This section develops the formal equilibrium analysis of the veiling model. It establishes three key thresholds—q, q₁, and q₂—that delineate distinct D1 equilibria: (i) a pooling equilibrium for low religious prevalence (q ≤ q₁) where both types veil minimally; (ii) a hybrid equilibrium for intermediate q (q₁ < q < q₂) where religious types mix between full veiling and no veiling with a mixing weight b that decreases continuously with q; and (iii) a Pareto‑dominant separating equilibrium for high q (q ≥ q₂) where each type chooses its optimal veiling level. Lemmas 2‑4 prove existence, uniqueness, and monotonicity of these equilibria, deriving incentive‑compatibility conditions and belief updates via Bayes’ rule. The section then derives comparative statics for the optimal veiling levels (\overline{v}r, \overline{v}s) under observable behavior (Table 1) and for the additional veiling level v_r that may arise when behavior is unobservable (Table 2). Proposition 4 shows how these veiling choices respond to parameters p, λr, λs, and q. The dynamic evolution of the religious share q over time is modeled by a recursive function h(q) (Eq. 31) and its variant \tilde h(q) under a veiling ban. Propositions 5 and 6 prove existence of interior equilibria, monotonic convergence to a stable religious equilibrium, and the counter‑intuitive effect that a temporary ban can increase the long‑run proportion of religious types. Throughout, the analysis relies on envelope theorems, sign‑analysis of cross‑partial derivatives, and Lyapunov stability arguments.
Topics
Equilibrium structure and thresholds (q₁, q₂) in the veiling model
Establishes three D1 equilibria distinguished by the religious share q: (i) a pooling equilibrium for low q (q ≤ q₁) where both types veil minimally; (ii) a hybrid equilibrium for intermediate q (q₁ < q < q₂) in which religious types randomize between full veiling and no veiling with a mixing weight b that strictly decreases with q; (iii) a Pareto‑dominant separating equilibrium for high q (q ≥ q₂) where each type selects its perfect‑information optimal veiling level. Lemmas 2‑4 prove existence, uniqueness, and monotonicity of these equilibria, derive the incentive‑compatibility conditions, and specify belief updates via Bayes’ rule.