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Homework Assignment 1
Due 1/23/24
You may work with others to complete this assignment, and may submit a single hard copy
for a group of no more than 4. You are, of course, welcome to compete this individually.
Consider a very similar game to the one we played in class. Countries (denoted by i) are
endowed with income Yi , which is public knowledge. Each country secretly chooses how to allocate their income between consumption, ci , and covert defensive counter-terrorism spending,
di . If a country is not attacked they receive utility Ui = ci . If a country is attacked, their utility
is lower, but the damage is mitigated by defense spending. An attacked country’s utility is
ci . Notice that setting defense spending below 1 will yield significant pain if attacked.
Ui = did−1
i
There is a single foreign terrorist group, which will attack one country each year. The terrorist
receives benefit Bi = dcii from the attack. Thus, countries with higher levels of consumption
and lower levels of defense are more appealing targets to the terrorists. Terrorists select their
target before learning di
1) Write the budget constraint for a country with income Yi .
Yi =
2) Write the expected utility for a country which believes they will be attacked with probability pi .
E [U (Yi , ci , di , pi )] =
3) Using those two equations, find your first order condition.
dE [U ]
=
dd
4) Find optimal defense spending for a country based on their income and perceived risk of
attack.
d∗i (Yi , pi ) =
5) If the terrorists use a mixed strategy, and attack each country with probability pi , what
triple equality will make them equally happy no matter which country they end up attacking?
6) Let Y1 = 8, Y2 = 10, Y3 = 12. What is the probability the terrorists will attack each
country?
7) Explain why these results are different from what we found in class.
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