Finite-panel stability of the sparse composition-equilibrium branch

Establish stability for the finite-panel regular composition equilibria on the sparse branch, rather than only for the limiting rescaled best-response map, by proving the required convergence of the sparse incentive function in the C^1 sense.

Background

The paper constructs a sequence of finite-panel regular composition equilibria whose fidelity rates satisfy NxN∗→c∗(κ)N x_N^*\to c^*(\kappa) near the boundary κ=4/5\kappa=4/5. It proves that the corresponding positive fixed point of the limiting rescaled best-response map is locally stable. This establishes existence and limiting localization of the sparse branch, but it does not establish stability for each finite panel size.

Finite-panel stability would require more than convergence of the incentive function's level: it would require C1C^1 convergence of the rescaled best-response or sparse incentive function, so that derivatives of the finite-panel maps could be controlled. The authors explicitly state that this convergence has not been established, leaving finite-NN stability unresolved.

References

The theorem deliberately asserts stability only for the limiting rescaled map. A finite-$N$ stability claim would require $C1$ convergence of the sparse incentive function rather than convergence of its level, and we have not established that. It is unnecessary for existence and for localising the branch, and we prefer to say so rather than to assert more than the proof delivers.

— Anonymous Accountability  (2609.19771 - Lukyanov et al., 17 Sep 2026) in Section 6, immediately after the proof of Theorem 6.2 (Theorem \ref{thm:sparsebranch}), before Section 6.1