Continuity and approximability of competitive spectral radii

Marianne Akian; Stéphane Gaubert; Loïc Marchesini; Ian Morris

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arXiv · 2505.22468

Continuity and approximability of competitive spectral radii

Marianne Akian·Stéphane Gaubert·Loïc Marchesini·Ian Morris

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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.

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BibTeX RIS

Marianne Akian, Stéphane Gaubert, Loïc Marchesini, Ian Morris. 2025-05-28. Continuity and approximability of competitive spectral radii. https://arxiv.org/abs/2505.22468

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