Advanced Microeconomics
Advanced Microeconomics
ECON5200 - Fall 2014
Advanced Microeconomics
Introduction
I
What you have done:
- consumers maximize their utility subject to budget constraints
and …rms maximize their pro…ts given technology and market
prices;
- no strategic behavior.
I
What we will do:
- in many interesting situations, agents’optimal behavior
depends on the other agents’behavior;
- strategic behavior.
I
Game theory provides a language to analyze such strategic
situations;
I
Countless number of examples! Auctions, Bargaining, Price
competition, Civil Con‡icts. . .
Advanced Microeconomics
Introduction
Road map
I
Static Game:
1. With Complete Information (I);
2. With Incomplete Information (II).
I
Dynamic Game:
1. With Complete Information (II-III);
2. With Incomplete Information (III).
Advanced Microeconomics
Strategic Games with Complete Information
Strategic Game with Pure Strategies
I
N players with i 2 I ;
I
s2S
∏
i =1,..,N
Si pure strategy pro…le, si 2 Si ;
I
ui (s ) payo¤;
I
G
hI , fSi gi , fui (s )gi i strategic formof …nite game with
pure strategy.
Advanced Microeconomics
Strategic Games with Complete Information
Strategic Game with Mixed Strategies
I
I
σ 2 ∆ (S )
ui ( σ ) =
∏
i =1,..,N
∑ ∏
∆ (Si ) mixed strategypro…le, σi 2 ∆ (Si );
σj (sj ) ui (sj ) expected utility;
s 2S j =1,..,N
I
G
hI , f∆ (Si )gi , fui (σ)gi i strategic form of …nite game
with mixed strategy;
I
Interpreting mixed strategies:
- as object of choice;
- as pure strategies of a perturbed game (see later in Bayesian
Games);
- as beliefs.
Advanced Microeconomics
Strategic Games with Complete Information
Equilibrium Concepts
I
Nash Equilibrium ) it is assumed that each player holds the
correct expectation about the other players’behavior and act
rationally (steady state equilibrium notion);
I
Rationalizability ) players’beliefs about each other’s actions
are not assumed to be correct, but are constrained by
consideration of rationality;
I
Every Nash equilibrium is rationalizable.
Advanced Microeconomics
Strategic Games with Complete Information
Rationalizability
De…nition
In G , si is rationalizableif there exists Zj
that:
Sj for each j 2 I such
1. si 2 Zi ;
2. every sj 2 Zj is a best response to some belief µj 2 ∆ (Z j ).
I
Common knowledge of rationality;
I
An action is rationalizable if and only if it can be rationalized
by an in…nite sequence of actions and beliefs.
Advanced Microeconomics
Strategic Games with Complete Information
Example (1 - Rationalizability - See notes!)
...
Advanced Microeconomics
Strategic Games with Complete Information
Strictly Dominance
De…nition
si is not strictly dominatedif it does not exist a strategy σi :
ui ( σ i , s i ) > ui ( s i , s i ) , 8 s
i
2S
i
Advanced Microeconomics
Strategic Games with Complete Information
Strictly Dominance
I
A unique strictly dominant strategy equilibrium (D, D ):
I
It is Pareto dominated by (C , C ). Does it really occur??
Advanced Microeconomics
Strategic Games with Complete Information
Iterative Elimination of Strictly Dominated Strategies
De…nition
Set S 0 = S, then for any m > 0 si 2 Sim i¤ there does not exist
any σi such that:
ui ( σ i , s i ) > ui ( s i , s i ) , 8 s
i
2 S mi
1
De…nition
For any player
\ i, a strategy is said to be rationalizable if and only if
∞
s i 2 Si
Sim .
m 0
Advanced Microeconomics
Strategic Games with Complete Information
Example (2 - Beauty Contest - See notes!)
...
Advanced Microeconomics
Strategic Games with Complete Information
Iterated Weak Dominance
I
There can be more that one answer for iterated weak
dominance;
I
Not for iterated strong dominance.
Advanced Microeconomics
Strategic Games with Complete Information
Example (3 - Cournot vs Bertrand Competition - Proposed
as exercise)
Example
n pro…t-maximizer-…rms produce qi quantity of consumption good
at a marginal cost equal to c > 0;
I
I
demand function is P = max f1
Q, 0g with Q 2
∑
qi ;
i =1...n
Find:
1. The rationalizable equilibria when n = 2;
2. The rationalizable equilibria when n > 2;
3. Compare your results with the Bertrand competition outcome.
Advanced Microeconomics
Strategic Games with Complete Information
Nash Equilibrium
De…nition
σi 2 ∆ (Si ) is a best responseto σ
ui ( σ i , σ i )
Let Bi (σ i )
i
2 ∆ (S i ) if:
ui (si , σ i ) for all si 2 Si
∆ (Si ) be the set of player’i best response.
De…nition
σ is a Nash equilibriumpro…le if for each i 2 I .
σ i 2 Bi ( σ i )
Advanced Microeconomics
Strategic Games with Complete Information
Nash Theorem
Theorem (Nash (1950))
A Nash equilibrium exists in a …nite game.
Theorem (Kakutani Fixed Point Theorem)
Let X be a compact, convex and non-empty subset of Rn , a
correspondence f : X ! X has a …xed point if:
1. f is non-empty for all x 2 X ;
2. f is convex for all x 2 X ;
3. f is upper hemi-continuous (closed graph).
Advanced Microeconomics
Strategic Games with Complete Information
Best Response Correspondence Example
Advanced Microeconomics
Strategic Games with Complete Information
The Kitty Genovese Problem/Bystander E¤ect
I
n identical people;
I
x > 1 bene…ts if someone calls the police;
I
1 cost of calling the police;
What is the symmetric mixed strategy equilibriumwith p equal
to the probability of calling the policy?
I
In equilibrium each player must be indi¤erent between calling
or not the police;
I
If i calls the police, gets x
I
If i doesn’t, gets:
1 for sure;
0 with Pr (1
x with Pr 1
p )n
(1
1
p )n
1
Advanced Microeconomics
Strategic Games with Complete Information
The Kitty Genovese Problem/Bystander E¤ect
I
Indi¤erence when:
x
1=x 1
(1
p )n
1
(1/x )1/(n
I
Equilibrium symmetric mixed strategy is p = 1
I
http://en.wikipedia.org/wiki/Murder_of_Kitty_Genovese
1)
Advanced Microeconomics
Strategic Games with Complete Information
Zero-Sum Game
De…nition
A N-player game G is a zero-sum game(a strictly competitive
game) if ∑ ui (s ) = K for every s 2 S.
i =1,..,N
Advanced Microeconomics
Strategic Games with Complete Information
Zero-Sum Game
De…nition
σi 2 ∆ (Si ) is maxminimizerfor player i if:
min
σ i 2 ∆ (S
i)
ui ( σ i , σ i )
min
σ i 2 ∆ (S
i)
ui σi0 , σ
i
for each σi 2 ∆ (Si )
A maxminimizer maximizes the payo¤ in the worst case scenario
Theorem
Let G be a zero-sum game. Then σ 2 ∆ (S ) is a Nash Equilibrium
i¤, for each i, σ is a maxminimizer.
Advanced Microeconomics
Strategic Games with Complete Information
Example (4 - All-Pay Auction - Proposed as exercise)
Two players submit a bid for an object of worth k;
I
bi 2 [0, k ] individual strategy space where bi is the bid;
I
The winner is the player with the highest bid;
I
If tie each player gets half the object, k/2;
I
Each player pays her bid regardless of whether she wins;
I
Find that:
1. No pure Nash equilibria exist;
2. The mixed strategy equilibrium is equal to the one represented
here below.
Advanced Microeconomics
Strategic Games with Complete Information
Example (4 - All-Pay Auction - Proposed as exercise)
Advanced Microeconomics
Extensive Form Games
Representation of a Game
I
Normal or strategic form;
I
Extensive form.
The Extensive form contains all the information about a game:
I
who moves when;
I
what each player knows when he moves;
I
what moves are available to him;
I
where each move leads.
whereas a normal form is a ‘summary’representation.
Advanced Microeconomics
Extensive Form Games
Extensive Form
De…nition
A treeis a set of nodes and directed edges connecting these nodes
such that:
1. for each node, there is at most one incoming edge;
2. for any two nodes, there is a unique path that connect these
two nodes.
De…nition
An extensive form game consists of i) a set of players (including
possibly Nature), ii) a tree, iii) an information set for each player,
iv) an informational partition, and v) payo¤s for each player at
each end node (except Nature).
Advanced Microeconomics
Extensive Form Games
Extensive Form
De…nition
An information setis a collection of points (nodes) such that:
1. the same player i is to move at each of these nodes;
2. the same moves are available at each of these nodes.
De…nition
An information partitionis an allocation of each node of the tree
(except the starting and end-nodes) to an information set.
De…nition
A (behavioral) strategyof a player is a complete contingent-plan
determining which action he will take at each information set he is
to move.
Advanced Microeconomics
Extensive Form Games
Extensive Form vs Normal Form