Optimal Bundling and Dominance

The optimal bundling problem is the design of bundles (and prices) that maximize the expected virtual surplus, constrained by individual rationality and incentive compatibility. I focus on a relaxed, constraint-free problem that maximizes the expected virtual surplus with deterministic allocations. I show that when the difference of two virtual value functions for any pair of stochastic bundles is single-crossing, the minimal optimal menu for the relaxed problem is the set of all undominated bundles, where dominance refers to the natural comparison of virtual values across the type space. Under single-crossing differences, this comparison simplifies to comparisons on boundary types, enabling an algorithm to generate the minimal optimal menu. I further show that when the difference of two virtual value functions for any pair of stochastic bundles is monotonic, there exists an endogenous order among deterministic bundles. This order justifies that the minimal optimal menu for the relaxed problem is also optimal for the nonrelaxed problem. I also characterize the conditions under which consumers' valuation functions and distributions allow the virtual value functions to exhibit monotonic differences. This characterization, combined with the dominance notion, solves related bundling problems such as distributional robustness, bundling with additive values, and the optimality of pure, nested, and relatively novel, singleton-bundle menus. As a side result, I discover new properties of single crossing and monotonic differences in convex environments.