It is common knowledge between Countries 1 and 2 that Country 1 plans to attack Country 2. The

attack can occur at one of two locations, C and D. The success or failure of the attack depends on three

factors: where Country 1's troops are amassed prior to the attack (near C or near D), where the attack

occurs, and which location is defended. Let x denote the fraction of Country 1's troops amassed near C,

and (1 - x) the fraction of Country 1's troops amassed near D ( 0 1 x ). The game is played as

follows: simultaneously, Country 1 chooses to attack either C or D, and Country 2 decides to defend

either C or D. There are only two possible outcomes, from the point of view of Country 1: Success and

Failure. Country 1 strictly prefers Success to Failure and Country 2 has the opposite ranking. The two

countries have von Neumann-Morgenstern preferences over lotteries involving these two outcomes. The

probability of a successful attack is determined as follows. Let z denote the fraction of Country 1's

troops amassed near the location where Country 1 attacks (so z = x if Country 1 attacks at C, and z =

1 x if Country 1 attacks at D). Then the probability of a successful attack is z if Country 2 does not

defend the location where Country 1 attacks, and 1

2

z if Country 2 does defend the location where

Country 1 attacks.

For parts (a)-(f) assume that the value of x is fixed and cannot be changed; furthermore, the value of

x is common knowledge between the two countries.

(a) form astrategic-form game that represents the situation described above.

(b) Is there a range of values of x for which Country 1 has a dominant strategy? If so, state the range and

specify whether it is strict or weak dominance.

(c) Is there a range of values of x for which Country 2 has a dominant strategy? If so, state the range and

specify whether it is strict or weak dominance.

(d) Are there ranges of values of x for which pure strategy Nash equilibria exist? If so, indicate the

ranges and specify the equilibrium strategies.

(e) Are there ranges of values of x for which a mixed strategy Nash equilibrium exists? If so, indicate

the ranges and specify the equilibrium strategies and corresponding payoffs.

(f) Draw a graph representing Country 1's payoff at the Nash equilibrium as a function of x.

(g) Now imagine that, instead of being fixed, the value of x is chosen by Country 1. Events occur in the

following order:

1. Country 1 decides how many troops to amass near each location (that is, it chooses x).

2. Country 2 observes the deployment of Country 1's troops (that is, it observes x).

3. Simultaneously, Country 1 chooses to attack either C or D, and Country 2 decides to defend

either C or D.

a) Roger lives a simple life: For breakfast, he eats eggs with coffee, and for dinner he

eats hot dogs with beer. In between he watches Fox News and earns his money with

maintaining a couple of thousand twitter bots. Since he likes everything to be in

order and simple, he puts his income into two pots: One with money for breakfast

and one with money for dinner. Eggs and coffee are paid only from the breakfast

pot; hot dogs and beer only from the dinner pot. That is,

p1x1 + p2x2 wB

p3x3 + p4x4 wD

with p1, p2, p3, p4, wB, wD > 0, where subscripts 1, 2, 3, 4, B, and D refer to eggs,

coffee, hot dogs, beer, breakfast, and dinner, respectively. As usual, pi

is the price

of one unit of commodity i, xi

is the quantity consumed of commodity i, and wB

and wD is the amount of money in his breakfast or dinner pot, respectively.

His utility function is given by

u(x1, x2, x3, x4) =

x

e

1x

c

2 + x

h

3x

b

4

a

with e, c, h, b, a > 0.

aa) Given wB and wD, derive step-by-step Roger's Walrasian demand functions

for eggs, coffee, hot dogs, and beer. Verify also second-order conditions.

ab) While watching Fox News, Roger heard about the government shifting money

earmarked for fighting drugs to the construction of the border wall. He suddenly thought whether it would be better for him to move one dollar from his

breakfast pot to the dinner pot. Find a condition on the primitives (i.e., parameters e, c, h, b, a, prices p1, p2, p3, p4, and budgets wB and wD) under which

moving a dollar from his breakfast pot and putting it in the dinner pot is

better for him.

ac) Suppose that the primitives are such that it is better for Roger to move a

dollar from his breakfast pot to his dinner pot. Suppose further that both

e + c 1 and h + b 1. Would it be better for Roger to skip breakfast

altogether and just spend all the money on dinner?

b) Verify for the case of Cobb-Douglas utility functions on R

2

+ that the Slutsky substitution matrix is negative semidefinite and symmetric