Chaos in the Cobweb Model
with a
New Learning Dynamic
George Waters
Illinois State University
Cobweb Model
• stability
• supply & demand slopes
Dynamic switching between
forecasting strategies
• Brock and Hommes (1997)
– MNL
• Sethi and Franke (1995)
– EGT
• Branch and McGough (2005)
Chaos
BNN
Brown, von Neumann and Nash
(1950)
Well behaved
• positive correlation
• inventiveness
• Nash stationarity
Dynamics under BNN and MNL
• interpretation of steady states
• stability analysis / bifurcations
Notation
qk,t – fraction using strategy k
pk,t – payoff to strategy k
population average payoff
H


t   q h,t 
h,t
h1
excess payoff



t
k,t 
k,t 
Excess Payoff Dynamics
(Sandholm 2006)
q k,t1 

q k,t 


k,t 
H

1



h,t 
h1




k,t 0 for 
k,t 0




k,t 0 for 
k,t 0
BNN
q k,t1 

q k,t 


k,t 

H

1



h,t 

h1







for

0
k,t 
k,t
k,t





0
for

0
k,t 
k,t

Desiderata
• Positive Correlation
– weak monotonicity condition
H

 q h,t h,t 0
h1
• Inventiveness
– positive excess payoff implies a
positive fraction of followers
Selection Dynamics
and Desiderata
Proposition: BNN satisfies
Positive Correlation and
Inventiveness.
MNL does not satisfy Positive
Correlation.
Imitative Dynamics (replicator)
do not satisfy Inventiveness.
MNL
q k,t1 
exp 

k,t 
H
 exph,t 
h1
• b is search intensity.
• q>0 for all k
Imitative Dynamics
q k,t1 q k,t

k,t 
H
 q h,th,t 
h1
w(p) is the weighting function
linear w(p) – replicator
Strategies cannot be reborn.
Alternatives
• Infinite search intensity in MNL
• Imitative dynamics with drift
• combining imitative and excess
payoff dynamics
• Best response dynamics
– Gilboa and Matsui (1991)
• Pairwise comparison dynamics
– Sandholm (2006)
– agents compare their payoff with
payoff of a random strategy
– scarcity of data
– 2 strategies – equivalent to BNN
Nash stationarity
• Steady states correspond to
Nash equilibria
• BNN satisfies
• MNL does not
• Imitative dynamics
– Lyapunov stable steady states
correspond to NE
Cobweb model doesn’t have NE.
Cobweb Model
• Quantity supplied by firms
determined by price
expectations and cost.
• Rational and Naïve predictors
• Linear demand determines
price.
• Payoffs are firm profits
– adjusted for cost of the predictor
S
e
p k,t1
 1

c
e
p k,t1
H
D
p t1   q h,t S
e
p h,t1
h1
S
e
p k,t1
e
bp k,t1
D
p t1 A Bp t1
Predictors
Rational
- cost C
- used by q
Naïve
- costless
- used by 1-q
e
p R,t1
p t1
e
p N,t1
p t
S&D:
A Bp t1 q t bp t1 
1 q t 
bp t
Price dynamics
p t1 

b
1q t 

bq t 
1
pt

b b/B
Ratio < 1 implies a stable steady
state at pt = 0.
Payoffs
b
2
 p t1 C #
2
b
2

#
N,t bp t
1 pt  pt .
2
2

R,t bp t
1
Excess Payoffs



1 q t 

R,t  
R,t 
N,t  #



.
N,t q t 
N,t 
R,t 
#
Payoff Difference
Function

p t , q t 
R,t 
N,t
2

p t , q t p t

B b b
1
2

2 bq t 
1
2
C
Evolution Function
(pt+1,qt+1) =F(pt,qt)
p t1 

b
1q t 

bq t 
1
pt
q t1 
qt
1 q t 
pt, qt 
1
for 
p t , q t 0
1 q t
1 
1 q t 
p t , q t 
for 
p t , q t 0
Steady States
Proposition
Let the parameters b, B 0.
i) The point 
p, q
0, 0is a steady state for C 0.
ii) If C 0, the set of points where p 0 are steady states.

iii) For C 0 and b 1, there is a 2-cycle of given by


p , q   C , b 1
 .
2b
2b
1
q
0.75
=0
=0
0.5
0.25
q*
0
-1
0
p
1
Stability
If C>0, (pt,qt) = (0,0) is stable iff

b 1
In a stable market, no incentive
to incur the cost of the rational
forecast.
Under MNL, this steady state
only exists for infinite search
intensity.
Stability of the 2-cycle

Define F
reflecting across q-axis

2-cycle is a steady state of F

F does not have a continuous
derivative along 
p t , q t 0.

Examine Jacobians on either side

Eigenvalues for both are complex
with modulus greater than one.
Stability of the 2-cycle
• Eigenvalue condition does not
guarantee (in)stability(!).
The 2-cycle is
exponentially unstable.
Chaos

• Bifurcations in b
• Different than MNL
– no saddles under BNN
– unstable two cycle has two
eigenvalues passing through -1
• Periodic attractors
• Strange attractors
0.5
q
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
p
0.2
0.4
0.6
0.8
1
0.5
q
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
p
0.2
0.4
0.6
0.8
1
0.5
q
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
p
0.2
0.4
0.6
0.8
1
1
p
0.5
0
-0.5
-1
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
1
0.8
q
0.6
0.4
0.2
0
- BNN
Weibull (1994)
q j,t1 

q j,t 

j,t 

H

1
h1




h,t 


 - BNN
For
 1
the 2-cycle is stable
but not asymptotically unstable
For MNL, dynamics depend on
search intensity.
Summary
• BNN is well behaved
– PC, I, NS
– The edges of the simplex are not
problematic.
• Cobweb model
– Stable steady states are easy to
interpret
– Dynamics don’t depend strongly
on .
Imitative Dynamics
• Similar steady states
• Another steady state at q=1
– all edges of the simplex are
steady states
– lacking Inventiveness
– if C>0, hard to justify
Stability
If C=0, (pt,qt) = (0,1) is the
unique stable steady state.
No reason to use the naïve
forecast.
Under MNL, both predictors have
equal population shares.
Counterexample
y t1 A
t
yt
0
A
t 
2
for t even
1
4
0
0
1
4
2
for t odd
0
eigenvalues of A
t are  1 , for all t, but
2
2 2n
n
0
n
0
2
0
2n
for t even
t 
 A
t1
2
2n
0
for t odd