Julien Lesca

LP Solvable Models for Multiagent Fair Allocation Problems

This paper proposes several operational approaches for solving fair allocation problems in the context of multiagent optimization. These problems arise in various contexts such as assigning conference papers to referees or sharing of indivisible goods among agents. We present and discuss various social welfare functions that might be used to maximize the satisfaction of agents while maintaining a notion of fairness in the distribution. All these welfare functions are in fact non-linear, which precludes the use of classical min-cost max-flow algorithms for finding an optimal allocation. For each welfare function considered, we present a Mixed Integer Linear Programming formulation of the allocation problem that can be efficiently solved using standard solvers. The results of numerical tests we conducted on realistic cases are given at the end of the paper to confirm the practical feasibility of the proposed approaches.

Service Exchange Problem

In this paper, we study the service exchange problem where each agent is willing to provide her service in order to receive in exchange the service of someone else. We assume that agent’s preference depends both on the service that she receives and the person who receives her service. This framework is an extension of the housing market problem to preferences including a degree of externalities. We investigate the complexity of computing an individually rational and Pareto efficient allocation of services to agents for ordinal preferences, and the complexity of computing an allocation which maximizes either the utility sum or the utility of the least served agent for cardinal preferences.

The Fair OWA One-to-One Assignment Problem: NP-Hardness and Polynomial Time Special Cases

We consider a one-to-one assignment problem consisting of matching n objects with n agents. Any matching leads to a utility vector whose n components measure the satisfaction of the various agents. We want to find an assignment maximizing a global utility defined as an ordered weighted average (OWA) of the n individual utilities. OWA weights are assumed to be non-increasing with ranks of satisfaction so as to include an idea of fairness in the definition of social utility. We first prove that the problem is NP-hard; then we propose a polynomial algorithm under some restrictions on the set of admissible weight vectors, proving that the problem belongs to XP.

Object Allocation via Swaps along a Social Network

This article deals with object allocation where each agent receives a single item. Starting from an initial endowment, the agents can be better off by exchanging their objects. However, not all trades are likely because some participants are unable to communicate. By considering that the agents are embedded in a social network, we propose to study the allocations emerging from a sequence of simple swaps between pairs of neighbors in the network. This model raises natural questions regarding (i) the reachability of a given assignment, (ii) the ability of an agent to obtain a given object, and (iii) the search of Pareto-efficient allocations. We investigate the complexity of these problems by providing, according to the structure of the social network, polynomial and NP-complete cases.