Question

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 qi. You plan on selecting a site to protect, with probability pi 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 {qi} and {pi} represent the attacker’s and defender’s strategy, respectively.

1) What is the probability that you successfully prevent the attack, given strategies {qi}, {pi}?? (5 points)

2) If you knew {q1,…,qN} in advance, how should you choose {pi} to maximize the probability you successfully prevent an attack? (5 points)

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? (5 points)

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. (10 points)

In the questions that follow, we imagine that a successful attack on site i will cost the defender Ci.

5) What is the expected or average cost of an attack, given strategies {qi}, {pi}? (5 points)

6) If you knew {q1,…,qN}  in advance, how should you choose {pi} to minimize the expected cost of an attack? (5 points)

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 {qi} to maximize the expected cost of an attack? (Assume that your strategy is going to leak to your opponent.) (5 points)

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 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. (10 points)

Bonus (20 points MAX): Restricting ourselves to two sites, site A and site B, suppose that a successful attack on site i gives a reward of Ri to the attacker, at cost Ci to the defender. if the attacker wants to maximize their expected reward, and the defender wants to minimize their expected 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, RB}, {CA, CB} relate to each other.

(Please answer all questions)

Answer 1

1)For every 'i' the probability of attacking will be  and probability of preventing is . So, the probability of attack and defend on 'i' will be .

If it comes true. the attack will be prevented so total probability would be sum of all  in range of ( 1 to n ).

2)i is Successfully attacked site  means, we failed to prevent that site.

 For i, probability of getting not defended is .

 For i probability of getting attacked is 

 From above, expected cost will be calculated it is  for all i from 1 to n

3)To prevent the attack we should choose the same values for from 1 to n.

It ensume that the site which has highest probability of getting attacked has also highest probability of getting defended. It maxi mising overall chance of the attack prevented.

4)Consider summation  from 1 to n.

Let assign .

 The site which has the highest expected cost if an attack is occuted also has highest probability to defended.

 The probability of defend is distiduted in ratio of the expected cases that would be incurred if attached.

5)We assigned   probabilities to all n sites to be attacked because in into the probability of getting attacked will remain highest.

6)As above consider equal  probabilities  to attack all 'n' sites, because defender will try to minimize the cost he would defend in ratio of cost incurred. So, expected cost of all sites will be same.

7)If the attacker would attack all of them with same probabilities of 1/n and defender would defend them with same  probabilities of 1/n.

8)The defender would defend them in ratio of cost incurred if particular site is attacked and the attacker would attack all of them with equal  probabilities of (1/n) as the expected cost of all is same.