ = µ/σ . The statio-nary distribution of the Fokker–Planck equation (5.43) is an integrable function which solvesthe first order differential equation(5.44) ddw (cid:16) w δ g ( w ) (cid:17) + γδ (cid:18) − (cid:16) ¯ w L w (cid:17) δ (cid:19) w δ g ( w ) = 0 . We solve (5.44) with respect to h ( w ) = w δ g ( w ) by separation of variables. The function h ( w ) solves the differential equation(5.45) dh ( w ) dw + γδ (cid:18) w − ¯ w δL w δ (cid:19) h ( w ) = 0 . If h ( w ) (cid:54) = 0 , equation (5.45) is clearly equivalent to(5.46) h ( w ) dh ( w ) dw = − γδ (cid:18) w − ¯ w δL w δ (cid:19) , that can be rewritten as(5.47) ddw log h ( w ) = − γδ (cid:26) ddw log w ¯ w L + 1 δ ddw (cid:20)(cid:16) ¯ w L w (cid:17) δ − (cid:21)(cid:27) . In this way we find that the unique solutions to (5.44) are the functions(5.48) g ∞ ( w ) = g ∞ ( ¯ w L ) (cid:16) ¯ w L w (cid:17) δ + γ/δ exp (cid:26) − γδ (cid:18)(cid:16) ¯ w L w (cid:17) δ − (cid:19)(cid:27) . If we fix the mass of the steady state (5.48) equal to one, the consequent probability density isa particular case of the generalized Gamma distribution, usually named Amoroso distribution[1], as given in (1.3). Note that (5.48) corresponds to values of β = − δ < in the expressionof the generalized Gamma distribution (1.3). The remaining parameters of the equilibriumstate (5.48) are given by(5.49) α = 1 + 1 δ + γδ , θ = ¯ w L (cid:16) γδ (cid:17) /δ . The limit δ → in the Fokker–Planck equation (5.43) corresponds to the drift term inducedby the value function (4.23). In this case, the equilibrium distribution (5.48) takes the formof a lognormal density [45]. It is remarkable that the lognormal density is achieved as limitcase when looking at the distribution of alcohol consumption [33], where the Fokker–Plancktype equation has been obtained resorting to the value functions (4.22). Hence, the lognormaldensity appears as the limit case of both generalized Gamma distributions characterized bypositive values of the parameter β , and the present case of Amoroso distributions, given bynegative values of the parameter β .The case δ = 1 in (5.48) corresponds to an inverse Gamma distribution. In this case α = 2 + γ , and θ = γ ¯ w L . The steady state of the distribution of social rank coincideswith the steady state of wealth distribution discussed in Section 3. Hence, in the grazinglimit, the inverse Gamma distribution appears as steady state of two different kinetic models,characterized by two different value functions (4.22) with δ = 1 , and, respectively (3.9).Note that for all values of δ > the moments are expressed in terms of the parameters ¯ w L , σ , µ and δ denoting respectively the target level of social rank, the variance σ of the randomeffects and the values δ and µ characterizing the value function Ψ (cid:15)δ defined in (4.24).It is interesting to remark that, in the case of the inverse Gamma distribution, the meanvalue of the equilibrium density (5.48) is always less than ¯ w L . In this case in fact M = ¯ w L γγ + 1 . Numerics
In this Section, we perform several numerical experiments with the aim of describing thebehaviors of the social climbing Boltzmann model (4.19) and to quantify the goodness of itsFokker-Planck counterpart. In more details, we intend to show the effects of different choicesof the parameter appearing in the value function Ψ (cid:15)δ defined by (4.24). The shape of both thedistribution function in time and its final steady state are indeed modified when µ/σ , (cid:15) and δ are modified. The first quantity determines the intensity of the interaction with respect tothe intensity of the random effects in the climbing dynamic. The second quantity determinesthe rate of change of the social status of an agent due to a single interaction while the lastparameter is responsible for the final shape of the social distribution in a given population. OCIAL CLIMBING AND AMOROSO DISTRIBUTION 21
Figure 6.4.
Test 1. Convergence to the Fokker-Planck steady state for theBoltzmann model as a function of time with δ = 0 . . Top left γ = 5 , top right γ = 2 , bottom left γ = 1 and bottom right γ = 2 / .6.1. Trend to equilibrium.
In this first test, we analyse the rate of trend to equilibriumfor the Boltzmann model (4.19) starting from a uniform distribution f ( w, t = 0) over thedomain [0 , . The convergence to the steady state Amoroso distribution for various values ofthe parameters γ = µ/σ are shown in Figure 6.4. The random effects are taken into accountby sampling η (cid:15) from a uniform distribution with variance σ . The average level of well-being ¯ w L is fixed equal to while the parameter δ which characterizes the exponent of the steadystate is fixed equal to . . The collision frequency is w δ , the number of samples is and ε = 10 − . We clearly observe that for all tested situations the Boltzmann model convergestowards the corresponding Amoroso equilibrium distribution (5.48).6.2. Wealth and Social climbing.
In this part, we compare the different behaviors of theBoltzmann model for wealth distribution described by equation (3.8) and the Boltzmannmodel for social climbing (4.24). The main difference between the two models relies in theconvexity/concavity properties of the two values functions acting on the lower values of thewealth (here we identify wealth and social rank value). The first value function, describingvariation of wealth, is always concave while the second, describing the social level, has aninflection point for ¯ w < ¯ w L . This corresponds to the empirical observation that for peoplebelonging to the lower level of the social ladder, the change of status is very difficult. Forthe tests reported in Figure 6.5, we have chosen δ = 1 , which gives an inverse Gammadistribution as common steady state solution for the Fokker-Planck equation (5.43). Thecollision frequency of the wealth model is chosen accordingly to [38] as ( vw ) δ where v is Figure 6.5.
Test 2. Comparison of the wealth and the social climbing modelsin the case δ = 1 , γ = 1 and (cid:15) = 0 . . From top left to bottom right the imagesshow the trend to the steady state solution for the two models. The initialand the final Fokker-Planck states are also shown.sampled from a uniform distribution around ¯ w L . Moreover γ = 1 , ¯ w L = 1 while the scalingparameter (cid:15) = 0 . is such that the models (3.8) and (4.24) are far from the correspondingFokker-Planck limits.This has been done to highlight the difference between the two dynamics, since, as shownhere and in [38], both share the same steady state in the limit (cid:15) → and consequently thedifferences in this limit setting would be very small. In Figure 6.6 the same curves are shownzooming around the lower wealth level where as expected the differences exhibited by the twomodels are larger. Each image in Figure 6.5 and 6.6 reports the initial distribution, the finalFokker-Planck distribution, the distribution of the social status f S ( t, w ) and of the wealth f W ( t, w ) at different times from top left to bottom right. The same results are shown inFigure 6.7 and 6.8 where γ = 2 / , the other parameters remaining unchanged. As expected,the agents belonging to the lower social class are the ones for which the two models give largerdifferences.6.3. Wealth and Social climbing: convergence to to the Fokker-Planck dynamicsand Pareto tails.
In this last test, we investigate the differences between the Boltzmann andthe Fokker-Planck dynamics for the social climbing interaction (4.24) and the wealth model(3.8), respectively. The final equilibrium state is in both cases represented by the same inverseGamma distribution, i.e. δ = 1 . This means that Pareto tails are expected to appear in thissteady state limit when (cid:15) → . The Figure 6.9 shows on the left the steady state distributions OCIAL CLIMBING AND AMOROSO DISTRIBUTION 23
Figure 6.6.
Test 2. Comparison of the wealth and the social climbing modelsin the case δ = 1 , γ = 1 and (cid:15) = 0 . . From top left to bottom right the imagesshow the trend to the steady state solution for the two models. The initial andthe final Fokker-Planck states are also shown. Zoom around the lower wealthlevels.for wealth and social status for (cid:15) = 0 . and (cid:15) = 0 . as well as the corresponding inverseGamma distribution (5.48) equilibrium state. On the right hand side, we plot the resultsin log-log scale, which highlights the Pareto tails index. The different behaviors of the twomodels far from the Fokker-Planck dynamics are visible, both for the agents possessing alower wealth level as well as for the wealthy part of the population living in the tail of thedistribution. 7. Conclusions
The statistical distribution of social rank in a multi-agent society has been described inthis paper by resorting to classical methods of collision-like kinetic theory. The main goalof our analysis was to provide an explanation of the emergence of steady states in the formof generalized Amoroso distributions, a family of distributions with polynomial tails thatrepresent at best the formation of a social elite. The macroscopic behavior is consequent to the
Figure 6.7.
Test 2. Comparison of the wealth and the social climbing modelsin the case δ = 1 , γ = 2 / and (cid:15) = 0 . . From top left to bottom right theimages show the trend to the steady state solution for the two models. Theinitial and the final Fokker-Planck states are also shown.choice made at the microscopic level, choice that takes into account the essential features of thehuman behavior related to the phenomenon of social climbing. The kinetic modeling is similarto the one introduced in [45], subsequently generalized in [33, 46], in which the human behaviorhas been shown to be responsible of the formation of a macroscopic equilibrium in the formof probability distributions with thin tails, like the lognormal distribution or the Gamma andWeibull ones. From this point of view, the present results can be considered as an extensionof the kinetic description of [33, 46, 77], which allows to classify, at a microscopic level, themain differences in the elementary interaction which produce a whole class of generalizedGamma distributions, that range from the classical Gamma density to the lognormal one.Well-known arguments of kinetic theory allow to model these phenomena by means of aFokker–Planck equation with variable coefficients of diffusion and drift. Interestingly enough,for this class of Fokker–Planck equations a lot of mathematical results can be proven, includingthe exponential convergence towards equilibrium [77].A relevant part of this analysis relies in a detailed comparison of the main rules of socialclimbing with thats of wealth distribution, and helps to share some light on possible improve-ments of the latter by taking into account in a more substantial way both the individual andsocial aspects of human behavior present in the former. The numerical comparison of the twomodels reveals indeed a marked difference between the kinetic description in the low part ofthe profiles, difference that disappears only in the grazing asymptotics. OCIAL CLIMBING AND AMOROSO DISTRIBUTION 25
Figure 6.8.
Test 2. Comparison of the wealth and the social climbing modelsin the case δ = 1 , γ = 2 / and (cid:15) = 0 . . From top left to bottom right theimages show the trend to the steady state solution for the two models. Theinitial and the final Fokker-Planck states are also shown. Zoom aroung thelower wealth levels. -1 -4 -2 Figure 6.9.
Test 3. Asymptotic behavior of the Fokker-Planck model andthe Boltzmann wealth and social climbing models with δ = 0 . , γ = 1 , ¯ w L = 1 .The right image is in log-log scale. Acknowledgement
This work has been written within the activities of GNFM group of INdAM (NationalInstitute of High Mathematics), and partially supported by MIUR project “Optimal masstransportation, geometrical and functional inequalities with applications”.
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