The exact three-player threshold for maximal multiplicity of feedback Nash equilibria in the symmetric scalar discounted linear-quadratic game

Abstract

Cavalagli, Bemporad and Zanon (IEEE Control Systems Letters textbf(10) (2026), 919–924) bound the number of hyperbolic feedback Nash equilibria of the symmetric N-player scalar discounted linear-quadratic game xₜ₊₁=axₜ+Σᵢ u_(i,t) and show that the maximal number 2^N-2 is attained when |a| ≥ (N-1)sqrtσ+√(σ+1/γ), a condition that is also necessary when N is even; for odd N their proof stops with the remark that obtaining a necessary condition would mean solving their inequality (21), "which cannot be done in closed-form". We settle the smallest odd case, N=3. In the aggregate coordinate g=(k+σ/k)/2 the relevant auxiliary function becomes a linear term plus a single square root, its minimisation over the negative axis reduces to the cubic (1-c)w³-3(1+c)w²+4(1+c)=0 with c=γσ, which has exactly one root bar w ∈ (1,2) for every c>0, and the threshold is a^(*)(σ,γ)=4sqrtσ/((2-bar w)√(1+bar w)). The maximal-multiplicity criterion of the source at N=3 holds if and only if agea^(*); equivalently, with the cubic root eliminated, if and only if X ≥ 5c+1 and (X+2-2c)³ ≥ 27(1+c)²X for X=γ a²; and a^(*)<2sqrtσ+√(σ+1/γ) strictly for all parameters, so the source's sufficient condition is never necessary at N=3. A root of the auxiliary function is shown to yield stable equilibria under the source's standing assumption, rational parameter points on both sides of the threshold are exhibited, and a missing factor γ in the quadratic the source displays just above its equation (13) is documented (equation (13) itself is unaffected). Every statement is verified in Lean 4 against Mathlib.

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