Continuity and approximability of competitive spectral radii
Marianne Akian, Stéphane Gaubert, Loïc Marchesini, Ian Morris
TL;DR
This work addresses extending the joint spectral radius to a two-player setting, defining the competitive spectral radius as the value of an escape-rate game on cone-preserving operators. It casts the problem as a nonlinear eigenproblem on a cross-section of a cone via a Shapley operator and distance-like functions, then develops a discretization-based method to approximate the value with provable bounds. The main contributions are a Lipschitz (indeed, 1-Lipschitz) continuity result with respect to parameter changes in the matrix-sets under a strict positivity cone condition, and a scalable numerical scheme (RVI-KM) that achieves arbitrary precision with complexity $O(|\mathcal{A}||\mathcal{B}|/h^{2d})$ in dimension $d$. The approach is illustrated on an age-structured population dynamics model, demonstrating both theoretical insights and practical computability for two-player matrix product growth problems.
Abstract
The competitive spectral radius extends the notion of joint spectral radius to the two-player case: two players alternatively select matrices in prescribed compact sets, resulting in an infinite matrix product; one player wishes to maximize the growth rate of this product, whereas the other player wishes to minimize it. We show that when the matrices represent linear operators preserving a cone and satisfying a "strict positivity" assumption, the competitive spectral radius depends continuously - and even in a Lipschitz-continuous way - on the matrix sets. Moreover, we show that the competive spectral radius can be approximated up to any accuracy. This relies on the solution of a discretized infinite dimensional non-linear eigenproblem. We illustrate the approach with an example of age-structured population dynamics.