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# Tough Apps: Travelling Games

### Solutions

1. If x >= 3 and y <= 4, then setting the price at \$300 is best for the hotels, though only x people will get rooms. Otherwise, if x + y >= 3, then charge \$200 to get \$800 for the two hotels. Otherwise, set the hotel prices to \$100.

2. No. You capture no more customers than if you set the price to \$300 and receive less revenue.

3. We already saw in the Warm-Ups that you would charge \$100 if x = y = z = 2 to guarantee at least \$300 in revenue. Any other price would give a lesser guarantee (though the possibility of more). If x >= 5, then each hotel may as well charge \$300 as each will get at least two customers at that price. If x <= 4 and x + y >= 5, then \$200 is the best price (because charging \$300 would guarantee you only one customer and charging \$200 gurantees at least two).

4. There are no x, y, z for which it is worthwhile to charge something between \$200 and \$300 unless you know the prices of the other. If, on the other hand, you knew the other hotel was charging \$300 and x = 3 and y = 5, then you could charge say \$280 and get all the \$300 customers.

5. If x >= 3, then a price of \$300 will guaranteee \$900. If x + y >= 2, then \$200 will guarantee \$600. Your competitor can't charge even \$200 unless x + y >= 5.

6. With monopoly power, you know all customers will come to you, so you can set the price through direct calculation. In a competitive market, you have to assume the worst case. If you can convince customers to come to you first, then you can charge the most per room.

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