# Question 3: Purely Continuous Simultaneous-move Games

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## Question 3: Purely Continuous Simultaneous-move Games

Two carts selling coconut milk (from the coconut) are located at 0 and 1, 1 mile apart on the beach in Rio de Janeiro. (They are the only two coconut-milk carts on the beach.) The cartsβCart 0 and Cart 1βcharge prices π0 and π1, respectively, for each coconut. One thousand beachgoers buy coconut milk, and these customers are uniformly distributed along the beach between carts 0 and 1. Each beachgoer will purchase one coconut milk in the course of her day at the beach, and in addition to the price, each will incur a transport cost of 0.5 Γ π2, where π is the distance (in miles) from her beach blanket to the coconut cart. In this system, Cart 0 sells to all of the beachgoers located between 0 and π₯, and Cart 1 sells to all of the beachgoers located between π₯ and 1, where π₯ is the location of the beachgoer who pays the same total price if she goes to 0 or 1. Location π₯ is then defined by the expression:

π0 + 0.5π₯2 = π1 + 0.5(1 β π₯)2

The two carts will set their prices to maximize their bottom-line profit figures, B; profits are determined by revenue (the cartβs price times its number of customers) and cost (each cart incurs a cost of \$0.25 per coconut times the number of coconuts sold).

1. For each cart, determine the expression for the number of customers served as a function of π0 and π1. (Recall that Cart 0 gets the customers between 0 and π₯, or just π₯, while Cart 1 gets the customers between π₯ and 1, or 1 β π₯. That is, cart 0 sells to π₯ customers, where π₯ is measured in thousands, and cart 1 sells to (1 β π₯) thousand.)

2. Write the profit functions for the two carts. Find the two best-response rules for each cart as a function of their rivalβs price.