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]
