Handbook of critical and indigenous methodologies

Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

We study the problem of fairly allocating a set of indivisible goods to a set of people from an algorithmic perspective. fair division has been a central topic in the economic literature and several concepts of fairness have been suggested. The criterion that we focus on is envy-freeness. In our model, a monotone utility function is associated with every player specifying the value of each subset of the goods for the player. An allocation is envy-free if every player prefers her own share than the share of any other player. When the goods are divisible, envy-free allocations always exist. In the presence of indivisibilities, we show that there exist allocations in which the envy is bounded by the maximum marginal utility, and present a simple algorithm for computing such allocations. We then look at the optimization problem of finding an allocation with minimum possible envy. In the general case the problem is not solvable or approximable in polynomial time unless P = NP. We consider natural special cases (e.g.additive utilities) which are closely related to a class of job scheduling problems. Approximation algorithms as well as inapproximability results are obtained. Finally we investigate the problem of designing truthful mechanisms for producing allocations with bounded envy.

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On approximately fair allocations of indivisible goods

Impossibility theorems suggest that the only efficient and strategyproof mechanisms for the problem of combinatorial assignment - e.g., assigning schedules of courses to students - are dictatorships. Dictatorships are mostly rejected as unfair: for any two agents, one chooses all their objects before the other chooses any. Any solution will involve compromise amongst efficiency, incentive and fairness considerations.This paper proposes a solution to the combinatorial assignment problem. It is developed in four steps. First, I propose two new criteria of outcome fairness, the maximin share guarantee and envy bounded by a single good, which weaken well-known criteria to accommodate indivisibilities; the criteria formalize why dictatorships are unfair. Second, I prove existence of an approximation to Competitive Equilibrium from Equal Incomes in which (i) incomes are unequal but arbitrarily close together; (ii) the market clears with error, which approaches zero in the limit and is small for realistic problems. Third, I show that this Approximate CEEI satisfies the fairness criteria. Last, I define a mechanism based on Approximate CEEI that is strategyproof for the zero-measure agents economists traditionally regard as price takers. The proposed mechanism is calibrated on real data and is compared to alternatives from theory and practice: all other known mechanisms are either manipulable by zero-measure agents or unfair ex-post, and most are both manipulable and unfair.

/pdf/the-combinatorial-assignment-problem-approximate-competitive-wf6yzn45q6.pdf

The combinatorial assignment problem: approximate competitive equilibrium from equal incomes

The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.

/pdf/handbook-of-computational-social-choice-12i1xbv1b8.pdf

Handbook of Computational Social Choice

The challenge of dividing an asset fairly, from cakes to more important properties, is of great practical importance in many situations. Since the famous Polish school of mathematicians (Steinhaus, Banach, and Knaster) introduced and described algorithms for the fair division problem in the 1940s, the concept has been widely popularized. This book gathers into one readable and inclusive source a comprehensive discussion of the state of the art in cake-cutting problems for both the novice and the professional. It offers a complete treatment of all cake-cutting algorithms under all the considered definitions of "fair" and presents them in a coherent, reader-friendly manner. Robertson and Webb have brought this elegant problem to life for both the bright high school student and the professional researcher.

Cake Cutting Algorithms: Be Fair If You Can

Near Exact and Envy Free Cake Division.

A characterization and hereditary properties for partition graphs

Approximating fair division with a limited number of cuts

A general partition graph (gpg) is an intersection graph G on a set S so that for every maximal independent set M of vertices in G, the subsets assigned to the vertices in M partition S. These graphs have been characterized by the presence of special clique covers. The Triangle Condition T for a graph G is that for any maximal independent set M and any edge uv in G M, there is a vcrtex W E M so that uvw is a triangle in G. Condition T is necessary but not sufficient for a graph to be a gpg and a computer search has found the smallest ten counterexamples, one with nine vertices and nine with ten verticcs. Any non-gpg satisfying Condition T is shown to induce a required subgraph on six vertices, and a method of generating an infinite class of such graphs is described. The main result establishes the equivalence of the following conditions in a chordal graph G: (i) G is a gpg (ii) G satisfies Condition T (iii) every edge in G is in an end-clique. The result is extended to a larger class of graphs.

https://ajc.maths.uq.edu.au/pdf/16/ocr-ajc-v16-p285.pdf

Jack M. Robertson

Papers

Cake Cutting Algorithms: Be Fair If You Can

Near Exact and Envy Free Cake Division.

A characterization and hereditary properties for partition graphs

Approximating fair division with a limited number of cuts

When are chordal graphs also partition graphs