STV4217B Rational Choice Models and International Conflict Exam 2023 PDF

EXAM STV4217B Rational Choice Models and International Conflict Autumn 2023 15:00 20 December 2023 In...

EXAM STV4217B Rational Choice Models and International Conflict Autumn 2023 15:00 20 December 2023 Instructions: You have three hours to take the exam. Please allocate your time carefully. Answer all the questions. Question 1 ( 20%) Explain briefly the meaning of the following game-theoretical concepts: (a) Decision node (b) Best Response (c) Weakly dominated strategy (d) Common knowledge (e) Information set (f) Incomplete information (g) Perfect information (h) Pareto optimal (i) Subgame (j) Bayesian perfect equilibrium 1 Solution: (a) Decision node A point in the game tree where a player must make a decision. (b) Best Response (c) Weakly dominated strategy A strategy that is strictly worse than another strategy in one contingency and not better in any contingency. (d) Common knowledge A piece of information is common knowledge if both (all) players know it, know that all players know it, know that all players know that all players know it, and so on ad infinitum. (e) Information set An information set conveys the information a player has about previous moves in the game when the player concerned must make a decision. (f) Incomplete information The players’ strategic types (strategy sets and preferences) are not known by at least one player (g) Perfect information The history of the game is always common knowledge. (h) Pareto optimal (i) Subgame A subgame starts with a singleton, encompasses all subsequent decision nodes, and does not cut across any information set. (j) Bayesian perfect equilibrium Two definitional criteria: 1. A set of strategies that are best responses to each other for every subgame, given the players’ beliefs. 2 The players’ beliefs are updated along the equilibrium path, using Bayes’ rule (wherever possible). Page 2 Question 2 ( 2 5%) Suppose State 2 threatens State 1 with war unless State 1 closes certain facilities that State 2 suspects could enable State 1 to produce weapons of mass destruction. A war will end either with victory for State 1 or with victory for State 2. The probability of victory for 1 is p1 and the probability of victory for 2 is p2 = 1 − p1. The situation is shown in Figure 1. Suppose p1 = c1c+c1 2 , where p1 is the probability of victory for state 1, c1 is the military capability of state 1, and c2 is the military capability of state 2. State 1 Y ield Stand F irm State 2 -50,60 Give Go to W ar In Nature 0,0 V ictory f or V ictory f or State 2 State 1 -80,40 20,-30 Figure 1: Question 2 Game Tree 1. What is p2 in terms of c1 and c2 ? 2. Find the subgame-perfect equilibrium of the game when it is known that the military capability of State 2 is three times that of State 1, c2 = 3c1. 3. Find the subgame-perfect equilibrium of the game when it is common knowledge that the military capabilities of state 1 and state 2 are equal, c1 = c2. 4. Based on this model, how does the balance of power effect the likelihood of war? Please explain your answer. Page 3 Solution: 1. c1 c2 p2 = 1 − p1 = 1 − = c1 + c2 c1 + c2 2. Let c2 = 3c1. Then, c2 3c1 3 p2 = = = c1 + c2 c1 + 3c1 4 And 1 p2 = 4 1 3 90 EUs2 (W ar) = p1 (−30) + p2 (40) = (−30) + (40) = 4 4 4 EUs2 (GI) = 0 State 2 prefers W ar to GI. 1 3 EUs1 (SF ) = p1 (20) + p2 (−80) = (20) + (−80) = −55 4 4 EUs1 (Y ield) = −50 State 1 prefers Y ield to SF. SP N E = {(Y ield, W ar)} 3. Let c2 = c1. Then, c1 c1 1 p1 = = = = p2 c1 + c2 c1 + c1 2 1 1 EUs2 (W ar) = p1 (−30) + p2 (40) = (−30) + (40) = 5 2 2 EUs2 (GI) = 0 1 1 EUs1 (SF ) = p1 (20) + p2 (−80) = (20) + (−80) = −30 2 2 EUs1 (Y ield) = −50 SP N E = {(SF, W ar)} 4. Open response. Increased balance of war increases the likelihood of conflict. Page 4 Question 3 ( 2 5%) Suppose that Country 1 can violate an international norm and Country 2 can impose sanctions if this occurs. If Country 1 chooses not to, nothing happens. If Country 1 violates the norm, Country 2 can impose harsh sanctions on Country 1. If this happens, Country 1 has to deal with the sanctions. For Country 2, there is a cost of imposing the sanctions, c (e.g., loss of trade). This cost can be high, c = −15, or low, c = −5. Country 2 can also do nothing. In this case, Country 1 benefits from violating the norm with no consequences. The situation is shown in Figure 2. Country 1 Don′ t violate V iolate norm norm Country 2 0,0 Do Harsh nothing Sanctions 40, −10 −10, −c Figure 2: Question 3 Game Tree 1. Find the subgame-perfect equilibrium of the game both when the cost is high c = −15 and when the cost is low c = −5. 2. Show the game in normal form when the cost is high c = −15. 3. When the cost is high (c = −15), does the game have any Nash equilibria other than the subgame- perfect equilibrium? If so, list them. 4. When the cost is high (c = −15), if there are any Nash equilibria besides the subgame-perfect equi- librium, explain in words why this outcome is not credible. Would such a Nash equilibrium be an “effective threat”? Now suppose that Country 1 is uncertain whether the cost of sanctions to Country 2 is high or low (i.e., there is now incomplete information). While Country 2 knows the cost of sanctions, Country 1 does not. Country 1 believes that the cost of sanctions to Country 2 is high (c = −15) with probability p and low (c = −5) with probability 1 − p. 5. Redraw the game tree to account for the uncertainty of Country 1. 6. Find the perfect Bayesian equilibria for possible values of p. 7. Are they separating or pooling equilibria? 8. For at least one equilibrium, use Bayes’ rule to find the posterior belief of Country 1 at the end of the game about whether the cost of sanctioning was high c = −15 given the actions of Country 2. Bayes’ Rule: P (B|A) · P (A) P (A|B) = P (B|A) · P (A) + P (B|¬A) · P (¬A) Page 5 Solution: 1. When the cost is low (c = −5), SP N E = {DV N, HS} and when the cost is high (c = −15), SP N E = {V N, DN }. 2. Normal form: Country 2 DN HS Country 1 DVN 0,0 0,0 VN 40, -10 -10,-15 3. Yes, N E = {(V N, DN ), (DV N, HS)} where SP N E = {V N, DN }. 4. The NE (V N, DN ) is not credible because Country 2 would not rationally do nothing. They would deviate, given the order of play. This threat would not be an “effective threat” because Country 1 is not going to choose not to violate the norm because of the threat of sanctions. 5. Revised game tree: Nature p 1−p Country 1 Country 1 DV N DV N VN VN Country 2 Country 2 0,0 0,0 DN HS DN HS 40, −10 −10, −15 40, −10 −10, −5 6. Country 2 with low costs plays HS and with high costs DN. Then, Country 1 violates the norm given: EU (V N ) = p(40) + (1 − p)(−10) = 50p − 10 EU (DV N ) = 0 1 ⇒p≥ 5 1 1 P BE = {(DV N, (DN, HS); p < ), (V N, (DN, HS); p ≥ )} 5 5 7. These are separating because the types of Country 2 are taking different actions. 8. After the game: For both PBE: P r(DN |High)P r(High) P r(High|DN ) = P r(DN )P r(High) + P r(DN |Low)P r(Low) 1·p 1 = = =1 1 · p + 0 · (1 − p) 1 Page 6  P r(HS|High)P r(High) P r(High|HS) = P r(HS|High)P r(High) + P r(HS|Low)P r(Low) 0·p 0 = = =0 0 · p + 1 · (1 − p) 1−p Page 7  U SA Bargaining Range USA’s value for war p Low cost Soviet’s value for war 0 0.45 ′ 1 (p − CUSA ) (p + CUSSR ) (p + CUSSR ) Soviet’s ideal USA’s ideal U SSR Figure 3: Question 4 Geometric Model Question 4 ( 2 5%) 1. Consider Figure. Given a known probability of winning the war, p = 0.45, and the corresponding values of p − CU SA and p + CU SSR , constituting the respective costs of war for the USA and the USSR. Would you expect war under these conditions? Why? 2. Now consider a situation, referring to Figure , in which the costs of war for the Soviet Union are uncertain. They are either p + CU SSR or p + CU′ SSR. If the USA does not know the costs of war for the USSR, what are the implications? Would you expect war? Why? 3. Set p − CU SA = 0.35, p + CU SSR = 0.55, and p + CU′ SSR = 0.65. Q is the probability that the USSR has high costs of war, p + CU SSR = 0.55. Given this game of one-sided incomplete information, what are the subgame perfect equilibria? 4. During the height of the Cold War, both the USSR and USA had the capability to destroy the other side after a first-strike attack. This was referred to as Mutual Assurred Destruction (MAD). Referring to Figure , what happens to the costs of war for both sides in this situation? Would you expect war? Why? Page 8 Solution: 1. In this situation, the players have complete information. In other words, both players know the respective probabilities of victory and the respective costs of war. It is, therefore, common knowledge where the bargaining range is. Given the bargaining range, both players will prefer a bargained peace over war. Under conditions of complete information, war is unlikely. 2. The US does not know whether the USSR faces low cost or high cost war. In other words, this is a game of one-sided incomplete information. War is possible (maybe even likely). The US does not know whether the USSR has a low cost of war (p + CU SSR ) or a high cost of war (p + CU′ SSR ). Nonetheless, as the US wants to maximize its utility, it will be tempted to make as big a demand as possible. If the US believes that the USSR has a high cost of war, it will make a demand just short of (p + CU′ SSR. But if the USSR has low costs of war, and the US makes such a high demand, the USSR will go to war. Only if the USSR has a high cost of war, will they accept the high demand from the US. The information problem facing the US is compounded by the USSR’s incentive to signal its resolve and strong relative capabilities by bluffing that it faces low costs of war. 3. To determine the Subgame Perfect Equilbria, we need to establish the probability of the USSR facing high costs of war. The US can determine this by setting the expected utility of a high demand against the expected utility of a low demand. Given the parameters provided, we can determine the following: EUU S (high demand) = q(0.65) + (1 − q)(0.35) EUU S (low demand) = 0.55 0.55 < q(0.65) + (1 − q)(0.35) 0.55 − 0.35 < 0.65q − 0.35q 0.20 < 0.30q 2/3 < q The subgame perfect equilibria is the US making the larger offer when (2/3 < q). When (2/3 > q), the US should make the offer that both types will accept (.55). In other words, the SPE is: {(high demand, (war, peace bargain); q > 2/3} and {(low demand, (war, peace bargain); q < 2/3}. 4. Under the conditions of Mutual Assurred Destruction (MAD), it is complete information that a first- strike would lead to a response that assures the destruction of the country that launched the attack. We can presume that the costs of war are thus so large that there is no value for war. The US would be at 0 and the USSR at 1, both at the extremes of the bargaining range. This leaves the entire field as a bargaining range. Given these conditions, we would not expect war. Page 9