Probabilistic Serial Mechanism for Multi-Type Resource Allocation

TL;DR

This paper introduces a new mechanism called LexiPS for multi-type resource allocation with indivisible items under lexicographic preferences, overcoming an impossibility result and ensuring efficiency, envy-freeness, and strategyproofness.

Contribution

It extends the probabilistic serial mechanism to multi-type settings with indivisible items under lexicographic preferences, achieving desirable fairness and efficiency properties.

Findings

LexiPS satisfies sd-efficiency and sd-envy-freeness.

MPS satisfies lexi-efficiency and is sd-envy-free under strict preferences.

MPS characterized by leximin-ptimality and no-generalized-cycle condition.

Abstract

In multi-type resource allocation (MTRA) problems, there are p $\ge$ 2 types of items, and n agents, who each demand one unit of items of each type, and have strict linear preferences over bundles consisting of one item of each type. For MTRAs with indivisible items, our first result is an impossibility theorem that is in direct contrast to the single type (p = 1) setting: No mechanism, the output of which is always decomposable into a probability distribution over discrete assignments (where no item is split between agents), can satisfy both sd-efficiency and sd-envy-freeness. To circumvent this impossibility result, we consider the natural assumption of lexicographic preference, and provide an extension of the probabilistic serial (PS), called lexicographic probabilistic serial (LexiPS).We prove that LexiPS satisfies sd-efficiency and sd-envy-freeness, retaining the desirable properties of PS. Moreover, LexiPS satisfies sd-weak-strategyproofness when agents are not allowed to misreport their importance orders. For MTRAs with divisible items, we show that the existing multi-type probabilistic serial (MPS) mechanism satisfies the stronger efficiency notion of lexi-efficiency, and is sd-envy-free under strict linear preferences, and sd-weak-strategyproof under lexicographic preferences. We also prove that MPS can be characterized both by leximin-ptimality and by item-wise ordinal fairness, and the family of eating algorithms which MPS belongs to can be characterized by no-generalized-cycle condition.