A certified two-point equilibrium range for the reciprocal-xi contest

Abstract

We give an unconditional equilibrium characterization for Ewerhart’s reciprocal-xi contest on an explicit prize interval. Let f be the normalized inverse Fourier transform of ξ(1/2)/ξ(1/2+u) and let G(y) = ∫₀ʸ f(t) dt. The unique equilibrium is zero effort for 0 < W ≤ 1/f(0). Above this threshold and through W* = Y/G(Y), where Y = 36591/80000, each player uses an explicit two-point distribution on {0,y}, with WG(y) = y. We prove W* > 0.94817826. Uniqueness holds among all Borel mixed strategies, including asymmetric profiles. The proof reduces uniqueness to a strict payoff inequality, establishes that inequality analytically for small y, and extends it to Y by a uniform finite certificate. The retained Lean project checks the density assumptions, certificate bounds, endpoint comparison, and connection to the original game. This gives a bounded-prize result without the Riemann hypothesis assumed in the known all-prize uniqueness theorem.

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