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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posted:10/10/2011
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