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
Beyond the representative agent approach for asset markets
1) agents are not identical but rather they are strongly heterogeneous, which is relevant, as pointed out by El Farol problem in market interaction.
2) Agents are a finite large number - and not a continuum -
which allows to eveluate the effect of market impact.
We find results for a non-trivial thermodynamic limit.
The model is quite similar to the minority game in the end, but at odds with it it has a direct economic interpretation.
All the intuitions due to our analytic approach to the MG prove useful and the behavior is largely the same.

2. The asset market model:
State: w = 1,…,W; W = aN
Agents: i=1,…,N
Asset N units
price: p
information: m=kiw
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

3. Asymmetric information (M=2)w
Market
Internal representation of the states of the market
Each agent has his own partition of Omega
kiw

4. Market information efficiency:
Agents
zim ?
price pw w
pw = Rw
return Rw
Def:
Key question is how agents transfer information into prices
by their behavior. If prices are equal to returns,
all information has been incorporated int prices.
In this case the market is information efficient beause even with ful information (know omega) an agent cannot gain.
Introduce a measure of deviation from efficiencyl
H=Sw (pw - Rw)2
H= pw = Rw for all w (efficient market)

5. Market’s equilibria (static)
(*)
Competitive
equilibrium
price taker
agents
Nash
equilibrium
strategic
agents
Naively
Static equilibria
Competitive
equilibrium
Nash
equilibrium
* ui = agent’s utility = agent’s expected payoff =

6. Two stages “process” RW “fast” process “slow” process
adjustment to r w/N1/2 R
H = distance to Rw

7. Results for : - equilibria are the solution {zim } of the problem
h=0, competitive eq.
h=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
Note:

8. Analytical results (*)
Market’s efficiency (s=1)
Phase diagram for h=0
inefficient
phase (H>0) a
H/a
efficient
phase (H=0) a s
phase transition
(*) using statistical mechanics of disordered systems: M=2.

9. Phase transition for h=0
Density plot of H in the space {zim}
H=Hmin
H= Hmin =0 a ac
Dependence on
prior beliefs!

10. Dynamics: adaptive learning
repeated game, w drawn random at each t=1,2,...
ci(U)
Scores Uim(t)
Investment zim(t)=ci[Uim(t)]
Reinforcement
h=0 price takers, h=1 sophisticated agents U
The reinforcement is just given by the derivative of u_i
w.r.t. z_i

11. H/a a a 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 a<ac
(*) =

12. Dynamics of the wealth of agents
Agents have a finite wealth wi and zim < wi
wealth is updated as:
how agents choose depend on utility
h=0 price takers
log utility
linear utility

13. Results with dynamics of wi
Distance in W space
Distance in strategy space wi
H/a a a

14. 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 H=0
payoffs=0
Not unique eq. (H=0)
No phase transition
payoffs>0
Unique equilibrium