A Synergistic Relationship
Two individuals are in a synergistic relationship. If both devote more effort to the relationship, they are both better off. They simultaneously choose effort levels a1 and a2 respectively, which are nonnegative numbers. Individual i's preferences for i = 1, 2 are represented by the payoff function πi(ai, aj) = ai(c + aj - ai), where c > 0 is a constant.
a. Find a Nash equilibrium of this game.
b. Is the Nash equilibrium efficient or inefficient? That is, is there another strategy profile that is not a Nash equilibrium that gives both players higher payoffs? Explain why or why not. Are there externalities involved in this game? Explain.