Auctions with Heterogeneous Items and Budget Limits

TL;DR

This paper investigates the design of auction mechanisms with heterogeneous items and budget constraints, establishing impossibility results for deterministic mechanisms and providing randomized solutions under certain conditions.

Contribution

It demonstrates fundamental limitations of deterministic mechanisms in complex auction settings and introduces randomized mechanisms that achieve desired properties for specific cases.

Findings

No deterministic mechanism exists for divisible items with these properties.

Randomized mechanisms can achieve the properties for indivisible items with single-dimensional valuations.

Impossibility results are independent of computational complexity.

Abstract

We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either divisible or indivisible items. The impossibility results hold for public budgets, while the mechanism allows private budgets, which is in both cases the harder variant to show. While all positive results are polynomial-time algorithms, all negative results hold independent of complexity considerations.