Simultaneous Games

1
Simultaneous Games
McCain- Schumer Calculus Solution
2
Profits

• Pharma
• 40 (y / (y x) ) - y
• GPhA
• 30 (x / (y x) ) x
• The idea now is to make an hypothesis about what
  your opponent is doing
• This will allow you to figure out what is the
  best thing that you can do against him, given
  your hypothesis

3
Best Response Pharma

• Fix the level of lobby by GPhA at X
• Max 40 (y / (y X) ) y
• First order condition
• 40 ( ((y X) - y ) / (y X)2 ) 10
• 40 ( X / (y X)2 ) 1
• 40 X (y X)2
• (40 X)1/2 - X y
• For each value of X we have the optimal y

4
Best Response GPhA

• Fix the level of lobby by Pharma at Y
• Max 30 (x / (x Y) ) x
• First order condition
• 30 ( ((x Y) - x ) / (x Y)2 ) 10
• 30 ( Y / (x Y)2 ) 1
• 30 Y (x Y)2
• (30 Y)1/2 - Y x
• For each value of Y we have the optimal x

5
Graph
6
Solving the system

• 30 Y (X Y)2
• 40 X (Y X)2
• Dividing both terms
• Y ( 4 / 3 ) X
• Replacing
• 40 X (( 4 / 3 ) X X)2
• 40 X (( 4 / 3 ) 1)2 X2
• 7.3469 X
• Replacing
• Y ( 4 / 3 ) 7.3469
• Y 9.7959