There are two countries that are battling in two locations: location A and

location B. Suppose that each country has 2 divisions in its army. Divisions

cannot be subdivided, so a country must choose either to put both divisions

at Location A, both at Location B, or to put one division at location A and

one at location B. If a country has more divisions at one location, then they

win that location. If the countries have the same number of divisions at a

location, then each country wins that location with a probability of 1/2. A

country wins if it wins both locations and then it gets a payoff of 1 and the

other country gets a payoff of 1.

If the countries split which locations they

win, then the war is a stalemate and they each get a payoff of 0.

a. Find all the pure and mixed strategy equilibria of this game (hint: writing

out the payoff matrix for the normal form will make this fairly easy).

b. Now suppose that both countries have three divisions (so they can put

three divisions at one location or two at one location and one at the other

location). Find all the pure and mixed equilibria of the game.

c. Consider a variation of the game where one country has three divisions

and the other has only two divisions. Draw the normal form payoff matrix

for the game and enter the expected payoffs.

d. Find all of the pure and mixed strategy equilibria for the game in c).