A$.$ Suppose Russia is deciding to Invade or Not Invade its neighbor Ukraine. The U$.$S$.$ has to decide to be Tough or Make Concessions. They will make their decisions simultaneously. Their payoffs are as follows:

U$.$S$./$ Russia Not Invade Invade

Be Tough $1.$ U$.$S$.$ gets $5$

Russia gets $42.$ U$.$S$.$ gets $7$

Russia gets $3$

Make Concessions $3.$ U$.$S$.$ gets $3$

Russia gets $54.$ U$.$S$.$ gets $1$

Russia gets $9$

What is the Nash Equilibrium of this Game?

$(5$ Marks$)$

B$.$ Two gas stations, A and B$,$ are locked in a price war. Each player has the option of raising its price $($R$)$ or continuing to charge the low price $($C$).$ They will choose strategies simultaneously. If both choose C$,$ they will both suffer a loss of $$100.$ If one chooses R and the other chooses C$,($i$)$ the one that chooses R loses many of its customers and earns $$0,$ and $($ii$)$ the one that chooses C wins many new customers and earns $$1,000.$ If they both choose R the price war ends and they each earn $$500.$

Draw the payoff matrix for this game. Does either player have a dominant strategy? How many Nash equilibria does this game have?

$(10$ Marks$)$

C$.$ Two competing firms must choose their quantity of production simultaneously. Each firm can either choose high quantity of $3$ or a Low quantity of $2.$ The price for both firms is $9-$ Q$,$ where Q is the sum of both quantities. Costs are zero, the profit is simply price times quantity. For example, if firm $1$ chooses High and firm $2$ chooses Low then price is $9-(3+2)=4$; payoff for firm $1$ is $12$ while payoff for firm $2$ is $8.$

Draw the complete matrix for this game. Find all Nash equilibria. If this game were instead played sequentially, would there be a first$-$mover advantage? Briefly explain.