Question $1$

Coca$-$Cola spends around $$3$ billion per year in advertising $($AE $=$ advertising expenditure$)$ with $$20$ billion in annual profits.$$ PepsiCo spends around $$2\mathrm{.}5$ billion with $$16\mathrm{.}5$ billion in profits.$$ If both companies stopped advertising altogether, their profits would reduce by $2\%$ in lost customers but save the costs of advertising.$$ However, if only one company reduces their advertising $($but the other does not$),$ the reduced$-$advertising company$$s profits would fall by $20\%$ and the other company$$s profits would rise by $10\%$ as they$$d take away consumers from their competitor. This results in the following payoff matrix:

$$Coca$-$ColaHigh advertising $($AE $=$ $$3)$Low advertising $($AE $=$ $$0)$Pepsi$$High advertising $($AE $=$ $$2\mathrm{.}5$b$)$$$16\mathrm{.}5$b$$18\mathrm{.}1$b$$30$b$$27$bLow advertising $($AE $=$ $$0$b$)$$$15\mathrm{.}7$b$$18\mathrm{.}6$b$$33$b$$32\mathrm{.}4$b

Coca$-$Cola's best response to Pepsi's high advertising expenditure is to spend $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

Coca$-$Cola's best response to Pepsi's low advertising expenditure is to spend $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

Pepsi's best response to Coca$-$Cola's high advertising expenditure is to spend $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

Pepsi's best response to Coca$-$Cola's low advertising expenditure is to spend $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

Paired outcomes are listed as $($player $1\text{'}$s strategy, player $2\text{'}$s strategy$)$:

The Nash Equilibrium is $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

The socially$-$preferred$/$highest joint payoff outcome is $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

Coca$-$Cola's preferred outcome is $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

Pepsi's preferred outcome is $$high AElow AE$($high AE$,$ high AE$)($high AE$,$ low AE$)($low AE$,$ high AE$)($low AE$,$ low AE$).$

$10$ points $$

Question $2$

Using the payoff matrix below, find the Nash Equilibrium:

$1\backslash 2$iiiiiiivI$(6,6)(8,8)(10,2)(18,0)$II$(5,6)(10,5)(15,2)(14,3)$III$(4,5)(12,6)(5,6)(8,9)$IV$(3,5)(14,7)(0,4)(16,6)$