Nash Equilibria Strategies and Equivalent Single-Objective Optimization Problems. The Case of Linear Partial Differential Equations

In this paper we study the existence and uniqueness of Nash equilibria (solution to competition-wise problems, with several controls trying to reach possibly different goals) associated to linear partial differential equations and show that, in some cases, they are also the solution of suitable single-objective optimization problems (i.e. cooperative-wise problems, where all the controls cooperate to reach a common goal). We use cost functions associated with a particular linear parabolic partial differential equation and distributed controls, but the results are also valid for more general linear differential equations (including elliptic and hyperbolic cases) and controls (e.g. boundary controls, initial value controls,...).


Introduction
Nash equilibria are solutions of a noncooperative multiobjective optimization strategy first proposed by Nash (see [1]).Since it originated in game theory and economics, the notion of player is often used.For an optimization problem with N objectives or functionals J i to minimize, a Nash strategy consists in having N players or controls v i , each optimizing his own criterion.However, each player has to optimize his criterion given that all the other criteria are fixed by the rest of the players.When no player can further improve his criterion, it means that the system has reached a Nash equilibrium state.
To the best of our knowledge, [5] and [6] are the first articles dealing with the theoretical and numerical study of Nash equilibria for differential games associated to partial differential equations.Following [5] we deal here with a general linear case with N cost functions and controllers and show how, in some cases, the Nash equilibria (solution to differential games associated to multiobjective optimization problems with several noncooperative controllers), are also the solution of single-objective optimization problems (where all the controllers cooperate to reach a common goal).We use cost functions associated with linear parabolic partial differential equations and distributed controls, but other kinds of linear differential equations (e.g.elliptic, hyperbolic,..) and controls (e.g.boundary controls, initial value controls,...) can be also used, using the same technique.
The fact that a noncooperative game (i.e a competition-wise problem) can be seen as a (cooperative) single-objective optimization problem (i.e. a noncompetitionwise problem) is very interesting, not only because of the curious noncooperativecooperative equivalence, but also because of the huge amount of software to compute solutions and literature written about the latter kind of problems, that could be used in the framework of, apparently, a different type of problems.
In Section 2 we formulate the problem and give an optimality system providing a necessary and sufficient condition for the Nash equilibria.The existence and uniqueness of Nash equlibria is studied in Section 3. In Section 4 we show the equivalence, in some cases, between the noncooperative multiobjective differential games defining the Nash equlibria and suitable (cooperative) singleobjective optimization problems.Finally, in Section 5 give a summary of the major results of the paper.

Formulation of the Problem
Let us consider T > 0, Ω ⊂ R d a bounded and smooth open set with d ∈ {1, 2, 3}, and two subsets Γ 1 , Γ 2 ⊂ ∂Ω such that ∂Ω = Γ 1 ∪ Γ 2 .We define Finally, we consider the functionals J i : U → R, with i ∈ {1, ..., N }, given by with f, g 1 , g 2 , y 0 , y i,d and y i,T being smooth enough functions and χ ω : Ω → R the characteristic function (with values 1 in ω and 0 in Ω \ ω) for any ω ⊂ Ω.This generalizes the typical examples in the literature of 2 controls (instead of N ), This case is a competitionwise problem, with each control (or player) trying to reach (possibly) different goals over a common domain.In some sense this is the case where the behavior of the solution y associated to a Nash equilibrium is most difficult to forecast.
Remark 1 Most of the results to follow are also valid for more general linear operators such as, for instance, The technique is also valid for different type of controls such as, for instance, boundary or initial controls.
Now, given i ∈ {1, ..., N }, for every (w 1 , ..., ) is a solution of the coupled (optimality) system: In the linear case studied here, this system of equations is a necessary and sufficient condition for u to be a Nash equilibrium.In general this system is only a necessary condition, although in some nonlinear cases (see, e.g.[7]), the functionals are convex and system (2) is also a sufficient condition.

Existence and uniqueness of solution of Nash Equlibria
It is obvious that is an affine mapping of U. Therefore, there exist a linear continuous mapping A ∈ L(U, U) and a vector b ∈ U such that Let us identify mapping A: For every v = (v 1 , ..., v N ) ∈ U, the linear part of the affine mapping in relation ( 3) is defined by where and y = y(v) is the solution of Proposition 1 Mapping A : U → U is linear and continuous.Furthermore, if min i∈{1,...,N } {α i } is sufficiently large, it is also U-elliptic, i.e., there existe C > 0 where (•, •) and || • || represent the canonical scalar product and norm of the Hilbert space U, respectively.
Proof: It is obvious that A is a linear mapping and it is easy to show that it is continuous (see [8]).
Let us consider v = (v 1 , ..., v N ) ∈ U and w = (w 1 , ..., w N ) ∈ U. We have then (Av, w) = (α Let us focus on the term ωi×(0,T ) p i (v)w i dxdt, following the approach in [5] and [9].We have ωi×(0,T ) Since the mapping v → y(v) is linear and continuous from U to C([0, T ]; L 2 (Ω)) (see, e.g., [8]), it is easy to prove there exist a constant c > 0 such that proves that A is U-elliptic in that case and completes the proof.
Let us identify b: The constant part of the affine mapping (3) is the function b ∈ U defined by b = (p 1 χ ω1 , ..., p N χ ωN ), where and y is the solution of that, for any v ∈ U, y(v) = y(v) + y and p i (v) = p i (v) + p i .
Theorem 1 If min i∈{1,...,N } {α i } is sufficiently large, there exist a unique Nash equilibrium of the problem defined in Section 2.
Proof: As showed above, the Nash equilibria are characterized by the solutions of ( 2), which are also characterized by the solutions u ∈ U of where a a(•, and Proposition 1 proves that mapping a(•, •) is bilinear, continuous and, if min is sufficiently large, it is also U-elliptic.Furthermore, mapping L is (obviously) linear and continuous.Thus, by the (well-known) Lax-Milgram Theorem, system (2) has a unique solution or, equivalently, there exists a unique Nash equilibrium of the problem defined in Section 2, if min i∈{1,...,N } {α i } is sufficiently large.
The discretization of the problem considered above and the development of suitable algorithms to get a numerical solution approximating the Nash equilibra can follow the approaches in [5] and [9].

Equivalent single-objective control problems
In this section we will show that, in some cases, the solution of noncooperative differential games defining Nash equilibria, are the solution of suitable optimization problems, where all the controls cooperate to minimize a suitable single-objective cost function.
Let us consider the subfamily of problems defined in Section 2, for which ρ i = ρ and η i = η, for all i ∈ {1, ..., N }.Therefore, in this case the functional J i , with i ∈ {1, ..., N }, is given by with y = y(v) being the solution of (1).As in the general case studied in Section 2, a Nash equilibrium is a N -tuple (u 1 , ..., u N ) ∈ U = U 1 × • • • × U N solution of (2), where Therefore, system (2) is equivalent in this case to and now Av = (α 1 v 1 + pχ ω1 , ..., α N v N + pχ ωN ), and y is the solution of (4).
Theorem 2 There exist a unique Nash equilibrium of the problem defined in Section 4.

Proof:
The proof follows the one of Theorem 1, taking into account that in this case A is unconditionally U-elliptic.
The discretization of the problem considered above and the development of suitable algorithms to get a numerical solution approximating the Nash equilibra are given in [5], where numerical examples are also showed.
Theorem 3 The (unique) Nash equilibrium u = (u 1 , ..., u N ) ∈ U of the problem defined in Section 4 is the (unique) solution of the following optimal control problems:

Conclusions
This paper studies Nash equilibria of noncooperative differential games with several players (controllers), each one trying to minimize his own cost function defined in terms of a general class of linear partial differential equations.We give results of existence and uniqueness of Nash equilibria and show how, in some cases, the corresponding Nash equilibria (solution to competition-wise problems, with each control trying to reach possibly different goals), are also the solution of suitable single-objective optimization problems (i.e.cooperative-wise problems, where all the controls cooperate to reach a common goal).A natural question arises: Are there Nash equilibria associated to nonlinear problems than can be also characterized as the solutions of single-objective problems?This is an open problem for interested researchers.