E QUILIBRE DE N ASH & TRANSPORT OPTIMAL Adrien Blanchet – TSE (GREMAQ, Universit´ e Toulouse 1 Capitole) Modelisation with optimal transport (ANR TOMMI) In collaboration with P . Mossay & F. Santambrogio and G. Carlier E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 1 / 22
I NTRODUCTION P LAN 1 I NTRODUCTION 2 T HE MODELS Model I: one type of agent Model II: agent with types 3 M AIN RESULTS 4 C ONNEXION WITH OPTIMAL TRANSPORT 5 I DEA OF THE PROOF 6 D ISCUSSIONS E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 2 / 22
I NTRODUCTION N ON - COOPERATIVE GAMES G AME THEORY The players choose actions in a given set. The payoff of the agent i depends on her action a i and the actions of all the other players a − i . We denote the payoff Π( a i , a − i ) . A player can also play in mixed strategy , i.e. to play a strategy e j with a probability x j . This mixed strategy is thus given by a vector ( x 1 , · · · , x N ) . 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 that x ∗ = Argmax x Π( 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). N ASH (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 ). E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 3 / 22
I NTRODUCTION C ONTINUUM OF PLAYERS V ON N EUMANN -M ORGENSTERN (1944) “ An almost exact theory of a gas, containing about 10 25 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. ” S CHMEIDLER (1973) “ Non-atomic games enable us to analyze a conflict situation where the single player has no influence on the situation. ” → Existence of an equilibria in a non-atomic game with an arbitrary finite number of pure strategies. See also [Mas-Colell, 1984]. E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 4 / 22
T HE MODELS P LAN 1 I NTRODUCTION 2 T HE MODELS Model I: one type of agent Model II: agent with types 3 M AIN RESULTS 4 C ONNEXION WITH OPTIMAL TRANSPORT 5 I DEA OF THE PROOF 6 D ISCUSSIONS E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 5 / 22
T HE MODELS M ODEL 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” in Pierre-Louis Lions’ terminology). C OST FUNCTION The agent has to take action in a compact metric action space Y . Given an action distribution ν ∈ P ( Y ) the agent taking action y incurs the cost Π( y , ν ) := V [ ν ]( y ) . E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 6 / 22
T HE MODELS M ODEL I: ONE TYPE OF AGENT E XTERNALITIES R IVALRY /C ONGESTION The utility of the agent decreases when the number of players who choose the same action increases. Examples: Consumption of the same public good (motorway game), Food supply in an habitat decreases with the number of its users (ex. Sticklebacks (Milinsky)). I NTERACTIONS 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. E XTERNALITIES IN [B ECKMANN , 1976]’ S MODEL Congestion: the agents benefit from social interactions but there is a cost to access to distant agents, Interaction: more populated areas lead to higher competition for land. E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 7 / 22
T HE MODELS M ODEL I: ONE TYPE OF AGENT A N URBAN REGION MODEL Let K be a convex domain of R d and ν the density of agents. We assume that ν is a probability density. I NDIRECT COST FUNCTIONAL Consider � V [ ν ]( y ) := f [ ν ( y )] + φ ( | y − z | ) ν ( z ) d z + A ( y ) . � �� � K � �� � � �� � congestion amenities interaction 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 symmetric continuous function. A is an external potential. We assume that A is a continuous function bounded from below. N ASH 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 . E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 8 / 22
T HE MODELS M ODEL II: AGENT WITH TYPES G ENERALISATION Consider now that the agents have a given type x in a compact metric space X . Given an action distribution ν ∈ P ( Y ) , the type- x agent taking action y incurs the cost Π( x , y , ν ) . Assume C OST IN A SEPARABLE FORM Π( x , y , ν ) := c ( x , y ) + V [ ν ]( y ) . N ASH 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 . E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 9 / 22
M AIN RESULTS P LAN 1 I NTRODUCTION 2 T HE MODELS Model I: one type of agent Model II: agent with types 3 M AIN RESULTS 4 C ONNEXION WITH OPTIMAL TRANSPORT 5 I DEA OF THE PROOF 6 D ISCUSSIONS E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 10 / 22
M AIN RESULTS M AIN RESULTS E XISTENCE AND UNIQUENESS [B., M OSSAY & S ANTAMBROGIO , 2012] AND [B. & C ARLIER , 2012] There exists a unique Nash equilibrium. Our results apply to P OTENTIAL GAMES ( SEE [M ONDERER -S HAPLEY , 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 ] = δ E δν . Under the assumptions: E displacement convex and coercive. Ex.: φ convex symmetric and the congestion function satisfies the Inada condition. c satisfies a generalised Spence-Mirrlees condition i.e. for every x , y �→ ∇ x c ( x , y ) is injective. Ex. c smooth and strictly convex. For sake of simplicity, we assume from now on c ( x , y ) = | x − y | 2 . 2 E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 11 / 22
C ONNEXION WITH OPTIMAL TRANSPORT P LAN 1 I NTRODUCTION 2 T HE MODELS Model I: one type of agent Model II: agent with types 3 M AIN RESULTS 4 C ONNEXION WITH OPTIMAL TRANSPORT 5 I DEA OF THE PROOF 6 D ISCUSSIONS E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 12 / 22
C ONNEXION WITH OPTIMAL TRANSPORT O PTIMAL TRANSPORT : THE M ONGE -K ANTOROVICH DISTANCE C ONNECTION WITH OPTIMAL TRANSPORT Let γ ∈ P ( X × Y ) be a Nash equilibrium of second marginal ν . Then γ is a solution to the Kantorovich problem, i.e. γ is a solution to �� min c ( x , y ) d γ ( x , y ) =: W c ( µ, ν ) Π X γ = µ, Π Y γ = ν X × Y Proof: Let η be of first marginal µ and second marginal ν then we have �� �� c ( x , y ) d η ( x , y ) ≥ ( ϕ ( x ) − V [ ν ]( y )) d η ( x , y ) X × Y X × Y � � �� = ϕ ( x ) d µ ( x ) − V [ ν ]( y ) d ν ( y ) = c ( x , y ) d γ ( x , y ) . X Y X × Y P URITY OF THE EQUILIBRIUM If µ does not give weight to points then any Nash equilibrium is pure. E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 13 / 22
I DEA OF THE PROOF P LAN 1 I NTRODUCTION 2 T HE MODELS Model I: one type of agent Model II: agent with types 3 M AIN RESULTS 4 C ONNEXION WITH OPTIMAL TRANSPORT 5 I DEA OF THE PROOF 6 D ISCUSSIONS E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 14 / 22
I DEA OF THE PROOF M AIN IDEA V ARIATIONAL PROBLEM � 1 � 2 W 2 inf 2 ( µ, ν ) + E [ ν ] (1) ν ∈P ( Y ) where � � �� A ( x ) d ν + 1 E [ ν ] = F ( ν ( x )) d x + K 2 φ ( | x − y | ) ν ( x ) ν ( y ) d x d y . 2 K K andwhere F is an antiderivative of f and the Monge-Kantorovich distance is defined by �� | x − y | 2 W 2 2 ( µ, ν ) := min d γ ( x , y ) 2 Π X γ = µ, Π Y γ = ν X × Y E QUIVALENCE 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. E QUILIBRE DE N ASH & TRANSPORT OPTIMAL LJK G RENOBLE – 3-4 OCTOBRE 2013 15 / 22
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