On the Fair and Efficient Allocation of Indivisible Commodities
On the Fair and Efficient Allocation of Indivisible Commodities

Author: Richard Engelbrecht-Wiggans

Publisher:

Published: 1978

Total Pages: 280

ISBN-13:

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Auctions and fair division problems are situations in which commodities are to be allocated fairly and efficiently. While a variety of schemes exist for fairly allocating finely divisible homogeneous commodities, most schemes are not applicable to the problem of allocating indivisible items. This paper considers the problem of fairly allocation sets of indivisible objects. 'Dollars, ' a finely divisible, homogeneous, transferrable commodity, are used to evaluate individuals preferences and to transfer value among individuals. This introduction of dollars has several implications; the main result is that fair allocation problems may be viewed as two smaller problems. First auction the goods among the individuals and then divide the resulting revenue according to the chosen definition of fairness. Several existing fair allocation schemes are reviewed; examples illustrate some difficulties associated with their use. Kuhn's definitions of 'fairness' are presented and two extensions are considered for the case where individuals have different shares in the collection of goods.

Efficiency and Fairness in the Allocation of Indivisible Goods
Efficiency and Fairness in the Allocation of Indivisible Goods

Author: Akina Ikudo

Publisher:

Published: 2021

Total Pages: 147

ISBN-13:

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This dissertation studies the efficient and fair allocation of indivisible goods without monetary transfer. It is a collection of three papers and uses school-choice programs as a motivating example. I provide theoretical results that can guide the design of new allocation systems as well as tools that can be used to enhance existing systems. In Chapter 1, I analyze how information disclosure affects social welfare using a stylized model. In my model, the utility of agents consists of a vertical "quality" component and a horizontal "idiosyncratic taste" component. The exact qualities of the objects are unknown to the agents, and the social planner seeks an information-disclosure policy that will maximize the total utility. The results show that (1) the optimal disclosure policy hides small differences in quality and reveals large differences in quality, (2) more information is disclosed when the valuations of the quality are heterogeneous, and (3) the Immediate Acceptance mechanism is more conducive for information disclosure than the Deferred Acceptance mechanism. In Chapter 2, I study the collocation of groups of students in school-choice programs. In particular, I examine when and how stochastic assignment matrices can be decomposed into lotteries over deterministic assignments subject to collocation constraints. I first show that---regardless of the number of pairs of twins in the student body---twin collocation can be maintained in a decomposition if one extra seat can be added to each school. I then propose a decomposition algorithm based on Column Generation that can incorporate a wide variety of constraints including collocation constraints. In Chapter 3, I propose a new notion of fairness that combines the concept of rank values and the maximin principle. An assignment is rank-egalitarian undominated (REU) if there is no other assignment that is equally or more egalitarian for any set of rank values. I show that each REU assignment can be generated as a solution to a linear programming problem that maximizes the weighted sum of expected rank values of the worst-off agents. I also provide an algorithm that generates special subsets of REU assignments that are practically important.

Fair Allocation
Fair Allocation

Author: H. Peyton Young

Publisher: American Mathematical Soc.

Published: 1985-12-31

Total Pages: 188

ISBN-13: 9780821867402

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This collection of six papers provides a valuable source of material on the real-world problem of allocating objects among competing claimants. The examples given show how mathematics, particularly the axiomatic method, can be applied to give insight into complex social problems. Originally presented as an AMS Short Course, these papers could serve as a suitable text for courses touching on game theory, decision sciences, economics, or quantitative political science. Most of the material is accessible to the mathematically mature undergraduate with a background in advanced calculus and algebra. Each article surveys the recent literature and includes statements and sketches of proofs, as well as unsolved problems which should excite student curiosity. The articles analyze the question of fair allocation via six examples: the apportionment of political representation, the measurement of income inequality, the allocation of joint costs, the levying of taxes, the design of voting laws, and the framing of auction procedures. In each of these examples fairness has a somewhat different significance, but common axiomatic threads reveal broad underlying principles. Each of the topics is concerned with norms of comparative equity for evaluating allocations or with standards of procedures for effecting them; it is this focus on normative properties which suggests that a mathematical analysis is appropriate. Though game theory provides a useful tool in many of these allocation problems, the emphasis here is on standards rather than strategy and equity rather than rationality, an approach which more accurately mirrors real-world social problems.

Stability and Perfection of Nash Equilibria
Stability and Perfection of Nash Equilibria

Author: Eric van Damme

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 354

ISBN-13: 3642582427

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I have been pleased with the favourable reception of the first edition of this book and I am grateful to have the opportunity to prepare this second edition. In this revised and enlarged edition I corrected some misprints and errors that occurred in the first edition (fortunately I didn't find too many) and I added a large number of notes that give the reader an impression of what kind of results have been obtained since the first edition was printed and that give an indication of the direction the subject is taking. Many of the notes discuss (or refer to papers discussing) applications of the refinements that are considered. Of course, it is the quantity and the quality of the insights and the applications that lend the refinements their validity. Although the guide to the applications is far from complete, the notes certainly allow the reader to form a good judgement of which refinements have really yielded new insights. Hence, as in the first edition, I will refrain from speculating on which refinements of Nash equilibria will survive in the long run. To defend this position let me also cite Binmore [1990] who compares writing about refinements to the Herculean task of defeating the nine-headed Hydra which grew too heads for each that was struck off. It is a pleasure to have the opportunity to thank my secretary, Marjoleine de Wit, who skilfully and, as always, cheerfully typed the manuscript and did the proofreading.