Axiomatic Analysis of Problems in Economic Theory by Andrew Mackenzie Submitted in Partial ...

200 Pages·2017·7.71 MB·English

Axiomatic Analysis of Problems in Economic Theory
by
Andrew Mackenzie
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
2017
ii
To my parents
iii
Table of Contents
Biographical Sketch v
Acknowledgments vi
Abstract vii
Contributors and Funding Sources viii
List of Figures ix
Chapter 1 A Foundation for Probabilistic Beliefs
with or without Atoms 1
1.1 Introduction 1
1.2 Model and Main Results 12
1.3 Proof Sketch for Theorem 1 19
1.4 Proof Sketch for Theorem 2 33
1.5 Proof Sketch for Theorems 3 and 4 35
1.6 Proof Sketch for Theorem 5 43
1.7 Conclusion 44
Appendix 1.1 46
Appendix 1.2 49
Appendix 1.3 55
Appendix 1.4 62
Appendix 1.5 67
Appendix 1.6 79
Appendix 1.7 87
Appendix 1.8 92
Appendix 1.9 104
Chapter 1 References 112
Chapter 2 A Game of the Throne of Saint Peter 120
2.1 Introduction 120
iv
2.2 Historical Analysis 125
2.3 Scrutiny Design 132
2.4 Deadlock 139
2.5 Conclusion 144
Appendix 2 147
Chapter 2 References 153
Chapter 3 Symmetry and Impartial Lotteries 159
3.1 Introduction 159
3.2 The Setting 164
3.3 Results and Discussion 169
3.4 Conclusion 174
Appendix 3.1 176
Appendix 3.2 182
Chapter 3 References 190
v
Biographical Sketch
The author was born in Norwalk, Connecticut, USA. He attended Union Col-
lege and graduated summa cum laude with a Bachelor of Science degree in Mathe-
matics and Economics. He began doctoral studies in Economics at the University
of Rochester in 2011. He was awarded a Robert L. and Mary L. Sproull Fellow-
ship in 2011 by the University of Rochester. He pursued his research in economic
theory under the direction of Professor William Thomson.
The following publications were a result of work conducted during doctoral
study:
• Mackenzie, A.(2017). “Afoundationforprobabilisticbeliefswithorwithout
atoms.” Revised and resubmitted to Theoretical Economics.
• Mackenzie, A. (2015). “Symmetry and impartial lotteries.” Games and Eco-
nomic Behavior 94, 15-28.
vi
Acknowledgments
It has truly been a privilege, and a delight, to spend these years with the
Department of Economics at the University of Rochester. I would like to thank
Yu Awaya, Narayana Kocherlakota, Asen Kochov, Romans Pancs, and especially
Paulo Barelli, Srihari Govindan, and Christian Trudeau, for their significant con-
tributions to my development as a scholar. Most of all, I am extraordinarily grate-
ful to four people for their monumental support over the course of this degree’s
completion: my parents, Joanne Green Mackenzie and Kenneth Albert Mackenzie;
my partner, Kayleigh Dunn; and my advisor, William Thomson.
vii
Abstract
The purpose of this thesis is to demonstrate the application of the axiomatic
method to a series of problems in economic theory. In these problems, an axiom
is a mathematical property that captures an idea whose implications are of inter-
est, and the analysis involves the logical deduction of the joint implications of a
collection of these axioms.
InChapter1, Iinvestigatewhensomeone’sbeliefs—theircomparisonsofevents
on the basis of relative likelihood—may be represented with a probability mea-
sure. Here, an axiom expresses a property of beliefs that captures some aspect
of idealized rationality. I propose axioms that allow for events that cannot be
divided into two less-likely events, or atoms, and prove that these axioms suffice
for the representation.
In Chapter 2, I investigate how the Roman Catholic Church should elect the
pope. Here, an axiom expresses a property of an election rule that, through
historical analysis, I identify as an objective of the church. According to my
analysis, the church should overturn a 1945 change of Pope Pius XII to reinstate
the rule of Pope Gregory XV.
In Chapter 3, I investigate how a prize should be randomly assigned to a
candidate based on nominations from the candidates. Here, an axiom expresses
a property of a nomination rule that is normatively appealing. My analysis gives
a complete description of the rules that treat participants symmetrically without
providing incentives for dishonest nomination, according to two novel notions of
symmetric treatment.
viii
Contributors and Funding Sources
This work was supported by a dissertation committee consisting of Professor
William Thomson (advisor) and Professor Paulo Barelli of the Department of Eco-
nomics and Professor Guy Arie of the Simon Business School. All work conducted
for the dissertation was completed by the student independently. Graduate study
was supported by a Robert L. and Mary L. Sproull Fellowship from the University
of Rochester.
ix
List of Figures
Figure Title Page
Figure 1 Strict third-order atom-swarming. 3
Figure 2 The results and their relationships. 20
Figure 3 Supercabinet construction. 31
Figure 4 Half-Equivalence Lemma Euler diagram. 63
Figure 5 The ballot layout of Pope Gregory XV (1622)
as printed in 1692. 131
1
1 A Foundation for Probabilistic Beliefs
with or without Atoms
1.1 Introduction
1.1.1 Executive summary
From the doctor’s choice of treatment, to the employer’s choice of job applicant,
to the investor’s choice of portfolio, to the mortal’s choice of religion, and beyond:
much behavior, including much of the economic behavior we observe and strive to
model, is the selection of an action with uncertain consequences. Our standard
model is founded on the postulate that such choices, when made by someone
who is rational, can be decomposed into (1) beliefs about the relative likelihood
of events, and (2) tastes among outcomes (Ramsey, 1931). This article revisits
a classic question: when are such beliefs consistent with standard probability
theory?1
More precisely, suppose we are given a nonempty set of states S, a σ-algebra
of events A ⊆ 2S with S ∈ A, and a qualitative probability (cid:37) on A: a binary
relation on A, consisting of comparisons of events on the basis of relative likeli-
hood, satisfying minimal probabilistic requirements (Bernstein, 1917; de Finetti,
1937; Koopman, 1940; Savage, 1954). When does (cid:37) admit representation by a
σ-measure2 µ : A → [0,1]?
A necessary condition is monotone continuity (Villegas, 1964; Arrow, 1970): if
B ⊇ B ⊇ ..., and for each i ∈ N, B (cid:37) A, then ∩B (cid:37) A.3 On the appeal of this
1 2 i i
axiom, Arrow writes: “The assumption of Monotone Continuity seems, I believe
correctly, to be the harmless simplification almost inevitable in the formalization
1MachinaandSchmeidler(1992)callthisquestionthefirstoftwolinesofinquiryculminating
in the modern theory of subjective probability.
2Inthisarticle, ameasureisafinitely-additiveprobabilitymeasure, σ-additivityiscountable-
additivity, and a σ-measure is a σ-additive measure.
3Villegas (1964) and Arrow (1970) give different statements that are logically equivalent; I
owe this particularly elegant statement to a referee.