Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Lecture 10. Learn to Play Maximum Revenue
Auction
Xiaotie Deng
AIMS Lab
Department of Computer Science
Shanghai Jiaotong University
November 28, 2016
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
1
Prior in Myerson Auction
2
Knowledge Among Multiple Agents
3
Can We Learn Games?
4
Epistemology of Games
5
Repeated Meryeson Auction
6
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Prior in Myerson Auction
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Myerson Auction
1 item to sell to n bidders, each with a private value
vi ∈ Di = D[ai , bi ].
Distribution D = ×ni=1 Di is a Prior probability distribution, a
common knowledge to all.
Myerson auction is a revenue maximizer among all truthful
mechanism to sell the item
It derives a threshold price for each buyer based on the
distribution D.
Allocation the item to the higher virtual bid (price minus
threshold).
Charge it the maximum of the second highest bid and the
threshold.price.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Information Retrieval of Probability Distribution
BIEN Methodology: Peshkin L, Pfeffer A. Bayesian
information extraction network. arXiv cs/0306039, 2003.
Chickering D M. Learning Bayesian networks is
NP-complete//Learning from data. Springer, 1996: 121-130.
Conditional Probability: Sutton C, McCallum A. An intro. to
conditional random fields for relational learning. Intro. to
statistical relational learning. MIT Press, 2006.
Machine Learning: Lavelli A, Califf M E, Ciravegna F, et al.
Evaluation of machine learning-based information extraction
algorithms: criticisms and recommendations. Language
Resources and Evaluation, 2008, 42(4): 361-393.
Stock Market Investment: Hrus̆ová, Ivona: How rewarding is
technical analysis? Evidence from Central and Eastern
European Stock Markets.
Thesis.
Charles
University
Prague
Xiaotie Deng
Lecture
10. Learn to
Play Maximum in
Revenue
Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Can We Learn Myerson Auction?
1
Two Tasks
Exploration: Find out the distribution.
Exploitation: Extract maximum revenue.
2
Would the Buyers Cheat?
One buyer of uniform distribution [0,1]: cheating report [0,1/2]
and setting bid b = v /2.
Optimal auction price 1/4.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Knowledge among Multiple Agents
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Epistemic Modal Logic
Reasoning about knowledge.
knowledge and belief
capture the semantics of knowledge
Common Knowledge:
SG ϕ: someone in G knows ϕ.
EG ϕ: everyone in G knows ϕ.
Level k knowledge EGk+1 := EG [EGk ϕ].
i
Common Knowledge CG ϕ := ∩∞
i=1 EG ϕ.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Common Knowledge Example
Reference: Halpern J Y, Moses Y. Knowledge and common
knowledge in a distributed environment[J]. Journal of the ACM
(JACM), 1990, 37(3): 549-587.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Epistemic Logic Common Knowledge
Halpern J Y, Moses Y. Knowledge and common knowledge in
a distributed environment[J]. Journal of the ACM (JACM),
1990, 37(3): 549-587.
Aumann R J. Backward induction and common knowledge of
rationality[J]. Games and Economic Behavior, 1995, 8(1):
6-19.
Monderer D, Samet D. Approximating common knowledge
with common beliefs[J]. Games and Economic Behavior, 1989,
1(2): 170-190.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Informal Use of Common Knowledge
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Can We Learn Games?
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Can We Learn Distribution?
”I am familiar with the math behind basic machine learning
instruments, neural networks, svm/knnc and trees, but i don’t feel
that i can apply either of these to solve this problem. What
algorithms can i apply?”
https://www.reddit.com/r/MachineLearning/comments/37vbfw/
predict a probability distribution/
Answer:
Hidden Markov model
Bayesian network
Parameterized statistical model.
REFERENCES: https://en.wikipedia.org/wiki/Sufficient statistic
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Statistical Model
Defined by a pair (S, P), where S is the sample space, and P
is a set of probability distributions on S, often parameterized.
Example distributions, each determined by one or more
parameters: Bernoulli, uniform, Poisson, Gaussian,
exponential, Gamma, power law.
REFERENCES: https://en.wikipedia.org/wiki/Statistical model
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Epistemology of Games
Extracted from “Stanford Encyclopedia of Philosophy”
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Bayesian Epistemology
Advocates for Bayesian inferences
to justify the rule of inductive logic.
David Miller: Critical Rationalism
Falsification: “it presupposes what it attempts to justify”
https://en.wikipedia.org/wiki/Bayesian inference
David Miller: Critical Rationalism
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Computational Epistemology
A study of the intrinsic complexity of inductive problems (wiki)
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Epistemic Game Theory
Interactions with other agents bring in higher order
information.
Their concerns are the information and choices of other
players, and vice versa
Techniques for higher order are applied to understand the
game setting.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Key Features in Epistemic Game Theory
Bayesian perspective view on decision making.
Uncertainty about opponents’ strategies takes center stage,
especially higher order information (what are the other players
thinking)
Solution concepts.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Sources of Uncertainty
Stages of decision making: ex ante, ex interim, ex post.
ex interim: the players have made their decisions, but they are
still uninformed about the decisions and intentions of the other
players.
Incomplete information, perfect recall, mixed strategies.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Modeling Games
Information, belief, knowledge, probability and possible worlds.
Type space
Common Knowledge: KI = ∩i∈I Ki , KI0 (E ) = E ,
i
KIi+1 = KI (KIi (E )), CI (E ) = ∪∞
i=1 KI (E )
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Fundamentals
Removal of strictly dominated strategies, backward induction,
rationalizability principle.
Nash equilibrium, cautious belief, unawareness (Ui (E ))
Paradox of self-reference.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Challenges
Epistemic GT’s goal?
“Much of the work in epistemic game theory can
be viewed as an attempt to use precise
representations of the players’ knowledge and
beliefs to help resolve some of the confusion
alluded to in the above quote.”
Alternative agents’ choice rule: minregret.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Repeated Meryeson Auction
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
One Round Auction
Buyers’ values follow known distributions.
The auctioneer can determine a threshold for each buyer,
assign the highest virtual bidder to win and request a payment
equal to the threshold and the second highest bid.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Some Examples
Uniform Distribution [0, a] where 0 < a ≤ 1
Power Law Distribution
Negative Exponential
Doubly Exponential
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Myerson Auction: Uniform
IID uniform in [0, 1]: the probability density function 1[0,1]
virtual bid vi −
1−Fi (t)
fi (t)
= vi −
1−vi
1
= 2vi − 1.
Threshold price v¯i = 1/2: derived from 2v¯i − 1 = 0.
~ −i |Xi ]].
Buyer’s utility ui (vi ) = EX [Pro[Xi > max X
Question: Would Buyer i be able to improve by lying about
its distribution to [0, α] for some α ∈ (0, 1)?
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Lying on Distribution in Myerson Auction: Uniform
Lying on Distribution [0, α]: always bid α ∗ vi .
The first buyer cheats in two player case.
Virtual value φ1 = α2
Winning if α ∗ v1 > max{v2 , α2 }.
Winning probability at value t: α ∗ t ∗ 1t∈[1/2,1]
R α∗t
Winning utility at value t > 12 : (t − α2 ) ∗ α2 + α/2 (t − τ )dτ .
which is equal to 21 (t − α2 )2 − 12 (1 − α)2 t 2
The expected utility by cheating is
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
(Ex Ante) Optimal Lying Uniform Distribution
The expected utility gain by cheating is maximized by 0.132408 at
α = 0.600668.
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Uniform
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Power Law
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction
Outline
Prior in Myerson Auction
Knowledge Among Multiple Agents
Can We Learn Games?
Epistemology of Games
Repeated Meryeson Auction
Distribution Cheating Game Equilibrium (D., Xiao, Zhu, 2016)
Exponential
Xiaotie Deng
Lecture 10. Learn to Play Maximum Revenue Auction