 Strict Nash Equilibrium
In a Nash equilibrium, a player might be indifferent between his
equilibrium action and some other action, given the action of the other
players.
Definition: The action profile
in a strategic game is a Strict Nash
Equilibrium if for every player :
( )
(
)
for every action
of player .
As it is clear the difference here with the definition on page 19 is the
strict inequality.
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 37
Example:
L
C
R
U
-4 , -4
3,2
-1 , -2
M
-4 , 1
2 , -1
2,0
D
-6 , 2
2,3
2,3
The game has 3 NE in pure strategies: (M , L), (U , C), and (D , R).
(M , L) and (D , R) are non-strict (weak) NE.
(U , C) is the strict equilibrium.
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 38
Example: Guessing Two-third of the Average (Os Exercise 34.1)
Players:
Strategy:
{
}
{
}
Rule: The player who has the closest
to of the average is the
winner.
Payoffs:
{
What is the NE of this game?
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 39
Example: A Synergistic Relationship (Os Example 39.1)
Players:
{
}
Strategy:
Payoffs:
; The individual’s level of effort
(
)
FOC:
The best response function:
( )
Game Theory
(
)
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 40
a2
BR1
BR2
c
c/2
c/2
Game Theory
c
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
a1
Page 41
The BR functions:
(
)
(
)
So, the unique and strict NE of the game is:
(
Game Theory
)
(
)
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 42
 Cournot’s Model of Oligopoly
We have seen the Cournot’s model with just two firms before, here we
revisit the model this time with
Players: N
{
};
firms.
firms all produce a homogenous good.
Strategy:
Output: The total output is ∑
and the price is determined by
this inverse demand function:
{
( )
Payoffs:
For simplicity assume
( )
Show that the symmetric NE of the game is
Game Theory
(
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
).
Page 43
 Bertrand’s Model of Oligopoly
We revisit the Bertrand’s model of duopoly where firms compete in
prices. The main difference with the Cournot’s model is that the strategic
variable is price instead of quantity.
Players: N
{
};
firms all produce a homogenous good.
Strategy:
Payoffs: ( ( ) is the demand function at price )
( )
{
Game Theory
(
( )
)
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 44
In the simplest version:
,
( )
{
,
(
( )
).
Then
(
)(
)
(
)(
)
{
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 45
The best response function is (
( )
is the monopolistic price):
{ |
}
{ |
}
{
And the only NE of the game is:
(
)
(
)
The standard way to find the NE is through BR functions; however, in
many cases we can use alternative ways to find the NE in an easier
manner.
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 46
 Electoral Competition
Voters are distributed over a turf of political ideas (say from Left to
Right).
Usually distribution function considered to be uniform.
Variants of this game:
When two ice-cream sellers try to sell ice-cream in a beach where
consumers are distributed on the beach.
Two supermarkets compete in order to attract consumers located on a
linear city.
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 47
Players: N
{
};
Strategy:
candidates standing in an election
; Each candidate chooses a platform
Output: Each voter votes for the candidate who has chosen the closest
platform to his position
Preferences:
(
)
(
)
(
)
We study two cases:
1) Two candidates (median voter)
2) Three candidates; each have the option of staying out, and would
prefer staying out to entering and then losing.
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 48

 Auctions
Auctions are a widely used mechanism to allocate resources.
Many different types can be considered based on:
- Simultaneous or sequential bidding?
- The rule determining the winner?
- How much the winner should pay?
- Private value or common value (with asymmetric information sets)?
- How many units of the object are auctioned off?
- How to break ties?
- Is there a reserve price (Is the reserve price common knowledge)?
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 49
 Second-Price Sealed-Bid Auction
Players: N
{
} ; people all bidding for the same indivisible
object simultaneously
Each player has a valuation for the object
(for simplicity
)
Strategy:
; Each bidder submits a non-negative bid
Output: The player with the highest bid is the winner and pays the
second highest bid
̅
Pay-offs:
(
̅
)
{ (
̅
)
̅
(
)
̅
̅
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 50
Nash equilibria:
1)
2)
3)
... You find the other Nash equilibria
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 51
 First-Price Sealed-Bid Auction
Players: N
{
};
people all bidding for the same indivisible
object simultaneously
Each player has a valuation for the object
(for simplicity
)
Strategy:
; Each bidder submits a non-negative bid
Output: The player with the highest bid is the winner and pays his own
bid
̅
Pay-offs:
(
̅
)
{ (
̅
)
(
)
̅
̅
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 52
Suppose we change the tie-break rule slightly and say in case of a tie the
bidder with the smallest index is the winner.
Nash equilibria:
1)
2)
... Many other equilibria
However, we can show that in all NE of the game player 1 obtains the
object (you are asked to show this in PS).
We will revisit the auctions later for some other fascinating results ...
Game Theory
Dr. F. Fatemi
Graduate School of Management and Economics – Sharif University of Technology
Page 53