Construct a game with joint decisions by augmenting the Cournot duopoly game in the following way: Suppose the two firms have a profit-sharing pact that is enforced by the government (legal collusion). First, the firms simultaneously select quantities, q1 and q2 , at a marginal cost of 10.
The price is determined from the inverse demand curve, p = 100 - q1 - q2 , and the total revenue p(q1 + q2) is deposited into a joint bank account. Then the firms must negotiate over how to share the revenue. Model the negotiation as a joint decision over firm l's share of the revenue, m, so that agreement yields a payoff of m - 10q1 to firm 1 and p(q1 + q2) - m - 10q2 to firm 2. If they disagree then neither firm obtains any revenue, so firm 1 gets -10q1 and firm 2 gets -10q2 .