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1. Sonic the hedgehog wants to purchase running shoes from Dr Robotnik.   It is commonly known that Sonic values the shoes at v and  Dr  Robotnik’s cost of procuring the shoes is zero.

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)  Formulate this situation as a game, define the appropriate equilibrium notion

and solve for all the equilibria of this game.

Answer:   This is an extensive form game with Robotnik’s strategies p e R+ and Sonic’s strategy b  : R+   → {0, 1}.  The payoffs are UR   = pb(p) and US  = (v - p)b(p).  We will look for SPNE. There is a unique one in which

p*  = v and b(p) = I{p < v}                                                                 □

(b) Suppose that in order for Dr. 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 2 v .

Answer: This is a hold-up problem. In a subgame in which Robotnik makes an offer, the equilibrium found in the previous bullet point holds. The equi- librium value of the Robotnik’s offer to Sonic is zero, so if cost of travel is positive, Sonic does not visit Robotnik.  It does not matter how c compares

to v .                                                                                                         □

(c) Suppose there are a hundred clones of Dr.   Robotnik, each with his own laboratory, indexed by i e {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 .

Answer: In a symmetric pure strategy SPNE, every Robotnik sets the price equal to v .  The only possible deviation is downward and it is clearly not profitable.  To show that there are no other equilibria, consider some other price vector p =  (p1 , p2 , ..., p100 ).   Let p = min{p1 , p2 , ..., p100 }.   Clearly, p < v .  If pi  = p, Robotnik i can increase his price by min{c, v - p} and be strictly better off in a history in which Sonic visits him.                             □

2. Two chefs are competing for a position at a restaurant called “Food for Thought” . 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 wins. 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.

(a) Suppose the realized prices for the flour are publicly observed before the chefs

make their choices. 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.

Answer: Chef’s payoff if he wins the contest is v - pi x. The game is a first price auction in which the chefs valuations are v/pi .   In equilibrium, since

v/p1  > v/p2 , both chefs bake a cake of size v/p2                                                          □

(b) Suppose the 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.

Answer: This is a first price auction with private valuations v/pi . Let F be a c.d.f. for v/pi : F (v/p) = 2 - p. Since v/2 be the lowest possible valuation. By envelope condition, the size of the cake multiplied by the probability of chef i winning

v/pi

xi (v/pi )F (v/pi ) = v/pi [F (v/pi )] -     [F (x)]dx

v/2

Therefore, the equilibrium cake size is

v/pi

xi (v/pi ) = v/pi -                  dx

v/2

(c)  Continue assuming that the chefs privately observe the realization of their prices.  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.

Answer: This is an all-pay auction with private valuations v/pi . Let F be a c.d.f. for v/pi : F (v/p) = 2 - p. Since v/2 be the lowest possible valuation. By envelope condition,

v/pi

xi (v/pi ) = v/pi [F (v/pi )] -     [F (x)]dx

v/2

□

3.  Consider the following modified beer-and-quiche game

3,-1                                x,0           y,1                                 7,0

2

Q

p                    1 - p     

1

strong                   weak

B

5,-1                                9,0           1,1                                 5,0

(a) Suppose y = 3 and x = 7. Find all weak sequential equilibria of this game.

Answer:  There are no separating equilibria in this game:  Case 1:  strong type plays B .  In this case player 2 plays N and weak type wants to deviate from Q to B.

Case 2:  strong type plays Q.  In this case player 2 plays D and weak type wants to deviate from B to Q.

There is a pooling equilibrium in which both types of player 1 play B, as long as p < 1/2:  Payoff from playing Q for strong type is 3 and for weak type is 1. If player 2 is playing aD + (1 - a)N, both types will choose B as long as

9 - x      1

4         2

and

5(1 - a) + a 2 y  年÷  a < 1/2

So as long as the probability of N is above a half both types want to play B. This will happen when the belief of the agent 2 satisfies

1 2 2(1 - µ)  年÷  µ 2 1/2

where µ = Pr{strong|B}

The consistency of beliefs implies that µ = p, hence this equilibria exists iff p 2 1/2.

There is also an equilibrium in which both types play Q: Consider a belief µ 2 1/2 for which the best reply by player 2 is (D,n,d’).   Note that this equilibrium can occur for any value of p because consistency of the belief has  no bite.                                                                                                     □ (b) Suppose y = 3. Find all values of x such that there is an equilibrium in which

both types of player 1 play B .

Answer:  From the previous derivations, this equilibrium exists as long as

x < 9.                                                                                                       □

(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.

Answer:   Equilibrium in which strong type player B and weak type plays Q exists if y 2 5. The beliefs are µ = 1. Player 2’s strategy is (N,n,d’).

Equilibrium in which strong type player Q and weak type plays B exists if

y < 1.The beliefs are µ = 0. Player 2’s strategy is (D,n,d’).                     □

4.  Consider an outbreak of an infectious disease that is transmitted in a human- to-human  interaction.   There  is a society of N  individuals.   Each  individual i

independently chooses a level of an economic activity ai   2 0.   The economic activity is valuable for the individual, but it facilitates the spread of the infection. The payoff of an individual i is

-f ╱ aj \ + ai ,

where the first  part is the expected cost of being infected which is increasing

N

in the total economic activity in the society      ai , and the second part is the

i=1

individual benefit of the economic activity.  Assume that f is twice continuously differentiable, f\ (x) > 0 and f\\ (x) > 0 for all x.  Also, assume that f\ (0) = 0

and  lim f\ (x) = o

x→&

(a)  Formulate this situation as a game. Define the equilibrium notion that is the

most appropriate. Find all symmetric equilibria in this game.

Answer: This is a game in normal form and we will look for NE. The best

f\ (ai +z aj ) = 1

j i

Thus, the symmetric equilibrium is ai(*)  = a*  = f\ N(一1)(1) .                               □

(b)  Find the symmetric Pareto-efficient outcome in this game.  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.

Answer: Social planner’s solution is

afb  =  < a*

Agents impose negative externalities on each other which are corrected by a partial lockdown imposed by the social planner.                                        □

(c) Suppose that 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 individual is publicly observable. The payoff of agent i who is of type t e {V, I} is

-θt f ╱  aj \ + ai .

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.

Answer: First, note that there is no upper bound on ai , hence the first order condition for I individuals will hold:

f\-1  ╱ 、

M

This means that the constraint ai  2 0 will be binding for individuals V and aV(*)  = 0.

Note that MaI(*)  > Na* . Individuals of type I under invest in the public good (low level of infection in the society) and crowd out the economic activity by the vulnerable individuals.                                                                         □