Truth and Envy in Capacitated Allocation Games

TL;DR

This paper investigates auction mechanisms for capacitated allocation games, demonstrating conditions under which VCG mechanisms are envy-free and exploring the limitations of Walrasian pricing in heterogeneous capacity settings.

Contribution

It extends the understanding of incentive-compatible and envy-free mechanisms to general capacities, identifying when VCG mechanisms work and highlighting limitations of Walrasian pricing.

Findings

VCG with Clarke Pivot payments is envy free for homogeneous capacities.

No incentive compatible Walrasian pricing exists for all capacities.

Some cases allow mechanisms that are both incentive compatible and envy free.

Abstract

We study auctions with additive valuations where agents have a limit on the number of goods they may receive. We refer to such valuations as {\em capacitated} and seek mechanisms that maximize social welfare and are simultaneously incentive compatible, envy-free, individually rational, and have no positive transfers. If capacities are infinite, then sequentially repeating the 2nd price Vickrey auction meets these requirements. In 1983, Leonard showed that for unit capacities, VCG with Clarke Pivot payments is also envy free. For capacities that are all unit or all infinite, the mechanism produces a Walrasian pricing (subject to capacity constraints). Here, we consider general capacities. For homogeneous capacities (all capacities equal) we show that VCG with Clarke Pivot payments is envy free (VCG with Clarke Pivot payments is always incentive compatible, individually rational, and has no positive transfers). Contrariwise, there is no incentive compatible Walrasian pricing. For heterogeneous capacities, we show that there is no mechanism with all 4 properties, but at least in some cases, one can achieve both incentive compatibility and envy freeness.