Price of Anarchy for Mean Field Games

The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics are presented.


Introduction
The concept of the 'price of anarchy' was introduced to quantify the inefficiency of selfish behavior in finite player games [9][10] [13] [16] [17] [18]. In this report, we extend the notion of price of anarchy to mean field games (MFG). Mean field games were introduced by Lasry and Lions [14] and Caines and his collaborators [12] to describe the limiting regime of large symmetric games when the number of players, N , tends to infinity. A mean field game equilibrium characterizes the analogue of a Nash equilibrium in the N = ∞ regime. Thus, as in the finite player case, it is possible that the mean field game equilibrium is inefficient. In fact, in the paper of Balandat and Tomlin [2], they present a numerical example that shows that mean field game equilibria are not efficient, in general. The suboptimality of a mean field game equilibrium is also illustrated numerically for a congestion model in a paper of Achdou and Laurière [1]. More recently Cardaliaguet and Rainer gave in [5] a partial differential equation based thorough analysis of the (in)efficiency of the mean field game equilibria.
In this report, the goal is to define the price of anarchy in the context of mean field games, and to compute it for a class of linear quadratic mean field game models, which can be solved explicitly. In fact, we consider an even more general class of games by allowing for interaction between the players through their controls, in addition to interaction through their states. This is often referred in the literature as extended mean field game, or mean field game of control. We compare the social cost of a mean field game equilibrium to the cost incurred when the players execute a strategy computed centrally. We consider a system of N players whose private states are denoted at time t by X 1 t , X 2 t , · · · , X N t . To keep the presentations simple, we assume the state space is R. We denote by µ N t the empirical distribution of the states, namely: We assume that these states evolve in continuous time under the influences of controls α 1 t , α 2 t , · · · , α N t ∈ A, where the set of admissible controls, A, will be defined later. Let ν N t denote the empirical measure of the controls: We also assume that if and when interactions between these states and controls are present, they are of a mean field type, i.e. through µ N t and ν N t . The time evolution of the state for player i is given by the Itô dynamics: We work over the interval [0, T ] limited by a finite time horizon T ∈ R + . We assume the drift function b : [0, T ] × R × P(R) × A × P(A) ∋ (t, x, µ, α, ν) → R is Lipschitz in each of it's inputs.
For the sake of simplicity, we assume that the volatility, σ, is a positive constant.

Cost Functionals
We assume that we are given two functions f : [0, T ] × R × P(R) × A × P(A) ∋ (t, x, µ, α, ν) → R and g : R × P(R) ∋ (x, µ) → R which we call running and terminal cost functions, respectively. We assume f and g are Lipschitz in each of their arguments. The goal of player i is to minimize their expected cost as given by:

Social Cost
We restrict ourselves to Markovian control strategies α = (α t ) 0≤t≤T given by feedback functions in the form α t = φ(t, X t ) and we let A denote the set of such controls. If the N players use distributed Markovian control strategies of the form α i t = φ(t, X i t ), we define the cost (per player) to the system as the quantity J (N ) φ : We shall compute this social cost in the limit N → ∞ when all the players use the distributed control strategies given by the same feedback function φ identified by solving an optimization problem in the limit N → ∞. We take the social cost to be the limit as N → ∞ of J (N ) φ , namely: if we use the notation < ϕ, ρ > for the integral ϕ(z)ρ(dz) of the function ϕ with respect to the measure ρ. Now if we assume that in the limit N → ∞ the empirical distributions µ N t converge toward a measure µ t , and thus ν N t = 1 N N i=1 δ φ(t,X i t ) also converges toward a measure ν t , then the social cost of the feedback function φ becomes: with the expectation, E, disappearing when the limiting flows µ = (µ t ) 0≤t≤T and ν = (ν t ) 0≤t≤T are deterministic.
We would like to evaluate SC(φ) in the N = ∞ regime directly, without having to construct the deterministic measure flows µ and ν as limits of the finite player empirical measures. To do this, we assume that propagation of chaos holds and that the states of the N players become asymptotically independent in the limit as N → ∞. We consider a representative agent whose state is given by X φ = (X φ t ) 0≤t≤T , the continuous time solution of the stochastic differential equation of McKean-Vlasov type: controlled by φ. Then we can identify µ as the law of a representative agent using the feedback function φ, i.e. µ t = L(X φ t ), and similarly, we can identify ν as the law of the control, such that ν t = L(φ(t, X φ t )). Thus, in the N = ∞ regime, we rewrite the social cost as: where X φ satisfies equation (1). For the remainder of the paper, we work in the N = ∞ regime.
As mentioned earlier, φ should be identified by solving an optimal control problem. We consider two distinct problems: • φ is a feedback function providing a mean field game equilibrium. We detail more precisely what is meant by φ providing a mean field game equilibrium in section 1.1. • φ is the feedback function minimizing the social cost SC(φ), without having to be a mean field game equilibrium, in which case we use the notation SC M KV for SC(φ). This is a control problem of McKean-Vlasov type, which is detailed more precisely in section 1.2.
The two problems are detailed more precisely in sections 1.1 and 1.2. In section 1.3, we define the price of anarchy based on these two problem formulations. The class of linear quadratic models is explored in section 2, where we provide some theoretical results on the price of anarchy for this class of games. This includes our main result, Theorem 2, which provides a sufficient and necessary condition to have no price of anarchy. In section 2, we also prove some limiting cases and show numerical results. We conclude in section 3.

Nash Equilibrium: Mean Field Game Formulation
The goal of this subsection is to articulate what is meant by a feedback function providing a mean field game equilibrium. To begin, we define what we call the mean field environment. By symmetry of the players, we suppose all of the players in the mean field game use the same feedback function, φ. Then the mean field environment specified by φ is characterized by L(X φ t ) 0≤t≤T and L(φ(t, X φ t )) 0≤t≤T where the dynamics of (X φ t ) 0≤t≤T are given by equation (1). Since we search for a Nash equilibrium, we consider a representative agent who wishes to find their best response, φ ′ , to the mean field environment specified by φ, in which case their state is given by solving the standard stochastic differential equation: Consider the function: The best response for the representative agent in the mean field environment specified by φ is the feedback function minimizing this cost, namely φ * = arg inf φ ′ S(φ ′ , φ). Assuming the minimizer is unique (which will be the case for the models we consider), this defines a mapping Φ : φ → φ * . If there is aφ such that Φ(φ) =φ, then the players are in a mean field game equilibrium. Thus, the search for a feedback function providing a mean field game equilibrium can be summarized as the following set of two successive steps: Define the mapping Φ(φ) := φ * .
When these two steps can be taken successfully, we say thatφ provides a mean field game equilibrium. Note that Xφ ,φ = Xφ and therefore S(φ,φ) = SC(φ) gives the social cost for the mean field game equilibrium provided byφ. Notice that there could possibly be many feedback functions providing a mean field game equilibrium. Let N denote the set of all such feedback functions providing mean field game equilibria, as detailed above, i.e.

Centralized Control: Optimal Control of McKean-Vlasov Type
The goal of this subsection is to articulate how to compute the cost associated with the control problem of McKean-Vlasov type, SC M KV . The central planner considers the following control problem: Thus, the cost of the solution to the optimal control problem of McKean-Vlasov is given by: Remark 1. We are not concerned with uniqueness for the control of McKean-Vlasov type problem, because SC M KV = SC(φ 1 ) = SC(φ 2 ) is still well defined even if there are two different optimal feedback functions φ 1 and φ 2 minimizing SC(φ).

Price of Anarchy
We have described two approaches to compute the optimal feedback function φ. In the mean field game formulation, we require φ ∈ N , where N denotes the set of feedback functions providing mean field game equilibria. In the optimal control of McKean-Vlasov type formulation, the optimal control to be adopted by all players is computed by a central planner, who optimizes the social cost function SC(φ) directly. Thus, we necessarily have: In other words, there is a 'price of anarchy' associated with allowing players to choose their controls selfishly. We thus define the price of anarchy (denoted P oA) as the ratio between the worst case cost for a mean field game equilibrium and the optimal cost computed by a central planner:

Price of Anarchy for Linear Quadratic Extended Mean Field Games
The class of linear quadratic extended mean field games is a class of problems for which explicit solutions can be computed analytically, and thus, we can compute the price of anarchy explicitly.
To the best of our knowledge, the case of linear quadratic extended mean field games has not been explored in the literature, as well as computing the price of anarchy for this class of games.
To begin, we need to describe in more detail the two problems that will be used to compute the price of anarchy: the linear quadratic extended mean field game, and the linear quadratic control problem of McKean-Vlasov type with dependence on the law of the control. To specify the problems, we only need to specify the drift and cost functions, b, f , and g introduced in section 1. For the linear quadratic models, we take the drift to be linear: b(t, x, µ, α, ν) = b 1 (t)x +b 1 (t)μ + b 2 (t)α +b 2 (t)ν, whereμ denotes the mean of the measure µ, namely,μ = R xdµ(x), and similarly forν. We take the running and terminal costs to be quadratic: Remark 2. Ifb 2 (t) ≡ 0 andr(t) ≡ 0, then we have the standard mean field game or control problem of McKean-Vlasov type. (See Theorem 1 for assumptions that provide existence and uniqueness.)

Linear Quadratic Extended Mean Field Games
To solve the linear quadratic extended mean field game (LQEMFG), we begin by considering the reduced Hamiltonian for this problem: and whenever the flowsμ = (μ t ) 0≤t≤T andν = (ν t ) 0≤t≤T are fixed, we consider for each control process α = (α t ) 0≤t≤T the adjoint equation: According to the Pontryagin stochastic maximum principle, a sufficient condition for optimality is ∂ α H(t, X t ,μ t ,α t ,ν t , Y t ) = 0. Thus, we find the optimal control: When solving the fixed point step, we identifyν t = E(α t ). By taking the expectation, we find: .
The solution of the mean field game equilibrium problem is given by the solution to the FBSDE system: with initial condition X 0 = ξ, a random variable with finite mean and variance, and terminal condition Y T = (q T +q T )X T −q T s T EX T . This is a linear FBSDE of McKean-Vlasov type, which can be solved explicitly under mild assumptions (or at least in the case of time-independent coefficients which we will consider later.
and v M F G t denote the solutions for this problem as described in the appendix so that: provide a solution to the LQEMFG problem. Then from the appendix, we have: with terminal condition η M F G T = q T +q T , and where the dot is the standard ODE notation for a derivative. And thus, we obtain explicit x) is the feedback function specified by this solution, namely, from equation (3), we have: Then we can compute the social cost as described in section 1.1: where we have used the fact that:

Linear Quadratic Control of McKean-Vlasov Type Involving the Law of the Control
To solve the linear quadratic optimal control problem of McKean-Vlasov type involving the law of the control (LQEMKV), we begin with the reduced Hamiltonian, which is the same as in the LQEMFG problem: Since we requireν t to be equal to E(α t ) throughout the optimization, it is not sufficient to minimize the Hamiltonian with respect to the α input alone in order to guarantee optimality. A sufficient condition for control problems of McKean-Vlasov type involving the law of the control is derived in [6]. Since we consider a Hamiltonian that depends on the means ofμ andν instead of the full distributions, the sufficient condition reduces to the following (see section 4 in [6]): where the adjoint equation is given by: and where (X,Ỹ ,α) denotes an independent copy of (X, Y , α). In the present LQ case, the sufficient condition can be used to solve for: with: So the solution of the optimal control problem of McKean-Vlasov type is given by the solution to the FBSDE system: As in the previous section, this is a linear FBSDE of McKean-Vlasov type, which can be solved explicitly under mild assumptions (or at least in the case of time-independent coefficients which we and v M KV t denote the solutions for this problem as described in the appendix so that: provide a solution to the LQEMKV problem. Then from the appendix, we have: with initial conditionx M KV and where the dot is the standard ODE notation for a derivative. And thus, we obtain explicit solutions forx M KV t and v M KV t : Then SC M KV = SC(φ M KV ) where φ M KV is the feedback function specified by this solution, namely, from equation (11), we have: Then we can compute the social cost, denoted SC M KV , as described in section 1.2: where we have used the fact that:

Theoretical Results
For the remainder of the paper, we assume the coefficients are independent of time and non-negative: and therefore, Also, it will be convenient to denote: and to make the following observations: Theorem 1. Assume the following: Then there exists a unique solution to the LQEMFG problem, and there exists a unique solution to the LQEMKV problem. And therefore, P oA = SC M F G SC M KV where SC M F G and SC M KV are given by equations (10) and (18), respectively.
Remark 3. Note that existence in Theorem 1 follows from the explicit construction in Appendix A, because the above conditions provide existence to the solutions of the Riccati equations. Uniqueness comes from the connection between LQEMFG or LQEMKV and deterministic LQ optimal control. (See section 3.5.1 in [8]).
Proof. Comparing the FBSDE systems (4) and (12), and using the fact that

Remark 4.
Recall from Remark 2 that in the standard mean field game,b 2 =r = 0, and thus, λ = 1. Although Proposition 1 is a simple result, we will see shortly in Corollary 3 that in the case when λ = 1, the sufficient condition given by equations (22)-(24) is also a necessary condition to have P oA = 1. We can see that in the standard mean field game setting, Proposition 1 is similar to Theorem 3.4 in [5] which characterizes the global efficiency of mean field game equilibria in the case of a separated Hamiltonian. See also Remark 6.1 in [15], where it is noted that the mean field game and control of McKean-Vlasov type problems are the same for a particular model of flocking. Using the observations in equation (20), we can rewrite: In the following, we intend to simplify the explicit solutions (25) and (26) for the social costs in the LQEMFG and LQEMKV problems. First, consider the quantity then using integration by parts for the first term in the bracket: and together with equation (6) yields: Finally, we arrive at: .
If we denote: and use the terminal condition forη M F G T , then equation (25) can be rewritten as: Similarly, equation (26) can be rewritten as: Let's denote the (weighted) difference between the solutions of the Riccati equations associated Proposition 2. Under assumption (21), the difference in the social costs in the LQEMFG and LQEMKV problems can be represented by: for the Riccati equations (5) and (13), respectively, are well defined under assumption (21) (see Appendix A). We notice that ∆η t defined in (29) satisfies the following linear first-order differential equation: with coefficients: (27) and (28) that: where we use equation (6) for the fourth equality.

Remark 5.
We can see directly from Proposition 2 that the social cost in the LQEMFG problem is larger than (or possibly equal to) the social cost in the LQEMKV problem. This result is consistent with the definition of the price of anarchy in section 1.3.
Note that we can write: It will be useful for us to note here the scalar Riccati equations associated with with: (73) in Appendix A) the existence and uniqueness for u t which can be expressed by: Under assumption (21), the above conditions on B u , C u , and D u are satisfied, and we have δ − u < 0 < δ + u , u t > 0 for all t ∈ [0, T ), and u T ≥ 0. We have analogous expressions for w t and η t , in terms of δ ± w and δ ± η , respectively. Note that B u = B w =: B.
It will also be useful to compute the derivative of u t with respect to time t from the explicit form in equation (36): Theorem 2. Assume (21) and the initial condition ξ satisfies E(ξ) = 0. Let A u , A w , B, C u , C w , D u , and D w as defined in equation (35).
• Whenb 1 > 0, we have P oA = 1 if and only if: • Whenb 1 = 0, then A u = A w and we have P oA = 1 if and only if: Proof. From an analogous equation to (36) for w t , we know that under assumption (21) Suppose now that P oA = 1. Then u t = w t for all t ∈ [0, T ] and clearly: Now, if we take the difference between the two Riccati equations (32) and (33), and by using u t = w t for all t ∈ [0, T ], we obtain: . Thus, the time derivatives of u t and w t should be zero. From equation (37), and the fact that δ + u − δ − u > 0, we deduce: Corollary 3. Assume (21) and λ = 1. Then the sufficient condition (equations (22)-(24)) from Proposition 1 is also a necessary condition to have P oA = 1.
We study in the following the variation of P oA by letting only one of the coefficients tend to zero or to infinity. In order to make the following computations easier to follow, we repeat equations (30), (28), (8), and (9), which we recall is equivalent to equation (17), using the above notations. Assuming (21), we have: Also for convenience, recall the definition from equation (19): In the following propositions, we utilize the following assumption to make their proofs simpler. Proof. First, we consider r → ∞. For every given r > 0, we have existence and uniqueness of the solutions u r t , w r t and η r t to the scalar Riccati equations (32)-(34). Note that we have added the superscript r to emphasize the dependence on this parameter.
When r → ∞, we have: Let u r→∞ : [0, T ] → R be the solution to the linear first-order differential equation: Then we have: It is easy to show directly from their explicit solutions (see equation (36)) that for every time t ∈ [0, T ], lim r→∞ u r t = u r→∞ t , and thus, lim r→∞ B r u r t = 0. Next, our goal is to bound the u r t uniformly over t ∈ [0, T ] for large r. Note that A u < 0, B r , C u,r , λ r→∞ , C u,r→∞ , D u,r , D u,r→∞ > 0, and δ −,r u < 0 < δ +,r u . Let ǫ > 0. Then there exists a r * > 0 such that max{B r , C u,r , D u,r } ≤ max{C u,r→∞ , D u,r→∞ } + ǫ =: ζ for r ≥ r * . Thus, we can deduce that for r ≥ r * , and for every t ∈ [0, T ]: From equation (43)   Otherwise, lim b 2 →∞ P oA > 1.
Proof. When b 2 → ∞, we have: Moreover, we notice that: (36), for all t ∈ [0, T ), we deduce: Similarly, for all t ∈ [0, T ): and thus for all t ∈ [0, T ]: Then, by the same technique in inequality (45), there exists a b * ,η,lower 2 > 0 and m η > 0 such that for all b 2 ≥ b * ,η,lower 2 and all t ∈ [0, T ]: From equation (37), we see that By the same argument for w b 2 t and η b 2 t , there exists a b * ,upper 2 ≥ b * ,u,upper 2 such that: and such that the 0 (E(ξ)) 2 and thus: with: and: Fix ǫ > 0. In the following, we show that I b 2 1 ≤ ǫ and I b 2 2 ≤ ǫ for large b 2 . First, consider and all s ∈ [0, T /2] we have: Thus, for any t ∈ [0, T /2] and b 2 ≥ b * ,I 1 2 : Therefore, where the last inequality comes from the definition of ζ 2 . Next, consider Hence, there exists a b * ,I 2 2 ≥ max{b * ,upper 2 , b * ,u,lower 2 } such that for all b 2 ≥ b * ,I 2 2 : Since the proof holds for arbitrary ǫ > 0, and c = (q +q(1 − s) 2 )(r +r(1 −s) 2 ) > 0 is independent of b 2 and ǫ, we conclude: and thus, from equation (31): Then c u = c w . We want to show that lim b 2 →∞ P oA b 2 > 1. To do so, we will show that where c num and M den are two constants independent of b 2 . We assume in the following that b 2 ≥ b * ,basic Step 1: We derive a lower bound for (b 2 +b 2 )∆SC b 2 by adapting the techniques used in inequality (49). We have shown that for every t which implies that for all t ∈ [0, T /2]: Thus, similar to inequality (49), for all b 2 ≥ b * ,num 2 , we deduce: Step 2: We derive an upper bound for b 2 SC M KV,b 2 . From equation (42), we have: We derive the following two results which are useful for deriving upper bounds for J b 2 1 and J b 2 2 . First, let κ 3 := 2mη r+r > 0 and l(b 2 , t) : Next, using equation (34), integration by parts, and since −A η , η b 2 t , C η , l(b 2 , t) ≥ 0 we deduce: After rearranging terms, we obtain: First, consider J b 2 1 . From equation (44) and inequalities (46) and (53), we have that for all b 2 ≥ b * ,basic 2 and for all t ∈ [0, T ]: Let b * ,J 1 ≥ b * ,basic such that for all b 2 ≥ b * ,J 1 2 : Then for all b 2 ≥ b * ,J 1 2 : Next, consider the quantity J b 2 2 , which can be written as: u du ds dt.
Case 2: Now, let's assumeb 2 > 0. As b 2 → 0, we have: and A u , (A w , C w , D w ), (A η , C η , D η ) are independent of b 2 . Moreover, we have: Thus, from equation (36) we deduce that for every fixed time t ∈ [0, T ], lim b 2 →0 u b 2 t = 0. Similar to Proposition 3, we can derive a uniform bound for u b 2 t over [0, T ] for small b 2 . Indeed, for any fixed ǫ > 0 there exists a b * 2 > 0 such that for any b 2 ≤ b * 2 : From equation (43), the assumption E(ξ) = 0, and by the bounded convergence theorem, we derive that for any fixed t ∈ [0, T ]: It can also be shown that x M F G,b 2 t ≤ |E(ξ)|e (b 1 +b 1 )T for any t ∈ [0, T ] and b 2 > 0.
t is strictly positive over [0, T ). It is easy to check that w b 2 t is also uniformly bounded over [0, T ] for small b 2 . Hence, from equation (41) and the bounded convergence theorem, we deduce: Since B η,b 2 → 0, A η < 0, C η > 0, and D η > 0, using the same argument shown in Proposition 3, we deduce that η b 2 t is uniformly bounded over [0, T ] for small b 2 and for all t ∈ [0, T ]: From equation (44) and the bounded convergence theorem, for all t ∈ [0, T ]: Proposition 6. Assuming Assumption 1, then: P oA > 1.
Remark 6. Consider Assumption 1 and the case whenb 2 tends to zero. We have 0 < SC M KV,b 2 →0 < ∞, and therefore, limb 2 →0 P oAb 2 = 1 if and only if limb 2 →0 ∆SCb 2 = 0. Since we can pass the limit as in equation (66), we have an analogous result as Theorem 2 but for the limiting coefficients A u , A w , Bb 2 →0 , C u,b 2 →0 , C w , D u,b 2 →0 , and D w . Therefore, the assumption in Proposition 4 Case 2, , which is equivalent to D u,b 2 →0 = D w , is sufficient, but not necessary, in order to have lim b 2 →0 P oA b 2 > 1.

Numerical Results
The price of anarchy for the class of linear quadratic extended mean field games that we consider is given by the ratio of the two quantities given by equations (25) and (26), which are explicit, up to evaluating integrals. Using the simple rectangle rule to estimate integrals, we numerically compute the price of anarchy when the coefficients are time-independent, non-negative, and satisfy Assumption 1. In particular, when we allow for full interaction (i.e. through the states and the controls), we choose the following default values: q = 1,q = 1, s = 0.5, r = 1,r = 1,s = 0.5, q T = 1,q T = 1, s T = 0.5.
Unless otherwise stated, the parameters stay at these default values. For results involving only interaction through the states, we setb 2 = 0 andr = 0. For results involving only interaction through the controls, we setb 1 = 0,q = 0, andq T = 0. Figures 1-5 show the price of anarchy as we vary one parameter at a time for each of three interaction cases: full interaction (i.e. through the states and the controls), interaction only through the states, and interaction only through the controls. The results show various limiting behaviors, such as some of the cases proved in the previous section. Figure 1 confirms as in Proposition 3 that lim r→∞ P oA = 1 and limr →∞ P oA = 1. Propositions 4 and 5 are confirmed in Figure 2. In the case of full interaction, we have q+q(1−s) r+r(1−s) = q+q(1−s) 2 r+r(1−s) 2 and we see that lim b 2 →∞ P oA = 1. In the cases of interaction only through the states or interaction only through the controls, then q+q(1−s) r+r(1−s) = q+q(1−s) 2 r+r(1−s) 2 , and we see that lim b 2 →∞ P oA > 1, confirming Proposition 4. For Proposition 5, when there is only interaction through the states, thenb 2 = 0 and we see that lim b 2 →0 P oA = 1. When there is full interaction or only interaction through the controls, thenb 2 > 0 and we see that lim b 2 →0 P oA > 1. For Proposition 6, Figure 3 confirms that limb 2 →0 P oA > 1 and limb 2 →∞ P oA = 1. Note that the condition r+r(1−s) 2 r+r(1−s) = q T +q T (1−s T ) 2 q T +q T (1−s T ) is not satisfied for the full interaction case, and is therefore a sufficient, but not necessary, assumption. This agrees with the conclusion of Remark 6. In Figures 4 and 5, we note that Proposition 7 is confirmed. The condition b 2 b 2 +b 2 · r+r(1−s) 2 r+r(1−s) · (q T +q T (1 − s T )) = q T +q T (1 − s T ) 2 is satisfied for all three interaction cases and we see that lim b 1 →0 P oA > 1, limb 1 →0 P oA > 1, lim b 1 →∞ P oA = 1, and limb 1 →∞ P oA = ∞. Thus, the numerical computations confirm the results presented in Propositions 3-7.

Conclusion
We defined the price of anarchy (P oA) in the context of extended mean field games as the ratio of the worst case social cost when the players are in a mean field game equilibrium to the social cost as computed by a central planner. Since the central planner does not require that the players be in   a mean field game equilibrium, the central planner will realize a social cost that is no worse than that of a mean field game equilibrium. Thus, P oA ≥ 1.
We computed the price of anarchy for linear quadratic extended mean field games, for which explicit computations are possible. We identify a large class of models for which P oA = 1 (see Proposition 1 and Corollaries 1 and 2), as well as giving a sufficient and necessary condition to have P oA = 1 (see Theorem 2 and Corollary 3). We also derive some limiting cases where P oA → 1 as certain parameters tend to zero or to infinity (see . The numerics support our theoretical results. then it has a unique solution: with δ ± = −A ± (A) 2 + BC. Furthermore, if B → 0 and A = 0, we can deduce that the limiting solution of the scalar Riccati equation coincides with the linear first-order differential equation: ρ t − 2Aρ t + C = 0, with terminal condition ρ T = D, namely: If B → 0 and A = 0, the limiting solution of the scalar Riccati equation coincides with the linear first-order differential equation:ρ t + C = 0 with terminal condition ρ T = D, namely: Hence, returning to the linear FBSDE (70), forη t , we use: LGEMKV problems, if we assume the coefficients are non-negative, we see that these conditions are exactly assumption (21).