# Buy-many mechanisms are not much better than item pricing

## TL;DR

This paper demonstrates that simple item pricing mechanisms are nearly as effective as complex multi-item mechanisms, with revenue within a logarithmic factor, challenging the belief that complexity yields significantly higher revenue.

## Contribution

It proves that the revenue gap between simple item pricing and complex mechanisms is at most logarithmic, even under general and correlated valuation settings, and shows this bound is tight.

## Key findings

- Revenue of any multi-item mechanism is at most O(log n) times the revenue of item pricing.
- This logarithmic bound holds even with correlated valuations and randomized mechanisms.
- Simple item pricing mechanisms are nearly optimal compared to complex mechanisms in broad settings.

## Abstract

Multi-item mechanisms can be very complex offering many different bundles to the buyer that could even be randomized. Such complexity is thought to be necessary as the revenue gaps between randomized and deterministic mechanisms, or deterministic and simple mechanisms are huge even for additive valuations. We challenge this conventional belief by showing that these large gaps can only happen in restricted situations. These are situations where the mechanism overcharges a buyer for a bundle while selling individual items at much lower prices. Arguably this is impractical in many settings because the buyer can break his order into smaller pieces paying a much lower price overall. Our main result is that if the buyer is allowed to purchase as many (randomized) bundles as he pleases, the revenue of any multi-item mechanism is at most O(logn) times the revenue achievable by item pricing, where n is the number of items. This holds in the most general setting possible, with an arbitrarily correlated distribution of buyer types and arbitrary valuations. We also show that this result is tight in a very strong sense. Any family of mechanisms of subexponential description complexity cannot achieve better than logarithmic approximation even against the best deterministic mechanism and even for additive valuations. In contrast, item pricing that has linear description complexity matches this bound against randomized mechanisms.