This work studied the optimal reinsurance design in a duopolistic market comprising two types of insurers and two reinsurers under asymmetric information, where reinsurers cannot directly observe insurers' risk types. We modeled reinsurers as risk-neutral agents maximizing expected net profit, subject to individual rationality, incentive compatibility, and convex preference constraints. We introduced the principle of Pareto optimality to formulate the objective function in a multi-agent setting. Applying the Lagrange dual approach, we derived optimal reinsurance menus for all cases. Under the Value-at-Risk (VaR) risk measure, we identified the globally optimal reinsurance menu by comparative analysis and provided its closed-form solution. Furthermore, we compared exponential and Pareto distributions with identical expected losses to study tail risk effects.