Modelling Altruism and Selfishness in Welfare Dynamics · K. Sigmund, The Calculus of Sel shness (Princeton Univ. Press, 2011). M. Dol n, M. Lachowicz Modelling Altruism and Sel shness

Mathematical StructuresModeling altruism and sel�shness in welfare dynamics

Simulations

Modelling Altruism and Sel�shness

in Welfare Dynamics

M. Dol�n*, M. Lachowicz

* Dip. di Ingegneria Civile e Matematica Applicata - Univ. di Messina

Dep. of Mathematics and Applied Mechanics - Univ. of Warsaw

SIMAI 2014 - MS Complex SystemsTaormina, 8 luglio 2014

M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...


Simulations

Progetto di ricerca (GNFM) 2013:

Politiche del welfare e dinamiche sociali in una societá vista come

un sistema evolutivo complesso.

MD, M. Lachowicz, Modeling altruism and sel�shness in welfare

dynamics: the role of non linear interactions, Math. ModelsMethods Appl. Sci., 24 (2014) in stampa.

MD, M. Lachowicz, Modeling DNA thermal denaturation at the

mesoscopic level, Cont. Disc. Dyn. Syst. B (2014) in stampa.



Simulations

Outline

1 Mathematical Structures

2 Modeling altruism and sel�shness in welfare dynamics

3 SimulationsSimulations Case I - rôle of the thresholdSimulations Case II - rôle of the initial conditionsSimulations Case III - Interplay between initial conditions andthresholdSimulations Case IV - The rôle of �sel�shness'.Emergent behaviors - Research perspectives



Simulations

Outline






Simulations

Outline






Simulations

Hallmarks

Stochastic binary interactions between agents: living entitiesat each interaction play a game with an output that istechnically related to the state of the interacting entities andto global quantities.The output of the game is given in probability.

Nonlinear e�ects on strategies.

Emergent behaviors.



Simulations

Complex systems

Social systems:

complex evolutive systems

including nonlinear interactions

and learning phenomena

where emergent behaviors appear.



Simulations

Microscale

State at the microscale → activity variable: wealth status u.



Simulations

Discrete wealth status (ui = i)

Ip=poor cluster︷︸︸︷−n ? ? ? −1

I0=neutral︷︸︸︷0

Iw=wealthy cluster︷︸︸︷1 ? ? ? n

Overall socio-economic system Iu = Ip ∪ Io ∪ Iw



Simulations

Mesoscale

Discrete probability distribution function:

f(t, u) = {fi (t, u)}i∈Iu fi : [0,T ]× Iu → [0, 1], (1)

with constraint ∑i∈Iu

fi (t) = 1, ∀t ∈ [0,T ]. (2)



Simulations

Macroscale

Social gap:

N[f(t)] :=∑i∈Iw

fi (t)−∑i∈Ip

fi (t). (3)

Overall wealth:M[f(t)] =

∑i∈Iu

ifi (t). (4)

Notation W = W[f(t)] = (N[f(t)],M[f(t)]).



Simulations

Summary of the Model Framework

Modeling the collective welfare dynamic of a large system ofinteracting individuals:

Active particles: individuals or groups of individuals.

Microscopic state discrete: wealth status (activity alone,spatial homogeneity).

Discrete probability distribution over the microscopic state.

Stochastic interactions.

Interactions conditioned not only by the micro-state of theinteracting pairs, but also by global quantities.

Variable threshold triggering the interactions.

Constant number of active particles.



Simulations

Model equations

The evolution of the probability distribution is obtained by abalance of particles within elementary volumes of the space of themicroscopic states, where the dynamics of in�ow and out�ow ofparticles is related to interactions at the microscopic scale:

dfidt = Ji [W;µ] =

∑h,k∈Iu

ηhk [W;µ]Bihk [W;µ]fhfk − fi∑k∈Iu

ηik [W;µ]fk ,

dµdt = ϕ(W;µ).



Simulations

Microscopic pair interactions:

µ[W] threshold variable

triggers the interaction rules.

ηhk [W;µ] encounter ratefrequency of interactions of the test particle belonging to theh-th socio-economic class with a �eld particle of the k-thsocio-economic class.

Bihk [W;µ] transition probabilitiesprobability for the test particle belonging to the h-thsocio-economic class to shift to the i-th class due to anencounter with a �eld particle of the k-th socio-economic class.



Simulations

E�ect of the threshold

Pairwise interaction between particles (belonging to uh and uk) isof competition or of cooperation, depending on the threshold µ[W].

competition (|h− k | < µ) cooperation (|h− k | ≥ µ)

a) Competition b) Cooperation



Simulations

1 Competitive interaction (|h − k | < µ[W]):

1 h = −n , n: Bihk = δi,h;

2 h < k : Bihk = β0δi,h−1 + (1− β0)δi,h;

3 h > k : Bihk = (1− (β0 + β1N))δi,h + (β0 + β1N)δi,h+1.

1 Cooperative interaction ( |h − k| ≥ µ[W]):

1 h ∈ Ip, k ∈ Iw : Bihk = (1− (β0+β1N))δi,h+(β0+β1N)δi,h+1;

2 h ∈ Iw , k ∈ Ip: Bihk = β0δi,h−1 + (1− β0)δi,h.



Simulations

Constant threshold

The welfare policy of governments can substantially in�uence thedynamics of the parameter µ.For instance, if a government has a strong in�uence on thepopulation of a nation, it has the power of selecting a certain valueof the parameter and keep it constant for a certain time interval:

dµ

dt= 0 ⇒ µ = µ0 = constant.



Simulations

Variable threshold

dµ

dt= ε( 1

nM[f] + γ N[f]

)(1− µ

2n

)µ,

(ε small dimensionless parameter). Namely the rate of increasing ofµ grows with M and with positive values of N, while it iscontrasted by negative values of M. back

Simulations put in evidence, not only the dynamics of the "shape"of wealth distribution, but also the dynamics of the moments andthe rôle of the parameter γ, which measures the weight of'sel�shness'.

K. Sigmund, The Calculus of Sel�shness (Princeton Univ. Press, 2011).



Simulations

Variable threshold

Dynamics totally left to internal competition/con�icts of thepopulation. Wealthy people can show a trend to increase µ, whilepoor classes attempt to obtain the opposite e�ect.

1 When µ tends to its maximum admissible value 2n it cannotincrease any longer, while when µ tends to zero it cannotdecrease to negative values;

2 The quantities µ, M, and N have been referred to theirmaximal values, respectively, 2n, n and 1;

3 The dimensionless parameter ε is the ratio between theencounter rate that generates the dynamics of µ (namely ηµ)and the encounter rate that generates the social dynamics (η).Since ηµ is much slower than η, ε is a small parameter;

4 γ is a dimensionless parameter that weights the in�uence of Nwith respect to M.



Simulations

Simulations Case I - rôle of the thresholdSimulations Case II - rôle of the initial conditionsSimulations Case III - Interplay between initial conditions and thresholdSimulations Case IV - The rôle of �sel�shness'.Emergent behaviors - Research perspectives

Outline






Simulations


Case I - Initial condition

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5N = 0.3

M = 0.7

Initial probability distribution vs. social classes with prevalent

middleclass.



Simulations


Case I - Low constant threshold

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

time0

2

4

NM

On the left: asymptotic trend in the case of constant thresholdµ0 = 3.On the right: corresponding evolution of N and the �rst momentM.



Simulations


Case I - High constant threshold

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

time0

0.5

1

1.5

NM

On the left: asymptotic trend in the case of constant thresholdµ0 = 9.On the right: corresponding evolution of N and the �rst momentM.



Simulations


Case I - Solution in time

Initial probability distribution vs. social classes with low threshold

µ0 = 3.



Simulations


Case I - Solution in time

Initial probability distribution vs. social classes with high threshold

µ0 = 9.



Simulations


Case I - Variable threshold (µ0 = 3).

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

0

0.5

1

1.5

time2

10

NM

threshold

On the left: asymptotic trend in the case of variable thresholdµ0 = 3.On the right: corresponding evolution of N and the �rst momentM.



Simulations


Case I - Variable threshold (µ0 = 9).

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

0

0.5

1

1.5

time9

10

NM

threshold

On the left: asymptotic trend in the case of variable thresholdµ0 = 9.On the right: corresponding evolution of N and the �rst momentM.



Simulations


Remark

Simulations show that a signi�cant presence of middle classes and alow value of the threshold increases the overall wealth and hencereduces the presence of poor classes. On the other hand, theopposite trend is observed for high values of µ.The case of variable µ enlarges this trend with a more importantsuppression of middle classes and higher concentrations on theextreme conditions. Moreover, high values of µ reduce the overallwealth and radicalize the division between poor and wealthyclusters.



Simulations


Outline






Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.2

0.4

0.6N = − 0.5

M = − 1.7


poor cluster



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.2

0.4

0.6

time−4

−2

0

NM

Medium constant threshold. On the left: asymptotic trend inthe case of constant threshold µ0 = 5.On the right: the corresponding time evolution of N and the �rstmoment M.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5N = 0.3

M = 0.7

c


middle classes.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

time0

1

2

NM

Medium constant threshold. On the left: asymptotic trend inthe case of constant threshold µ0 = 5.On the right: the corresponding time evolution of N and the �rstmoment M.



Simulations


Case II - Solution in time

12

34

56

78

910

11

0.2

0

time

socio.economic classes

prob

abili

ty d

istr

ibut

ion


middle classes and medium threshold µ0 = 5.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5N = 0.5

M = 1.7


wealthy cluster.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

time0

1

2

NM

Medium constant threshold. On the left: asymptotic trend inthe case of constant threshold µ0 = 5.On the right: the corresponding evolution of N and the �rstmoment M.



Simulations


Remark

Simulations show that a in the case of a medium constantthreshold during the observation time, initial conditions a�ectstrongly the asymptotic trend when initial prevalent poor cluster orinitial prevalent wealthy is considered, while the di�erence withprevalent middleclass is not strong. Moreover, simulations showthat the initial condition with a prevalent middleclass gives the bestincreasing rate for the overall wealth of the society.



Simulations


Outline






Simulations


In this set of simulations we compare each asymptotic trend withconstant threshold (top of each �gure) to the corresponding onewith variable threshold (bottom of each �gure), for three di�erentinitial social distributions: prevalent poor cluster, prevalentmiddleclass and prevalent wealty cluster; in all cases µ0 = 5 andγ = 0.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.2

0.4

0.6

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.2

0.4

0.6

time−4

−2

0

−4

−2

0

NM

Initial condition with prevalent poor cluster. Top: asymptotictrend in the case of constant threshold µ0 = 5.Bottom: asymptotic trend in the case of variable threshold with thesame initial conditions.



Simulations


−5 −4 −2−3 −2 −1 0 1 2 3 4 50

0.5

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

0

1

2

time0

1

2

NM

Initial condition with prevalent middleclass. Top: asymptotictrend in the case of constant threshold µ0 = 5.Bottom: asymptotic trend in the case of variable threshold with thesame initial conditions.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

0

1

2

time0

1

2

NM

Initial condition with prevalent wealth cluster. Top:asymptotic trend in the case of constant threshold µ0 = 5.Bottom: asymptotic trend in the case of variable threshold with thesame initial conditions.



Simulations


Remark

Simulations show that, independently of the initial conditions, thequalitative trend of the overall wealth status of the society is betterin the case of a constant threshold with respect to the case ofvariable threshold. Again, as in the case study II, the best'performance' is observed when an initial condition with prevalentmiddleclass is considered.



Simulations


Outline






Simulations


In this last set of simulations we enlight the rôle of 'sel�shness'(represented by the parameter γ go ) on the asymptotic trends ofthe initial probability distributions of particles on thesocio-economic classes. We make two simulations using for each ofthem the same initial conditions for the probability distribution,namely with prevalent middleclass and for the variable threshold.The �rst asymptotic trend is then obtained for γ = −1 representingan 'altruistic' society and the second one for γ = 1 representing a'sel�sh' society.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5N = 0.3

M = 0.7


middle classes.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

0

1

2

3

time0

5

NM

threshold

Altruism. On the left: asymptotic trend in the case of variablethreshold and 'sel�shness' γ = −1.On the right: the corresponding evolutions of N, the �rst momentM and the threshold µ.



Simulations


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

0

1

2

3

time

5

10

NM

threshold

Sel�shness. On the left: asymptotic trend in the case of variablethreshold and 'sel�shness' γ = 1.On the right: the corresponding evolutions of N, the �rst momentM and the threshold µ.



Simulations


Remark

Simulations show that in the same conditions for the initialprobability distribution and for the variable threshold the introducedparameter of 'sel�shness' determine a positive rate of the overallwealth of the society in the case of minimal 'sel�shness' and anegative rate in the case of maximal 'sel�shness'.



Simulations


Outline






Simulations


Case I – Rôle of the threshold

Low constant threshold

Increasing wealth

High constant threshold ˅ Variable threshold

Decreasing wealth

Case II - Rôle of the initial conditions

Prevalent middleclass

Max. positive rate of wealth

Case III – Interplay between initial conditions and threshold

Prevalent Prevalent

middleclass ˅ wealthy cluster

˄

Constant Variable

threshold ˅ threshold

unchanged

wealth

Prevalent poor cluster

˄

Constant Variable

threshold ˅ threshold

different

asymptotic

“shapes”

Case IV - Rôle of “selfishness”

Altruism

Increasing wealth

Selfishness

Decreasing wealth



Simulations


Kinetic Theory of Active Particles

G. Ajmone Marsan, N. Bellomo, A. Tosin,Complex systems and society - Modeling and simulations, SpringerBriefs (Springer, 2013).

N. Bellomo, D. Knopo�, J. Soler,On the di�cult interplay between life, complexity, andmathematical sciences, Math. Models Methods Appl. Sci. 23(10)(2013) 1861�1913.

N. Bellomo, M. A. Herrero, A. Tosin, On the dynamics of socialcon�icts looking for the Black Swan, Kinet. Relat. Models, 6(2013) 459�479.



Simulations


Discrete Systems and Developments

J. Banasiak, M. Lachowicz, Methods of Small Parameter in

Mathematical Biology, Birkhäuser (2014).

M. L. Bertotti, M. Delitala,Conservation laws and asymptotic behavior of a model of social

dynamics, Nonlinear An. Real World Appl., 9 (2008) 183�196.

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D. Knopo�, On the modeling of migration phenomena on smallnetworks, Math. Models Methods Appl. Sci., 23 (3) (2013)541-563.



Simulations



Documents

Modelling Altruism and Selfishness in Welfare Dynamics · K. Sigmund, The Calculus of Sel shness (Princeton Univ. Press, 2011). M. Dol n, M. Lachowicz Modelling Altruism and Sel shness