A Simple Mechanism for a Budget-Constrained Buyer

We study a classic Bayesian mechanism design setting of monopoly problem for an additive buyer in the presence of budgets. In this setting a monopolist seller with $m$ heterogeneous items faces a single buyer and seeks to maximize her revenue. The buyer has a budget and additive valuations drawn independently for each item from (non-identical) distributions. We show that when the buyer's budget is publicly known, the better of selling each item separately and selling the grand bundle extracts a constant fraction of the optimal revenue. When the budget is private, we consider a standard Bayesian setting where buyer's budget $b$ is drawn from a known distribution $B$. We show that if $b$ is independent of the valuations and distribution $B$ satisfies monotone hazard rate condition, then selling items separately or in a grand bundle is still approximately optimal. We give a complementary example showing that no constant approximation simple mechanism is possible if budget $b$ can be interdependent with valuations.