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851-0585-04L – Modeling and Simulating
Social Systems with MATLAB
Lecture 11 – Evolutionary Stability and Replicator
Dynamics
Karsten Donnay and Stefano Balietti
Chair of Sociology, in particular of
Modeling and Simulation
© ETH Zürich |
2012-05-14
Schedule of the course
Introduction to
MATLAB
20.02.
27.02.
05.03.
12.03.
19.03.
26.03.
Working on
projects
(seminar
thesis)
02.04.
23.04.
Introduction to
social-science
modeling and
simulations
30.04.
07.05.
14.05.
No lecture, but we will be in
the room to supervise you
21.05.
28.05.
2012-05-14
Handing in seminar thesis
and giving a presentation
K. Donnay & S. Balietti / [email protected] [email protected]
2
Final presentation schedule
 Project presentation 15’ + 5’ (for Q&A)
 3 slots available:



Tuesday, May 29th
Wednesday, May 30th
Wednesday, May 30th
9.15 – 12.15
9.15 – 12.15
16.15 – 18.15
 Registration for final presentation is binding; if you
do not want to obtain credits, do not register!
 Sign up for a presentation slot this week in class
or via mailing list [email protected]
2012-05-14
K. Donnay & S. Balietti / [email protected] [email protected]
3
Final presentation schedule
 Presentations take place in the seminar room of
the offices of the Chair of Sociology, in particular
Modeling and Simulation
 The seminar room is located on the first floor, first
door on the right. Room C 1
 Enter without ringing the bell, but please keep
silence in the corridor
 http://www.soms.ethz.ch/box_feeder/howtoreach.pdf
2012-05-14
K. Donnay & S. Balietti / [email protected] [email protected]
4
Final presentation schedule
 Reminder:
Your reports are due midnight of Friday May 25th
2012
 Submission is through GIT:
simply upload your report, source code, videos
etc. before the deadline and we will retrieve it.
In case of submission problems, please contact
us beforehand.
2012-05-14
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5
Goals of Lecture 11: students will
1. Consolidate knowledge acquired during lecture 10,
through brief repetition of the main points.
2. Review Schelling’s segregation model and its main
results from a classical perspective
3. Understand the motivation and the rules behind a discrete
time replicator dynamics equation
4. Receive a brief introduction to stability analysis in
evolutionary models
5. Revisit Schelling’s segregation model from an
evolutionary perspective
6. Reflect about the main message of the course
2012-05-14
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6
Repetition: statistical testing

Statistical tests tell us how sure we can be that our
simulation reproduces an empirical observation

Testing represent the closure of the modeling cycle

Statistical test of your simulation results is NOT a
requisite for passing this course
2012-05-14
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7
Repetition: scientific writing


All claims must be well justified

Most of the time scientific writing is “standardized”
Correlation is not causation
2012-05-14
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8
Modeling Population Dynamics
In many American cities
racial segregation can be
traced back to separated
neighborhood for people
of different race.
However, when
interviewed most of the
people said they prefer to
live in integrated
neighborhoods.
Clark(1991)
2012-05-14
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9
Modeling Population Dynamics

In large populations, how do persistent structures of
interaction evolve in the absence of deliberate design?

Understanding individual’s preferences and beliefs may
allow us the prediction of individual behavior

To explain aggregate outcomes we cannot simply sum up
the predicted individual behaviors

The actions taken by each individual typically affect the
constraints, beliefs, or preferences of others.
2012-05-14
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10
Schelling’s Segregation Model


Cellular automata

Individuals do not want be part of the minority

Individuals can relocate themselves if they are not happy
with their current situation

Result: local myopic interaction determines global
segregation pattern.
Individuals prefer integration
2012-05-14
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11
Schelling’s Segregation Model: demo
2012-05-14
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12
Evolutionary models

Evolutionary models of population mimic the dynamics of
biological system under the combined influence of:
1. Chance
2. Inheritance
3. Natural Selection
4. Path-dependence (historical contingency)
2012-05-14
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13
Evolutionary models

Individuals are bearer of behavioral rules (traits) which
can spread to the whole population or remain confined in a
niche.

The transmission can take place through different
mechanisms:

Differential replication: genetic or cultural

From parents: vertical transmission:

From others: oblique transmission

From members of the group: horizontal
2012-05-14
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Evolutionary models
1. Individual implements a strategy dictated
by its genetic or behavioral trait
Loop for 1:N
2. Individual play the strategy against
1.
The Environment
2.
Other randomly paired individuals
3. A certain payoff is granted
Learning
Natural Selection
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4. An individual can update his strategy
5. Replication relative to payoff
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15
Replicator Equation

Gives complete account of out-of-equilibrium dynamics

Discover evolutionary irrelevant equilibria


Some non-equilibrium states are of substantial importance
Population is hierarchically structured and differential
replication ca take place at more than one level
2012-05-14
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Discrete Time Replicator Equation

Assume a population is divided in two groups: x and y

At each time step individuals are randomly paired.

They expected payoff E(B) for the group would depend on:

The relative frequency p of the group in the population

The relative payoff π for the strategy of each group
E(Bx)  p (x, x)  (1 p) (x, y)
E(By)  p (y, x)  (1 p) (y, y)
2012-05-14
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Discrete Time Replicator Equation

The expected population frequency for trait x at time t+1
can be written as the sum:
p' p  p(1 p) (by  bx)
p(1 p) (bx  by)


where ω is the fraction of population in “update state”, and
β is a positive constant that renormalize small differences in
payoff
2012-05-14
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Discrete Time Replicator Equation

And the population differential would be:
p  p(1 p)(bx  by)

Δp will be the greater:

The greater the fraction of population in “update-state”: ω

The greater the sensitivity of the population to small differences in

payoff: β

The greater the possibility to meet other population groups: p -> 1/2

The greater the difference in payoff: bx – by
2012-05-14
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19
Discrete Time Replicator Equation

The replicator equation can be generalized to n traits:
p  p (bx  b)

where b_hat is the average population payoff.

P.D. Taylor and L.B. Jonker, Evolutionary stable strategies and game
dynamics. Mathematical Biosciences, 40 (1978), pp. 145–156.

This update rule is also called payoff monotonic update.

2012-05-14
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Stability Analysis
p  p (bx  b)


For each value of p the replicator equation gives us the
velocity and direction of change of the state.

We are interested in studying those values of p such that
Δp = 0

which identify a stationary state or a fixed point.
2012-05-14
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Stability Analysis
p  p (bx  b)


Apart from trivial cases (which?), Δp takes the sign of
bx – b_hat, which reflects the monotonic update rule.

Intuitively, in order to be stable a state needs a
corrective feedback mechanism which counteract a
change in Δp

In this case, a state is stable if its derivative on respect
to p is negative.
2012-05-14
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22
Stability Analysis

An equilibrium p* is said to be Lyapunov stable if for any
small enough perturbation, the system will not move further
away.
2012-05-14
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23
Stability Analysis

An equilibrium p* is said to be asymptotically stable if for
any small enough perturbation the system can return to p*

An asymptotically stable state is also Lyapunov stable
2012-05-14
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Visual Classification of Fixed Points
Stable node
Unstable focus
2012-05-14
Unstable node
Stable focus
Elliptic point
(stable limit cycle)
Saddle point
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Neighborhood Segregation as an
Evolutionary Process
Samuel Bowles. Microeconomics. Behavior, Institutions, and
Evolutions. Princeton University Press. 2006
2012-05-14
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Assumptions
 A neighborhood is composed by two ethnic groups (X,Y)
 Both have a preference for integrated neighborhoods, but
none of them want to be part of the minority.
 Their preference is reflected in the price that they would
pay for a house.
Px( p)  0.5( p  )  0.5( p   ) 2  K
Py( p)  0.5( p   )  0.5( p   ) 2  K
 Where δ is the differential preference for one’s own trait, K
is the unbiased price of the house
2012-05-14
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27
Assumptions
 At each time step a fraction α of the population is
selling a house
 Prospective buyers from outside the neighborhood
visit it proportionally to the current composition
 Buyers and sellers are randomly matched
 Each seller meets just one buyer per period
 A sale is concluded if the buyer’s evaluation of the
house is higher than the one of the seller.
2012-05-14
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28
The Replicator Dynamic Equation
 We can write the replicator equation as follows:
p' p  p(1 p) (Py  Px)
 (1 p) p (Px  Py)
Probability that X is in
update-state, meet Y and
have higher (lower)
differential payoff
p  p'p  p(1 p)(Px  Py)
2012-05-14
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Stability Analysis
 We are now looking if a change in Δp is selfcorrecting.
dp
0
dp
 Unfortunately, in our case the inequality does not hold
 that although p = 0.5 is a fixed point, is not
 That means
a stable one. Small perturbations will drive the system
into a new state
2012-05-14
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30
Stability Analysis: delta = 0.1
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Stability Analysis: delta = 0
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Stability Analysis: delta = 0.5
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Thank you very much!
2012-05-14
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34
The Take Home Message of the Course
 Finding good questions is as important as finding
good answers.
“The formulation of a
problem is often more
essential than its solution
which may be merely a
matter of mathematical or
experimental skills.”
2012-05-14
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Albert Einstein
(14 Mar 1879 –
18 Apr 1955)
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The Take Home Message of the Course
 Simulation is a relatively new and heterogeneous
research method:

networks, grids, continuous space, dynamical systems,
rational agents, evolutionary models etc.
 Simulation focuses on micro-mechanisms
Vs
U.S.A.
50 states
2012-05-14
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The Take Home Message of the Course
 Emergence:

2012-05-14
“Interactions among objects at one level give rise to
different types of objects at another level.”
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Goals of the course: students have… ?
1. Acquired firm understanding of the basics of MATLAB.
2. Attained practical knowledge of MATLAB necessary to run
computer simulations.
3. Learned how to implement (simple) models of various social
processes and systems, replicating and extending
established models of the literature.
4. Developed independence in adequately individuating and
selecting further literature (internet, books, paper…) to
expand your knowledge of MATLAB
5. Became confident in presenting scientific results in
academic context (still in progress…).
2012-05-14
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38
(Quality?) Survey




Completely anonymous
Takes 5 minutes
We really read the results
Results are used for planning next semester
course
 http://www.surveymonkey.com/s/WBRTWDZ
2012-05-14
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39
References
 WAV Clark. Residential preferences and
neighborhood racial segregation: a test of the
Schelling segregation model. Demography
(1991)
 Samuel Bowles. Microeconomics. Behavior,
Institutions, and Evolutions. Princeton University
Press. 2006
 http://www.radicalcartography.net
2012-05-14
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40
References
 P.D. Taylor and L.B. Jonker, Evolutionary stable
strategies and game dynamics. Mathematical
Biosciences, 40 (1978), pp. 145–156.
 V. Nemytsky and V. Stepanov, Qualitative
Theory of Differential Equations (Princeton
University, Princeton, 1960)
2012-05-14
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41