A Simple Mechanism for a Budget-Constrained Buyer

TL;DR

This paper analyzes simple auction mechanisms for a monopolist selling multiple items to a single budget-constrained buyer, showing approximate optimality under certain conditions and highlighting limitations when budgets are interdependent with valuations.

Contribution

It demonstrates that selling items separately or as a bundle achieves a constant fraction of optimal revenue when the buyer's budget is known or independent, and identifies limitations with interdependent budgets.

Findings

Selling separately or bundling yields constant-factor approximation when budget is known.

Approximate optimality holds under independent budget and valuation distributions with monotone hazard rate.

No constant approximation mechanism exists if budget and valuations are interdependent.

Abstract

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.