Each member ∈ [1 ] chooses her level of ≥ 0 to maximize her utility = √− P 6= − a function that is twice-differentiable and concave. Assuming ≥ 0 to be the Lagrangian parameters, the optimal level of # satisfies the necessary and sufficient first-order conditions 1 2 − 1 2 − + = 0 and the Kuhn-Tucker conditions = 0 for any ∈ [1 ], thus ...