Essays in Matching and Resource Allocation Problems by Azar Abizada Submitted in Partial ...

128 Pages·2012·0.47 MB·English

Essays in Matching and Resource Allocation Problems
by
Azar Abizada
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
2012
ii
To my parents, Dilruba Piriyeva and Etibar Abizade.
iii
Curriculum Vitae
The author was born in Baku, Azerbaijan on July 19, 1984. He attended
Boˇgazi¸ci University from 2001 to 2006, and graduated with a Bachelor of Arts
in Business Administration. He came to the University of Rochester in the Fall
of 2006 and began graduate studies in Economics. He received a University of
Rochester Fellowship during the years 2006 to 2011. He received the Master of
Arts degree from the University of Rochester in 2009 and pursued his research in
Microeconomic Theory under the supervision of Professor William Thomson.
iv
Acknowledgments
First and foremost, I am deeply grateful to my advisor William Thomson for
his encouragement, constant support and motivations throughout my studies, for
his unending patience, and insightful and detailed comments on all of my work. I
have been extremely fortunate and privileged to have him as my advisor.
I am really indebted to Paulo Barelli for his motivations especially in early
years of my studies, his support throughout the program, and insightful discus-
sions. I am also thankful to G´abor Vir´ag for his useful comments and support.
I am grateful to my coauthor Siwei Chen not only for the pleasure of working
with him, but also for his friendship. I also want to thank Srihari Govindan,
Asen Kochov, Romans Pancs, John Duggan, Fuhito Kojima, Onur Kesten, Utku
¨
Unver,JamesSchummer,NigarHashimzade,BattalDoˇgan,EunJeongHeo,Paula
Jaramillo, C¸aˇgatay Kayi, Rodrigo Velez, Yoichi Kasajima, Vikram Manjunath,
Ayse Yazici, Karol Szwagrzak, Nanyang Bu, Eray Cumbul, Sanan Mirzayev for
their valuable feedbacks and comments. I am also grateful to Rajab Davudov,
Samir Nabiyev, and Goshgar Hajili for their moral support and friendship that
was a source of my strength.
Finally, and most importantly, my deepest gratitude goes to my parents Dil-
ruba Piriyeva and Etibar Abizade, and my sister, Aysel Abizade. Without their
support I would never be able to complete this dissertation. I want to thank them
for their continual understanding, inspiration, constant encouragement, their pa-
tience and love.
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Abstract
This thesis is a collection of essays on matching and resource allocation problems.
In Chapters 1 and 2, we study college admission. Each college has strict
preferences over sets of students, and each student has preferences over colleges.
In Chapter 1, we allow monetary transfers, e.g., students may receive stipends
from colleges. We specify a fixed budget for each college. The preferences of the
students are over college-stipend bundles. We define pairwise stability. We con-
struct an algorithm. The rule defined through this algorithm is not only pairwise
stable but also incentive compatible for the students.
In Chapter 2, we consider a central exam students should take. We allow
different students to receive the same scores. We define and study an eligibility
criterion. Students are divided into two groups: those who are eligible to apply
to colleges, and those who are not. Eligibility respects the students’ scores. We
introduce three notions of stability that respect eligibility. We define three new
rules, each of which satisfies a different notion of stability. We also study an
incentive compatibility requirement.
Chapters 3 and 4 investigate the implications of a robustness principle for
different resource allocation problems. In Chapter 3, we study the problem of
assigning a set of objects, e.g., houses or tasks, to a group of individuals having
preferencesovertheseobjects. Wemayobservecaseswhentherearemoreorfewer
of objects to be allocated than was planned. We formulate these changes in two
new robustness properties. We characterize the family of rules that satisfy either
of these properties, together with efficiency, fairness and incentive compatibility
requirements. They are the sequential priority rules.
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In Chapter 4, we study the problem of allocating a divisible good among a
group of people. Each person’s preferences are single-peaked. Here too, we may
observe the cases when there might be more or fewer of resources to be allocated
than was planned. We formulate changes in a robustness property. We show
that there is a unique rule that is efficient and fair, and satisfies our composition
property. This rule is the uniform rule.
Table of Contents
Foreword 1
Chapter 1 Pairwise stability and strategy-proofness for college ad-
missions with budget constraints 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Paths to Pairwise Stability . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.7 Concluding Remarks and Discussions . . . . . . . . . . . . . . . . 46
Chapter 2 Stability and strategy-proofness for college admission
problems with an eligibility criterion 49
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 3 Allocating tasks when some may get canceled or addi-
tional ones may arrive 71
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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Chapter 4 Two characterizations of the uniform rule based on a
robustness property 92
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography 114
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List of Tables
Table Title Page
2.1 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . 70
4.1 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . 112
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List of Figures
Figure Title Page
1.1 Preferences of a student. . . . . . . . . . . . . . . . . . . . . . . . 7
4.1 Shifting operation on preferences. . . . . . . . . . . . . . . . . . . 98
4.2 Symmetry operation on preferences. . . . . . . . . . . . . . . . . . 100