Numerical approaches and modeling of multiple agents
in uncertain environment
Gabriel Turinici
CEREMADE,
Université Paris Dauphine
joint work with Julien SALOMON, Min SHEN, Agnés BIALECKI,
Eléonore HAGUET
ITN 2012, Iasi, July 2-7 2012
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
1 / 49
Outline
1
Motivation and introduction to Mean Field Games (MFG)
2
Theoretical results of Lasry-Lions
3
Some numerical approaches
4
General monotonic algorithms (J. Salomon, G.T.)
Related applications: bi-linear problems
Framework
Construction of monotonic algorithms
5
Technology choice modelling (A. Lachapelle, J. Salomon, G.T.)
The model
Numerical simulations
6
Liquidity source: heterogeneous beliefs and analysis costs
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
2 / 49
Mean field games: introduction
• MFG = model for interaction among a large number of agent / players
... not particles. An agent can decide, based on a set of preferences and
by acting on parameters ( ... control theory).
Note: in standard rumor spreading (or opinion making) modeling agent is
supposed to be a mechanical black-box, not the case here. This situation
is included as particular case.
• distinctive properties: the existence of a collective behavior (fashion
trends, financial crises, real estates valuation, etc.). One agent by itself
cannot influence the collective behavior, it only optimizes its own decisions
given the environmental situation.
References: Lasry Lions CRAS notes (2006), Lions online course at College
de France. Further references latter on.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
3 / 49
Mean field games: introduction
• Nash equilibrium: a game of N players is in a Nash equilibrium if, for
any player j supposing other N − 1 remain the same, there is no decision
of the player j that can improve its outcome.
• MFG = Nash equilibrium equations for N → ∞. All players are the
same.
• Agent follows an evolution equation involving some controlling action.
Its decision criterion depend on the others, more precisely on the density of
other players.
• Will consider here stochastic diff. equations, but deterministic case is a
particular situation and can be treated.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
4 / 49
Mathematical framework of MFG
What follows is the most simple model that shows the properties of MFG
models. Cf. references for more involved modeling. Xtx = the
characteristics at time t of a player starting in x at time 0. It evolves with
SDE:
dXtx = α(t, Xtx )dt + σdWtx , X0x = x
(1)
• α(t, Xtx ) = control can be changed by the agent/ player.
• independent brownians (!)
• m(t, x) = the density of players at time t and position x ∈ E ; E is the
state space. Optimization problem of the agent: fixed T = finite horizon
Z T
x
x
x
x
inf E
L(Xt , α(t, Xt )) + V (Xt ; m(t, ·))dt + V0 (XT ; m(T , ·))
(2)
α
0
static case (infinite horizon):
Z T
1
x
x
x
x
inf lim inf E
L(Xt , α(t, Xt )) + V (Xt ; m(t, ·))dt + V0 (XT ; m(T , ·))
α T →∞
T 0
(3)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
5 / 49
Mathematical framework of MFG: examples
Example: choice of a holiday destination.
Particular case: deterministic, no dependence on the initial condition, no
dependence on the control. Each individual minimizes distance to an ideal
destination and a term depending on the presence of others:
V0 (y ; m) = F0 (y ) + F1 (m).
Question: what is the solution ? XTx will be chosen as the minimum of
y 7→ F0 (y ) + F1 (m(y )). Then m is the distribution of such XTx .
COUPLING between m and XTx !!
Particular case: F0 (y ) = y 2 on R. Origin is the most preferred point for all
individuals, distance increases slowly in neighborhood, fast outside. Take
F1 (m) = cm.
Modelization: c > 0 = crowd aversion, c < 0 = propensity to crowd.
Remark: all points y in the the support of m have to be minimums of V0 !
2
Solution: c > 0: semi-circular distribution m(y ) = (λ−yc )+
c < 0: Dirac masses at minimum of F0 .
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
6 / 49
Outline
1
Motivation and introduction to Mean Field Games (MFG)
2
Theoretical results of Lasry-Lions
3
Some numerical approaches
4
General monotonic algorithms (J. Salomon, G.T.)
Related applications: bi-linear problems
Framework
Construction of monotonic algorithms
5
Technology choice modelling (A. Lachapelle, J. Salomon, G.T.)
The model
Numerical simulations
6
Liquidity source: heterogeneous beliefs and analysis costs
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
7 / 49
Theoretical results of Lasry-Lions
Nash equlibrium for finite N. Agent
hR k minimizes
i
T
1
k
1
N
J (α , ..., α ) = lim inf T →∞ T E 0 L(Xtk , αtk ) + F k (Xt1 , ..., XtN )dt
The set of decisions (αk )k is a Nash equilibrium if ∀k, ∀αk :
J k (α1 , ...αk−1 , αk , αk+1 , ..., αN ) ≤ J k (α1 , ...αk−1 , αk , αk+1 , ..., αN ), (4)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
8 / 49
Theoretical results of Lasry-Lions
Here F k is symmetric in the other N − 1 variables and moreover all agents
are the same i.e. F k does not P
depend on k:
1
k
1
N
k
F (Xt , ..., Xt ) = V (X ; N−1 `6=k δX ` )
Define: H(x, ξ) = supα hξ, αi − L(x, α); ν = σ 2 /2.
Limit for N → ∞: static case; the optimality equations converge (up to
sub-sequences) to solutions of the MFG system
Z
+div(αm) − ν∆m = 0,
α=−
m = 1, m ≥ 0
(5)
∂
H(x, ∇u)
∂p
(6)
Z
−ν∆u + H(x, ∇u) + λ = V (x, m),
u = 0.
(7)
RUniqueness: when V is a strictly monotone operator i.e.
(V (m1 ) − V (m2 ))(m1 − m2 ) ≤ 0 implies V (m1 ) = V (m2 ).
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
9 / 49
Theoretical results of Lasry-Lions
Limit for N → ∞: finite horizon case (i.e. finite T ); the optimality
equations converge (up to sub-sequences) to solutions of MFG system
∂t m + div(αm) − ν∆m = 0,
Z
m(0, x) = m0 (x),
m = 1, m ≥ 0
(8)
(9)
∂
H(x, ∇u)
∂p
∂t u + ν∆u − H(x, ∇u) + V (x, m) = 0,
Z
u(T , x) = V0 (x, m(T , ·)),
u = 0.
α=−
(10)
(11)
(12)
Remark: these are not necessarily the critical point equations for an
optimization problem ! But will be in some particular cases studied latter.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
10 / 49
Outline
1
Motivation and introduction to Mean Field Games (MFG)
2
Theoretical results of Lasry-Lions
3
Some numerical approaches
4
General monotonic algorithms (J. Salomon, G.T.)
Related applications: bi-linear problems
Framework
Construction of monotonic algorithms
5
Technology choice modelling (A. Lachapelle, J. Salomon, G.T.)
The model
Numerical simulations
6
Liquidity source: heterogeneous beliefs and analysis costs
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
11 / 49
Mean field games notations (reminder)
• Mean field games: limits of Nash equilibriums for infinite number of
players (P.L.Lions & J.M.Lasry)
• equation for each player dXtx = αdt + σdWtx , α(t, x) = control
• m(t, x) = the density of players at time t and position x ∈ Q
• evolution equation
∂
m(t, x) − ν∆m(t, x) + div (α(t, x)m(t, x)) = 0,
∂t
m(0, x) = m0 (x).
• We consider the optimisation
setting: minα J(α)
o
RT n
R
J(α) := Ψ(m(·, T )) + 0 Φ(m(t, ·)) + Q L(x, α)m(t, x)dx dt
2
• Φ, Ψ can be linear, concave, ... Typical L : L(x, α) = α2 .
Rq: MFG equations are critical point equations for the functional J;
relationship with individual level: ∇m Φ = V , ∇m Ψ = V0 , L = h.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
12 / 49
Numerics of MFG : literature overview
• (in)finite horizon: finite-difference discretization: approximation
properties, existence and uniqueness, bounds on the solutions. ”Mean
Field Games: Numerical Methods” Y. Achdou & I. Capuzzo-Dolcetta
• Y. Achdou & I. Capuzzo-Dolcetta: Newton method for the coupled
direct-adjoint critical point equations (finite horizon, cx case)
• O. Gueant: study of a prototypical case: solution, stability (09),
quadratic Hamiltonian (11)
• solution of the MFG equations from an optimization point of view (A.
Lachapelle, J. Salomon, G. Turinici, M3AS 2010)
• Lachapelle & Wolfram (2011) (congestion modelling)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
13 / 49
Outline
1
Motivation and introduction to Mean Field Games (MFG)
2
Theoretical results of Lasry-Lions
3
Some numerical approaches
4
General monotonic algorithms (J. Salomon, G.T.)
Related applications: bi-linear problems
Framework
Construction of monotonic algorithms
5
Technology choice modelling (A. Lachapelle, J. Salomon, G.T.)
The model
Numerical simulations
6
Liquidity source: heterogeneous beliefs and analysis costs
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
14 / 49
Optimal control of a Fokker-Plank equation (G. Carlier &
J. Salomon)
Evolution equation :
∂t ρ − 2 ∆ρ + div (v ρ) = 0
(13)
ρ(x, t = 0) = ρ0 (x)
(14)
• goal: minimize w.r. to v the functional (for some given V (·)) :
Z Z
Z
E (v ) =
ρv 2 dxdt + ρ(x, 1)V (x)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
15 / 49
Control of the time dependent Schrödinger equation
∂
Ψ(x, t) = (H0 − (t)k µ(x))Ψ(x, t)
i ∂t
Ψ(x, t = 0) = Ψ0 (x)
(15)
• vectorial case (rotation control, NMR):
∂
i ∂t
Ψ(x, t) = [H0 + (E1 (t)2 + E2 (t)2 )µ1 + E1 (t)2 · E2 (t)µ2 ]Ψ(x, t).
H0 = −∆ + V (x), unbounded domain
Evolution on the unit sphere: kΨ(t)kL2 = 1, ∀t ≥ 0.
• evaluation of the quality of a control through a objective functional to
minimize
RT
J() = −2<hψtarget |ψ(·, T )i + 0 α(t)2 (t)dt
RT
J() = kψtarget − ψ(·, T )k2L2 − 2 + 0 α(t)2 (t)dt
RT
J() = − hΨ(T ), OΨ(T )i + 0 α(t)2 (t)dt
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
16 / 49
General monotonic algorithms (J. Salomon, G.T.)
state X ∈ H , control v ∈ E , H, E = Hilbert/ Banach spaces.
• ∂t Xv + A(t, v (t))Xv = B(t, v (t))
• minv J(v ), J(v ) :=
RT
0
F t, v (t), Xv (t) dt + G Xv (T ) .
• F , G : C 1 + concavity with respect to X (not v !)
∀X , X 0 ∈ H, G (X 0 ) − G (X ) ≤ h∇X G (X ), X 0 − X i
∀t ∈ R, ∀v ∈ E , ∀X , X 0 ∈ H :
F (t, v , X 0 ) − F (t, v , X ) ≤ h∇X F (t, v , X ), X 0 − X i.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
17 / 49
Direct-adjoint equations and first lemma
∂t Xv + A(t, v (t))Xv = B(t, v (t))
X (0) = X0
∂t Yv − A∗ t, v (t) Yv +
∇
F
t,
v
(t),
X
(t)
=0
v
X
Yv (T ) = ∇X G Xv (T ) .
Lemma
Suppose that A, B, F are differentiable everywhere in v ∈ E , then there
exists ∆(·, ·; t, X , Y ) ∈ C 0 (E 2 , E ) such that, for all v , v 0 ∈ E
0
Z
J(v ) − J(v ) ≤
T
∆(v 0 , v ; t, Xv 0 , Yv ) ·E v 0 − v dt
(16)
0
Proof: cf. refs.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
18 / 49
Well-posedness
0
Z
J(v ) − J(v ) ≤
T
∆(v 0 , v ; t, Xv 0 , Yv ) ·E v 0 − v dt
(17)
0
Remark: useful factorisation because can test at each step if J goes the
right way; also can choose v 0 (t ∗ ) = v (t ∗ ) if pb.
Remark: ∆(v 0 , v ; t, X , Y ) has an explicit formula once the problem is
given; also note the dependence on Yv any not Yv 0 .
Lemma
Under hypothesis on A, B, F , G , θ > 0
∆(v 0 , v ; t, X , Y ) = −θ(v 0 − v )
(18)
has an unique solution v 0 = Vθ (t, v , X , Y ) ∈ E .
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
19 / 49
Well-posedness
Theorem (J. Salomon, G.T. Int J Contr, 84(3), 551, 2011)
Under hypothesis ...
• the following eq. has a solution:
∂t Xv 0 (t) + A(t, v 0 )Xv 0 (t) = B(t, v 0 )
(19)
0
v (t) = Vθ (t, v (t), Xv 0 (t), Yv (t))
(20)
Xv 0 (0) = X0
(21)
• ∃ (θk )k∈N such that v k+1 (t) = Vθk (t, v k (t), Xv k+1 (t), Yv k (t))
• J(v k+1 ) − J(v k ) ≤ −θk kv k+1 − v k k2L2 ([0,T ]) ;
• if v k+1 (t) = v k (t) : ∇v J(v k ) = 0.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
20 / 49
Outline
1
Motivation and introduction to Mean Field Games (MFG)
2
Theoretical results of Lasry-Lions
3
Some numerical approaches
4
General monotonic algorithms (J. Salomon, G.T.)
Related applications: bi-linear problems
Framework
Construction of monotonic algorithms
5
Technology choice modelling (A. Lachapelle, J. Salomon, G.T.)
The model
Numerical simulations
6
Liquidity source: heterogeneous beliefs and analysis costs
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
21 / 49
The Model : framework
• large economy: continuum of consumer agents
• time period: [0, T ]
• any household owns exactly one house and cannot move to another one
until T
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
22 / 49
The Model : the agents
• arbitrage between insulation and heating. A generic player (agent) has an
insulation level x ∈ [0, 1] (x = 0: no insulation, x = 1: maximal insulation)
• controlled process of the agent: dXtx = σdWt + vt dt + dNt (Xtx ),
X0x = x; v is the control parameter (insulation effort), the noise level σ is
given.
• note that Xt is a diffusion process with reflexion, in the above equality,
dNt (Xt ) has the form χ{0,1} (Xt )~
ndξt ( ξ = local time at the boundary
{0, 1} = ∂[0, 1] cf. Freidlin)
• initial density: X0 ∼ m0 (dx)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
23 / 49
The Model : the costs
An agent of the economy solves a minimization problem composed of
several terms:
• Insulation acquisition cost: h(v ) :=
v2
2
x
• Insulation maintenance cost: g (t, x, m) := c1 +cc20m(t,x)
increasing in x
decreasing in m : economy of scale, positive externality. The agents
should do the same choice, stay together. The higher is the number of
players having chosen an insulation level, the lower are the related costs.
• Heating cost: f (t, x) := p(t)(1 − 0, 8x) where p(t) is the unit heating
cost (unit price of energy, say)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
24 / 49
The model - The minimization problem and MFG (1)
• Define the aggregate state cost:
Z 1
Φ(m) :=
p(t)(1 − 0, 8x) +
0
c0 x
c1 + c2 m(t, x)
m(t, x)dx
and V = Φ0 .
• In the model, the agents have rational expectations, i.e they see m as
given; we can write the individual agent’s problem:
hR
i
(
T
inf E 0 h(v (t, Xtx )) + V [m](Xtx )dt
v
adm
dXt = vt dt + σdWt + dNt (Xt ), X0 = x
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
25 / 49
The model - The minimization problem and MFG (2)
• We already know that it is linked with the optimal control problem:

RT R1
h(v (t, x)) + Φ(mt )(t)dt

 v inf
adm 0 0
2
∂ m − σ2 ∆m + div(vm) = 0 , m|t=0 = m0 (.) ,

 t0
m (., 0) = m0 (., 1) = 0
2
• Finally, if ν := σ2 , a Mean field equilibrium (Nash equilibrium with an
infinite number of players) corresponds to a solution of the following
system:

 ∂t m − ν∆m + div(vm) = 0 , m|t=0 = m0
∇u = v
(22)

u2
0
∂t u + ν∆u + v · ∇u − 2 = Φ (m) , v |t=T = 0
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
26 / 49
The model - externality & scale effect
The MFG framework is interesting to describe a situation which lives
between two economical ideas: positive externality and economy of scale
• positive externality: positive impact on any agent utility NOT
INVOLVED in a choice of an insulation level by a player
• economy of scale: economies of scale are the cost advantages that a firm
obtains due to expansion (unit costs decrease)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
27 / 49
Criticism of the model:
• stylised from the ”industrial” point of view
• not realistic (heating price, maintenance...)
• transition effect (continuous time, continuous space)
• atomised agent (her/his action has no influence on the global density,
micro-macro approach)
• non-cooperative equilibrium with rational expectations
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
28 / 49
Numerical simulations
• Optimization method: Monotonic algorithm

k+1 − ν∆mk+1 + div(v k+1 mk+1 ) = 0 , mk+1 (x, 0) = m

0
 ∂t m
k −∇u k
v k+1 = (θ−1/2)v
(θ+1/2)

k+1 2

k+1
∂t u
+ ν∆u k+1 + v k+1 · ∇u k+1 − (u 2 ) = Φ0 (mk+1 ), v k+1 (T ) = 0
(23)
• Discretization of the PDEs: Godunov scheme (to preserve the positivity
of the density m)
• The costs:
heating: f (t, x) = p(t)(1 − 0, 8x)
x
insulation: g (t, x, m) = 0.1+m(t,x)
• 1st example: p(t) constant / same choices
• 2d example: p(t) reaching a peak (non constant) / multiplicity of
equilibria
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
29 / 49
Numerical results - First case
• the initial density of the householders is a gaussian centered in
1
2
• the time period and the noise are respectively T = 1 and ν = 0.07
• the energy price is constant (p(t) ≡ 0, 3.2 and 10)
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
30 / 49
Figure : Numerical results : p(t) ≡ 0. Since the cost of energy is null all agents
choose to heat their house, move to this choice together.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
31 / 49
Figure : Numerical results : p(t) ≡ 3.2. Cost of energy is intermediary, agents
keep their status.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
32 / 49
Figure : Numerical results : p(t) ≡ 10. Cost of energy is high, agents choose to
better insulate, all have the same behavior.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
33 / 49
Numerical results - Second case
the initial density of the agents is an approximation of a Dirac in 0.1
(i.e agents are not equipped in insulation material)
the energy price is not a constant parameter, we look at the following
case: the price first reaches a peak and then decreases to its initial
level.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
34 / 49
Figure : Numerical results - p(t). Question: In such a case, can we find two
Mean Field equilibria, the first related to the expectation of a higher insulation
level, the second to the expectation of heating ?
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
35 / 49
Figure : Numerical results - One of the two equilibria: the energy consumption
equilibrium. Agents expect that everybody will keep a low insulation level so there
are no gains in insulating.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
36 / 49
Figure : Numerical results - One of the two equilibria: the insulation equilibrium.
Agents expect that everybody will better insulate, which makes insulating
attractive.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
37 / 49
Multiplicity of equilibria - Incentive policy
• we found an insulation-equilibrium and an energy
consumption-equilibrium
• from the ecological point of view: the best is the insulation-equilibrium
• incentive public policies could steer towards the ”best” equilibrium (from
a certain point of view) when the solution is not unique.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
38 / 49
Outline
1
Motivation and introduction to Mean Field Games (MFG)
2
Theoretical results of Lasry-Lions
3
Some numerical approaches
4
General monotonic algorithms (J. Salomon, G.T.)
Related applications: bi-linear problems
Framework
Construction of monotonic algorithms
5
Technology choice modelling (A. Lachapelle, J. Salomon, G.T.)
The model
Numerical simulations
6
Liquidity source: heterogeneous beliefs and analysis costs
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
39 / 49
Liquidity from heterogeneous beliefs and analysis costs
• Why do agents trade ? Here: heterogenous beliefs and expectations
• Liquidity : many definitions (bid/ask spread, rapidity to recover price
after shock, max volume traded at same price etc). (cf. works by M.
O’Hara) Here: trading volume.
• Heterogeneous beliefs: (back to Keynes) then works on asset pricing,
CAPM equlibrium etc.
• previous works: Grossman-Stiglitz : a costless estimation with low
precision and a costly high precision estimation (used to argue the
inefficiency of the markets)
• previous works: José. A. Scheinkman and Wei Xiong: questions on the
incentive to become trader and the sub-optimal equilibrium in terms of
“grey” resources
• ... and so on. Key works in the literature: insider trading, noise traders,
• specific investigation of this work: cost of obtaining precise estimation
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
40 / 49
Heterogeneous beliefs and liquidity: the model
• One security of ”true” value V .
• each agent has its own estimation (random variable) V Ã of mean VA
and variance V 2 σ 2 (Ã). The precision on à is B = 1/σ 2 . The agent
cannot change its A but can change σ 2 (Ã). Precision can be improved
paying f (B) and/or waiting for the estimation to converge or new data to
be revealed.
• Agents are distributed with density ρ(A); the mean of this distribution
is taken to be 1 (overall neutrality).
• INITIAL TIME (t = 0): agent pays research costs f (B) depending on
the precision (s)he wants and trade θopt (A, P) units i.e. V · θopt (A, P) =
size of the position of agent at A.
• FINAL TIME (t = 1): agent expects to end its position at price V Ã
• Each agent has a CARA utility function −e λ(gain) whose optimums are
the same, for normal variables, as those of E(gain) − λ2 var (gain). Note
gain is function of θ,B (thus also mean and variance).
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
41 / 49
Heterogeneous beliefs and liquidity: theoretical results
• Price at t = 0 is V P and clears the market i.e. the overall balance is
null (implicit equation for P):
Z
θopt (A, P)ρ(A)dA = 0
(24)
• is the solution uniqueR ?
• is the total demand θopt (A, p)+ ρ(A)dA a decreasing function of the
price p ? Note: if the answer is yes (and total supply an increasing
function) then the solution of (24) is unique and the unique solution is
also the
R level that maximizes
R the total transaction volume which is
min{ θopt (A, p)+ ρ(A)dA, θopt (A, p)− ρ(A)dA}
Note: P Ris not necessarily equal to 1 even if the mean
E(A)(= Aρ(A)dA) = 1.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
42 / 49
Heterogeneous beliefs and liquidity: theoretical results
Technical framework: Mean Field Games by Lasry & Lions; Nash
equilibrium. x = (A, θ, B)T , total time = 1.
d(A, θ, B)T = (0, αθ , αB )T dt + 0dWt ,
m(A, θ, B, t)
= ρ(A)δ(θ − θ0 )δ(B − B0 ), e.g . B0 = 0 (25)
t=0
R1
Mean profit for agent at x is mean(θ, B) = V θ(A − P) − 0 L(B, αB (t))dt,
with L(B, αB ) = +∞ if B < 0 and f (αB ) otherwise ; variance of the
profit is variance(θ, B) = θ2 V 2 /B+ . Thus agent in x optimizes:
1
Z
f (αB (t))dt − λθ(T )2 V 2 /B+ (T ).
V θ(T )(A − P) −
(26)
0
To this we add the equilibrium condition above (eqn.(24)).
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
43 / 49
Heterogeneous beliefs and liquidity: theoretical results
Mean profit for agent at price p is E(θ, B) = V θ(A − p) − f (B)
Variance of the profit is variance(θ, B) = θ2 V 2 /B.
For some price p agent optimizes with respect to θ, B (and potentially for
any given A and p) :
λ
V θ(A − p) − f (B) − θ2 V 2 /B.
(27)
2
Optimal values Bopt (A, p) and θopt (A, p) form aggregate supply and
demand at price p:
Z ∞
1
(A − p)− Bopt (p, A)ρ(A)dA
(28)
S(p) =
2V λ 0
Z ∞
1
D(p) =
(A − p)+ Bopt (p, A)ρ(A)dA
(29)
2V λ 0
Recall trading volume TV (p) = min{S(p), D(p)}
Question: are S(p) and D(p) (strictly) monotonic so as to define a unique
price and a maximal trading volume; are them continuous, ...
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
44 / 49
Heterogeneous beliefs and liquidity: theoretical results
Function f (B): it is the “research cost” to reach the precision B.
Conditions for f : f (0) = 0, l.s.c., limx→∞ f (x)/x 1+β > 0 (for some
β > 0).
Theorem (M Shen, G.T. 2011)
Under assumptions above on function f and some on ρ(A)
for general, possibly non equilibrium price P, offer and demand are
monotone with respect to P.
an equilibriumR price P that clears the market (eqn.(24)) exists and is
unique: P =
∞
ABopt (A,P)ρ(A)dA
R0 ∞
.
0 Bopt (A,P)ρ(A)dA
Proof. : write the critical point equations and use assumptions on f ; for
last two formulas obtain θ as function of B, A and use the balance
equation.
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
45 / 49
Heterogeneous beliefs and liquidity
Theorem (M Shen, G.T. 2011)
Under assumptions on functions f if ρ is symmetric around p 1 then
(liquidity is maximal for p = p 1 i.e.) P = p 1 .
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
46 / 49
Heterogeneous beliefs and liquidity
For smooth, convex functions f the relative market price P is solution to
the equation:
Z ∞
2
1
0 −1 (A − P)
ρ(A)dA = 0
(30)
(A − P)(f )
2V λ 0
2λ
The trading volume TVf is
Z ∞
2
P
0 −1 (A − P)
TVf =
(A − P)+ (f )
ρ(A)dA
2λ 0
2λ
(31)
Theorem (anti-monotony of trading volume)
Let f , g be two information cost functions such that g 0 (b) ≥ f 0 (b) for any
b ∈ R+ . Then the trading volume satisfies TVf > TVg .
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
47 / 49
Heterogenous beliefs and liquidity
• interesting technical “detail”: several functions f (B) give the same S(p)
and D(p); in fact replacing f by its twice Legendre-Fenchel transform f ∗∗
the S(p) and D(p) are the same (under assumptions on ρ).
Note that this is specific to expressing cost as function of precision B and
not variance σ 2 = 1/B i.e., f (1/B)∗∗ 6= f ∗∗ (1/B) .
Application: Grossman-Stiglitz
f (B1 ) = 0, f (B2 ) = c with B2 > B1 , f (B) = ∞ for any B > B2 .
( (A−p)B
1
b
if |A − p| < V (B2λc
B1
λV
2 −B1 )
, Bopt (p, A) =
θopt (p, A) =
(A−p)B2
2λcb
B2
if |A − p| ≥
λV
Gabriel Turinici (U Paris Dauphine)
V (B2 −B1 )
Modeling multiple agents
ITN 2012, Iasi
48 / 49
Perspectives
• introduce evolution for A (non-ergodic), possibly depending on B
• introduce stochastic dynamics for A, B, θ
• introduce evolution equation for V ; usual form dV /V = µdt + σV dWtV
Gabriel Turinici (U Paris Dauphine)
Modeling multiple agents
ITN 2012, Iasi
49 / 49