4. Trade War: Two countries, A and B, decide how much to "protect" their domestic market from A, B = {0, 1,2}. The payoff to each country i is π = imports by choosing a barrier to trade 2+x₁-x²³, resp. π = 2+xß-x². (a) Find a Nash equilibrium of this game! (b) Suppose that this game is repeated 1,000 times with additive payoffs. Is there a Nash equilibrium in which countries do not set barriers, i.e. choose x₁ = 0, in the first stage? Explain! Now the game is repeated infinitely often, payoffs are additive and future payoffs are discounted at rate 8. (c) Show that for large enough values of d there exists a subgame-perfect equilibrium in which coun- tries always choose A = B = 0 on the equilibrium path! Specify the off-equilibrium path actions in this equilibrium! Show exactly how large d needs to be for these strategies to constitute an equilibrium! (d) State an example of a different subgame-perfect equilbrium where the maximal barrier x = 2 or B = 2 is chosen sometimes on the equilibrium path, but not always? Be careful to define the strategies properly using the "on" and "off equilibrium path" terminology. You do not need to prove that your proposed strategy profile is indeed an equilibrium. (e) Now suppose that is too small to make the strategies defined in part (c) an equilibrium. Show that there may still be an equilibrium in which both countries always choose an intermediate barrier x = x = 1 on the equilibrium path! Specify the off-equilibrium path actions and show exactly how large d needs to be for these other strategies to constitute an equilibrium!