Consider two cigarette companies, PM and BA$.$ If neither company advertises, the two companies split the market and earn $$50$ million each. If they both advertise, they again split the market, but profits of each company are lower by $$10$ million since each company must bear the cost of advertising. Yet if one company advertises while the other does not, the one that advertises attracts customers from the other. In this case, the company that advertises earns $$60$ million while the company that does not advertise earns only $$30$ million.

Question:

$(1)$ Draw a matrix of strategies and payoffs for the two companies. $6$ points

$(2)$ What is the Nash equilibrium in this game? Why? $4$ points