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EQUILIBRE DE NASH & TRANSPORT OPTIMAL
Adrien Blanchet – TSE (GREMAQ, Universite Toulouse 1 Capitole)
Modelisation with optimal transport (ANR TOMMI)
In collaboration with P. Mossay & F. Santambrogio and G. Carlier
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 1 / 22
INTRODUCTION
PLAN
1 INTRODUCTION
2 THE MODELS
Model I: one type of agentModel II: agent with types
3 MAIN RESULTS
4 CONNEXION WITH OPTIMAL TRANSPORT
5 IDEA OF THE PROOF
6 DISCUSSIONS
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 2 / 22
INTRODUCTION
NON-COOPERATIVE GAMES
GAME THEORY
The players choose actions in a given set. The payoff of the agent i depends on her action ai andthe actions of all the other players a−i . We denote the payoff Π(ai , a−i).A player can also play in mixed strategy, i.e. to play a strategy ej with a probability xj . This mixedstrategy is thus given by a vector (x1, · · · , xN).If the strategy y of Player 2 is known we say that Player 1 is in best reply against y if her action x∗
is such thatx∗ = ArgmaxxΠ(x, y) .
A pair (x, y) is a Nash equilibirum if each agent is in best reply against the other player’s action(i.e. all the agents have no incentive to relocate).
NASH (1950)
“The theory of non-cooperative games is based on the absence of coalitions in that it is assumed
that each participant acts independently, without collaboration and communication from any of the
others. ”
→ Existence of equilibria in a non-cooperative n-persons game (n ∈ N).
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 3 / 22
INTRODUCTION
CONTINUUM OF PLAYERS
VON NEUMANN-MORGENSTERN (1944)
“An almost exact theory of a gas, containing about 1025 freely moving particles, is incomparably
easier than that of the solar system, made up of 9 major bodies.”
“It is a well known phenomenon in many branches of the exact and physical sciences that very
great numbers are often easier to handle than those of medium size. This is of course due to the
excellent possibility of applying the laws of statistics and probabilities in the first case.”
“When the number of participants becomes really great, some hope emerges that the influence of
every particular participant will become negligible, and that the above difficulties may recede and a
more conventional theory become possible.”
SCHMEIDLER (1973)
“Non-atomic games enable us to analyze a conflict situation where the single player has noinfluence on the situation.”
→ Existence of an equilibria in a non-atomic game with an arbitrary finite number of purestrategies. See also [Mas-Colell, 1984].
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 4 / 22
THE MODELS
PLAN
1 INTRODUCTION
2 THE MODELS
Model I: one type of agentModel II: agent with types
3 MAIN RESULTS
4 CONNEXION WITH OPTIMAL TRANSPORT
5 IDEA OF THE PROOF
6 DISCUSSIONS
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 5 / 22
THE MODELSMODEL I: ONE TYPE OF AGENT
A MODEL WITH ONE TYPE OF AGENT
Consider a non-cooperative anonymous game with a continuum of agents (= “mean field game” inPierre-Louis Lions’ terminology).
COST FUNCTION
The agent has to take action in a compact metric action space Y . Given an action distributionν ∈ P(Y ) the agent taking action y incurs the cost
Π(y , ν) := V [ν](y) .
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 6 / 22
THE MODELSMODEL I: ONE TYPE OF AGENT
EXTERNALITIES
RIVALRY/CONGESTION
The utility of the agent decreases when the number of players who choose the same actionincreases.
Examples:
Consumption of the same public good (motorway game),
Food supply in an habitat decreases with the number of its users (ex. Sticklebacks (Milinsky)).
INTERACTIONS
The utility of the agents increases because some other agents play a similar action.
Examples:
Location to go shopping,
Quality of a product in a differentiated industry.
EXTERNALITIES IN [BECKMANN, 1976]’S MODEL
Congestion: the agents benefit from social interactions but there is a cost to access to distantagents,
Interaction: more populated areas lead to higher competition for land.
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 7 / 22
THE MODELSMODEL I: ONE TYPE OF AGENT
AN URBAN REGION MODEL
Let K be a convex domain of Rd and ν the density of agents. We assume that ν is a probabilitydensity.
INDIRECT COST FUNCTIONAL
Consider
V [ν](y) := f [ν(y)]︸ ︷︷ ︸
congestion
+
∫
K
φ(|y − z|)ν(z) dz
︸ ︷︷ ︸
interaction
+ A(y)︸ ︷︷ ︸
amenities
.
where
f is the competition for land. We assume that f is an increasing function.
φ is the travelling cost. We assume that φ is a non-negative and radially symmetriccontinuous function.
A is an external potential. We assume that A is a continuous function bounded from below.
NASH EQUILIBRIUM
The probability ν ∈ P(Y ) is a Nash equilibrium if:
{V [ν](y) = V ν-a.e. y ,
V [ν](y) ≥ V a.e. y ∈ Y .
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 8 / 22
THE MODELSMODEL II: AGENT WITH TYPES
GENERALISATION
Consider now that the agents have a given type x in a compact metric space X . Given an actiondistribution ν ∈ P(Y ), the type-x agent taking action y incurs the cost
Π(x, y , ν) .
Assume
COST IN A SEPARABLE FORM
Π(x, y , ν) := c(x, y) + V [ν](y) .
NASH EQUILIBRIUM
The probability γ ∈ P(X × Y ) is a Nash equilibrium if:
its first marginal is µ ,
its second marginal ν is such that there exists a function ϕ such that
{Π(x, y , ν) = ϕ(x) γ-a.e. (x, y),
Π(x, y , ν) ≥ ϕ(x) a.e. (x, y) ∈ X × Y .
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 9 / 22
MAIN RESULTS
PLAN
1 INTRODUCTION
2 THE MODELS
Model I: one type of agentModel II: agent with types
3 MAIN RESULTS
4 CONNEXION WITH OPTIMAL TRANSPORT
5 IDEA OF THE PROOF
6 DISCUSSIONS
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 10 / 22
MAIN RESULTS
MAIN RESULTS
EXISTENCE AND UNIQUENESS [B., MOSSAY & SANTAMBROGIO, 2012] AND [B. & CARLIER, 2012]
There exists a unique Nash equilibrium.
Our results apply to
POTENTIAL GAMES (SEE [MONDERER-SHAPLEY, 1996] FOR A FINITE NUMBER OF PLAYERS)
There exists a functional E such that V [ν] is the first variation of E i.e.
V [ν] =δE
δν.
Under the assumptions:
E displacement convex and coercive. Ex.: φ convex symmetric and the congestion functionsatisfies the Inada condition.
c satisfies a generalised Spence-Mirrlees condition i.e. for every x , y 7→ ∇x c(x, y) isinjective. Ex. c smooth and strictly convex.
For sake of simplicity, we assume from now on
c(x, y) =|x − y |2
2.
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 11 / 22
CONNEXION WITH OPTIMAL TRANSPORT
PLAN
1 INTRODUCTION
2 THE MODELS
Model I: one type of agentModel II: agent with types
3 MAIN RESULTS
4 CONNEXION WITH OPTIMAL TRANSPORT
5 IDEA OF THE PROOF
6 DISCUSSIONS
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 12 / 22
CONNEXION WITH OPTIMAL TRANSPORT
OPTIMAL TRANSPORT: THE MONGE-KANTOROVICH DISTANCE
CONNECTION WITH OPTIMAL TRANSPORT
Let γ ∈ P(X × Y ) be a Nash equilibrium of second marginal ν. Then γ is a solution to theKantorovich problem, i.e. γ is a solution to
minΠX γ=µ,ΠY γ=ν
∫∫
X×Y
c(x, y) dγ(x, y) =: Wc(µ, ν)
Proof: Let η be of first marginal µ and second marginal ν then we have
∫∫
X×Y
c(x, y) dη(x, y) ≥
∫∫
X×Y
(ϕ(x) − V [ν](y)) dη(x, y)
=
∫
X
ϕ(x) dµ(x) −
∫
Y
V [ν](y) dν(y) =
∫∫
X×Y
c(x, y) dγ(x, y) .
PURITY OF THE EQUILIBRIUM
If µ does not give weight to points then any Nash equilibrium is pure.
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 13 / 22
IDEA OF THE PROOF
PLAN
1 INTRODUCTION
2 THE MODELS
Model I: one type of agentModel II: agent with types
3 MAIN RESULTS
4 CONNEXION WITH OPTIMAL TRANSPORT
5 IDEA OF THE PROOF
6 DISCUSSIONS
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 14 / 22
IDEA OF THE PROOF
MAIN IDEA
VARIATIONAL PROBLEM
infν∈P(Y )
{1
2W2
2 (µ, ν) + E[ν]
}
(1)
where
E[ν] =
∫
K
F (ν(x)) dx +
∫
K
A(x) dν +1
2
∫∫
K2φ(|x − y |)ν(x)ν(y) dx dy .
andwhere F is an antiderivative of f and the Monge-Kantorovich distance is defined by
W22(µ, ν) := min
ΠX γ=µ,ΠY γ=ν
∫∫
X×Y
|x − y |2
2dγ(x, y)
EQUIVALENCE BETWEEN EQUILIBRIUM AND MINIMISER
γ ∈ P(X × Y ) is a Nash equilibrium if and only if
ν is a minimiser of (1),
γ is a solution to the Kantorovich problem.
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 15 / 22
IDEA OF THE PROOF
OPTIMAL TRANSPORT: PURITY OF THE EQUILIBRIUM
Let µ ∈ P(X) and ν ∈ P(Y ). A measurable function T : X → Y pushes-forward µ onto ν, andwe denote T#µ = ν, if
∀ζ ∈ C0b (Y ),
∫
X
ζ [T (x)] dµ(x) =
∫
Y
ζ(y) dν(y) .
BRENIER’S THEOREM (CPAM, 1991)
There exists a unique optimal transport map T solution to the Kantorovich problem. Moreover it isa solution to the Monge problem
infT :T#µ=ν
∫
X
|x − T (x)|2 dµ(x) = W22(µ, ν) .
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 16 / 22
IDEA OF THE PROOF
OPTIMAL TRANSPORT: UNIQUENESS
DISPLACEMENT INTERPOLATION, SEE MCCANN (ADV. MATH., 1997)
Let T be the optimal transport map which transports ρ0 dx onto ρ1 dy . The displacement
interpolation between ρ0 and ρ1 is
ρt = [(1 − t)id + tT ] #ρ0 .
DISPLACEMENT CONVEXITY
A functional G is displacement convex if for all ρ0 ∈ P(X), ρ1 ∈ P(Y )
G[ρt ] ≤ (1 − t)G[ρ0] + tG[ρ1].
CRITERIA OF DISPLACEMENT CONVEXITY, SEE MCCANN (ADV. MATH., 1997)
Assume that
F (0) = 0 and r 7→ rd F (r−d ) is convex non-increasing,
A is convex then V [ν] is displacement convex.
φ is convex then W[ν] is displacement convex.
then E is displacement convex.
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 17 / 22
DISCUSSIONS
PLAN
1 INTRODUCTION
2 THE MODELS
Model I: one type of agentModel II: agent with types
3 MAIN RESULTS
4 CONNEXION WITH OPTIMAL TRANSPORT
5 IDEA OF THE PROOF
6 DISCUSSIONS
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 18 / 22
DISCUSSIONS
COMPUTATION OF THE EQUILIBRIUM
A PARTIAL DIFFERENTIAL EQUATION FOR THE EQUILIBRIUM
Let u be a solution to the following Monge-Ampere equation
µ(x) = det(D2u(x)) exp
(
−|∇u(x)|2
2+ x · ∇u(x) − u(x) −
∫
Y
φ(∇u(y),∇u(z)) dµ(z)
)
then ϕ(x) = u(x) + |x|2/2 is the optimal transport which transport µ onto ν so that ν = ϕ#µ.
0 2 4 6 8 10 12 14 160.0
0.5
1.0
1.5
2.0
2.5
3.0
FIGURE: The distribution µ in dash line and ν in the case f (x) = x8 , φ(z) = 10−4|z|2 and A = (x − 10)4 .
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 19 / 22
DISCUSSIONS
WELFARE ANALYSIS
SOCIAL WELFARE
∫∫
X×Y
Π(x, y , ν) dγ =
∫∫
X×Y
|x − y |2
2dγ +
∫
Y
[
f [ν(y)] +
∫
Y
φ(y , z) dν(z) + A(y)
]
dν(y).
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.50.0
0.1
0.2
0.3
0.4
0.5
0.6
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.50
100
200
300
400
500
600
FIGURE: Left: the optimum (red) and the equilibrium (blue). Right: tax at the equilibrium. Cost of anarchy ∼ 1.8.
TAX TO RESTORE EFFICIENCY
Tax[ν](y) = ν(y)f [ν(y)] − F [ν(y)] +1
2
∫
Y
φ(y , z) dν(z) .
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 20 / 22
DISCUSSIONS
DYNAMICAL PERSPECTIVE
The agents start with a distribution of strategies and adjust it over time by choosing
MINIMISING SCHEME
νk+1 ∈ argminν
{1
2τW2
2 (νk , ν) + E[ν]
}
.
This scheme converges in some sense to the
CONTINUOUS EVOLUTION EQUATION
∂ν
∂t+ div (−ν∇V [ν]) = 0,
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0
0.5
1.0
1.5
2.0
2.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
FIGURE: Convergence and stabilisation toward the equilibrium in the case of a logarithmic congestion, cubic
interaction, and a potential A(x) := (x − 5)3 with 1l[0,1] as initial guess (left) and made of two bumps (right).
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 21 / 22
DISCUSSIONS
Merci pour votre attention
EQUILIBRE DE NASH & TRANSPORT OPTIMAL LJK GRENOBLE – 3-4 OCTOBRE 2013 22 / 22