Periodic limiting prior for binomial Bayesian games

Prove that for the binomial Bayesian games {N,x_A,x_B} with 0 < x_A < x_B < 1, the equilibrium prior P^*_N(x_A,x_B) converges, as N \to \infty, to a periodic function P^{*,\varphi}_\loopedsquare(x_A,x_B) depending on the phase factor \varphi = 2\pi N x_0^*(x_A,x_B) \bmod 2\pi, and establish the first-order approximation P^*_N(x_A,x_B) = P^{*,\varphi}_\loopedsquare(x_A,x_B) + \mathcal{O}(1/N).

Background

For Bayesian games with binomial sampling, the authors derive asymptotic integral representations suggesting that the equilibrium prior exhibits phase-dependent oscillations as N increases, governed by the phase \varphi = 2\pi N x_0^*. Numerical computations show oscillatory convergence consistent with a periodic limit.

The conjecture formalizes this observation by positing a periodic limiting function P^{*,\varphi}_\loopedsquare with an explicit implicit characterization (via asymptotic fixed-point equations for log-odds), together with an \mathcal{O}(1/N) error term. A rigorous proof of this asymptotic behavior is presently lacking.