Truthful and Fair Resource Allocation

229 Pages·2013·1.34 MB·English

Truthful and Fair Resource Allocation
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Lai, John Kwang. 2013. Truthful and Fair Resource Allocation. Doctoral dissertation, Harvard
University.
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Truthful and Fair Resource Allocation
A dissertation presented
by
John Kwang Lai
to
The School of Engineering and Applied Sciences
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Computer Science
Harvard University
Cambridge, Massachusetts
April 2013
(cid:13)c 2013 John Kwang Lai
All Rights Reserved.
Thesis advisor Author
David C. Parkes John Kwang Lai
Truthful and Fair Resource Allocation
Abstract
How should we divide a good or set of goods among a set of agents? There are various
constraints that we can consider. We consider two particular constraints. The first is
fairness – how can we find fair allocations? The second is truthfulness – what if we do
not know agents valuations for the goods being allocated? What if these valuations need
to be elicited, and agents will misreport their valuations if it is beneficial? Can we design
procedures that elicit agents’ true valuations while preserving the quality of the allocation?
We consider truthful and fair resource allocation procedures through a computational
lens. We first study fair division of a heterogeneous, divisible good, colloquially known as
the cake cutting problem. We depart from the existing literature and assume that agents
have restricted valuations that can be succinctly communicated. We consider the problems
of welfare-maximization, expressiveness, and truthfulness in cake cutting under this model.
In the second part of this dissertation we consider truthfulness in settings where pay-
ments can be used to incentivize agents to truthfully reveal their private information. A
mechanism asks agents to report their private preference information and computes an al-
location and payments based on these reports. The mechanism design problem is to find
incentive compatible mechanisms which incentivize agents to truthfully reveal their private
information and simultaneously compute allocations with desirable properties. The tradi-
tional approach to mechanism design specifies mechanisms by hand and proves that certain
desirable properties are satisfied. This limits the design space to mechanisms that can be
written down and analyzed. We take a computational approach, giving computational pro-
cedures that produce mechanisms with desirable properties. Our first contribution designs
a procedure that modifies heuristic branch and bound search and makes it usable as the
allocation algorithm in an incentive compatible mechanism. Our second contribution draws
anovelconnectionbetweenincentivecompatiblemechanismsandmachinelearning. Weuse
this connection to learn payment rules to pair with provided allocation rules. Our payment
rules are not exactly incentive compatibility, but they minimize a measure of how much
agents can gain by misreporting.
iii
Contents
Abstract iii
Acknowledgements ix
1 Introduction 1
1.1 Fairness and Cake Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Cake Cutting under Restricted Valuations . . . . . . . . . . . . . . . 4
1.1.2 Welfare Maximization in Cake Cutting . . . . . . . . . . . . . . . . . 6
1.1.3 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Truthfulness and Mechanism Design . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Truthful Cake Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Combinatorial Auctions . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.3 Computational Approaches to Mechanism Design . . . . . . . . . . . 14
1.2.4 Monotone Branch and Bound Search . . . . . . . . . . . . . . . . . . 16
1.2.5 Learning Payment Rules . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Bibliographic Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Resource Allocation and Mechanism Design 22
2.1 Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Concrete Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Payments and Quasi-Linear Utility . . . . . . . . . . . . . . . . . . . 23
2.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Truthful Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Truthfulness and the Revelation Principle . . . . . . . . . . . . . . . 29
2.2.3 Classic Mechanism Design Results . . . . . . . . . . . . . . . . . . . 32
iv
3 Cake Cutting 35
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Normalization of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Models of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1 Classic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2 Complexity of Cake Cutting . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.3 Direct Revelation Model . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Families of Valuation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5.1 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . 43
4 Welfare Maximization and Cake Cutting 44
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Computing Welfare Maximizing Fair Allocations . . . . . . . . . . . . . . . 45
4.2.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Piecewise Constant Valuations . . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Piecewise Linear Valuations . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.4 General Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Properties of Maxsum Fair Allocations . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Pareto Efficiency of Maxsum Allocations . . . . . . . . . . . . . . . . 59
4.3.2 Maxsum EQ vs. Maxsum EF Allocations . . . . . . . . . . . . . . . 64
4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Towards More Expressive Cake Cutting 76
5.1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 PUML Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Proportionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.1 Algorithmic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.2 Impossibility Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4 Proportionality and Envy-Freeness . . . . . . . . . . . . . . . . . . . . . . . 82
5.4.1 An Algorithmic Skeleton . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.2 Finding Fair Filtering, Point pairs . . . . . . . . . . . . . . . . . . . 84
v
5.4.3 Tying Things Together. . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.6 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Truthful Cake Cutting 90
6.1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4 Deterministic Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4.1 A Deterministic Mechanism . . . . . . . . . . . . . . . . . . . . . . . 95
6.4.2 The Two Agent Mechanism . . . . . . . . . . . . . . . . . . . . . . . 97
6.4.3 Exact Allocations and Maximum Flows . . . . . . . . . . . . . . . . 100
6.4.4 Polynomial Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4.5 Fairness, Efficiency, Truthfulness . . . . . . . . . . . . . . . . . . . . 104
6.5 Randomized Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.7 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Combinatorial Auctions 115
7.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3 Computational Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . 118
7.4 Single-Minded CAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8 Monotone Branch and Bound Search 121
8.1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.4 Ironing, Discretization and a First Approach . . . . . . . . . . . . . . . . . 125
8.5 Branch-and-Bound Search for Single-Minded CAs . . . . . . . . . . . . . . . 128
8.6 Optimized Sensitivity Checking for BnB . . . . . . . . . . . . . . . . . . . . 131
8.6.1 Impact of a Change in Value on a Search Node . . . . . . . . . . . . 132
8.6.2 Isolating Major Changes and Defining get-sensitivity . . . . . . . . . 134
vi
8.6.3 Correctness of get-sens-single-state . . . . . . . . . . . . . . . . . . . 135
8.6.4 Implementing get-sens-single-state . . . . . . . . . . . . . . . . . . . 137
8.6.5 Hot Restart and Inference . . . . . . . . . . . . . . . . . . . . . . . . 137
8.6.6 Linear Program Caching, Parallelization . . . . . . . . . . . . . . . . 140
8.7 Making Branch-and-Bound Search more Monotone . . . . . . . . . . . . . . 141
8.7.1 Input Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.7.2 Fractional Bucketing . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.8 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.8.1 Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.8.2 Effectiveness of Optimized Sensitivity . . . . . . . . . . . . . . . . . 146
8.8.3 Analysis of Search Changes . . . . . . . . . . . . . . . . . . . . . . . 147
8.8.4 Hard Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.9 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.9.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9 Learning Payment Rules 151
9.1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.4 Payment Rules from Multi-Class Classifiers . . . . . . . . . . . . . . . . . . 158
9.4.1 Mechanism Design as Classification . . . . . . . . . . . . . . . . . . . 158
9.4.2 Example: Single-Item Auction . . . . . . . . . . . . . . . . . . . . . 159
9.4.3 Perfect Classifiers and Implementable Outcome Rules . . . . . . . . 160
9.4.4 Approximate Classification and Approximate Strategyproofness . . . 162
9.5 A Solution using Structural Support Vector Machines . . . . . . . . . . . . 163
9.5.1 Structural SVMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.5.2 Structural SVMs for Mechanism Design . . . . . . . . . . . . . . . . 168
9.6 Applying the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.6.1 Multi-Minded Combinatorial Auctions . . . . . . . . . . . . . . . . . 172
9.6.2 Combinatorial Auctions with Positive k-wise Dependent Valuations. 177
9.6.3 The Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . . 184
9.7 Experimental Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.7.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.7.2 Single-Item Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9.7.3 Multi-Minded CAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
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9.7.4 Combinatorial Auctions with Positive k-wise Dependent Valuations. 195
9.7.5 The Egalitarian Assignment Problem. . . . . . . . . . . . . . . . . . 197
9.8 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10 Conclusions 202
10.1 Brief Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.1.1 Cake Cutting with Restricted Valuations . . . . . . . . . . . . . . . 203
10.1.2 Computational Approaches to Mechanism Design . . . . . . . . . . . 204
10.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Bibliography 209
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Acknowledgements
I could not have completed this dissertation without the guidance and support of many
people. First and foremost, thank you to my advisor David Parkes. David, you are the
best advisor anyone could ask for and a true inspiration. Thank you for all of the brilliant
research insights and guidance. Thank you for believing in me, advocating on my behalf,
and being caring, supportive, and understanding throughout my journey. I would like
to thank Ariel Procaccia, who introduced me to cake cutting and has been an incredible
collaborator, and the other members of my committee Yiling Chen and Vince Conitzer for
their time and valuable feedback. I am grateful for the EconCS group at Harvard; I could
not have asked for a more nurturing environment. Specific thanks to Michal Feldman, for
ourcollaborationsandhersupport, toSvenSeukenandHaoqiZhang, whostayeduplateat
conferences to give feedback on my talks, and to Andrew Mao, who has been a great friend
andalwayswillingtohelp. I’vebeenfortunatetocollaboratewithmanytalentedresearchers
including Moritz B¨acher, Steven Brams, Ioannis Caragiannis, Tanmoy Chakraborty, Yuga
Cohler, Paul Du¨tting, Felix Fischer, Pitchayut Jirapinyo, Ian Kash, David Kurokawa, Ben
Lubin, Jamie Morgenstern, and Aviv Zohar. I was generously supported by a National
Defense Science and Engineering Graduate Fellowship during my time in graduate school.
Outside of research, I have been blessed with a supportive community of friends and
family. Adi, Dennis, Ian, Matt, Naimish, Nick, and Steve, thanks for your unwavering
friendship and support. James and Jerry, thanks for the brotherly love. Mom and Dad,
thanksforallthesacrificesyou’vemadeandstillmakeonourbehalf. Yourdedication, love,
and care are amazing. Wendy, Chris, and his little brother, I love you. Thank you for the
joy you bring to my life.
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