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 integer entries. Also, keep the following (potentially) us
ECON 401: Auctions and Revenue Equivalence
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1st-price sealed-bid auction: how should we bid?
one object is for sale;
suppose there are n bidders;
the b
ECON 401: Auctions
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Auctions
the English auction is strategically equivalent to the 2nd price
sealed-bid auction.
the Dutch auction is strategically eq
ECON 401: Auctions
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English Auction
one object is for sale;
the price starts from a low price;
the auctioneer counts up the price;
all bidders signal to q
ECON 401: Auction: introduction
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What is an auction?
http:/www.youtube.com/watch?v=ACRbgKA0YrU
http:/www.youtube.com/watch?v=Tpl7k84-dm4&feature=relate
ECON 401: SPNE and BNE
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nd SPNE backward induction: example
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Player 1 strategy: "L" and "R";
s
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
1,1
F
0,1
2,0
2,1
consider three cases: 1) A is played i
ECON 401: Solution Concepts
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solution concepts
a solution for a game predicts what will happen (or what will not
happen) in this game.
We will talk about
ECON 401: Theorem of Khun and Tucker
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An example
a rm uses one iput x . 0/ to produce two outputs y . 0/ , z . 0/
by the following technology:
y2 C z
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 cdf function
F ( );
bidders submit bids (his price to bu
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 F ( ). We consider the the
following auction formats.
A
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 normal-form game,
then in any Nash equlibrium, player i m
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 transitivity;
(b) An negatively transitive strict preference
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
y i : y
y i both : x
x:
y and : y
x:
1. Is the followi
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. An agent tries to maximize her utility u (x1 ; x2 ; x3 )
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 also global optima?
(a) f (x; y ) =
x
1+x2 +y 2
2. Let
ECON 401: Revenue Equivalence
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revenue equivalence
Suppose there are n bidders, and biddersvalues vi are i.i.d. distributed on
[0, 1] following the cdf
ECON 401: Auctions
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auction with reserve price: second-price auction
consider two bidders, and biddersvalues are i.i.d. distributed
uniformly on [0,1];
ECON 401: Strategies
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normal-form games Vs extensive-form games
each normal-form game can be represented by an extensive-form
game.
each extensive-form g
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 F ( ). We consider the the
following auction formats.
A
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 penny, the two players a fair coin to decide whether
ip
INFORMATION AND KNOWLEDGE
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Knowledge
K : knowledge function;
K : 2 n f?g ! 2
K (E ) = f! 2
: P (! )
Eg
interpretation: for any event E , K (E ) is the set o
INFORMATION AND KNOWLEDGE
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Information
( ; ; P ) describes an agent information:
s
: states;
: prior;
P : a partition of
. More precisely, P is partitional
INFORMATION AND KNOWLEDGE
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Information
( ; ; P ) describes an agent information:
s
: states;
: prior;
P : a partition of
. More precisely, P is partitional
INFORMATION
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Preliminaries
: a nite set of states;
Every subset of
(i.e., the state space) is called an event;
There is a prior (or prior belief) on
;
every
UNCERTAINTY, STATES AND BAYESRULE
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Robert J. Aumann
2005 Nobel Prize laureate in Economics.
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Agre
REVEALED PREFERENCE
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Revealed Preference
Theorem
If c ( ) satises Sen property and , there exists some rational
s
that
c ( ) = c ( ; ):
Proof. Given c ( ), de
REVEALED PREFERENCE
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Revealed Preference
Let 2X denote the power set of X , i.e., 2X is the set of all subsets of
X.
Consider c : 2X n f?g ! 2X :
(i) c (A)