Final_Exam_Review_Key

1 Introduction to International Relations Spring 2023 Final Exam Practice Questions Key Below you can find a number of practice problems of similar style, content, and difficulty to what might appear on your final exam (these actually may be somewhat more difficult than the exam). As with the midterm review, this set of problems is not exhaustive of the content you are liable for but meant to guide your study in the right direction. As the final exam is cumulative, questions may be drawn from lectures/readings/recitations from any point in the semester – although there will be a natural bias towards content arising in the second half of the course. Answers to all questions should be short and to the point, one or two sentences generally being sufficient. 1. What is the fundamental puzzle of war, according to Fearon? What five assumptions must hold to generate this puzzle, and which three of these assumptions does Fearon favor as his rationalist explanations for war? The fundamental puzzle of war is that it occurs despite its risks and costs; an ex ante bargaining range should exist. This result holds under the assumptions of rationality, risk neutrality/aversion, complete information, binding contracts, and perfect issue divisibility. The last three of these, when loosened, are Fearon’s rationalist explanations for war. 2. What is a Nash equilibrium? What is a best response? What is the name of the technique for solving extensive form games of complete information (the type of game tree we have seen most frequently in class)? A Nash equilibrium is a simultaneous best response. A best response is when, holding the behavior of everyone else constant, you are doing what is best for you. Backwards induction. 3. One of the first major formal models of conflict introducing domestic politics to explain patterns of war was Fearon’s 1994 model of audience costs. Under what conditions do audience costs arise? Are threats more credible when audience costs are high or low? Are threats made more frequently when audience costs are high or low? Audience costs arise when a leader makes a threat and then backs down when resisted. Threats are more credible when audience costs are high, and threats are made more frequently when such costs are low.

2 4. Consider the following model of aid-for-policy deals in which Leader 1 offers some ? resources (out of the budget ? 1 ) to Leader 2 (in addition to their budget of ? 2 ) in return for some (public good-type) policy worth 𝜎 1 utility to the winning coalition of Leader 1 (? 1 ) and a (public bad-type) disutility of 𝜎 2 to the winning coalition of Leader 2 ? 2 . This offer can be either accepted or rejected, rejection leading to the entire leader’s budget being spent on private goods for their coalition and acceptance leading to the deal described above. a) Under what condition does Leader 2 accept the aid-for-policy deal? Solve this condition for ? 2 . Are large or small coalition leaders more likely to accept a given aid-for-policy deal? Leader 2 accepts when 𝑅 2 +𝑥 ? 2 − 𝜎 2 ≥ 𝑅 2 ? 2 , i.e. ? 2 ≤ 𝑥 𝜎 2 . Small coalition leaders are more willing to accept. b) Given the above, what is the optimal accepted offer that Leader 1 can make to Leader 2, i.e., that which leaves them with the largest amount of resources for themselves? Call this value ? ∗ . Are large or small coalition leaders cheaper to buy off? In other terms, conditional on receiving aid do large or small coalition leaders get more? Leader 1 makes Leader 2 indifferent, keeping the surplus for themselves: ? ∗ = ? 2 𝜎 2 . Small coalition leaders are cheaper to buy off; conditional on receiving aid large coalition leaders get more. c) Under what conditions is Leader 1 willing to make such an accepted offer? Solve this condition for ? 1 and substitute in the optimal offer found in part b. Are large or small coalition leaders more likely to provide aid? Leader 1 makes such an offer when 𝑅 1 −𝑥 ∗ ? 1 + 𝜎 1 ≥ 𝑅 1 ? 1 ; i.e. ? 1 ≥ 𝑥 ∗ 𝜎 1 ; i.e. ? 1 ≥ ? 2 𝜎 2 𝜎 1 . Large coalition leaders are more likely to provide aid.

4 c) Show that if beliefs are dogmatic (the prior belief in the hypothesis is either zero or one) that no updating occurs (the posterior equals the prior). Pr(𝐻|?) = Pr(?|𝐻) × 0 Pr(?|𝐻) × 0 + Pr(?|¬𝐻) × 1 = 0; Pr(?|𝐻) × 1 Pr(?|𝐻) × 1 + Pr(?|¬𝐻) × 0 = 1 d) Suppose that the probability of the evidence under ¬𝐻 is zero while the probability of the evidence under 𝐻 is positive. What is the posterior belief in 𝐻 after observing this ‘critical test’ ? ? Pr(𝐻|?) = Pr(?|𝐻) Pr(𝐻) Pr(?|𝐻) Pr(𝐻) + 0 × Pr(¬𝐻) = Pr(?|𝐻) Pr(𝐻) Pr(?|𝐻) Pr(𝐻) = 1 6. When solving for a Perfect Bayesian Equilibrium of the form we have encountered in class, what are the three essential steps? Check the consistency of posterior beliefs given the presumed behavior of types. Check the best response of the actor with incomplete information given these beliefs. Check the best responses of each type of the other player, given the behavior of the actor with incomplete information.

5 7. Consider the following audience cost model, presented in recitations. For our purposes we may omit the story underlying the model and instead simply recall that ? ̅ 𝑖 > ? ̂ 𝑖 for both leaders, that 0 < ? < 1 , and that −𝛼 > 0 . a) Suppose that audience costs are sufficiently low (−𝛼 ≥ ? ̅ 1 > ? ̂ 1 ) . Is the following strategy profile a Perfect Bayesian Equilibrium? {Threaten/Don’t Fight, Threaten/Don’t Fight; Resist; Pr(Strong|Threaten) = 𝜃} Yes. Both types threaten so posterior = prior. Player 2 resists because 1-x>1. Given resistance of Player 2, Player 1 prefers to threaten and then back down since – a>0. b) Now suppose that audience costs are sufficiently high (? ̅ 1 > ? ̂ 1 > −𝛼) . Notice that this implies that both the strong and weak type will fight if they are resisted. Write down the expected utility calculation Leader 2 needs to perform to decide whether or not to resist when threatened and solve this for 𝜃 . Leader 2 will resist if the following condition holds: 𝜃? ̂ 2 + (1 − 𝜃)? ̅ 2 ≥ 1 − ? This implies that Leader 2 will resist a threat when the probability of facing a strong type leader is sufficiently low: 𝜃 ≤ 1 − ? − ? ̅ 2 (? ̂ 2 − ? ̅ 2 )

7 d) When are they willing to make such an offer? i.e. under what conditions is Player 2 willing to make this optimal offer to Player 1? Does war ever occur in the second period? They make such an offer when 1 − ? 1 + 𝛿(1 − ? 2 ∗ ) ≥ 1 − ? 1 + (1 − ?)𝜆𝛿 which implies that 𝜆 ≤ 1 after a bit of algebra. Thus war never occurs in the second period. e) Given this result, what is the present value for each player of the deal being accepted in the first period? When is this better, for Player 1, than going to war in the first period? Solve this condition for ? 1 and call the indifference point ?̃ 1 . The present value for player 1 is ? 1 + 𝛿? 2 ∗ = ? 1 + ?𝛿𝜆 . They prefer this over going to war today when it is the case that ? 1 + ?𝛿𝜆 ≥ ?(𝜆 + 𝛿) which implies that ? 1 ≥ ?̃ 1 ≡ ?(𝜆 + 𝛿) − ?𝛿𝜆 . f) Recall that there is only one unit of land to divide up today. When is it the case that the minimal demand of Player 1 in the first period exceeds this constraint (i.e. ?̃ 1 > 1)? Solve this condition for ? and call the threshold value ? ̃. The budget constraint is violated when ?(𝜆 + 𝛿) − ?𝛿𝜆 > 1 which implies that ? > ? ̃ ≡ 1+?𝛿𝜆 𝜆+𝛿 g) Given this condition, is war more likely to occur when the shift in Player 1’s power across periods is large or small? Is conflict more likely when actors are patient or impatient? Conflict is more likely when the shift in power is large (a small q increases the range of p for which conflict occurs). Conflict is more likely when actors are patient (suppose 𝛿 = 0 ; then conflict would never occur since it would require ? > 1 𝜆 > 1 , which is impossible.

8 9. Consider the situation where you are a leader determining whether or not to put additional effort into a war effort. If you do not put in such effort, suppose that you lose the war with certainty and receive a payoff of 𝑅 ? where ? is your budget and ? is the size of your domestic winning coalition. If you do put in the effort, leaving you with some budget ? < ? , suppose that you win with certainty yielding a payoff of ? ? + ? , this last term being the public good of victory. a) When is it preferable to put in additional war effort? Solve this condition for ? . Leaders put in additional effort when it is the case that ? ? + ? ≥ 𝑅 ? which implies that ? ≥ 𝑅−? ? . b) Given the above, are large or small coalition leaders more likely to put in extra war effort? Large coalition leaders are more likely to put in extra war effort. 10. Consider the generic prisoner’s dilemma game given below. Here C stands for the cooperative payoff, D for the payoff when cooperation breaks down, T for the temptation payoff of defecting while your opponent cooperates, and S for the sucker payoff when you cooperate when your partner defects. Payoffs are ordered T>C>D>S. Cooperate Defect Cooperate C,C S,T Defect T,S D,D Recall that the value of an infinitely repeated payoff stream, starting today and going on forever, has a present value of ? = 𝑢 1−𝛿 where S is the stream value, u is the per-period utility, and 𝛿 is the discount factor. Suppose that you are Player 1 and know that Player 2 is playing a “grim trigger” punishment strategy such that if you defect against them they will defect against you forever while if you do not defect they will cooperate forever. Under what condition on 𝛿 do you prefer to cooperate with Player 2 rather than Defect against them today (receiving the temptation payoff) followed by mutual defection in every following period? Is conflict (defection) more likely when actors are patient or impatient?

10 12. Suppose that you have to decide whether or not to contribute effort to the production of a public good. The benefit of such a good is worth ? , the cost of contributing effort is ? . If you do contribute to the public good, suppose that the benefit is realized with probability 𝑛+1 𝑁 where 𝑁 is the total number of actors and 𝑛 is the total number of contributors, other than yourself. If you do not personally contribute, suppose that the probability of the public good being produced is 𝑛 𝑁 instead. a) Under what conditions do you participate in the creation of the public good? Simplify this expression. You contribute if it is the case that 𝑛+1 𝑁 ? − ? ≥ 𝑛 𝑁 ? , or, ? 𝑁 ≥ ? . b) Is your participation more likely when 𝑁 is large or small? Small. c) Is your participation more likely when ? is large or small? Large. d) Is your participation more likely when ? is large or small? Small. e) Compare two situations, one in which 𝑁 is small and the other in which 𝑁 is large (i.e. different degrees of multilateralism). To encourage participation, do the costs/benefits have to be smaller/larger in the large 𝑁 case relative to the small 𝑁 case? When N is large, one needs either higher benefits or smaller costs of compliance to generate participation. Likewise, if N is small, participation may obtain even under relatively smaller benefits and higher costs of compliance. This resembles the problem of multilateral versus bilateral cooperation where (holding the benefits constant) multilateral agreements are generally shallower (lower cost of compliance) and bilateral agreements are deeper (higher costs of compliance).