modelling altruism and selfishness in welfare dynamics · k. sigmund, the calculus of sel shness...
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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 ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
Simulations
Complex systems
Social systems:
complex evolutive systems
including nonlinear interactions
and learning phenomena
where emergent behaviors appear.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
Simulations
Microscale
State at the microscale → activity variable: wealth status u.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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)
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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)]).
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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).
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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).
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
Case I - Solution in time
Initial probability distribution vs. social classes with low threshold
µ0 = 3.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
Case I - Solution in time
Initial probability distribution vs. social classes with high threshold
µ0 = 9.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6N = − 0.5
M = − 1.7
Initial probability distribution vs. social classes with prevalent
poor cluster
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.5N = 0.3
M = 0.7
c
Initial probability distribution vs. social classes with prevalent
middle classes.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
Initial probability distribution vs. social classes with prevalent
middle classes and medium threshold µ0 = 5.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.5N = 0.5
M = 1.7
Initial probability distribution vs. social classes with prevalent
wealthy cluster.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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
middle classes.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
−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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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'.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
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.
D. Knopo�, On a mathematical theory of complex systems onnetworks with application to opinion formation, Math. ModelsMethods Appl. Sci., 24 (2) (2014) 405-426.
D. Knopo�, On the modeling of migration phenomena on smallnetworks, Math. Models Methods Appl. Sci., 23 (3) (2013)541-563.
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...
Mathematical StructuresModeling altruism and sel�shness in welfare dynamics
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
M. Dol�n, M. Lachowicz Modelling Altruism and Sel�shness ...