Problem 2: Strategic Defense There are N sites that need protection (number them 1 to N). Someone is going to pick one of them to attack, and you must pick one to protect. Suppose that the attacker is going to attack site i with probability q.. You plan on selecting a site to protect, with probability p of selecting site i. If you select the same site to protect that the attacker chooses to attack, you successfully defend that site. The choice of {q.} and {p} represent the attacker's and defender's strategy, respectively. 1) What is the probability that you successfully prevent the attack, given strategies (qi}, {pi}?? 2) If you knew {91.qN} in advance, how should you choose {p.} to maximize the probability you successfully prevent an attack? 3) If you are the attacker, and you know that the defender is going to choose the best strategy they can to maximize the probability of preventing an attack, how should you choose your strategy to maximize the probability of a successful attack? 4) Questions 2.1, 2.2, 2.3 address the probability of a successful defense from the perspective of the attacker thinking about the best possible defender. Consider as well the perspective of the defender thinking about the best possible attacker. Re-do 2.1, 2.2, 2.3 from this perspective, then argue what the 'final' strategies for each player will be in this game. In the questions that follow, we imagine that a successful attack on site i will cost the defender C,. 5) What is the expected or average cost of an attack, given strategies (q.}, {p.}? 6) If you knew {q1....,qn} in advance, how should you choose {p.} to minimize the expected cost of an attack? 7) If you were the attacker, and knew that your opponent was trying to minimize the expected cost of your attack, how should you choose {q} to maximize the expected cost of an attack? (Assume that your strategy is going to leak to your opponent.) 8) Questions 2.5, 2.6, 2.7 address the problem of the expected cost of an attack from the perspective of the attacker thinking about the best possible defender. Consider as well the the perspective of the defender thinking about the best possible attacker. Re-do 2.5, 2.6, 2.7 from this perspective, then argue what the 'final' strategies for each player will be in this game. Bonus Restricting ourselves to two sites, site A and site B, suppose that a successful attack on site i gives a reward of R, to the attacker, at cost C, to the defender. if the attacker wants to marimize their expected reward, and the defender wants to minimize their erpected cost, what strategies should they follow, and why? What if they had the opportunity to negotiate beforehand, how would that change things? Note, this will depend heavily on how {RA, Rp}, {CA. CR} relate to each other.