Mechanism Design with Approximate Types Zeyuan Allen Zhu

101 Pages·2012·1.2 MB·English

Mechanism Design with Approximate Types
by
Zeyuan Allen Zhu
B.S. in Mathematics and Physics, Tsinghua University (2010)
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2012
(cid:13)c Zeyuan Allen Zhu, MMXII. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
Author ..............................................................
Department of Electrical Engineering and Computer Science
January 6, 2012
Certified by..........................................................
Silvio Micali
Professor
Thesis Supervisor
Accepted by.........................................................
Leslie A. Kolodziejski
Chairman, Department Committee on Graduate Theses
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Mechanism Design with Approximate Types
by
Zeyuan Allen Zhu
Submitted to the Department of Electrical Engineering and Computer Science
on January 6, 2012, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
In mechanism design, we replace the strong assumption that each player knows his
own payoff type exactly with the more realistic assumption that he knows it only
approximately: each player i only knows that his true type θ is one among a set K ,
i i
and adversarially and secretly chosen in K at the beginning of the game.
i
This model is closely related to the Knightian [20] notion of uncertainty in eco-
nomics, but we consider it from purely mechanism design’s perspective. In particular,
we study the classical problem of maximizing social welfare in auctions when players
know their true valuations only within a constant multiplicative factor δ ∈ (0,1).
For single good auctions, we prove that no dominant-strategy mechanism can
guarantee better social welfare than assigning the good to a random player. On the
positiveside,weprovidetightupperandlowerboundsforthesocialwelfareachievable
in undominated strategies, whether deterministically or probabilistically.
For multiple-good auctions, we prove that all dominant-strategy mechanisms can
guarantee only an exponentially small fraction of the maximum social welfare, and
the celebrated VCG mechanism (which is no longer dominant-strategy) guarantees,
in undominated strategies, at most a doubly exponentially small fraction.
For general games beyond auctions, we provide definitional foundations for this
new approximate-type model, and provide a universality result showing that all rea-
sonable (including Bayesian or Knightian) models of type uncertainty are equivalent
to our set-theoretic one, at least for the setting when the type space is “convex”.
This work was done in collaboration with Silvio Micali and Alessandro Chiesa.
Thesis Supervisor: Silvio Micali
Title: Professor
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To my beloved mother, Xiaoli Xu
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Acknowledgments
Tracing back to October 2009, when I first met Silvio Micali at Tsinghua University,
I was deeply attracted by his inspiring and idiosyncratic talk in mechanism design,
and entered this field. A year later in September 2010, I was luckily selected as
a student of this adventurous and knowledgeable professor, and joined the CSAIL
Theory Group. Since then, the work in this thesis has started and been under Silvio’s
careful and close supervision. I appreciate it more than I can say.
I would also like to thank Alessandro Chiesa who introduced me all secrets about
MIT, so that I can integrate into this big family without difficulty. He is the one
I spent the past 1.5 years discussing everything related or not related to this thesis
with, the one that I learned all advanced techniques of LATEX from, the one that I
complained the locomotive whistles near our office with, etc. Such collaboration is
more enjoyable and engaging than ever.
Beyond this thesis, I am fortunate to have collaborated extensively with Jonathan
Kelner, Martin Rinard, Sasa Misailovic and Christian Sommer, and more or less with
many others at MIT. I am also grateful for the support that was not research-related,
including that from our instructors, our students and our secretaries in the CSAIL
Theory Group; and also including that from my sister-like friend Xiaodan Jia, who
made me porridge and bought me balloons when I was extremely sick rushing before
two paper deadlines.
This thesis is fully supported by a 9-month Greater China Fellowship, a 3-month
Big George Ventures Fund, and a 4.5-month teaching assistantship for 6.854 at MIT.
Therefore, I would like to thank the Greater China Fund, Ray Sidney and David
Karger respectively to make my study at MIT possible.
Last but not least, I never forget to thank my dearest mother for her unceasing
support and unselfish dedication to my family, and my education, throughout the
past 23 years and 204 days. For all of the above, and the unnamed, please allow me
to express my deepest gratitude, now and always.
Earlier Publications of this Work
Much of the material in this thesis has appeared in earlier publications done by the
author in joint work with Alessandro Chiesa and Silvio Micali. The result of single-
good auctions has appeared in ITCS 2012 [9]. The result of multi-good auctions has
appeared in part as an internal technical report [7]. The result of general games
has appeared in part as an MIT technical report [8]. The universality result of
approximate types has never published before.
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Contents
1 Introduction 11
1.1 Single-Good Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Our Model of Self Uncertainty . . . . . . . . . . . . . . . . . . 11
1.1.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 Our Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Multi-Good Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 General Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Universality of Approximate Types . . . . . . . . . . . . . . . . . . . 16
1.5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Single-Good Auctions 19
2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The Auction Context . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 The Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Solutions Concepts . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.4 The Performance Measure . . . . . . . . . . . . . . . . . . . . 22
2.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 The Inadequacy of Dominant-Strategy Mechanisms . . . . . . 23
2.2.2 The Power of Deterministic Undominated-Strategy Mechanisms 24
2.2.3 TheGreaterPowerofProbabilisticUndominated-StrategyMech-
anisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Single-Good Auctions:
Techniques of Independent Interest 29
3.1 The Undominated Intersection Lemma . . . . . . . . . . . . . . . . . 29
3.1.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 The Distinguishable Monotonicity Lemma . . . . . . . . . . . . . . . 33
3.2.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Single-Good Auctions:
Missing Proofs 39
4.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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4.2 Proof of Theorem 2b . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Proof of Theorem 3b . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Proof of Theorem 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Proof of Theorem 3a . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5.1 A Very Restricted Search . . . . . . . . . . . . . . . . . . . . . 44
4.5.2 Our Allocation Function . . . . . . . . . . . . . . . . . . . . . 46
4.5.3 Our Mechanism M(δ) . . . . . . . . . . . . . . . . . . . . . . . 53
opt
5 Multi-Good Auctions 55
5.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.1 A First Impossibility Result . . . . . . . . . . . . . . . . . . . 56
5.2.2 A Second Impossibility Result . . . . . . . . . . . . . . . . . . 56
6 Multi-Good Auctions:
Missing Proofs 59
6.1 Proof for Fact 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Proof for Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Proof for Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3.1 Proof Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3.2 The Bird Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3.3 Proof of the Bird Lemma . . . . . . . . . . . . . . . . . . . . . 66
6.3.4 The Hard Instance . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.5 Main Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7 General Games 79
7.1 Game: Context and Mechanism . . . . . . . . . . . . . . . . . . . . . 79
7.2 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.5 New Revelation Principle . . . . . . . . . . . . . . . . . . . . . . . . . 84
8 Universality of Approximate Types 87
9 Conclusion 93
A Performance Diagrams 95
B Our Optimal Mechanism M(δ) 97
opt
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