Strategic and Normative Analysis of Resource Allocation Problems by Vikram Manjunath ...

136 Pages·2011·1.28 MB·English

Strategic and Normative Analysis of Resource Allocation Problems
by
Vikram Manjunath
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor William Thomson
Department of Economics
Arts, Sciences and Engineering
School of Arts and Sciences
University of Rochester
Rochester, New York
2011
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To Troy and Milla.
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Curriculum Vitae
The author was born in Howell, Michigan, on the eleventh of February, 1982.
He attended Carnegie Mellon University from 2000 to 2003, and graduated with a
Bachelor of Science in Computer Science. He came to the University of Rochester
in the Fall of 2005 and began graduate studies in Economics. He received a
fellowship from the Department of Economics at the University of Rochester from
2005to2010. HereceivedaMasterofArtsdegreefromtheUniversityofRochester
in 2007 and pursued his research in Microeconomic Theory under the guidance of
Professor William Thomson.
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Acknowledgments
It is impossible for me to adequately express the debt that I owe my advisor
William Thomson for teaching me so much about more than just economics. That
being so, I offer this Sanskrit verse to my teacher:

!#" %$&' !#" )(*+" !#" -,(. /0123

!#" 4'5'67 82%& 69: ;< !#" = >/3
My mother, Shoba Manjunath, deserves all of the credit (but none of the
blame) for my being able to write this dissertation. It was she who taught me
to read, to write, and even to think. I thank both her and my father, Kudur
Manjunath, for their love and encouragement without which I would not have
overcome the many difficult periods of my graduate studies.
Finally, I express my gratitude to each of the following people for at least one
of these reasons:
1. Without their friendship and support I would not have had the opportunity
to write the “acknowledgments” page of a Ph.D. dissertation.
2. I have learned so much from the invaluable insights that they have shared
with me.
3. It has been a pleasure to work with them as a co-author.
Azer Abizade Paulo Barelli Seema Bhatt
Keri Donlon John Duggan Eun Jeong Heo
Michael Insler Paula Jaramillo Tasos Kalandrakis
Yoichi Kasajima Bettina Klaus Veena Manjunath
Romans Pancs Michelle Parnett Karol Szwagrzak
Ram Velu Gabor Virag
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Abstract
Economicsisthestudyofhowpeopleuseresources. Thesubjectofthisvolume
is then the study of economic problems.
When resources are scarce, there are limits to the ways in which they can
be used. In such an environment, people must choose a good way to allocate
resources. Defining what we mean by “good” depends on the economic context
and itself can be a challenging exercise.
Once we know what makes a solution to an allocation problem a good one, the
next task is to check that there is a good solution. If there is, a natural question
is to ask “what are all of the good solutions?”
This is the line of analysis that we apply to several economic situations.
In the first chapter, we study the problem of choosing a point in a one-
dimensional set over which people have single-dipped preferences. This applies
to situations such as dividing a budget between two alternative public goods, lo-
cating a “public bad,” and so on. The primary requirement that we examine is
that an allocation rule be “strategy-proof.” That is, it ought to be immune to
strategic manipulations. We characterize the class of strategy-proof rules.
In the second chapter, we consider the public choice between two alternatives
when people may be indifferent between them. We analyze the implications of
strategy-proofness and a more demanding version of strategy-proofness that deals
with manipulations by groups. We identify the classes of all rules that satisfy each
of these properties.
Inthethirdchapter,welookattheproblemofrationingadivisiblegoodamong
a group of people. Each person’s preferences are characterized by an ideal amount
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that he would prefer to receive and a minimum quantity that he will accept: he
finds any amount less than this threshold to be just as good as receiving nothing
at all. Further, any amount beyond his ideal quantity has no effect on his welfare.
We are interested in strategic properties, efficiency, and equity. We find these
three requirements to be incompatible and identify the classes of all rules that
satisfy each pair of them.
The fourth and fifth chapters differ in structure from the earlier chapters. We
analyze a novel model of “fractional matching.” In the fourth chapter we define
several normatively appealing properties of allocations and sort out their logical
relations.
In the fifth and final chapter, we take a new approach to matching and frame
fractional matching problems in a general equilibrium context. This allows us to
define a market-based solution to that has several desirable properties.
Table of Contents
Chapter 1 Efficient and strategy-proof social choice when prefer-
ences are single-dipped 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 More general sets of alternatives . . . . . . . . . . . . . . . . . . . 21
Chapter 2 Groupstrategy-proofnessandsocialchoicebetweentwo
alternatives 24
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Chapter 3 When too little is as good as nothing at all: Rationing a
disposable good among satiable people with acceptance
thresholds 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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viii
3.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 4 A chocolate box of cores for the probabilistic marriage
problem 64
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 A chocolate box of cores . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Logical relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Existence (or lack of it) . . . . . . . . . . . . . . . . . . . . . . . . 75
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 5 A market approach to fractional matching 78
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Existence of a core allocation . . . . . . . . . . . . . . . . . . . . 88
5.5 Nonemptiness of lim-DIP . . . . . . . . . . . . . . . . . . . . . . . 90
5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.7 Kinds of people . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Bibliography 108
Chapter A: Appendices to Chapter 1 115
A.1 The mapping ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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A.2 The Pareto-set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.3 Strong group strategy-proofness . . . . . . . . . . . . . . . . . . . 119
Chapter B: Appendices to Chapter 3 122
B.1 Proofs of some of the incompatibilities . . . . . . . . . . . . . . . 122
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List of Tables
Table Title Page
3.1 Rules and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 48