Nash Equilibrium in Discontinuous Games: A Caristi–Khamsi Proof under a Weak Robust Better-Reply Property

This note revisits Theorem 3.2 of Crettez et al. (2025), which establishes the existence of a pure-strategy Nash equilibrium in convex and compact games with possibly discontinuous payoffs under an F–π–ϖ weak robust better-reply correspondence property. Instead of the original fixed-point construction, the proof here proceeds via a single Caristi–Khamsi fixed point theorem for set-valued maps, under a mild additional lower semicontinuity assumption on the associated deviation–profit potential. The key step is to associate with each strategy profile a deviation–profit potential that measures the maximal unilateral gain from weakly robust deviations and to show that this potential decreases along the F–π–ϖ weak robust betterreply correspondence. The resulting Caristi inequality yields a fixed point of the induced better-reply map, which is a Nash equilibrium. The argument provides a structural reinterpretation of the result of Crettez et al. (2025), highlighting the potential-theoretic nature of the robustness condition and separating the economic structure of profitable deviations from the topological fixed-point machinery.