SUMMER TERM 1944
TAKE-HOME EXAMINATION
ECON0027: GAME THEORY
博弈论代考 Sonic can either accept it, decline it and travel to another (randomly chosen) laboratory, or decline the offer, give up the search and end the game.
Answer ALL questions. All questions carry equal weight.
1.Sonicthe hedgehog wants to purchase running shoes from Dr It is commonly known that Sonic values the shoes at v and Dr Robotnik’s cost of procuring the shoes is zero. 博弈论代考
Dr Robotnik chooses a price and makes an offer to Sonic. Sonic can either accept the offer or decline it. If Sonic accepts, he pays Dr. Robotnik’s price and gets his shoes. If he declines the offer, the trade does not occur and the game ends.
(a)For mulate this situation as a game, define the appropriate equilibrium notion and solve for all the equilibria of this game.
(b)Supposethat in order for Robotnik to make his offer, Sonic has to travel to his laboratory and meet him there. The cost of travel for Sonic is c > 0 (also commonly known). Find all equilibria of this game when c < v. Explain what would change in your analysis if c ≥ v. 博弈论代考
(c)Supposethere are a hundred clones of Robotnik, each with his own laboratory, indexed by i 1, 2, …100 . Each clone can sell shoes to Sonic (Sonic only needs one pair). Each Robotnik i secretly chooses a price in the beginning of the game. After the prices are chosen, Sonic is randomly placed in one of the hundred laboratories. At a laboratory, the price chosen by its owner is revealed to Sonic.
Sonic can either accept it, decline it and travel to another (randomly chosen) laboratory, or decline the offer, give up the search and end the game. If the offer is accepted, Sonic pays the price and gets the shoes. If Sonic declines the offer and travels to another laboratory, he faces a travel cost c. Solve for all pure strategy equilibria of this game. Explain how competition between Robotniks affects Sonic’s equilibrium payoff.
2.Twochefs are competing for a position at a restaurant called “Food for Thought”. 博弈论代考
The value of the position to each chef is equal to v. The competition takes a form of a contest in which one of the two chefs who bakes a bigger cake In order to bake a cake of size x > 0, chef i has to procure x kilograms of flour from the restaurant at a price pi.
The restaurant will charge a chef for the flour only if he wins the contest. The price of flour pi for each chef i is drawn randomly from a uniform distribution on [1, 2]. The prices for the two chefs are independent. The chefs choose the sizes of their cakes simultaneously to maximize the expected value of the position net of the expenses for the flour. 博弈论代考
(a)Supposethe realized prices for the flour are publicly observed before the chefs make their Let p1 < p2 and suppose that in the event the chefs bake the cakes of equal size, the chef who can source cheaper flour—i.e., chef 1— wins. Find a Nash equilibrium of this game.
(b)Supposethe chefs privately observe the realization of their prices: chef 1 observes only p1 and chef 2—only p2. Solve for a Bayes-Nash equilibrium of the game.
(c)Continue assuming that the chefs privately observe the realization oftheir In addition, suppose that the restaurant charges chefs for the flour independently of the contest outcome. Solve for a Bayes-Nash equilibrium of the game.
3.Consider the following modified beer-and-quiche game
(a)Supposey = 3 and x = 7. Find all weak sequential equilibria of this game.
(b)Supposey = 3. Find all values of x such that there is an equilibrium in which both types of player 1 play B.
(c)Suppose x = 7. Find all values of y such that there is an equilibrium in which different types of player 1 play different pure actions.
4.Consideran outbreak of an infectious disease that is transmitted in a human- to-human interaction. 博弈论代考
There is a society of N Each individual i independently chooses a level of an economic activity ai 0. The economic activity is valuable for the individual, but it facilitates the spread of the infection. The payoff of an individual i is
where the fifirst part is the expected cost of being infected which is increasing in the total economic activity in the society 
，ai , and the second part is the individual benefifit of the economic activity. Assume that f is twice continuously difffferentiable, f‘ (x) > 0 and f ”(x) > 0 for all x. Also, assume that f‘ (0) = 0
and 
(a)Formulatethis situation as a Define the equilibrium notion that is the most appropriate. Find all symmetric equilibria in this game.
(b)Find the symmetric Pareto-efficient outcome in this In order to do that, solve the social planner’s problem in which the social planner treats the population symmetrically. Compare your answer to your findings in question 4a. Explain the economic intuition behind this comparison. 博弈论代考
(c)Supposethat there are two types of individuals: V (vulnerable) and I (im- mune). The types differ by their valuation of the infection risk which is represented by parameters θV and θI (θV > 1 > θI). There are M individu-als of type I and N − M individuals of type V and the type of each individualis publicly observable. The payoff of agent i who is of type t ∈ {V, I} is
Find an equilibrium in which all individuals within a given type make the same choice (the choices may differ across types). Discuss the differences between this equilibrium and the one you found in question 4a.
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