Multi-asset Minority Game Complex Markets Meeting Marseille,6-7 - - PowerPoint PPT Presentation

Multi-asset Minority Game

Complex Markets Meeting Marseille,6-7 October 2006

G.Bianconi (ICTP), A. De Martino (Roma), F. F. Ferreira (Sao Paolo) and M. Marsili(ICTP)

Canonical Minority Game

The Minority Game is a synthetic market model devised to show how stylized facts emerge from speculators trading under minimal assumptions. The game introduce heterogeneity of speculators strategies and adapting behavior (learning of the agents) which generate cooperative phenomena.

Canonical Minority Game

Speculators φ=argmax γ Ui,γ 1 a 1

, i

± =

µ

1 − = φi 1 = φi

) t ( A a ) t ( U ) 1 t ( U

, i , i , i µ γ γ γ

− = +

1 a

1 , i

± =

µ − Ns

∑ ∑

= µ γ − = γ

γφ + =

N , 1 i i , i 1 , 1

2 / ) 1 ( a N 1 ) t ( A

Market in the state (or history) µ=1,...P

Where is the price? Where is the price?

1 ai + =

µ

1 ai − =

µ

2 / ) A N ( +

) t ( p 2 / ) A N ( −

) t ( p A N A N ) 1 t ( p − + = +

A )) t ( p log( )) 1 t ( p log( λ + = +

1. agent buy 1$ agent sell 1/p(t) units

• 2. Demand=

Supply=

• 3. Market clearing
• 4. Thus

MM Physica A 299 2001

Observables

Return Expected return given µ Volatility Predictability

) t ( A )) t ( p log( )) 1 t ( p log( λ + = +

µ λ A 1

2 2

))) t ( p log( )) 1 t ( p (log( − + = σ

∑

= µ

µ =

P ,.., 1 2

) t ( A P 1 H

Phase transition

• The model show a phase transition

– between a predictable phase (antisymmetric) in which the average price of the asset can be forecast given the available information – to a unpredictable phase (symmetric) in which the average price cannot be predicted by the available information in the system

• The symmetric phase correspond to a state in which

there is ergodicity breaking and there are many steady states of the dynamics corresponding to an efficient market. These states are selected by different initial conditions.

Phase transition

The parametr of the game is α=P/N For α<αc the system is in the unpredictable, or efficient phase which correspond also to the non-ergodic phase of the system (dependence on initial conditions)

• D. Challet, M. Marsili and R. Zecchina PRL (2000).

Figure from A. De Martino and M. Marsili (2006)

But in this coordinate context where to stylized fact came from?

In order to find stylized facts it is important to introduce two variations to the model

• 1. There are some producers in the market

which inject information and always play with fixed strategies.

• 2. Speculators are allowed not to play: there

is a new strategy (do not trade) for the speculators.

Gran-canonical Minority Game

• In the Gran-Canonical Minority Game agent are allowed

not to trade.

• At each time there is an incentive to trade/not to trade ε.

Each agent has one strategy played only is the corresponding payoff is positive. The update rule for the payoff function is given by

ε + − = +

µ

) t ( A a ) t ( U ) 1 t ( U

i i i

Grand-Canonical Minority Game

φi=θ(Ui) Market in the state (or history) µ=1,...,P

∑ ∑

+ = µ = µ

+ φ =

N , 1 Ns i i Ns , 1 i i i

a a ) t ( A

1 ai ± =

µ

i =

φ 1

i =

φ

1 ai ± =

µ

ε + − = +

µA

a ) t ( U ) 1 t ( U

i i i

Ns

Speculators Producers

Np

GC Minority Game-Observables

For there is no phase transition as ε->0 the number of active players diverges For ε=0 there is a phase transition with the appearence of an unpredictable non-ergodic phase

≠ ε

The parameter of the model are ns=Ns/P and np=Np/P

• D. Challet and M. Marsili PRE (2003).

Stylized facts in the Gran- Canonical Minority Game

Critical region Critical region

ns=Ns/P

ε

• 1. Finite size effects
• 2. And strong dynamical

effects before reaching equilibrium ns*

Critical line Critical line

• D. Challet and M. Marsili

PRE (2003).

Phase Transition between a Phase Transition between a predictable phase and an predictable phase and an unpredictable phase at unpredictable phase at ε=0 ε=0

10 10

1

10

2

ns

10

−6

10

−4

10

−2

10

H/N <φ>

H/N U0=−3000 H/N U0=3000 <φ> U0=−3000 <φ> U0=3000

At the phase transition At the phase transition 1.

• 1. H goes to zero

H goes to zero 2.

• 2. The number of active

The number of active speculators speculators <φ> depends on their prior depends on their prior beliefs beliefs 3.

• 3. The volatility also if

The volatility also if affected by prior affected by prior beliefs. beliefs.

In the critical regions there are In the critical regions there are non trivial dynamical effects non trivial dynamical effects

1000 2000 3000 4000

t/P

50 100

v(t)

1000 2000 3000 4000 5 10 15 20

v(t)

1000 2000 3000 4000 0.5 1 1.5 2

v(t)

ns=10 ns=40 ns=160

ε − ≈ / ) ns ns ( P T

* eq

*

) ( ns ns t v − ≈ ∆

500 1000 1500 2000

t/P

10 20

v(t)

500 1000 1500 2000 10 20

v(t)

500 1000 1500 2000 10 20

v(t)

ε+τ=0.01 ε+τ=0.001 ε+τ=0.0001

∑

∆ − −

∆ =

' t / ) ' t t ( 2

e ) ' t ( A 1 ) t ( v

Volatility and finite size effects in Volatility and finite size effects in the critical region the critical region

• The system in this phase

has significant finite size effects in the volatility

• There is a minimum in the

volatility as a function of ε

• in the infinite size limit

converges to the theoretical results.

σ 2

0.00 0.05 0.10

L=3000 L=6000 L=12000 theory

10

−6

10

−4

10

−2

ε

What role have speculators in a multi-asset market ?

Data Clustering

Given a certain structure of the correlations in the market which is the role of speculators which do not take into account risk considerations?

M.Marsili Quantitative Finance (2002) Minimal Spanning Trees Bonanno et. al. PRE (2002)

Minority Game with multiple assets

• How do speculators distribute their investments on

different assets with different amount of information?

• How do incentive of trading affect their behavior?
• How does risk and correlations affect the

composition of their portfolio?

2 2-

• asset Canonical Minority Game

asset Canonical Minority Game

∑

= µ γ

γφ + =

N , 1 i i s , i

2 / ) 1 ( a N 1 ) t ( A

Speculators φi=argmaxγUi,γ

Different information in the two assets.

• Asset γ=1 in the state

(or history) µ=1,...P1

• Asset γ=-1 in state

(or history) µ=1,...P2

1 − = φi 1 = φi

) t ( A a ) t ( U ) 1 t ( U

, i , i , i γ µ γ γ γ

− = +

1 a

1 , i

± =

µ −

1 a 1

, i

± =

µ N

Parameters and observables of the model

{ }

1 , 1 N P − ∈ γ = α

γ γ

2

s

A P 1 H H H

∑ ∑

µ γ γ γ γ γ γ

µ = = 2 2 2 2

A N 1

γ γ γ γ

= σ σ = σ

∑

Each market has a different information, thus the parameters of the game are The Lyapunov function of the game is There is one volatility for each asset There is one “magnetization” of the speculators

∑ φ

=

i i

N 1 m

Correlations

∑

φ − φ + =

− +

µ − µ + − + j , i j i , j , i

2 1 2 1 a a N 1 A A

a a a a

, j , i , j , i

≈ ≈

− + − +

µ − µ + µ − µ +

The correlation between the two asset is given by And it vanish since the strategies of the speculators in the two assets are uncorrelated This is an effect of the behavior of the trades aimed at detecting excess demand in the market with no consideration of the correlations between the assets Even if the strategies of each agent in the two asset were correlated

A A ≈

− +

− + − +

µ − µ + µ − µ +

≠

, i , i , i , i

a a a a

Exploitation of the available information

• The dynamics tries to exploit maximally the information

present in the system.

• This translates in the minimization of a Lyapunov function,

the predictability H

∑ ∑

µ γ γ γ

µ =

2

A P 1 H

Speculators tend to make the market as efficient as possible minimizing the excess demand trying to get to a unpredictable phase in which <A µ>=0 for every value of µ=1,…Pγ.

Static approach

Describes the equilibrium state of the Canonical Minority-Game as the

• ne minimizing H

defined on the variables In order to find H, the system is replicated and mediated over the different realization of the strategies. Thus we introduce a replicated partition function on the variables { }{ }

∫∫ ∏

µ γ

β − γ

=

) 1 , ( ) a , m ( H , a , i a i n

, i a i

e dm Z

i i

m φ =

∑ ∑ ∑

γ µ µ γ γ

γ γ

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ γ + =

2 i i , i

) m 1 ( a NP 1 H

Static approach

ψ ζ χ , ,

[ ]

) Q 1 ( 2 1 H 2 ) m ( H

2 2 2 2

− + = σ χ + α γ + α = ∑

γ γ γ

The volatility and the predictability can be calculated as a function of the parameters which are defined by the saddle point equations

mz 2 mz 2 ] 2 [ ) m ( zm

2 2

χ χ + α γα − = ψ χ χ + α α = ζ χ + α γ + α = χ

∑ ∑ ∑

γ γ γ γ γ γ γ γ γ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ζ ψ − ζ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ χ + α α − =

∑

γ γ γ

) z ( m m 2 ) z , m ( V

2

Dynamic approach-Batch MG

) t ( U ) t ( U ) t ( y

, i , i i − +

− =

∏ ∫

+ → = ψ

∑ φ

• ψ

t ) t ( ) t ( i

)) 1 t ( y ) t ( y ( W e )) ( y ( p ) t ( y d ) ( Z

t

r r r r

r r

Dynamical variable Generating functional Transition probability density operator

∏ ∑ ∑ ∑

= γ µ γ φ µ γ µ γ γ

γ γ γ

δ γ + − + δ = + →

N , 1 i j , , j , i i i

) a a P ) t ( y ) 1 t ( y ( )) 1 t ( y ) t ( y ( W

j

r r

Dynamical approach-continuation 2

Single agent effective dynamics

) t ( z 2 ) ' t ( G 2 n 1 ) t ( y ) 1 t ( y

1 ' t

+ γ + φ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = − +

− γ γ

∑∑

γ γ

α = / 1 n ) ' t ( ) t ( C ) ' t ( z ) t ( G

' t , t ' t , t

φ φ = ∂ φ ∂ =

with

∑

γ − γ γ γ − γ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + = Λ Λ =

1 1

) G 2 n 1 ( D n ) G 2 n 1 ( ) ' t , t ( ) ' t , t ( ) ' t ( z ) t ( z

[ ]

' t , t ' t , t

C ) ' t ( ) t ( 1 4 1 D + γφ + γφ + =

γ

∑

γ γ γ

χ + + γ + =

2 2

) n 2 ( ) c m 2 1 ( n z

Dynamic approach-continuation 3

+ − − − + +

< < φ = φ = χ + − = < − = φ < χ + = > = φ > z z z * y ~ n 2 2 z z 1 y ~ n 2 2 z z 1 y ~ t ) t ( y lim y ~ z 2 n 2 2 y ~

t ∞ → γ γ

= + γ + φ χ + − = ∑

) z z ( ) z z ( 2 ) z z ( ) z z ( ) z z ( * ) z z ( c ) z z ( ) z z ( ) z z ( * ) z z ( m

2

− θ − θ = χ + α χ α − θ + − θ − θ φ + − θ = − θ − − θ − θ φ + − θ =

+ − γ γ γ − + − + − + − +

∑

Phase diagram of the 2 asset Canonical Minority Game

• 0.4
• 0.2

0.2 0.4 0.6

m

0.005 0.01 0.015 0.02 0.025 0.03

H

• 0.4
• 0.2

0.2 0.4

α+−α−

0.05 0.1 0.15 0.2 0.25

σ

2

α++ α− = 0.5

0.1 0.2 0.3 0.4 0.5

α+

0.1 0.2 0.3 0.4 0.5

α− Ergodic Non-ergodic H=0 H>0

The speculators trade more in the asset with less information

2-asset Grand-Canonical Minority Game

Speculators Producers φi=argmaxγUi,γθ(Ui,γ) 2 Markets in the state (or history) µ+,µ−

∑ ∑

+ = µ γ = µ γ γ

+ φ =

N , 1 Ns i , i Ns , 1 i i , i

a a ) t ( A

1 a ,

i

± =

+

µ +

= φi 1 = φi

1 a ,

i

± =

+

µ +

ε + − = +

γ µ γ γ γ

A a ) t ( U ) 1 t ( U

, i , i , i

Ns

1 a ,

i

± =

−

µ −

1 a ,

i

± =

−

µ − 1 − = φi

2*Np

Static approach

Describes the equilibrium state of the Grand-Canonical Minority-Game as the one minimizing H In order to find H, the system is replicated and mediated over the different realization of the strategies. Thus we introduce a replicated partition function on the variables Finding an expression for H and σ2 which depends on the parameters

∑

γ φ γ

δ = π

i ,

i

{ }{ }

∫∫ ∏

µ γ γ

π β − γ γ

π =

) 1 , ( ) a , ( H , a , i a , i n

, i a , i

e d Z

K , ζ

( ) ( )

2 2 2 2 2

H 2 K H

γ γ γ γ γ γ γ γ

π − π + = σ π ε + ρ + π ε =

∑ ∑

Saddle point equations Saddle point equations

• The parameters are

fixed by the saddle point equations,

• The averages in the

expressions are performed by averaging over a effective potential and a auxiliary variables z

γ γ

ζ K ,

( )

γ γ γ γ γ γ γ γ γ γ

π α ζ − ε = χ + ε = ρ + π α = ζ z / 1 ) 1 ( K

2

∫ ∫∏ ∫∏ ∏

∞ ∞ − π β − γ γ π β − γ γ γ γ

π π π π = π

) z , ( V 1 ) z , ( V 1

e d e ) ( f d 2 Dz ) ( f

r r r r

r

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π − ζ π − π ε = π

γ γ γ γ γ γ γ γ

∑

K 2 z 2 K ) z , ( V

2

r r

Static approach 3

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π − ζ π − π ε = π

γ γ γ γ γ γ γ γ

∑

K 2 z 2 K ) z , ( V

2

r r

The effective potential V is a sum of the two Contributions coming from the two assets There agents

• which always play a single strategy
• which always trade in one asset or do

not trade,

• agents which always trade in one or in

the other asset,

• and finally, fickle agents which may

trade in one or another asset or do not trade. π−1 π1

1

1 1

≤ π + π

−

Phase diagram

0.1 0.2 0.3 0.4 0.5

α s

+

0.1 0.2 0.3 0.4 0.5

α s

• H>0

H=0 H+>0 , H-=0 H+=0 , H->0 Ergodic Non-Ergodic Non-Ergodic Non-Ergodic

r

2 1

= χ χ

We can parameterize the phase diagram H=0 in terms of a parameter r, given by With

∞ → χ ∞ → χ

2 1

The “magnetization” of the model

• 1
• 0,5

0,5 1

α s

+- α s

• 0,6
• 0,4
• 0,2

0,2 0,4 0,6

m

α s

++ α s

• =1

The magnetization of the speculators depend on the value of e. For e>0 when there is some incentive to trade, speculators trade more in the asset with less information For e<0 when there is some incentive not to trade the speculators trade more in the asset with more information

Non-ergodic region

0.1 0.2 0.3 0.4 0.5

α s

+

0.1 0.2 0.3 0.4 0.5

α s

• H>0

H=0 H+>0 , H-=0 H+=0 , H->0 Ergodic Non-Ergodic Non-Ergodic Non-Ergodic

• 0,4
• 0,2

0,2 0,4

α s

+- α s

• 0,2
• 0,15
• 0,1
• 0,05

0,05 0,1 0,15 0,2

m

∆U = 0 ∆ U = 5 ∆ U = 10 ∆ U = 20 α s

++ α s

• = 0.4

ε=0 In this model there is a asymmetric phase in which the system is non-ergodic

Conclusions

• We generalize the Minority Game to multiple assets

case.

• In the game the traders are only speculators which don’t

take into account the risk considerations, resulting in a speculative trading which doesn’t build up correlations.

• The role of incentives not to trade make them able to

detect the asset with less information

• In the model there is a presence of a non-ergodic region

where there is some predictability (respect to the

• utcome in one asset).