Evolution and market behavior with endogenous
investment rules
Giulio Bottazzi
Pietro Dindo
LEM, Scuola Superiore Sant’Anna, Pisa
GAM Workshop, 14-17 April 2010
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Research questions
Consider a market for a risky asset and an ecology of investment
strategies competing to gain superior returns. The open questions are:
⇒ which are the strategies surviving in the long run?
⇒ is it possible to establish an order relationship among them?
⇒ is a strategy dominating all the others?
Answers to these questions help to clarify specific issues (think of
financial markets) as well as general issues (“as if” point).
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Where do we stand?
On this issue
• Behavioral Finance (a survey is Barberis and Thaler, 2003)
Pros Ecology of strategies behaviorally grounded
Cons No wealth-driven strategy selection
Focus Market biases
• HAM Finance (a survey is Hommes, 2006)
Pros Focus on price feedbacks
Cons No wealth-driven strategy selection (mostly CARA), deterministic
Focus Stylized facts
• Evolutionary Finance (Kelly, 1956; Blume and Easley, 1992; a
survey is Evstigneev, Hens, and Schenk-Hoppe, 2009)
Pros Multi-asset stochastic general equilibrium framework
Cons Absence of price feedbacks (no endogenous investment rules)
Focus Market selection
⇒ Our approach: evolutionary finance with endogenous (price
dependent) investment rules.
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Framework
• Trading is repeated and occurs in discrete time
• Many assets in constant supply with uncertain dividends
• Market is complete
• Agents care about consumption, thus wealth
• A strategy is a portfolio of wealth fractions (CRRA)
• Walrasian market clearing
• Intertemporal budget constraint
• Market dynamics is formalized as a random dynamical system
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A toy market
• Two states of the world, s = 1, 2, which occur with probability π
and 1 − π. Bernoulli process ω = (. . . , ωt , . . . , ω0 ) ∈ Ω.
• Two (short-lived) Arrow’s securities, k = 1, 2, paying Dk ,s = δk ,s .
• Fraction of consumption is constant and uniform, α0 = c. All the
rest is invested.
• Define normalized prices ps,t =
Ps,t
Wt
so that p1,t + p2,t = 1 − α0 , ∀t.
• Two agents, i = 1, 2, with wealth fractions φt and 1 − φt .
• Endogenous strategies with one memory lag, L = 1,
1
- α1,t
= α11 (p1,t−1 ) describes the portfolio choice of the first agent,
2
- α1,t
= α12 (p1,t−1 ) describes the portfolio choice of the second agent.
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A toy market
⇒ Evolutionary finance literature shows that, among constant
investment rules, αs∗ = πs dominates and
Iπ (α) =
S
X
s=1
πs log
πs
αs
can be used to establish an ordering relationship.
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A toy market
Strategy i dominates strategy j, i > j, if
(
φjt
< ,
∀ > 0 , ∃T s.t. Prob
φit
Bottazzi-Dindo (LEM, Sant’Anna, Pisa)
)
∀t > T
=1.
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Two agents: the random dynamical system
Given xt = (φt , pt , qt = pt−1 ), the state of our market at time t, the
random dynamical system is the composition of the following maps
 1

α1 (qt )φt



with probability π



pt



φt+1 =
,




1 (q ))φ

(1−α
−α

t
t
0

1

with probability 1 − π
1−α0 −pt




pt+1 = α11 (pt )φt+1 + α12 (pt )(1 − φt+1 ) ,







qt+1 = pt .
That is, xt+1 = fπ (xt ) with probability π and xt+1 = f1−π (xt ) with
probability 1 − π, depending on the realization of ωt .
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Fixed points
Definition
Definition
The state x ∗ = (φ∗ , p∗ , q ∗ = p∗ ) is a deterministic fixed point of the
random dynamical system generated by the maps fπ and f1−π , that is,
ϕ(t, ω, x) = . . . fπ ◦ · · · ◦ f1−π . . . if it holds
ϕ(t, ω, x ∗ ) = x ∗
∀ω ∈ Ω
(1)
or, in terms of the maps, if it holds both
fπ (x ∗ ) = x ∗
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and
f1−π (x ∗ ) = x ∗ .
(2)
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Fixed points
In our toy market
Theorem
Fixed points of the random dynamical system that represents the toy
market dynamics are given by
x1∗ = (φ∗ = 1, p∗ = α11 (p∗ ), q ∗ = p∗ )
x2∗ = (φ∗ = 0, p∗ = α12 (p∗ ), q ∗ = p∗ )
∗
x1/2
= (φ∗ , p∗ = α11 (p∗ ) = α12 (p∗ ), q ∗ = p∗ )
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Fixed points on a plot: the Equilibrium Market Curve
1
E2
0.8
0.6
α1(p)
E1
E2
0.4
0.2
EMC
α11
α21
E2
0
0
0.2
0.4
0.6
0.8
1
p
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Local stability
Definition
Definition
A fixed point x ∗ of the random dynamical system ϕ(t, ω, x) is called
locally stable if limt→∞ ||ϕ(t, ω, x) − x ∗ || → 0 for all x in a neighborhood
U(ω) of x and for all ω ∈ Ω.
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Local stability
In our toy market
Theorem
Provided that the eigenvalues of the iterated map are inside the unit
circle the deterministic fixed point is locally stable (use Multiplicative
Ergodic Theorem and Local Hartman-Grobman Theorem). For fixed
points of the type (1, α11 (p∗ ), p∗ ) eigenvalues are
∂α11 (p) 1
2
(3)
µ = exp (Iπ (α ) − Iπ (α )) and λ =
∂p ∗
p
and for fixed points of the type (φ∗ , α11 (p∗ ) = α12 (p∗ ), p∗ )
1
2
∗ ∂α1 (p) ∗ ∂α1 (p) µ = 1 and λ = φ
+ (1 − φ )
∂p ∗
∂p p
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(4)
p∗
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Local stability on the EMC plot
1
U2
0.8
0.6
α1(p)
S1
U2
0.4
EMC
α11
α21
π
0.2
U2
0
0
0.2
0.4
0.6
0.8
1
p
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Ordering is not complete
Coexistence of stable equilibria
1
1
π
EMC
I
II
0.8
0.8
α1(p)
Wealth share
SII
0.6
I
II
0.9
0.4
SI
0.2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0
0.2
0.4
0.6
p
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1
0
500
1000
1500
2000
2500
3000
Time
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Ordering is not complete
Coexistence of stable equilibria
1
1
π
EMC
I
II
0.8
α1(p)
Wealth share
SII
0.6
I
II
0.8
0.4
SI
0.2
0.6
0.4
0.2
0
0
0
0.2
0.4
0.6
p
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0
500
1000
1500
2000
2500
3000
Time
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Ordering is not complete
Multiple unstable equilibria
1
1
π
EMC
III
0.8
IV
III
IV
0.9
UIV
Wealth share
0.8
α1(p)
0.6
0.4
0.2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
UIII
0
0
0
0.2
0.4
0.6
p
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50
100
150
200
250
300
350
400
450
500
Time
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Ordering is not complete
Multiple unstable equilibria
1
0.8
π
EMC
III
0.8
IV
p1
p2
0.7
UIV
0.6
Price
α1(p)
0.6
0.5
0.4
0.4
0.2
0.3
UIII
0
0.2
0
0.2
0.4
0.6
p
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1
50
100
150
200
250
300
350
400
450
500
Time
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Ordering is not transitive
I > III > V ∼ I
1
π
EMC
I
0.8
III
V
0.6
α1(p)
EV
EI/V
0.4
EIII
0.2
EV
0
0
0.2
0.4
0.6
0.8
1
p
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Ordering is not transitive
I > III
1
1
π
EMC
I
0.8
III
Wealth share
0.8
α1(p)
0.6
0.4
SI
UIII
0.2
I
III
0.9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0
0.2
0.4
0.6
p
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1
20
40
60
80
100
120
140
160
180
200
Time
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Ordering is not transitive
III > V
1
1
π
III
V
0.8
EMC
UV
Wealth share
α1(p)
0.6
UV
0.4
SIII
0.2
0
0.6
0.4
0.2
UV
0
III
V
0.8
0
0.2
0.4
0.6
p
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1
20
40
60
80
100
120
140
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180
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Time
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Ordering is not transitive
V ∼I
1
0.8
π
V
0.8 EMC
I
UV
Wealth share
α1(p)
0.6
SI/V
0.4
I
V
0.7
0.2
0.6
0.5
0.4
0.3
UV
0
0
0.2
0.2
0.4
0.6
p
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20
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Time
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Does it exist a dominant strategy?
Yes, but not strictly
1
1
UB
VIII
EMC
0.8
UNSTABLE
STABLE
0.8
0.6
p
α1(p)
0.6
0.4
0.4
0.2
0.2
0
0
φI*
0
0
0.2
0.4
0.6
0.8
1
1
φI
p
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Beyond toy market
Same type of results holds with I agents, L memory lag, S = K assets.
For x ∗ with φI = 1 and p∗ = αI (p∗ ), eigenvalues are
Λ = (µ1 , ..., µI−1 , λ1,1 , ..., λk ,l , . . . , λK −1,L ), with
!π k
K
Y
αki (p∗ )
µi =
,
(5)
I (p ∗ )
α
k
k =1
and, for a any given k , λk ,l one of the L solutions of the following
equation
L−1
X
L
λ +
λl (αkI )(L−1−l,k ) = 0 ,
(6)
l=0
where
(αkI )(0,k )
∂αkI =
∂pk ,
p∗
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(αkI )(l,k )
∂αkI =
∂pkl l = 1, . . . , L , k = 1, . . . , K −1 .
p∗
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Conclusion
• Many fixed points, located on the Equilibrium Market Curve,
whose local stability depends both on
- Entropy w.r.t. dividend payment process
- Price feedbacks being not too strong
⇒ No ordering relation based on market dominance can be
established
⇒ Constant investment rule that minimize entropy Iπ (α) is (locally)
dominating all others.
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