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ECON5111 Economics of Strategy

Problem Set 1

2.

Suppose ice cream consumers are distributed along a straight beach of 1km. Each consumer buys 1 ice cream from the closest ice cream cart. If carts are equidistant from a consumer, he will buy from each of them with equal probability. We have seen in lecture what the Nash equilibrium is if 2 carts simultaneously choose where to position themselves. Now suppose there are 3 ice cream carts         simultaneously deciding where to position themselves. Is it a Nash equilibrium for all 3 to position    themselves where the median consumer is located? (Explain in no more than two short sentences).

3.

There are 2 players. Each has an endowment of 10 dollars and chooses how many dollars to        contribute to a public pool. Call x1  and x2 the contributions of player 1 and player 2, respectively.

If x1 + x2 ≥ 5, then players 1 and 2 receive a payoff of 10 – x1 + 1 and 10 – x2 + 1, respectively. Otherwise they receive a payoff of 10 – x1  and 10 – x2, respectively.

a)    Find a Nash equilibrium of this game.

b)    Now suppose that if x1 + x2 ≥ 5, players 1 and 2 receive a payoff of 10 – x1 +10 and 10 – x2 + 10, respectively. Find a Nash equilibrium of this game.

4.

There are two firms producing a good, firm 1 and firm 2. Both have constant marginal cost of             production of 2. There are a total of 10 consumers, each of which will buy at most 1 unit of the good from whichever firm charges a lower price. If both firms charge the same price, each consumer will   choose randomly which firm to buy from (so on average each firm will sell to half of all customers).   The firms compete via Bertrand competition – simultaneously setting price – but they can only          choose prices that are integers, i.e. 1, 2, 3, 4, etc. This game has three pure strategy Nash equilibria. What are they?