1
Equilibria in Games with a Continuum of
Agents and Transport
Guillaume Carlier a .
Journée EDP en sciences sociales,
clotûre de la chaire de P. Markowich, 24 janvier 2014.
a CEREMADE,
Université Paris Dauphine
/1
2
Aim of this talk: illustrate the use of optimal transport theory
in equilibrium problems with a continuum of agents (not so
surprising since the very roots of Kantorovich work were in
decentralization of planning problems). Two parts:
• Matching for teams (joint with Ivar Ekeland, CEREMADE,
Dauphine),
• Cournot-Nash equilibria (joint with Adrien Blanchet,
Toulouse School of Economics).
/2
Introduction
3
Introduction
The basic optimal transport problem is as follows. We are given
two metric (compact, say) spaces X and Y , two Borel
probability measures µ ∈ P(X) and ν ∈ P(Y ), and a transport
cost function c ∈ C(X × Y ) (or simply lsc). One looks for the
cheapest way to transport µ to ν.
For a more detailed overview of optimal transport theory and
applications, see the books of Villani and Ambrosio, Gigli and
Savaré).
Introduction/1
Introduction
4
The image of µ by a Borel map T : X → Y is by definition
given by
T# µ(B) := µ(T −1 (B)), ∀B.
or equivalently (change of variables)
Z
Z
ϕ(y)T# µ(dy) =
ϕ(T (x))µ(dx), ∀ϕ ∈ C(X).
Y
X
A transport map between µ and ν is a map T such that
T# µ = ν (note that there might exist no such maps, e.g. when
µ is a Dirac mass and ν is not). Monge Problem
Z
inf
c(x, T (x))µ(dx).
T : T# µ=ν
X
In general quite complicated because of the awfully nonlinear
constraint (Jacobian equation in the euclidean and smooth
case).
Introduction/2
Introduction
5
This is why the problem was relaxed by Kantorovich in the 40’s.
A transport plan is a joint probability γ ∈ P(X × Y ) having µ
and ν as marginals, i.e.
γ(A × Y ) = µ(A), γ(X × B) = ν(B)
or, for all ϕ ∈ C(X), ψ ∈ C(Y )
Z
Z
Z
(ϕ(x) + ψ(y))γ(dx, dy) =
ϕµ +
ψν.
X×Y
X
Y
Introduction/3
Introduction
6
Denote by Π(µ, ν) the set of transport plans between µ and ν,
the Monge Kantorovich problem reads
Z
Wc (µ, ν) := inf
c(x, y)γ(dx, dy)
γ∈Π(µ,ν)
X×Y
• Linearity: existence of an optimal plan is for free,
• relaxation: provided µ has no atoms, Wc (µ, ν) coincides
with the infimum in Monge’s problem (which need not be
attained in general),
• there is a convenient dual formulation.
Introduction/4
Introduction
7
Dual formulation, Kantorovich duality formula, Wc (µ, ν)
coincides with
Z
Z
sup
ϕµ +
ψν : ϕ(x) + ψ(y) ≤ c(x, y)
(ϕ,ψ)
X
Y
since the constraint implies that
ψ(y) ≤ min{c(x, y) − ϕ(x)} := ϕc (y),
x∈X
one also has the (Kantorovich) duality formula
Z
Z
ϕµ +
ϕc ν
Wc (µ, ν) = sup
ϕ
X
Y
and the sup is achieved. Decentralization by prices principle
(Kantorovich was awarded the Nobel prize in economics in 76).
Introduction/5
Introduction
8
The Wasserstein space. Let µ and ν be probability measures on
Rd with finite second moments, the squared Wasserstein
distance between µ and ν is
Z
W22 (µ, ν) = inf
|x − y|2 γ(dx, dy).
γ∈Π(µ,ν)
Rd ×Rd
An important result of Yann Brenier says that whenever µ does
not charge Lipschitz hypersurfaces, there is a unique minimizer
that is given by a transport and characterized by
γ = (id, ∇φ)# µ with φ convex. Formally, related to the
Monge-Ampère equation:
det(D 2 φ)ν(∇φ) = µ, φ convex
with (rather tricky) boundary conditions.
Introduction/6
Matching for teams
9
Matching for teams
Market for houses, quality z ∈ Z. For one house z to be
available, need for one buyer and a team of producers (mason,
plumber, electrician). We shall denote by the index
i ∈ {1, · · · , I} the different populations (buyers, plumbers,
electricians, masons...). Contrary to the standard OT
framework, we are given measures on the populations and look
for a measure on Z by equilibrium requirements.
Matching for teams/1
Matching for teams
10
Agents in each population i are hetererogeneous, they are
characterized by a certain type xi ∈ Xi (metric compact) which
affects their cost, ci ∈ C(Xi × Z, R) with the interpretation that
ci (xi , z) is the cost for an agent of population i with type xi to
work in a team that produces good z. The distribution of type
xi in population i is known and given by some Borel probability
measure µi ∈ P(Xi ).
Matching for teams/2
Matching for teams
11
One then looks for an equilibrium and in particular an
equlibrium system of monetary transfers (paid by the buyer to
the producers). A system of transfers is a collection of
continuous functions ϕ1 , . . . ϕI : Z → R where ϕi (z) is the
amount paid to i by the other members of the team for
producing z. An obvious equilibrium requirement is that teams
are self-financed i.e.
I
X
ϕi (z) = 0, ∀z ∈ Z.
(1)
i=1
Given transfers ϕ1 , . . . ϕI , an agent from population i with type
xi ∈ Xi , gets a net minimal cost given by the so-called
ci -transform of ϕi :
ϕci i (xi ) := min{ci (xi , z) − ϕi (z)}
z∈Z
(2)
Matching for teams/3
Matching for teams
12
Agents are rational: they choose cost miniminizing qualities i.e.
a z ∈ Z such that
ϕci i (xi ) + ϕi (z) = ci (xi , z).
(3)
The last unknown is a collection of plans γi ∈ P(Xi × Z) such
that γi (Ai × A) represents the probability that an agent in
population i has a type in Ai and belongs to a team that
produces a quality in A. At equilibrium the first marginal of γi
should be µi (this is equilibrium on the i-th labor market) and
the second marginal of γi should not depend on i (this is
equilibrium on the quality good market), this common marginal
represents the equilibrium quality line.
Matching for teams/4
Matching for teams
13
Equilibrium: transfer system (ϕ1 , . . . ϕI ) ∈ C(Z, R)I ,
probability measures γi ∈ P(Xi × Z) and a probability measure
ν ∈ P(Z) such that
• teams are self-financed i.e. (1) holds,
• γi ∈ Π(µi , ν) for i = 1, . . . , I (equilibrium on the labor
markets and on the good market),
• (3) holds on the support of γi for i = 1, . . . , I (i.e. agents
choose cost minimizing qualities).
Matching for teams/5
Matching for teams
14
Variational characterization of equilibria
inf
ν∈P(Z)
J(ν) :=
I
X
(4)
Wci (µi , ν)
i=1
and its dual (concave maximization) formulation
)
( I Z
I
X
X
ci
ϕi = 0 .
ϕi (xi )µi (dxi ) :
sup
i=1
Xi
(5)
i=1
Theorem 1 (ϕi , γi , ν) is an equilibrium if and only if:
• ν solves (4),
• the transfers (ϕ1 , . . . ϕI ) solve (5),
• for i = 1, . . . , I, γi solves the Monge-Kantorovich problem
Wci (µi , ν).
This implies existence of equlibria.
Matching for teams/6
Matching for teams
15
These problems can be reformulated as (large) linear
programming problems. In an euclidean setting with quadratic
costs ci (xi , z) = λi |xi − z|2 , one finds equilibria as a solution of
the Wasserstein barycenter problem (which also finds
applications in image processing...)
inf
ν∈P2 (Rd )
I
X
λi W22 (µi , ν).
i=1
Under mild assumptions, existence, uniqueness and Lp
estimates, formally related to a system of Monge-Ampère
equations (C., Agueh).
Matching for teams/7
Matching for teams
16
Can be computed efficiently (C., Oberman, Oudet).
Wasserstein isobarycenter of 5 Gaussians:
Matching for teams/8
Cournot-Nash Equilibria
17
Cournot-Nash Equilibria
Continuum of agents, each of them has to choose a strategy,
individual cost depends on the distribution of strategies played
by the whole population. Typical example: doctors’ location
choice. Typically, two competing terms: congestion (driving
force for dispersion) and positive externalities (driving force for
concentration).
Cournot-Nash Equilibria/1
Cournot-Nash Equilibria
18
Setting: type space X (metric compact) endowed with a
probability measure µ ∈ P(X), action space Y (metric
compact). Cost: C(x, y, ν) where ν ∈ P(Y ) represents the
distribution of actions (anonymous game). Unknown:
γ ∈ P(X × Y ): γ(A × B) is the probability that an agent has
her type in A and takes an action in B. Then define
Definition 1 A Cournot-Nash equilibrium (CNE) is a
γ ∈ P(X × Y ) such that ΠX # γ = µ and
γ {(x, y) : C(x, y, ν) = min C(x, z, ν)} = 1
z∈Y
where ν := ΠY # γ.
Cournot-Nash Equilibria/2
Cournot-Nash Equilibria
19
Theorem 2 (Mas-Colell, 1984) In the regular case where
ν 7→ C(., ., ν) is continuous from (P(Y ), w − ∗) to C(X × Y )
then there exists CNE.
The assumption is extremely strong: rules out
congestion/purely local effects, what about uniqueness,
characterization, explicit or numerically computable solutions?
We shall restict ourselves to the separable case:
C(x, y, ν) = c(x, y) + V [ν](y)
(6)
and shall further impose that ν ∈ L1 (m0 ) with m0 a given
reference measure on Y (congestion). Can be viewed as a
simplified (static) version of the Mean-Field Games Theory of
Lasry and Lions.
Cournot-Nash Equilibria/3
Cournot-Nash Equilibria
20
Benchmark: ν ∈ P(Y ) ∩ L1 (m0 ) (m0 : fixed reference measure
according to which congestion is measured)
Z
V [ν](y) = f (y, ν(y)) +
φ(y, z1 , · · · , zm )dν ⊗m (z1 , · · · , zm ).
Y
Due to the first term, the previous fixed-point argument does
not work.
Domain
D := {ν ∈ L1 (m0 ) :
Z
|V [ν]|dν < +∞}.
Y
Cournot-Nash Equilibria/4
Cournot-Nash Equilibria
21
Connections with optimal transport
Again m0 ∈ P(Y ) fixed reference measure, D domain of the
cost, CNE are then defined by
Definition 2 γ ∈ P(X × Y ) is a Cournot-Nash equilibria if
and only if its first marginal is µ, its second marginal, ν,
belongs to D and there exists ϕ ∈ C(X) such that
c(x, y)+V [ν](y) ≥ ϕ(x) ∀x ∈ X and m0 -a.e. y with equality γ-a.e.
(7)
A Cournot-Nash equilibrium γ is called pure whenever it is of
the form γ = (id, T )# µ for some Borel map T : X → Y .
Cournot-Nash Equilibria/5
Cournot-Nash Equilibria
22
For ν ∈ P(Y ), let Π(µ, ν) denote the set of probability measures
on X × Y having µ and ν as marginals and let Wc (µ, ν) be the
least cost of transporting µ to ν for the cost c i.e. the value of
the Monge-Kantorovich optimal transport problem:
ZZ
Wc (µ, ν) := inf
c(x, y) dγ(x, y)
γ∈Π(µ,ν)
X×Y
let us also denote by Πo (µ, ν) the set of optimal transport plans
i.e.
ZZ
Πo (µ, ν) := {γ ∈ Π(µ, ν) :
c(x, y) dγ(x, y) = Wc (µ, ν)}.
X×Y
Cournot-Nash Equilibria/6
Cournot-Nash Equilibria
23
A first link between Cournot-Nash equilibria and optimal
transport is based on the following straightforward observation.
Lemma 1 If γ is a Cournot-Nash equilibrium and ν denotes its
second marginal then γ ∈ Πo (µ, ν).
Proof. Indeed, let ϕ ∈ C(X) be such that (7) holds and let
η ∈ Π(µ, ν) then we have
ZZ
ZZ
c(x, y) dη(x, y) ≥
(ϕ(x) − V [ν](y)) dη(x, y)
X×Y
X×Y
Z
Z
ZZ
=
ϕ(x) dµ(x) −
V [ν](y) dν(y) =
c(x, y) dγ(x, y)
X
Y
X×Y
so that γ ∈ Πo (µ, ν).
Cournot-Nash Equilibria/7
Cournot-Nash Equilibria
24
Similar to what happens in mean-field games: monotonicity
implies uniqueness (covers the case of pure congestion):
Theorem 3 If ν 7→ V [ν] is strictly monotone in the sense that
for every ν1 and ν2 in P(Y ), one has
Z
(V [ν1 ] − V [ν2 ])d(ν1 − ν2 ) ≥ 0
Y
and the inequality is strict whenever ν1 6= ν2 then all equilibria
have the same second marginal ν.
Cournot-Nash Equilibria/8
Cournot-Nash Equilibria
Proof.
25
Let (ν1 , γ1 , ϕ1 ), (ν2 , γ2 , ϕ2 ) be such that
V [νi ](y) ≥ ϕi (x) − c(x, y), i = 1, 2,
for every x and m0 -a.e. y with an equality γi -a.e., using the fact
that γi ∈ Π(µ, νi ), we get
Z
Z
Z
V [νi ]dνi =
ϕi dµ −
cdγi , i = 1, 2
Y
Z
X
Z
X×Y
Z
V [νi ]dνj ≥
ϕi dµ −
cdγj , for i 6= j
Y
X
X×Y
R
R
substracting, we get Y V [ν1 ]d(ν1 − ν2 ) ≤ X×Y cd(γ2 − γ1 ) and
R
R
V [ν2 ]d(ν2 − ν1 ) ≤ X×Y cd(γ1 − γ2 ) and monotonicity thus
Y
gives ν1 = ν2 .
Cournot-Nash Equilibria/9
A variational approach
26
A variational approach
R
Take V [ν](y) = f (y, ν(y)) + Y φ(y, z)dν(z) with f (y, .)
continuous nondecreasing (+ power or logarithm growth) and φ
continuous and symmetric i.e. φ(y, z) = φ(z, y). Then define
Rν
F (y, ν) := 0 f (y, s)ds and
Z
ZZ
1
E[ν] =
F (y, ν(y))dm0 (y) +
φ(y, z) dν(y) dν(z)
2
Y
Y ×Y
δE
δν
in the sense that for every (ρ, ν) ∈ D 2 , one has
Z
E[(1 − ε)ν + ερ] − E[ν]
lim
=
V [ν] d(ρ − ν).
ε
ε→0+
Y
then V [ν] =
Cournot-Nash Equilibria/10
A variational approach
27
Equilibria may be obtained by solving
inf Jµ [ν]
ν∈D
where Jµ [ν] := Wc (µ, ν) + E[ν].
(8)
Theorem 4 (Minimizers are equilibria) Assume that
X = Ω where Ω is some open bounded connected subset of Rd
with negligible boundary, that µ is equivalent to the Lebesgue
measure on X (that is both measures have the same negligible
sets) and that for every y ∈ Y , c(., y) is differentiable with ∇x c
bounded on X × Y . If ν solves (8) and γ ∈ Πo (µ, ν) then γ is a
Cournot-Nash equilibrium. In particular there exist CNE.
Optimality condition for (8) :

 ϕc + V [ν] ≥ 0
 ϕc (y) + ϕ(x) ≤ c(x, y)
with an equality ν-a.e.
with an equality γ-a.e. ,
(9)
Cournot-Nash Equilibria/11
A variational approach
28
If E is convex: equivalence between minimization and being an
equilibrium. If E strictly convex: uniqueness (of ν). The
congestion term is convex and forces dispersion whereas the
interaction term is nonconvex and rather fosters concentration.
It may be the case that the congestion term dominates so as to
make E convex but this is more the exception than the rule.
There is hidden convexity (as in McCann’s displacement
convexity) in the problem as we shall see now.
Cournot-Nash Equilibria/12
Hidden convexity: dimension one
29
Hidden convexity: dimension one
Intuition is easy to understand in dimension one: the functional
Jµ is not convex with respect to ν but it is with respect to T ,
the optimal transport map from µ to ν. Let us take
X = Y = [0, 1], m0 is the Lebesgue measure on [0, 1], µ is
absolutely continuous with respect to the Lebesgue measure,
and assume that V [ν] takes the form:
Z
V [ν](y) = f (ν(y)) + V (y) +
φ(y, z) dν(z)
[0,1]
the corresponding energy reads
Z 1
Z 1
Z
1
E(ν) :=
F (ν(y)) dy +
V (y) dν(y) +
φ dν ⊗2
2 [0,1]2
0
0
(with F ′ = f ).
Hidden convexity: dimension one/1
Hidden convexity: dimension one
30
Assume
• the transport cost c is of the form c(x, y) = C(x − y) where
C is strictly convex and differentiable,
• f is convex increasing (+growth condition),
• V is convex on [0, 1] and φ is convex, symmetric,
differentiable and has a locally Lipschitz gradient.
Hidden convexity: dimension one/2
Hidden convexity: dimension one
31
Let (ρ, ν) ∈ P([0, 1])2 then there is a unique optimal transport
map T0 (respectively T1 ) from µ to ν (respectively from µ to ρ)
for the cost c and it is nondecreasing. For t ∈ [0, 1], let us define:
νt := Tt # µ where Tt := ((1 − t)T0 + tT1 )
then the curve t 7→ νt connects ν0 = ν to ν1 = ρ. J :
P(Y ) → R ∪ {+∞} is called displacement convex if, for all
t ∈ [0, 1], J(νt ) ≤ (1 − t)J(ν) + tJ(ρ) (for every choice of
endpoints ν and ρ), it is called strictly displacement convex
when, in addition J(νt ) < (1 − t)J(ν) + tJ(ρ) when t ∈ (0, 1)
and ρ 6= µ.
We claim that Jµ is strictly displacement convex; take (ν, ρ) two
probability measures in the domain of E (which is convex by
convexity of F ), define νt as above.
Hidden convexity: dimension one/3
Hidden convexity: dimension one
32
By definition of Wc , νt and the strict convexity of C we have
Z 1
Wc (µ, νt ) ≤
C(x − ((1 − t)T0 (x) + tT1 (x))) dµ
0
≤ (1 − t)
Z
1
C(x − T0 (x)) dµ + t
0
= (1 − t)Wc (µ, ν) + tWc (µ, ρ)
Z
1
C(x − T1 (x))dµ
0
with a strict inequality if t ∈ (0, 1) and ν 6= ρ.
Hidden convexity: dimension one/4
Hidden convexity: dimension one
33
By construction
Z 1
Z 1
Z
V dνt =
V (Tt (x))dµ(x) =
0
0
1
V ((1−t)T0 (x)+tT1 (x))dµ(x)
0
which is convex with respect to t, by convexity of V . Similarly
Z
Z
φ(Tt (x), Tt (y)) dµ(x) dµ(y)
φ dνt⊗2 =
[0,1]2
[0,1]2
is convex with respect to t, by convexity of φ,
Hidden convexity: dimension one/5
Hidden convexity: dimension one
34
The convexity of the remaining congestion term is more
involved. Since νt = Tt # µ and Tt is nondecreasing, at least
formally we have νt (Tt (x))Tt′ (x) = µ(x), by the change of
variables formula we also have
Z 1
Z 1
Z 1 µ(x)
′
T
(x)
F (νt (y)) dy =
F (νt (Tt (x)))Tt′ (x) dx =
F
t
′ (x)
T
0
0
0
t
and we conclude by observing that α 7→ F (µ(x)α−1 )α is convex
and that Tt′ (x) is linear in t.
Hidden convexity: dimension one/6
Hidden convexity: dimension one
35
All this yields:
Theorem 5 Under the assumptions above, optima and
equilibria coincide and there exists a unique equilibrium (which
is actually pure).
This can be generalized to higher dimensions when the cost is
quadratic and the congestion term satisfies McCann’s condition
(power functions or logarithms are fine).
Hidden convexity: dimension one/7
A PDE for the equilibrium (quadratic cost)
36
A PDE for the equilibrium (quadratic cost)
For computational simplicity, take V = 0 and f (ν) = log(ν)
(satisfies McCann’s condition and ensures that the mass
remains positive everywhere). Optimality condition:
Z
φ(y, .)dν ⊗m = 0
(10)
log(ν(y)) + ϕc (y) +
Ym
Optimal transport map (Brenier) T = ∇u between µ and ν:
Monge-Ampère equation
µ(x) = det(D 2 u(x)) ν(∇u(x)),
∀x ∈ Ω
which has to be supplemented with the natural sort of
boundary condition
∇u(Ω) = Ω.
(11)
(12)
A PDE for the equilibrium/1
A PDE for the equilibrium (quadratic cost)
37
On the other hand ϕ(x) = 21 |x|2 − u(x), ϕc (y) = 12 |y|2 − u∗ (y) so
1
1
|∇u|2 − u∗ (∇u) = |∇u|2 − x · ∇u + u
2
2
substituting y = ∇u(x) in (10), using
Z
Z
φ(∇u(x), .) dν ⊗m =
φ(∇u(x), ∇u(x1 ), . . . , ∇u(xm )) dµ⊗m .
ϕc (∇u) =
Ym
Ωm
and eliminating ν thanks to (11), we get
1
µ(x) = det(D 2 u(x)) exp − |∇u(x)|2 + x · ∇u(x) − u(x) ×
2
Z
φ(∇u(x), ∇u(x1 ), . . . , ∇u(xm )) dµ⊗m (x1 , . . . xm ) .
exp −
Ωm
(13)
The equilibrium problem is therefore equivalent to a non-local
and nonlinear partial differential equation.
A PDE for the equilibrium/2
A PDE for the equilibrium (quadratic cost)
38
The problem can be solved numerically in dimension 1.
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Convergence and stabilisation toward the equilibrium in the case
of a logarithmic congestion, cubic interaction, and a potential
V (x) := (x − 5)3 with uniform measure on [0, 1] as initial guess
A PDE for the equilibrium/3