Limiting Behavior of LQ Deterministic Infinite Horizon Nash Games with Symmetric Players as the Number of Players goes to Infinity

A Linear Quadratic Deterministic Continuous Time Game with many symmetric players is considered and the Linear Feedback Nash strategies are studied as the number of players goes to infinity. We show that under some conditions the limit of the solutions exists and can be used to approximate the case with a finite but large number of players. It is shown that in the limit each player acts as if he were faced with one player only, who represents the average behavior of the others.

💡 Research Summary

The paper investigates a deterministic continuous‑time linear‑quadratic (LQ) Nash game involving a large number of symmetric players and studies the behavior of the linear‑feedback Nash equilibrium as the number of players M tends to infinity. Each player i controls a subsystem x_i∈ℝⁿ with input u_i∈ℝᵐ. The overall dynamics are written in a compact form
 ẋ = A x + B u, x = (x₁,…,x_M)ᵀ, u = (u₁,…,u_M)ᵀ,
where the matrices A and B are identical for all players, reflecting the symmetry of the model. The individual cost functional is
 J_i = ∫₀^∞ (x_iᵀQ x_i + u_iᵀR u_i) dt,
with Q≥0 and R>0, the same for every player.

The authors restrict attention to linear feedback strategies of the form
 u_i = –L x_i – L̄ z_i,
where z_i denotes the average state of the other players, i.e. z_i = (1/(M‑1))∑{j≠i} x_j. Substituting this ansatz into the dynamics yields a set of M coupled Riccati equations. By introducing the aggregate variable z = (1/M)∑{j=1}^M x_j, the multi‑player interaction can be reduced to a two‑player structure: each real player interacts with a single fictitious player whose state is the population average.

The closed‑loop matrix for the whole system is denoted Ā(M) = A – B R⁻¹ BᵀK(M), where K(M) solves the generalized Riccati equation
 Ā(M)ᵀK(M) + K(M)Ā(M) – K(M)B R⁻¹ BᵀK(M) + Q = 0.
Because Ā(M) depends linearly on 1/M, the authors set w = 1/M and expand K(w) as a power series:
 K(w) = K₀ + K₁ w + K₂ w² + … .
The coefficients K_n are obtained by applying the implicit function theorem to the Riccati map R(K,w) = 0. The derivative of R with respect to K at w = 0, denoted 𝓛₀, acts as a linear operator. If 𝓛₀ is invertible, each K_n satisfies a linear equation 𝓛₀(K_n) = F_n(K₀,…,K_{n‑1}), where F_n collects lower‑order terms. This recursive scheme yields a unique analytic solution K(w) in a neighbourhood of w = 0.

Stability of the closed‑loop system is ensured when the eigenvalues of A_cl = A – B R⁻¹ BᵀK(w) lie strictly in the left half‑plane. The paper proves that, under the invertibility of 𝓛₀ and the existence of a stabilizing K₀, the closed‑loop matrix remains stable for all sufficiently large M, guaranteeing finite costs and the existence of a Nash equilibrium in linear feedback strategies.

Taking the limit w → 0 (i.e., M → ∞) gives K(w) → K₀. The matrix K₀ coincides with the solution of the Riccati equation for a two‑player LQ game where the opponent is the “average player” representing the whole population. Consequently, each individual player, in the infinite‑population limit, behaves as if he were playing against a single fictitious opponent whose dynamics are the mean field of the other agents. This result mirrors the intuition behind mean‑field games but is derived in a purely deterministic setting and for matrix‑valued systems rather than scalar cases.

The main contributions of the work are:

1. Demonstrating that the coupled Riccati system for a symmetric LQ game can be reduced to a single Riccati equation involving the population average.
2. Providing a rigorous analytic expansion of the Riccati solution in powers of 1/M, based on the implicit function theorem and the invertibility of the linearized Riccati operator.
3. Establishing conditions under which the infinite‑player limit exists, is unique, and yields a stabilizing feedback law.
4. Interpreting the limit as a two‑player game against a fictitious “average” player, thereby linking deterministic large‑population LQ games to the broader mean‑field game literature.

These findings have practical implications for the design of decentralized control laws in large‑scale networks such as power grids, traffic systems, and communication networks, where each agent can safely approximate the influence of the rest of the population by a simple average term, leading to tractable and provably stable control strategies.