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Part $1$

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

$$$negative $1010.$

If one chooses R and the other chooses$$ C$,($i$)$ the one that chooses R loses many of its customers and earns

$$$$00,$

and$($ii$)$ the one that chooses C wins many new customers and earns

$$$$100100.$

If they both choose R the price war ends and they each earn

$$$$5050.$

Part $2$

Does either player have a dominant$$ strategy? Explain.

Part $3$

A$.$

The stations do not have dominant strategies because what works best depends on what the other station does.

B$.$

The stations have a dominant$$ strategy, which is to pick R because this maximizes their joint payoff.

C$.$

The stations have a dominant$$ strategy, which is for Station A to pick C and Station B to pick R because this is a best response regardless of what the other station does.

D$.$

The stations have a dominant$$ strategy, which is to pick C because this is a best response regardless of what the other station does.

E$.$

The stations have a dominant$$ strategy, which is for Station A to pick R and Station B to pick C because this maximizes their joint payoff.

Part $4$

What is the Nash$$ equilibrium?

Part $5$

A$.$

The Nash equilibrium is for both stations to pick R$.$

B$.$

The Nash equilibria are for both stations to pick R and for both stations to pick C$.$

C$.$

The Nash equilibria are for Station A to pick C and Station B to pick R and for Station A to pick R and Station B to pick C$.$

D$.$

The Nash equilibrium is for both stations to pick C$.$

E$.$

The game does not have a Nash equilibrium.