Lexicographic and Social Choice, Takeovers, and Fixed Points

Lexicographic and Social Choice, Takeovers, and

Fixed Points

Lexicographic and Social Choice, Takeovers, and Fixed Points

Henrik Petri

Dissertation for the Degree of Doctor of Philosophy, Ph.D., in Economics

Stockholm School of Economics, 2017

Lexicographic and Social Choice, Takeovers, and Fixed Points

© SSE and Henrik Petri, 2017 ISBN 978-91-7731-051-8 (printed) ISBN 978-91-7731-052-5 (pdf)

This book was typeset by the author using L^ATEX.

Printed by:

Ineko, Göteborg, 2017 Keywords:

Lexicographic preferences, bounded rationality, incomplete preferences, social choice, fixed points, leveraged tender offers, free-rider problem.

To Anna

Foreword

This volume is the result of a research project carried out at the Department of Finance at the Stockholm School of Economics (SSE).

This volume is submitted as a doctoral thesis at SSE. In keeping with the policies of SSE, the author has been entirely free to conduct and present his research in the manner of his choosing as an expression of his own ideas.

SSE is grateful for the financial support provided by the Jan Wallander and Tom Hedelius Foundation and the Tore Browaldh Foundation which has made it possible to carry out the project.

Göran Lindqvist Magnus Dahlquist

Director of Research Professor and Head of the Stockholm School of Economics Department of Finance

Stockholm School of Economics

Acknowledgements

To be a researcher I think it helps to be interested in problem solving. However, equally or more important is the ability to pose nuanced and relevant questions.

My advisor Mark Voorneveld has been a source of inspiration in this later respect.

I approached him in Fall 2014 to discuss a research problem that I was interested in.

Mark subtly, but generously proposed another research topic to work on, which later led to my first publication with him. I learnt a lot from that experience. I thank Mark for setting me on the road. I also thank him for critically assessing, and for providing detailed comments on, every aspect of the first five chapters of this thesis. I can assure you: all remaining errors are my own.

I thank Mike Burkart for guiding me through the first phase of the PhD program, for being open-minded and for encouraging me to pursue my interests in economic theory. I am grateful to Mike Burkart and his former student Samuel Lee for advising and co-authoring the sixth chapter of this thesis.

I am thankful for the scholarship provided by the Jan Wallander and Tom Hedelius Foundation and the Tore Browaldh Foundation, which has made it possible to complete this thesis.

I thank the faculty and staff at the Stockholm School of Economics for providing a friendly and inspiring research environment. I am indebted to Jungsuk Han for valuable guidance and encouragement during the Job Market. A particular thanks also goes to Ramin Baghai, Tomas Björk, Peter Englund, Vincent Maurin and Farzad Saidi. The administrative staff Anki, Anneli, Jenny and Hedvig, thank you for providing invaluable help throughout the PhD program.

I thank my fellow PhD students for making the PhD years pleasant and joyful.

Particular thanks goes to: Filipe Pires de Albuquerque, Tommy von Brömsen, Fatemeh Hosseini, Mats Levander, Jieying Li, Yingjie Qi, Valeri Sokolovski and Tomas Thörnqvist.

I am grateful to my friends outside academia. A special thanks to my childhood friends from Värmdö. You have given me, and continue to give me, a lot of fun memories.

My family means everything to me. I thank my parents Johan and Pia for their unconditional love and support and for believing in me. I thank my brother and sister, Gustav and Ebba, for making me proud. I thank my incredible and amazing fiancée Anna for her love and support throughout the writing of this thesis. I

thank her for giving birth to my beautiful daughter Wilma and I thank Wilma for being who she is: truly wonderful. Thank you for lighting up my days.

Sjöhaga, July 19, 2017 Henrik Petri

Contents

1 Characterizing lexicographic preferences 5

1.1 Introduction . . . 5

1.2 Notation and definition of lexicographic preferences . . . 7

1.3 Axioms and discussion . . . 10

1.4 Characterization of lexicographic preferences . . . 16

1.5 A comment on the robustness property . . . 18

1.6 Logical independence . . . 19

1.7 Relations with Fishburn’s characterizations . . . 21

1.A Appendix: Further examples of robust preferences . . . 26

References 28 2 Lexicographic Majority 31 2.1 Introduction . . . 31

2.2 The Model . . . 33

2.3 Axioms and discussion . . . 34

2.4 The characterization theorem . . . 37

2.5 Lexicographic majority as a search procedure . . . 43

2.6 A characterization in terms of weighted majority preferences . . . . 44

2.7 Concluding remarks . . . 47

2.A Appendix: Notation and preliminaries . . . 48

2.B Appendix: Proofs . . . 49

2.C Appendix: Pareto lemma . . . 54

2.D Appendix: Logical independence . . . 54

References 57 3 Choice by Incomplete Checklist 59 3.1 Introduction . . . 59

3.2 Model example . . . 62

3.3 The Model . . . 64

3.4 An alternative description of CIC . . . 67

3.5 Identification of CIC . . . 68

3.6 Checklist with ties. . . 70 ix

x

3.7 Rationalization by incomplete preferences . . . 72

3.8 Checklists of arbitrary length . . . 73

3.9 Relation to other models of choice . . . 78

3.A Appendix: Proofs . . . 82

References 93 4 Asymptotic properties of welfare relations 95 4.1 Introduction . . . 95

4.2 Notation and preliminaries . . . 97

4.3 Welfare aggregators and axioms. . . 99

4.4 Part 1: The conflict between finite anonymity and efficiency . . . . 102

4.5 Part 2: The conflict between bounded anonymity and efficiency principles . . . 107

4.6 Conclusion . . . 110

4.A Appendix: Proofs . . . 111

4.B Appendix: Bounded anonymity and weak Pareto . . . 119

References 120 5 No bullying! Brouwer’s fixed-point theorem 123 5.1 Introduction . . . 123

5.2 The no-bullying lemma and why it implies Brouwer . . . 125

5.3 Proof of the no-bullying lemma . . . 128

5.4 Concluding remarks . . . 131

5.A Appendix: The KKM Lemma . . . 133

References 135 6 Financing Takeovers: Dilution as Commitment 137 6.1 Introduction . . . 137

6.2 Model. . . 141

6.3 Post-takeover incentives and tender offer design . . . 143

6.4 Target shareholder wealth and social welfare . . . 154

6.5 Concluding remarks . . . 163

6.A Appendix: Proofs . . . 164

References 166

Introduction

This thesis consists of six independent theoretical papers in economic theory. Mark Voorneveld supervised the first five papers. The sixth paper was supervised by Mike Burkart. Although each paper analyses a separate problem, a common thread is preferences of agents. We model preferences by using binary relations. A binary relation%on a set of alternatives X is a subset of X×X. An interpretation is that x “is preferred to” or “at least as good as” y whenever (x, y) ∈%^{. The} first three papers are about lexicographic preferences and the properties that they satisfy. The fourth paper is about social welfare. In particular we ask if it is possible to aggregate preferences of individuals in a society to obtain an aggregated preference relation satisfying certain (desirable) properties. The fifth paper shows that Brouwer’s fixed-point theorem can be reduced to a ‘no-bullying’ lemma for agents with preferences over indivisible goods. The sixth paper analyzes the role of financing in tender offers with endogenous post-takeover value creation. Given certain assumptions on the raider’s preferences, we show that target shareholders as well as the raider benefit from the use of more debt in takeovers. A short introduction to each of the papers now follows.

Characterizing lexicographic preferences (with Mark Voorneveld)

We study lexicographic preferences. The purpose of the paper is to characterize lexicographic preferences on a product set X = ×_i∈IX_i of finitely many coordi- nates. The challenge, of course, is to do so without explicit reference to the order of importance of the different coordinates: the same set X allows for different lexico- graphic preferences by looking at different priorities of the coordinates. Informally, preferencesare lexicographic if we can find (a) an order on the coordinates i∈ _I and (b) for each coordinate i ∈ I, a preference_i on the coordinate set Xi such that x= (x_i)_i∈I is preferred to y= (y_i)_i∈I if and only if some coordinate makes a difference (x_{i i} y_ior y_{i i} x_i) and the first such coordinate makes x look better than y.

The main new axiom in our characterization is a robustness property. It roughly requires this: Suppose x is preferred to y; many of its coordinates indicate that the former is better and only a few indicate the opposite. Then the decision maker is allowed a change of mind turning one coordinate in favor of x to an indifference:

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even if one less argument supports the preference, the fact that we started with many arguments in favor of x suggests that such a small change is not enough to give rise to the opposite preference.

Lexicographic Majority

We propose a descriptive theory of lexicographic choice, lexicographic majority, where a decision maker’s ranking of attributes is not necessarily strict. Thus we allow an agent to assign the same rank to a set of attributes. If a decision maker assigns the same rank to a set of attributes, it is assumed that the simple majority heuristic is used, within the set, to discriminate between pairs of vectors. Our model is general enough to account for intransitive preferences. Moreover, it includes as special cases: pure lexicographic preferences, simple majority prefer- ences and lexicographic semiorders. We justify lexicographic majority preferences by providing an axiomatization in terms of behavioral properties. As a further justification we characterize lexicographic majority preferences as a subset of the class of weighted majority preferences.

Choice by Incomplete Checklist

We develop a choice theoretic model. A decision maker (DM) chooses alternatives from menus by proceeding sequentially through a checklist of properties. In contrast to the standard framework her ranking of properties is not necessarily complete. I.e. the DM may fail to rank certain pair of properties in order of importance. We characterize the model in terms of observed choices. As another main result of the paper we show that an important special case of our model is exactly equivalent to the maximization of an incomplete preference relation. We also relate the model to the choice model resulting from the maximization of a multi-utility (also called multi-criteria) function. Our results suggest a close rela- tionship between the degree of completeness of the order by which properties in a checklist are looked at, and the degree of completeness of the revealed preference relation maximizing the corresponding choice function.

Asymptotic properties of welfare relations

We introduce and discuss notions of efficiency in the aggregation of infinite utility streams. For any utility streams x and y, our efficiency criteria roughly require

CONTENTS 3 this: If a utility stream x dominates another utility stream y and if the asymptotic density of the set of coordinates in favor of x is strictly positive, then x is socially preferred to y. Asymptotic density is a notion designed to measure how large certain subsets of the natural numbers are. It formalizes the intuition that the set of even numbers is larger than the set of multiples of three, even though there is a one to one correspondence between them (both are countable). As a robustness check of the proposed efficiency axioms we explore the consistency of the axioms with notions of equity. Our main results characterize one period utility domains, i.e. the set of utilities Y attainable by each generation, admitting a social welfare aggregator with the desired properties.

No bullying! A playful proof of Brouwer⁰s fixed-point theorem (with Mark Voorneveld)

We give an elementary proof of Brouwer’s fixed-point theorem: in Euclidean spaceRⁿ , a continuous function from and to the unit simplex of nonnegative vectors with coordinates summing to one has a fixed point. The proof is accessible with a minimal mathematical background. The only mathematical prerequisite is a version of the Bolzano-Weierstrass theorem: a sequence in a compact subset of n-dimensional Euclidean space has a convergent subsequence with a limit in that set. Our main tool is a ‘no-bullying’ lemma for agents with preferences over indivisible goods. What does this lemma claim? Consider a finite number of children, each with a single indivisible good (a toy) and preferences over those toys. Let’s say that a group of children, possibly after exchanging toys, could bully some poor kid if all group members find their own current toy better than the toy of this victim. The no-bullying lemma asserts that some group S of children can redistribute their toys among themselves in such a way that all members of S get their favorite toy from S, but they cannot bully anyone.

Financing Takeovers: Dilution as Commitment (with Mike Burkart and Samuel Lee)

We analyze the role of financing in tender offers with endogenous post-takeover value creation. Optimal takeover financing, which determines the target firm’s capital and ownership structure, reflects the interaction of the exclusion benefit of debt in the face of the free-rider problem with financing constraints induced by moral hazard. Raising more debt to extract private gains requires the bidder

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to acquire more equity as commitment vis-à-vis lenders. Takeover leverage thus increases ownership concentration and post-takeover firm values. Besides enhanc- ing efficiency, takeover leverage can also be Pareto-improving, allowing the bidder to extract more gains, while target shareholders simultaneously enjoy a larger premium. Hence, target shareholders may prefer not to restrict the use of debt in takeovers.