Nash Demand Game
Nash Program (non cooperative games)
Demand Game
Π
Si  x | x  0

x
Π 1  x, y   

0
y S
x
 x, y   S
S
 x, y   Topics
3
1
Nash Demand Game
Nash Equilibria of
Π
Too many equilibria !!
Nash’s Solution:
Perturb the game’s payoffs,
Introduce noise (uncertainty)
and then reduce it to 0
(Perfect- trembling hand- Equilibrium)
S
Games with similar payoffs
should have similar equilibria
x
2
Nash Demand Game
Uncertainty about the boundary
p:
2
 0,1
iso-probability curves
0
S
1
3
Nash Demand Game
p:
2
 0,1
define a new game
Πp
Si  x | x  0
Π
0
S
p
1
 x, y  = xp  x, y 
Π 2p  x, y  = yp  x, y 
1
4
Nash Demand Game
Uncertainty about the boundary
p:
2
 0,1
or define p on a narrower
band of uncertainty
0
S
1
0
S
1
5
Nash Demand Game
Uncertainty about the boundary
p:
2
 0,1
Define a sequence of pn with
uncertainty bands converging to 0
0
S
1
0
S
1
6
Nash Demand Game
Π Π
p
0
S
1
0
S
1
7
Nash Demand Game
Π
Π p2 Π p1
Π
p
8
Nash Demand Game
Solving the game
max Π
x
p
1
Πp
 x, y 
= max xp  x, y 
x
x xp
p11xx,,yy  p px,
 xy , y 0  0
*
*
*
*
*
*
*
*
*


y yp
p22xx,,yy  p px,
 xy , y 0  0
*
*
*
*
*
*


x p1  x , y   y p2  x , y 
*
9
Nash Demand Game
Πp
Solving the game
x p1  x , y
*
*
y
*
x
  y p  x , y 
p  x , y 

???
p  x , y 
*
*
*
*
*
2
*
*
*
*
1
(x* , y* )
2
Denote:
p x , y
*
*
α
0
S
1
p(x, y) = α
10
Nash Demand Game
p1  x , y 
y

*
*
*
x
p2  x , y 
*
*
Solving the game
*
Πp
max xy
???
p(x, y)=α
L (x, y, λ) = xy - λ  p(x, y) - α 
L x (x, y, λ) = y - λp1 (x, y) = 0
L y (x, y, λ) = x - λp2 (x, y) = 0
y p1 (x, y)
=
x p2 (x, y)
0
S
1
p(x, y) = α
11
Nash Demand Game
Solving the game
Πp
A Nash Equilibrium of the perturbed game
is a Nash Barganing solution of
a bargaining problem
defined by the frontier
p(x, y) = α
0
S
(There may be multiple equilibria)
1
p(x, y) = α
12
Nash Demand Game
Solving the game
Πp
As the uncertainty band vanishes
the Nash Equilibria of the perturbed game
converge to the Nash bargaining the solution
of the unperturbed game
0
S
1
13
Nash Demand Game
The Nash Bargaining
Solution
Π
Π p2 Π p1
Π
p
14