5. 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 payo of 1 and the other country gets a payo of -1. If the
countries split which locations they win, then the war is a stalemate and they each
get a payo of 0.
(a) Find all the pure and mixed strategy equilibria of this game (hint: writing
out the payo 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 payo matrix for the
game and enter the expected payos.
(d) Find all of the pure strategy equilibria for the game in (c).