Asset market with heterogeneous agents

1
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

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The asset market model
State w 1,,W W aN
Agents i1,,N
Asset N units price p
information mkiw
w S i zim / N
investment zim
return Rw
payoff Rw zim /pw - zim
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
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Asymmetric information (M2)
w
Market
kiw
4
Market information efficiency
Agents zim
?
price pw
w
pw Rw
return Rw
Def
HSw (pw - Rw)2
H0 pw Rw for all w (efficient market)
5
Markets equilibria (static)
()
price taker agents
Competitive equilibrium
Nash equilibrium
strategic agents
Naively
Competitive equilibrium
Nash equilibrium
ui agents utility agents expected payoff
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Two stages process
fast process
RW
slow process adjustment to r w/N1/2
R
H distance to Rw
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Results for
- equilibria are the solution zim of the
problem
h0, competitive eq.
h1, Nash eq.

• agents minimize H
• agents payoff 0
• eq. not unique in zim
• eq. unique in pw

• agents payoff gt 0
• eq. unique in zim and in pw

Note
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Analytical results ()
Phase diagram for h0
Markets efficiency (s1)
inefficient phase (Hgt0)
a
H/a
efficient phase (H0)
a
s
phase transition
() using statistical mechanics of disordered
systems M2.
9
Phase transition for h0
Density plot of H in the space zim
HHmin
H Hmin 0
a
ac
Dependence on prior beliefs!
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Dynamics adaptive learning
repeated game, w drawn random at each t1,2,...
ci(U)

• Scores Uim(t)
• Investment zim(t)ciUim(t)
• Reinforcement
• h0 price takers, h1 sophisticated agents

U
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Results for adaptive learning (ci smooth enough)
Distance in W space
Distance in strategy space()
H/a
a
a

• agents converge to competitive or Nash
  equilibria
• dependence on initial conditions (prior beliefs)
  for altac

()
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Dynamics of the wealth of agents

• Agents have a finite wealth wi and zim lt wi
• wealth is updated as
• how agents choose depend on utility
• h0 price takers

log utility
linear utility
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Results with dynamics of wi
Distance in W space
Distance in strategy space
wi
H/a
a
a
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Conclusions

• Analytic approach to heterogeneous interacting
  agents
• Competitive equilibria not close to Nash
  equilibria
• Learning dynamics converges to equilibria
• Complex dynamics when wealth is updated

• Phase transition to H0
• payoffs0
• Not unique eq. (H0)

• No phase transition
• payoffsgt0
• Unique equilibrium