CS 573: Algorithmic Game Theory
Instructor: Chandra Chekuri
Lecture date: March 14, 2008
Scribe: James Lai
Contents
1 Bayesian Nash Equilibrium
1
2 Mechanisms Without Money
2.1 Single-Peaked Preferenc
Econ 401: Mathematical Economics
Recitation 02
Shasha Liu & Stephen Wol
25 September 2013
N.B. You may nd it computationally convenient to factor out common denominators so that
your matrices have int
ECON 401: Auctions and Revenue Equivalence
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1st-price sealed-bid auction: how should we bid?
one
ECON 401: Auctions
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Auctions
the English auction is strategically equivalent to the 2nd price
se
ECON 401: Auctions
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English Auction
one object is for sale;
the price starts from a low price;
the
ECON 401: Auction: introduction
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What is an auction?
http:/www.youtube.com/watch?v=ACRbgKA0YrU
h
ECON 401: SPNE and BNE
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nd SPNE backward induction: example
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Nov
ECON 401: Nash Equilibria
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Find all Pure-Strategy Nash Equilibrium
A
B
C
D
2,0
2,4
1,0
E
0,2
2,4
1
ECON 401: Solution Concepts
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solution concepts
a solution for a game predicts what will happen (or
ECON 401: Theorem of Khun and Tucker
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An example
a rm uses one iput x . 0/ to produce two outp
ECON 401: Assignment 8
due date: November 30, 2011
1. Consider the all-pay sealed-bid auction:
one object is for sale;
there are n bidders and the values are i.i.d. distributed on [0; 1] with the cd
ECON 401: Assignment 7
due date: November 23, 2011
1. An object is for sale via an auction. Suppose there are two bidders. Biddersprivate
values are i.i.d. distributed on [0; 1] with the cdf function
ECON 401: Assignment 6
due date: November 14, 2011
1. Determine whether the following statements are true or false. Explain why.
(a) Suppose a strictly dominant strategy exists for player i in some no
ECON 401: Assignment 5
due date: October 31, 2011
1. Are the following two statements true. If yes, prove them. If no, nd counterexamples.
(a) An asymmetric strict preference satises negatively transi
ECON 401: Assignment 4
due date: October 24, 2011
Starting with " ", we dene " " and " " as follows.
x
x
y i x
y and : y
y i both x
x:
y and y
x:
Starting with " ", we dene " " and " " as follows.
x
x
ECON 401: Assignment 3
due date: October 3, 2011
1. Solve the following optimization problems:
max (ln x + ln y )
s.t. x2 + y 2 1
x0
y0
2. Consider the utility maximization problem with three goods. A
ECON 401: Assignment 2
due date: September 21, 2011
1. Find and classify the critical points (local maximum, local minimum, neither) of the
each of the following functions. Are any of the local optima
ECON 401: Revenue Equivalence
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November 21, 2012
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revenue equivalence
Suppose there are n bidders, and biddersvalue
ECON 401: Auctions
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November 28, 2012
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auction with reserve price: second-price auction
consider two bidders, and b
ECON 401: Strategies
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normal-form games Vs extensive-form games
each normal-form game can be repre
ECON 401: Assignment 7
due date: November 28, 2012
1. An object is for sale via an auction. Suppose there are two bidders. Biddersprivate
values are i.i.d. distributed on [0; 1] with the cdf function
ECON 401: Assignment 6
due date: November 16, 2012
1. Describe the following game in an extensive-form game tree.
matching penny version E:
there are two players, 1 and 2;
prior to playing matching p
INFORMATION AND KNOWLEDGE
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Knowledge
K : knowledge function;
K : 2 n f?g ! 2
K (E ) = f! 2
: P (! )
E
INFORMATION AND KNOWLEDGE
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Information
( ; ; P ) describes an agent information:
s
: states;
: prior;
INFORMATION AND KNOWLEDGE
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Information
( ; ; P ) describes an agent information:
s
: states;
: prior;
INFORMATION
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Preliminaries
: a nite set of states;
Every subset of
(i.e., the state space) is called
UNCERTAINTY, STATES AND BAYESRULE
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Robert J. Aumann
2005 Nobel Prize laureate in Economics.
Xiong
REVEALED PREFERENCE
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Revealed Preference
Theorem
If c ( ) satises Sen property and , there exists some
REVEALED PREFERENCE
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Revealed Preference
Let 2X denote the power set of X , i.e., 2X is the set of a