Competitive division of a mixed manna

A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homogeneous of degree 1 and concave (and monotone), the competitive division maximizes the Nash product of utilities (Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles), unique, continuous, and easy to compute. We show that the competitive division of a mixed manna is still welfarist. If the zero utility profile is Pareto dominated, the competitive profile is strictly positive and still uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance, if all items are bads), the competitive profiles are strictly negative and are the critical points of the product of disutilities on the efficiency frontier. The latter allows for multiple competitive utility profiles, from which no single-valued selection can be continuous or resource monotonic. Thus the implementation of competitive fairness under linear preferences in interactive platforms like SPLIDDIT will be more difficult when the manna contains bads that overwhelm the goods

The Core of the Participatory Budgeting Problem

In participatory budgeting, communities collectively decide on the allocation of public tax dollars for local public projects. In this work, we consider the question of fairly aggregating the preferences of community members to determine an allocation of funds to projects. This problem is different from standard fair resource allocation because of public goods: The allocated goods benefit all users simultaneously. Fairness is crucial in participatory decision making, since generating equitable outcomes is an important goal of these processes. We argue that the classic game theoretic notion of core captures fairness in the setting. To compute the core, we first develop a novel characterization of a public goods market equilibrium called the Lindahl equilibrium, which is always a core solution. We then provide the first (to our knowledge) polynomial time algorithm for computing such an equilibrium for a broad set of utility functions; our algorithm also generalizes (in a non-trivial way) the well-known concept of proportional fairness. We use our theoretical insights to perform experiments on real participatory budgeting voting data. We empirically show that the core can be efficiently computed for utility functions that naturally model our practical setting, and examine the relation of the core with the familiar welfare objective. Finally, we address concerns of incentives and mechanism design by developing a randomized approximately dominant-strategy truthful mechanism building on the exponential mechanism from differential privacy

Quadratic forms and linear algebraic groups

Topics discussed at the workshop Quadratic Forms and Linear Algebraic Groups included besides the algebraic theory of quadratic and Hermitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic aspects pertaining to the theory of quadratic forms over function fields over local fields

Allocation in Practice

How do we allocate scarcere sources? How do we fairly allocate costs? These are two pressing challenges facing society today. I discuss two recent projects at NICTA concerning resource and cost allocation. In the first, we have been working with FoodBank Local, a social startup working in collaboration with food bank charities around the world to optimise the logistics of collecting and distributing donated food. Before we can distribute this food, we must decide how to allocate it to different charities and food kitchens. This gives rise to a fair division problem with several new dimensions, rarely considered in the literature. In the second, we have been looking at cost allocation within the distribution network of a large multinational company. This also has several new dimensions rarely considered in the literature.Comment: To appear in Proc. of 37th edition of the German Conference on Artificial Intelligence (KI 2014), Springer LNC

How to cut a pizza fairly: fair division with descreasing marginal evaluations.

Existential and constructive solutions to the classic problems of fair division are known for individuals with constant marginal evaluations. By considering nonatomic concave capacities instead of nonatomic probability measures, we extend some of these results to the case of individuals with decreasing marginal evaluations.