Asset market with heterogeneous agents
M. Marsili (INFM-SISSA Trieste)
J. Berg, R. Zecchina (ICTP, Trieste), A. Rustichini (Boston)
How does the trading behavior of agents
• eliminate arbitrages
• transfer information into prices
• make the market more (or less) efficient
The asset market model:
State: w = 1,…,W; W = aN
Agents: i=1,…,N
information: m=kiw
Asset N units
investment: zim
price: p w = S i zim / N
payoff:
Rw zim /pw -
zim
return: Rw
El Farol bar (Arthur ‘94) type problem, “minority” rewarded (Challet, Zhang ‘97)
Details: asymmetric information
ki : (1,…,w,…,W)
(1,…,m,…,M) random
returns Rw = R + rw/N1/2 , rw = gaussian random, stand. dev. = R s
w random uniform in (1,…,W)
Parameters: a, M, R, s
N 
Asymmetric information (M=2)
Market
w
kiw
Market information efficiency:
w
Agents
zim
price
pw
return
Rw
Def:
H=0
?
w
w
p =R
H=Sw (pw - Rw)2
pw = Rw for all w (efficient market)
Market’s equilibria (static)
Competitive
equilibrium
Nash
equilibrium
price taker
agents
dui ui
= m
m
dzi
zi pw
strategic
agents
dui
ui
= m

m
dzi
zi pw
(*)
ui pw
pw zim
pw 1
Naively

m
zi
N
Competitive
equilibrium
N


Nash
equilibrium
 Rw
 ki (w )
1

1
* ui = agent’s utility = agent’s expected payoff = w  w  zi
W
p

Two stages “process”
RW
Rw
“fast” process
p
w
pw =
1
N

i
ziki (w )  R
“slow” process
adjustment to r w/N1/2
R
H = distance to Rw
Results for N   :
- equilibria are the solution {zim } of the problem
W
w
w
R
p


min
{ zim } w =1

)
2



z )

N
N
i =1
m 2
i

,

1
p =
N
w
N
k i (w )
z
 i ,
i =1
1
NM
m
z
 i =R
i ,m
=0, competitive eq.
=1, Nash eq.
• agents minimize H
• agents payoff = 0
• eq. not unique in zim
• eq. unique in pw
• agents payoff > 0
• eq. unique in zim and in pw
w
w
Note: R - p  1
N
Analytical results (*)
Phase diagram for =0
Market’s efficiency (s=1)
H/a
a
inefficient
phase (H>0)
efficient
phase (H=0)
a
(*)
phase transition
using statistical mechanics of disordered systems: M=2.
s
Phase transition for =0
Density plot of H in the space {zim}
H=Hmin
H= Hmin =0
Dependence on
prior beliefs!
ac
a
Dynamics: adaptive learning
repeated game, w drawn random at each t=1,2,...
• Scores
Uim(t)
ci(U)
• Investment zim(t)=ci[Uim(t)]
• Reinforcement
U
m
w
w


z
(
t
)
R
R
m
m
i
  k (w ),m
U i (t  1) = U i (t )   w - 1 -  2
i
w
p

N
p


• =0 price takers, =1 sophisticated agents
Results for adaptive learning (ci smooth “enough”)
Distance in strategy space(*)
H/a
Distance in W space
a
a
• agents converge to competitive or Nash equilibria
• dependence on initial conditions (prior beliefs) for a<ac
(*) = 1
N

i
 zi - zi- 


2


2
Dynamics of the wealth of agents
• Agents have a finite wealth wi and zim < wi
• wealth is updated as:
 Rw

wi (t  1) = wi (t )   w - 1  ki (w ),m zim (t )
p

•how agents choose depend on utility
log utility
linear utility
zim = c i wi ,
c i : R  [0,1]
zim = min c i , wi , c i : R  R 
• =0 price takers
Results with dynamics of wi
Distance in W space
Distance in strategy space
H/a
wi
a
a
Conclusions
• Analytic approach to heterogeneous interacting agents
• Competitive equilibria not “close” to Nash equilibria
• Phase transition to H=0
• payoffs=0
• Not unique eq. (H=0)
• No phase transition
• payoffs>0
• Unique equilibrium
• Learning dynamics converges to equilibria
• Complex dynamics when wealth is updated