Game theory -  A Tutorial

Jean Walrand

Static: Cournot Game

Two companies produce quantities x and y, respectively, of a product whose unit selling price is
A - x - y and unit production cost is C. The selling price goes down with the quantity, as the market saturates.

Given y, the first company’s profit is

                         x(A - x - y - C) = x(B - x - y)

where B = A - C when it produces a quantity x. This profit is maximized by choosing x = x*(y) := (B - y)/2.  Similarly, given x,

the profit of the second company is maximized if it chooses to produce a quantity y*(x) = (B - x)/2. 

As the figure shows, these two best response functions have only one intersection: (x, y) = (B/3, B/3), which is the unique Nash equilibrium. The resulting profits are B²/9 for both firms.

Stackleberg Game. Assume that company 1 first chooses x, that company 2 then chooses the best response y*(x). Then the profit of the company 1 is equal to x(B - x - y*(x)) = x(B - x)/2. Anticipating this reaction of company 2, company 1 chooses the value of x that maximizes the profit above, i.e, x = B/2. The profit of company 1 is then B²/8 and that of company 2 is B²/16.

In this leader-follower game (aka Stackelberg), the leader has an advantage.