A Cournot game with inconsistent beliefs Each of two manufacturers i ∈ {I, II} must determine the quantity of a product xi to be manufactured (in thousands of units) for sale in the coming month. The unit price of the manufactured products depends on the total quantities both manufacturers produce, and is given by p = 2 - xI - xII.
Each manufacturer knows his own unit production cost, but does not know the unit production cost of his rival. The unit production cost of manufacturer i may be high (ci = 5/4) or low (ci = 3/4). Manufacturer i's profit is xi(p - ci). The first manufacturer ascribes probability 2/3 to the second manufacturer's costs being high (and probability 1/3 to the second manufacturer's costs being low).
The second manufacturer ascribes probability 3/4 to the costs of both manufacturers being equal to each other (and probability 1/4 to their costs being different). Answer the following questions:
(a) Describe this situation as a game with incomplete information.
(b) Prove that the beliefs of the manufacturers are inconsistent.
(c) Find all Bayesian equilibria in pure strategies in this game.