Bundling in Oligopoly: Revenue Maximization with Single-Item Competitors

TL;DR

This paper analyzes how a principal seller can maximize revenue by bundling products in an oligopoly setting with competing single-item sellers, establishing bounds and conditions for effective bundling strategies.

Contribution

It introduces a model of bundling in oligopoly with competing sellers, providing bounds on revenue and conditions under which bundling guarantees near-optimal revenue.

Findings

Grand-bundle pricing achieves a constant approximation to optimal revenue under certain conditions.

The model reduces to Bertrand competition when only single items are considered.

For some distributions, bundling does not guarantee near-optimal revenue.

Abstract

We consider a principal seller with $m$ heterogeneous products to sell to an additive buyer over independent items. The principal can offer an arbitrary menu of product bundles, but faces competition from smaller and more agile single-item sellers. The single-item sellers choose their prices after the principal commits to a menu, potentially under-cutting the principal's offerings. We explore to what extent the principal can leverage the ability to bundle product together to extract revenue. Any choice of menu by the principal induces an oligopoly pricing game between the single-item sellers, which may have multiple equilibria. When there is only a single item this model reduces to Bertrand competition, for which the principal's revenue is $0$ at any equilibrium, so we assume that no single item's value is too dominant. We establish an upper bound on the principal's optimal revenue at every equilibrium: the expected welfare after truncating each item's value to its revenue-maximizing price. Under a technical condition on the value distributions -- that the monopolist's revenue is sufficiently sensitive to price -- we show that the principal seller can simply price the grand-bundle and ensure (in any equilibrium) a constant approximation to this bound (and hence to the optimal revenue). We also show that for some value distributions violating our conditions, grand-bundle pricing does not yield a constant approximation to the optimal revenue in any equilibrium.