# Positive Feedback

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```					 Summary of previous lectures
1. How to treat markets which exhibit
’normal’ behaviour (lecture 2).

2. Looked at evidence that stock markets
were not always ’normal’, stationary nor
in equilibrium (lecture 1).

Is it possible to model non-normal markets?
From individual behaviour to
market dynamics

Describe how individuals interact with each
other.

Predict the global dynamics of the markets.

Test whether these assumptions and
predictions are consistent with reality.
El-Farol bar problem
• Consider a bar which has a music night
every Thursday. We define a payoff
function, f(x)=k-x, which measures the
‘satisfaction’ of individuals at the bar
attended by a total of x patrons.
• The population consists of n individuals.
What do we expect the stable patronage
of the bar to be?
Perfectly rational solution
El-Farol bar problem
• Imperfect information: you only know if you
got a table or not.

• You gather information from the
experience of others.
El-Farol bar problem
• If you find your own
’table’ then tell b
others about the bar.
If you have to fight
over a ’table’ then
don’t come back
• Interaction function

Schelling (1978) Micromotives and Macrobehaviour
Simulations of bar populations
7000
b=6
6000

5000
Beach
Population :x t

visitors                         4000

(at )                        3000

2000

time
1000

0
0   5   10   15    20     25      30   35   40   45   50
Time: t
n=4000 sites at the beach

Bk=1000 b=6
Simulations of bar populations
7000
b=6
6000

5000
Beach
Population :x t

visitors                          4000

(at )                         3000

2000

time
1000

0
0   5   10   15   20     25      30   35   40   45   50
n=4000 sites at the beach                     Time: t

Bk=1000 b=8
Simulations of bar populations
7000
b=6
6000

5000
Beach
Population :x t

visitors                         4000

(at )                        3000

2000
time
1000

0
0   5   10   15     20     25      30   35   40   45   50
n=4000 sites at the beach                      Time: t

Bk=1000 b=20
A derivation
Interaction function

The mean population on the next
generation is given by

where pk is the probability that k
individuals choose a particular site.

If pk is totally random (i.e. indiviudals
are Poisson distributed) then
b=6
25000

20000

at+1   15000

10000

5000

0
0   4000    8000   12000   16000

at
Simulations of bar populations
7000                                                 b=6                                        7000

6000                                                                                            6000

Population at time t+1:x t+1
5000                                                                                            5000
Population :x t

Beach 4000                                                                                                            4000

visitors
3000                                                                                                            3000

2000                                                                                            2000
(at )
1000                                                                                            1000

0                                                                                               0
0    5   10    15   20     25      30   35   40   45   50                                       0   1000   2000    3000      4000        5000   6000   7000
Time: t                       time                                                       Population at time t: x t

n=4000 sites at the beach

Bk=1000 b=6
Simulations of bar populations
b=6
7000                                                                                         7000

6000                                                                                                         6000
Beach

Population at time t+1:x t+1
visitors
5000                                                                                                         5000
Population :x t

4000                                                                                         4000
(at )
3000                                                                                         3000

2000                                                                                         2000

time
1000
1000

0
0                                                                                             0   1000   2000    3000      4000        5000   6000   7000
0   5   10   15   20     25      30   35   40    45   50
Population at time t: x t
Time: t
n=4000 sites at the beach

Bk=1000 b=8
Simulations of bar populations
b=6
7000                                                                                            7000

6000                                                                                            6000

Beach

Population at time t+1:x t+1
5000                                                                                                            5000
visitors
Population :x t

4000                                                                                            4000

(at )
3000                                                                                                      3000

2000                                                                                            2000

1000
time                                  1000

0                                                                                               0
0    5   10    15   20     25      30   35   40   45   50                                       0   1000   2000    3000      4000        5000   6000   7000
Time: t                                                                                  Population at time t: x t

n=4000 sites at the beach

Bk=1000 b=20
Period doubling route to chaos
Are stock markets chaotic?
Are stock markets chaotic?

Not really like the distributions we saw in lectue 1.
El-Farol bar problem

Arthur 1994
El-Farol bar problem

Arthur 1994
El-Farol bar problem

Arthur 1994
Minority game

Brain size is
number of bits in
signal (3)

Challet and Zhang 1997
Minority game

Challet and Zhang 1997
Minority game

Challet and Zhang 1998
Break
Do humans copy each other?
Asch’s experiment

Asch (1955) Scientific American
Asch’s experiment

Asch (1955) Scientific American
Asch’s experiment

Asch (1955) Scientific American
Milgram’s experiment
Milgram’s experiment
a

b

Hale (2008)
Milgram’s experiment

Milgram & Toch (1969)
Irrationality in financial experts
• Keynes beauty contest

• Behaviuoral economics (framing, mental
accounting, overconfidence etc.). Thaler,
Kahneman, Tversky etc.

• Herding? (less experimental evidence)
Consequences of copying
Summary
• Markets can be captured by some simple
models.

• These models in themselves exhibit
complex and chaotic behaviours.

• In pariticular, models of positive feedback
could be used to explain certain crashes.

```
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 views: 9 posted: 10/10/2011 language: English pages: 34