Statistical mechanics of systems ofheterogeneous interacting agents

Theory (some key points) : simplest Minority Game1) phase transition2) role of `market impact’3) optimal vs suboptimal solutions4) phase structure5) some more complicated but important things

research problems, general frameworkRecent directions (connection with A. Pagnani’s talks) (maybe)

http://chimera.roma1.infn.it/ANDREA

strict theory

these systems are out of equilibrium (microscopic dynamics violates detailed balance)

no Hamiltonian H, study dynamics

Ui(t + 1) ! Ui(t) = !aµ(t)i A(t) ! ! != "

!H

!Ui(t)+ noise

A(t) =!

j

!j(t)aµ(t)j ! A({!i(t)}) , !i(t) = !(Ui(t))

(dynamical generating functionalsa.k.a. path integrals)

dynamical generating funcionals?[Martin-Siggia-Rose ‘73, De Dominicis ‘78]

Problem : compute

m(t) =1

N

!

i

!!i(t)"

C(t, t′) =1

N

!

i

!!i(t)!i(t′)"

G(t, t!) =1

N

!

i

"

!"i(t)

!hi(t!)

#{

ex. Ui(t + 1) − Ui(t) = −aµ(t)i A[!(t)] − ! + hi(t)

!· · ·" =!

paths

· · · Prob{path}

· · · = disorder avgpath = {!(t)}t!0

equation of motion for !(t) = {!i(t)}

Luckily some important things can be understood without path integrals

Z[!] =⟨

ei

P

t!0

P

i!i(t)"i(t)

⟩

m(t) = !

i

N

!

i

"

lim!!0

∂Z[!]

∂ψi(t)

#

C(t, t!) = !

1

N

!

i

"

lim!!0

∂2Z[!]

∂ψi(t)∂ψi(t!)

#

Prob{path} = P [!(0)]!

t!0

W [!(t) ! !(t + 1)]

But the information process is Markovian...

W [!(t) ! !(t + 1)] =!

i

!(equation of motioni)

=!

i

"!

"!

eibUi(t)[equation of motion

i] d

#Ui(t)

2!

some game theory for the MG

N (odd) deductive agents, two possible actions (no information, no strategies), minority rule for payoffs

Optimal state :

!

N!1

2do a

N+1

2do !a

NNash ∼ eN!

, ! > 0

(Nash eq.)

Another Nash eq. : Prob{ai = ±1} = 1/2

...

Fluctuations : !2 =

!

i

(!a2

i " # !ai"2) =

!

i

(1 # !ai"2)

! !2

= N

for this Nash eq. ! !2 = 1

Prob{ai(t) = a} ! eaUi(t) , a = ±1

Ui(0) randomly sampled from q(U)

Ui(t + 1) ! Ui(t) = !!A(t)/N , A(t) =!

j

aj(t) , ! > 0

MG with inductive players

0 2 4 6s0

5

10

15!c(s)

2 4 6 8 10!

0

0.5

1

"2 /#2

s=1/2s=1

!2/N2

q(U) = G(0, s2)

New variable : y(t) = Ui(t) ! Ui(0)

N ! 1 " y(t + 1) # y(t) = #!$tanh[y(t) + U(0)]%0

Fixed point : y! such that !tanh[y! + U(0)]"0 = 0

!2/N = O(1) Fluctuations decrease withthe spread of i.c.’s

Linear stability : fixed point stable for ! < !c = 2N/!2

! > !c : new solution with !2/N2 = O(1)

Ui(t + 1) ! Ui(t) = !!A(t)/N

Fluctuations : !2 =!

i

(1 ! "ai#2) = N [1 ! "tanh2[y! + U(0)]#0]

Lesson is : the larger the spread of i.c.’s (heterogeneity), the smaller are the fluctuations and the more stable is the fixed point,

but fluctuations are horrible

Ui(t + 1) ! Ui(t) = !!A(t)/N , A(t) =!

j

aj(t) , ! > 0

i is in hereso why can’t they get to Nash?

remove self-interaction

! ! {0, 1}Ui(t + 1) ! Ui(t) = !

!

N[A(t) ! !ai(t)]

!Ui(t + 1)" # !Ui(t)" = #!

N[!

j

mj # !mi] mi = !ai"

= !

!

N

!H

!mi

H =1

2

!

"

i

mi

#2

!

!

2

"

i

m2

i

H =1

2

!

"

i

mi

#2

!

!

2

"

i

m2

i

! = 0 ! mi = 0

! = 1 : H is harmonic ! mi = ±1

!1 " mi " 1

! !2

= N

! !2 = 1 (odd N)

minima :

Market impact : basic idea

!out

g = !ag "A#

!in

g! ! "ag! #A$ " ag!ag! = !out

g! " 1

vg ! "Ug(t + 1) # Ug(t)$ = !in

g + 1 # fg

Agent with S strategies watching a MG wants to evaluate how good his trading strategies are

!A" # !A + ag!"

(time avg)!X"

Then goes in (and uses strategy g’) x =1

P

!

µ

xµ

!g!! ! "ag!!#A$ " ag!ag!! ! !out

g!! = !in

g!! + 1

Reducing the effects of market impact

vg ! "Ug(t + 1) # Ug(t)$ = !in

g + 1 # fg

vg = !in

g + 1 ! fg + "fg

Uig(t + 1) ! Uig(t) = !

1

Na

µ(t)ig A(t) +

!

N"g,egi(t)

Uig(n + 1) = Uig(n) !1

P

P!

µ=1

qµigQ

µ(n) +1

2(1 ! !g,egi(n))"ig(n)

!gi(n) = arg maxg

Uig(n)

!!ig(n)" = "

compare with the route choice game model :

10!2

10!1

100

101

c

0.0

0.2

0.4

0.6

0.8

1.0

!2

/N

random drivers

adaptive drivers, "=0

adaptive drivers, "=#2

! = !2

! = 0! = 0

fluctuations are drastically reduced also when the information is biased

`pessimistically’

Note

Uig(t + 1) ! Uig(t) = !aµ(t)ig

A(t)

N+

!

N"g,gi(t)

{ {

O(N!1/2) O(1/N)

But in the long term (average over information)

1

P

!

µ

A(t) !1

P

!

µ

A[µ(t)] = O(1)

cfr the role of Onsager reaction in spin glasses

?

Uig(t + 1) ! Uig(t) = !

1

Na

µ(t)ig A(t) +

!

N"g,egi(t)

Stat mech

A(t) =!

j

aµ(t)i,egj(t)

!µi =

aµi,1 ! aµ

i,2

2!

µi =

aµi,1 + a

µi,2

2y(t) =

Ui,1(t) ! Ui,2(t)

2

aµ(t)i,egi(t)

= !µ(t)i + si(t)"

µ(t)i

!gi(t) = arg maxg

Uig(t)

si(t) = sign[yi(t)] mi = !si"

H =1

P

!

µ

"

!

i

#

!µi + mi"

µi

$

%2

!

#

P

!

µ,i

("µi )2(1 ! m2

i )

Jij =1

P

!

µ

!µi !µ

j hi =2

P

!

µ

!µi

!

j

"µj

!

!

i,j

Jijmimj +!

i

himi " !!

i

Jii(1 " m2

i )

minimize H

H ! H(!)

H =1

P

!

µ

"

!

i

#

!µi + mi"

µi

$

%2

!

#

P

!

µ,i

("µi )2(1 ! m2

i )

!Aµ" =

!

j

!µj + mi"

µi

players minimize predictabilityH(0) =1

P

!

µ

!A|µ"2

H(1) =1

P

!

µ

!(Aµ)2" players minimize fluctuations

Q! =1

N

!

i

!si"2

! = !1

! = 0 ! = 0.7

Agents behave stochastically for ! < 1

! = P/N = 1/n

phase structure

0.01 0.1 1 10 100!

0

0.5

1

"

RS

RSB

ERGODICNON

ERGODIC

10!2 10!1 100 101 102

!0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

"(!)

AnalyticN=22N=20N=16

! = 1

Freezing

!(!)

!

!

!

! = P/N = 1/n

! = 1 :1

N

!

i

!si"2

= 1 # each agent uses one strategy

N (!=1)s.s. ! e

N!(")

Agents behave deterministically for ! = 1

research problems

1)

Compute critical indices

!

Critical line

n

100

101

102

103

104

A

10!5

10!4

10!3

10!2

10!1

100

P>(A

)

0 1000t

A

! = 0.01

! ! 2.8 for n = 20

P{A(t) > A} ! A!!

P{A(t) > A} Gaussian for small n but . . .

(dynamical RG?)

research problems

0.1 1 1e+01

ns

0

0.5

1

H/P

!2/P

0.0

0.5

1.0

1.5

<n

act>

! = 0

H/P

H =1

P

!

µ

!A|µ"2

Order parameter

predictable unpredictable

How does the game self-organize around the critical point?

2)

research problems

3)

MG with vector-valued information

µ ! {1, . . . , P} " µ = {µ1, . . . , µK} , µ! ! {1, . . . , P!}

fast/slow signals, strategies coupled to information streams

0

0.01

0.02

0.03

0.04

0.05

!a

0

0.01

0.02

0.03

0.04

0 400 800 1200 1600

!

t

btime

research problems

4)

Microscopic mechanism for the buildup of cross-correlations between stocks : diversification enhances correlations (?!)

References

MG mathematics : ACC Coolen, The math. theory of Minority Games

Review : De M-Marsili, physics/0606107 [J Phys A 2006]

Most recent : De M-Perez Castillo-Sherrington, physics/0611188 (JSTAT 2007) [general solution in the ergodic phase]