Game Theory Assignment-
Please write clearly and make sure to justify all your answers.
1. Two countries indexed by i = 1, 2 bargain for the division of one unit of territory. Country 1 makes an offer y1 = (y11; y12) in period 1, and country 2 accepts or rejects the offer. If country 2 accepts the offer, the territory is divided according to the offer and the game ends. If country 2 rejects the offer in period 1, then country 2 makes an offer in period 2, y2 = (y21; y22), and country 1 responds by accepting or rejecting. If country 1 accepts the offer in period 2, the territory is divided according to country 2's offer and the game ends. If country 1 rejects the offer in period 2, then the game ends and the territory is divided according to a status quo division (s, 1- s), 0 ≤ s ≤ 1. All offers yt, t = 1, 2; satisfy 0 ≤ yti ≤ 1; i = 1, 2, and y1t + y2t = 1. Each country's payoff is the share of the territory it receives at the end of the game minus any costs from war. In particular, each period an offer is rejected each country incurs a cost c, 0 ≤ c ≤ 1, due to war that rages over the territory. For example, if offer is rejected in the second period, payoff of country 2 is 1 - s - 2c, while payoff of country 2 from accepted offer in period 2 is y22 - c. Find a SPNE equilibrium of this game. Specifically:
(a) Specify what offers is country 1 willing to accept in period 2, that is, specify a threshold r2 where 0 ≤ r2 ≤ 1 such that in equilibrium after every history country 1 accepts offers y2 if and only if y21 ≥ r2. Make sure to specify an acceptable value for r2 for all values of s, c.
(b) Specify an offer y2 that country 2 is offering in equilibrium after every history in period 2, depending on the values of s, c.
(c) Determine what offers is country 2 willing to accept in period 1, that is, specify a threshold r1 where 0 ≤ r1 ≤ 1 such that in equilibrium after every history country 2 accepts offers y1 if and only if y12 ≥ r1. Make sure to specify an acceptable value for all values of s, c.
(d) Specify an offer y1 that country 1 is offering in equilibrium in period 1, depending on the values of s, c.
(e) Is war avoidable in equilibrium? Draw a graph with c on the horizontal axis and s on the vertical axis, indicating the region where various offers prevail. What is the two countries' payoff from equilibrium in each case? Does country 1 benefit when the cost of war, c, is higher? When her status quo share of the territory, s, is higher?