War, Peace, and World Order _ Midterm 1 Problem Sets + Midterm Practice

Problem Set 2: Expected Utility Principles ● actions: A = A 1 , A 2 , … , A n ● outcomes: X = X 1 , X 2 , … , X m ● utility for each outcome: U : X → R ● probability mapping: Pr ( X ∨ A ) = the probability of outcome X given action A ○ note: ∑ i m Pr ( X i ∨ A ) = 1 and Pr ( X i ∨ A ) > 0 ● expected value action of A : E [ U ( A ) ] = ∑ i m U ( X i ) Pr ( X i ∨ A ) the actor will pick the action with the highest → expected payoff A Simple Example Problem Given ● three nations: A , B , and J with strengths m A , m B , and m J (respectively) ● the probability that J defeats B : Pr J ∨ B = m J m B + m J ● the probability that J defeats a coalition of B and A : Pr J ∨ BA = m J m A + m B + m J The Situation ● nation J wants to take a resource from either A or B ● suppose J attacks B and a war starts ○ nation A has two actions: remain neutral or intervene ○ if A remains neutral then B wins with probability Pr ( Bwins ∨ Aneutral ) = m B m B + m J ○ if A intervenes then B wins with probability Pr ( Bwins ∨ Aintervenes ) = m B + m A m B + m A + m J ● A prefers that B wins the war because U A ( Bwins ) = 1 > U A ( Bloses ) = 0 but A pays cost k if it fights Expected Utilities ● E [ U A ( neutral ) ] = Pr ( Bwins ∨ neutral ) U A ( B wins ) + ( 1 − Pr ( Bwins ∨ neutral ) ) U A ( Bloses ) = m B m B + m J 1 + m B m B + m ● E [ U A ( intervene ) ] = Pr ( Bwins ∨ intervene ) U A ( B wins ) + ( 1 − Pr ( Bwins ∨ intervene ) ) U A ( Bloses ) = m B + m A m B + m A + m Next Steps ● suppose that m A = 10, m B = 1, m J = 5, and k = will A intervene or remain neutral? ⅕ → ● consider the problem from J’s perspective and suppose it pays a cost k for fighting, gets a payoff of 1 if it defects either A or B , and gets a payoff of 0 if it loses should → J attack the powerful nation A or the weak nation B ?

Problem Set 3: Falsification Hypotheses ● hypothesis: a sentence that describes the relationship between events, typically taking the form of an if/then statement ● if A is a sufficient condition for B if → A occurs then B occurs and vice-versa Falsification ● the hypothesis “if A then B ” is false if A is true and B is false ○ note: the statement is not falsified by the evidence B is true but A is false ● necessary condition: if B , then A ● sufficient condition: if A , then B Problem Set 4: Power Transition Theory + Game Theory ( Answers ) Problem Set 5: Rationalist Model of War ( Answers ) Midterm 1 Practice Questions ( Answers ) Falsification Examine the following statements carefully. Identify whether the statement constitutes a necessary or sufficient condition and explain why. Describe how to falsify the statement and give an example. 1) When targeted, democratic nations give in to economic sanctions. target = democratic nations democratic nation complies with economic sanctions → - having a democratic target is sufficient for sanctions to work - successful sanctions are necessary for a democratic target - falsification: a democratic target that doesn’t comply with sanctions 2) Possession of nuclear weapons ensures a nation’s victory in a dispute against a non-nuclear nation. nukes victory → - having nuclear weapons is sufficient for victory - victory is necessary for nuclear power - falsification: a nuclear nation losing a dispute against a non-nuclear nation 3) A nation only intervenes in an ongoing war if it’s allied with one of the belligerents. intervention alliance → - intervention is a sufficient condition for an alliance

U CHN ( intervene )= v m RUS + m CHN m RUS + m UKR + m CHN + m USA − k Given that the U.S. has intervened, Russia is less likely to win because the U.S. is powerful, so an intervention from China would have a much greater effect on Russia’s likelihood for victory. China’s intervention would even out the conflict so each side would have a similar chance of winning. So a U.S. intervention makes a Chinese intervention more likely. This should concern the U.S. because it would potentially need to fight both Russia and China. Power Transition Theory The GDP of the U.S. is currently about $21 trillion and the GDP of China is currently about $15 trillion. However, China is growing faster. GDP projections assume China is growing at a rate of 6% and the U.S. is growing at a rate of 3%. Suppose that GDP is the appropriate measure of power. A. In the context of Kugler and Organski’s Power Transition Theory, discuss the likelihood of war over the next 30 years based on the GDP projections in the graph. The key prediction of Power Transition Theory is that power parity is a necessary condition for war, so war can only occur if power is equal. But power parity doesn’t guarantee that war will occur. Based on GDP predictions, war would be possible/likely in ten years (around 2030), but it’s not guaranteed to occur. B. If conflict occurred between the U.S. and China in 2050, would this war support or falsify the Power Transition Theory? In 2050, China is predicted to be ~2x as strong as the U.S, so Power Transition Theory predicts peace. If war occurred, then the Power Transition Theory is falsified. C. In addition to GDP, what other systematically measurable factors contribute to a nation’s power? To name a few: population, military spending, military personnel, level of technology, healthcare, education (access + quality). Bayes’ Rule In October 2019, the White House announced the withdrawal of U.S. forces from Syria, stating “the United States Armed Forces will not support or be involved in the operation, and United States forces, having defeated the ISIS territory ‘Caliphate,’ will no longer be in the immediate area.” A major criticism of this policy was that Kurdish allies who had helped the U.S. defeat ISIS were abandoned and subsequently defeated by Turkey. The decision appears to have been made by President Trump, but many of his advisors disagreed. Suppose the cost of keeping U.S. troops in Syria was 1. Suppose Trump could have been one of two types: myopic or far-sighted. A myopic leader ignores the impact of current decisions on future outcomes and a far-sighted leader factors in the consequences of today’s policy on future outcomes. Suppose a far-sighted leader places a value of 2 on maintaining the trust of allies, while the myopic leaders gain nothing from maintaining this trust. ➢ Given that Trump withdrew, use Bayes’ rule to calculate the probability that Trump is a myopic leader. Trump could be a myopic leader (M) or a far-sighted leader (F). Let the prior probability that he is myopic be θ . For a myopic leader, the payoff of protecting an ally is 0 and the payoff from keeping troops in Syria is –1. This would lead to withdrawal (W). For a far-sighted leader, the payoff of protecting an ally is 2 – 1 = 1 > 0 and therefore the far-sighted leader will not withdraw. Thus: Pr ( M ) = θ , Pr ( W ∨ F ) = 0 , and Pr ( W ∨ M ) = 1 . So Bayes’ rule shows: Pr ( M ∨ W ) = Pr ( M ) Pr ( W ∨ M ) Pr ( W ) = θPr ( W ∨ M ) θPr ( W ∨ M ) + ( 1 − θ ) Pr ( W ∨ F ) = θ θ = 1 By withdrawing, Trump is showing that he’s a myopic leader — he doesn’t care about maintaining trust and is therefore willing and likely to abandon allies. ➢ Given this result, do you think China was emboldened in its attempts to claim sovereignty over disputed islands in the South China Sea? China likely anticipated that the U.S. would do little to help Taiwan and any other nations the U.S. was aligned with were

China to claim sovereignty over the islands in the South China Sea, so China was emboldened to claim sovereignty. Bayes’ Rule In March 2013, President Obama declared that if he discovered that Bashar al-Assad’s government used chemical weapons in Syria, it would be a game-changer. He forcefully stated his position, noting that “once we establish the facts, I have made it clear that the use of chemical weapons is a game changer.” In September 2013, UN inspectors produced evidence of chemical weapon usage. Obama had to decide whether to make good on his promise and he could be either a Hawk or a Dove. If Obama stayed out of Syria, then his payoff was 0. If he was a Hawk, then his payoff from intervening would be 1, but if he was a Dove, then his payoff would be –1. Suppose the prior probability that Obama was a Hawk was ½. ➢ Given that Obama did not intervene in Syria, calculate the probability that he was a Hawk. Let σ H = Pr ( Hawk intervenes ) and σ D = Pr ( Doveintervenes ) . p = 1 2 = prior probabilitythat Obamawasa Hawk . Hawks get a higher payoff from intervention, so σ H = 1 and σ H = 0 . Bayes’ rule shows: Pr ( Hawk ∨ nointervention ) = p ( 1 − σ H ) Pr ( nointervention ) = p ( 1 − σ H ) p ( 1 − σ H ) + ( 1 − p ) ( 1 − σ D ) = p× 0 p× 0 + ( 1 − p ) × 0 ➢ In 2014, Russia invaded Crimea. Based on the evidence from Syria, do you think President Putin should have been concerned about U.S. intervention? Obama’s reluctance to intervene in Syria suggested that he was a Dove. Given this information, Putin would have expected the U.S. to stay out of Ukraine if he invaded. Strategic Decisions (Game Theory) Consider the following strategic problem between Israel and Hamas. Hamas uses rocket attacks out of Gaza against Israel. Israel has missile defense systems, such as the Iron Dome, that can shoot down incoming rockets. To simplify, suppose there are two targets, A and B. Hamas can target either A or B and Israel’s defense system protects A or B. If Hamas attacks a defended target, then only half the damage is done. Suppose that the value of target A is 1 and the value of target B is v > 1. Hamas wants to maximize damage and Israel wants to minimize damage. Initially suppose the Hamas doesn’t know which target Israel is defending, as represented by the following normal form game: Hamas Attacks Target A Target B Israel Defends Target A -½, ½ - v , v Target B -1, 1 - v /2, v /2 A. Suppose the value of target B is much higher than target A — for example, v = 4. What is the Nash Equilibrium? If v /2 > 1, then Hamas has a dominant strategy to attack target B, and given that Hamas will attack target B no matter what Israel does, Israel should defend target B. The Nash Equilibrium is therefore ( B , B ). B. Suppose that the value of target B is only a little higher than target A — for example, v = 3/2. What is the Nash Equilibrium? If 1 > v /2, then Hamas wants to attack the unguarded target and Israel wants to defend the target that Hamas attacks. Therefore there’s no Nash Equilibrium. C. Suppose that Hamas can observe which target is defended before deciding which site to attack. Draw a game tree to represent this scenario and find the subgame perfect equilibrium. If Israel defends target A, Hamas will attack target B. If Israel defends target B, Hamas will attack target A if 2 > v and Hamas will attack target B if v > 2. Since defending A gives Israel a worse payoff, Israel will always defend B.