ECO 5341 An Example of a Contribution Game
Saltuk Ozerturk (SMU)
January 2016
Saltuk Ozerturk (SMU)
Contribution Game
An Example of a Contribution Game
An Example of a Contribution Game
Two roommates Robert and Larry each want to choose how
much time to allocate to keep the kitchen clean.
If Robert allocates xR hours and Larry allocates xL hours for
kitchen cleaning their payoffs are
πR (xR , xL ) = 100xR − 10xR xL − 10xR2 (for Robert)
πL (xR , xL ) = 100xL − 10xR xL − 10xL2 (for Larry)
Find the Nash equilibrium contributions xR∗ and xL∗ of each
player to keep the kitchen clean.
Saltuk Ozerturk (SMU)
Contribution Game
An Example of a Contribution Game
Deriving Robert’s best response function
Given any xL by Larry, Robert chooses xR to maximize
πR (xR , xL ) = 100xR − 10xR xL − 10xR2
First Order Condition yields
100 − 10xL − 20xR = 0
=⇒ xR∗ (xL ) = 5 −
xL
is Robert’s best response
2
Saltuk Ozerturk (SMU)
Contribution Game
An Example of a Contribution Game
Deriving Larry’s best response function
Given any xR by Robert, Larry chooses xL to maximize
πL (xR , xL ) = 100xL − 10xR xL − 10xL2
First Order Condition yields
100 − 10xR − 20xL = 0
xL∗ (xR ) = 5 −
xR
is Larry’s Best Response
2
Saltuk Ozerturk (SMU)
Contribution Game
An Example of a Contribution Game
The best response functions are
xR∗ (xL ) = 5 −
xL
xR
and xL∗ (xR ) = 5 −
2
2
The Nash Equilibrium pair (xR∗ , xL∗ ) solves the above pair of
equations.
x∗
xR∗ = 5 − L
2
x∗
1
⇒ xR∗ = 5 −
5− R
2
2
5 xR∗
10
+
⇒ 3xR∗ = 10 ⇒ xR∗ =
2
4
3
∗
x
1 10
10
xL∗ (xR∗ ) = 5 − R = 5 −
⇒ xL∗ =
2
2 3
3
⇒ xR∗ =
Saltuk Ozerturk (SMU)
Contribution Game