1. 851-0585-04L – Modeling and Simulating Social Systems with MATLAB
L – 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 |
© ETH Zürich |

2. No lecture, but we will be in the room to supervise you
Schedule of the course
Introduction to MATLAB
26.09.
03.10.
10.10.
17.10.
24.10.
31.10.
07.11.
14.11.
21.11.
28.11.
05.12.
12.12.
19.12.
Introduction to social-science modeling and simulations
Working on projects (seminar thesis)
No lecture, but we will be in the room to supervise you
Handing in seminar thesis and giving a presentation

3. Final presentation schedule
Project presentation 15’ + 5’ (for Q&A)
2 days available:
Monday, Dec. 19th and – 19.30
Tuesday, Dec. 20th
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

4. 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

5. Final presentation schedule
Reminder:
Your reports are due midnight of Friday December 16th 2011
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.

6. Goals of Lecture 11: students will
Goals of Lecture 11: students will
Consolidate knowledge acquired during lecture 10, through brief repetition of the main points.
Review Schelling’s segregation model and its main results from a classical perspective
Understand the motivation and the rules behind a discrete time replicator dynamics equation
Receive a brief introduction to stability analysis in evolutionary models
Revisit Schelling’s segregation model from an evolutionary perspective
Reflect about the main message of the course

7. 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

8. Repetition: scientific writing
All claims must be well justified
Correlation is not causation
Most of the time scientific writing is “standardized”

9. 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)

10. 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.

11. Schelling’s Segregation Model
Cellular automata
Individuals prefer integration
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.

12. Schelling’s Segregation Model: demo

13. Evolutionary models
Evolutionary models of population mimic the dynamics of biological system under the combined influence of:
Chance
Inheritance
Natural Selection
Path-dependence (historical contingency)

14.
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
Adaptive agents: from zero-intelligence to high cognitive abilities
Conformism: exposure
Reinforcement learning:
Best response updating:
Fictitious play

15. Evolutionary models Loop for 1:N An individual can update his strategy
Individual implements a strategy dictated by its genetic or behavioral trait
Individual play the strategy against
The Environment
Other randomly paired individuals
A certain payoff is granted
An individual can update his strategy
Replication relative to payoff
Loop for 1:N
Learning
Natural Selection

16. 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

17. 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

18. Discrete Time Replicator Equation
The expected population frequency for trait x at time t+1 can be written as the sum:
where ω is the fraction of population in “update state”, and β is a positive constant that renormalize small differences in payoff

19. Discrete Time Replicator Equation
And the population differential would be:
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

20. Discrete Time Replicator Equation
The replicator equation can be generalized to n traits:
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.

21. Stability Analysis
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.

22. Stability Analysis
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.

23. Stability Analysis
An equilibrium p* is said to be Lyapunov stable if for any small enough perturbation, the system will not move further away.

24. 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

25. Visual Classification of Fixed Points
Stable node
Unstable node
Stable focus
Elliptic point
(stable limit cycle)
Unstable focus
Saddle point

26. Neighborhood Segregation as an Evolutionary Process
Samuel Bowles. Microeconomics. Behavior, Institutions, and Evolutions. Princeton University Press. 2006

27. 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.
Where δ is the differential preference for one’s own trait, K is the unbiased price of the house

28. 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.

29. The Replicator Dynamic Equation
We can write the replicator equation as follows:
Probability that X is in
update-state, meet Y and have higher (lower) differential payoff

30. Stability Analysis
We are now looking if a change in Δp is self- correcting.
Unfortunately, in our case the inequality does not hold
That means that although p = 0.5 is a fixed point, is not a stable one. Small perturbations will drive the system into a new state

31. Stability Analysis: delta = 0.1

32. Stability Analysis: delta = 0

33. Stability Analysis: delta = 0.5

34. Thank you very much!

35. 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.”
Albert Einstein
(14 Mar 1879 –
18 Apr 1955)

36. 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
U.S.A. Vs
50 states

37. The Take Home Message of the Course
Emergence:
“Interactions among objects at one level give rise to different types of objects at another level.”

38. Goals of the course: students have… ?
Goals of the course: students have… ?
Acquired firm understanding of the basics of MATLAB.
Attained practical knowledge of MATLAB necessary to run computer simulations.
Learned how to implement (simple) models of various social processes and systems, replicating and extending established models of the literature.
Developed independence in adequately individuating and selecting further literature (internet, books, paper…) to expand your knowledge of MATLAB
Became confident in presenting scientific results in academic context (still in progress…).
NOT a computer course!!! Aims at

39. (Quality?) Survey Completely anonymous Takes 5 minutes
We really read the results
Results are used for planning next semester course

40. 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

41. 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)