micro economics question about sequential-move game. Please answer it with steps

Question

micro economics question about  sequential-move game. Please answer it with steps, thank you!

Consider the following two-player, sequential-move game. There arc only two gas stations near the Purdue campus, Smart Shop and Speedway. Smart Shop is thinking of implementing a third degree price discrimination scheme by giving Purdue students a special discount. Speedway must decide whether to offer student discounts as well. If both firms decide to offer student discounts, each expects profits to be $75k per week. If both firms decide not to offer student discounts, each expects profits to be $50k per week. However, if one firm offers students discounts and the other docs not, the firm that offers the discounts will cam $100k per week and the other firm profits will be S40k per week Show the extensive form of this game assuming Smart Shop moves first. Identify, graphically, the Nash Equilibrium outcomes Find the Sub game Perfect Nash Equilibrium outcomes What would be the Sub game Perfect Nash Equilibrium outcomes if Speedway moves first? Explain

Explanation / Answer

In this problem, two players are smart Shop (SS) and Speed way (SW). They are thinking about implementing third degree discrimination. They are deciding whether to offer students discount or not. If both of them allows discount, then each will make profit of $75 per week. If one of them allows discount and other does not allow it, then player offering discount will earn $100 per week and other will get $40 pere week. If none of them offer discount, then both will earn $50 per week.

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First consider extensive form of game. Here possible moves are shown by a decision tree like diagram. Assume SS moves first. It can either allow discount or refrain from it. Against each move, his rival SW can adopt two different moves. They are shown below:

To get the best solution compare the results shown at the end. Best profit for SS is $100 when it allows discount. So it will go for discount. Kowing this information SW will also go for discount to get the best possible result. So Final combination is both will allow discount and will earn $75each.

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Answer (b):

Nash equilibrium is observed when one player assumes that other will adopt a specific strategy. On the asis of it the player will decide his optimum strategy.

Suppose SS is deciding first. It has assumed that SW will go for discount. So he will also go for discount to get highest profit of $75, Thus the profit earned by them are ($75,$75).

Suppose he has assumed that SW go for no discount. Then he will go for maximizing his outcome. Thus profit of two players are ($140,$40).

Join the aove two points and extend it to get reaction curve of SS.

Now consider the decision of SW. He assumed that his rival SS will go for discount. So he will also allow discount so that both of them will earn $75 each. ($75, $75)

If SS is assumed not allowing discount. Then SW will allow discount to maximize profit. So outcome is ($40,$100)

Join them to get reaction curve of SW. Interesection of these two reaction curve will give Nash equilibrium outcome. It is $75,$75when both will allow discount.

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