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USC Economics Fall 2024 ECON 504 Problem Set - 2 DUE 10pm Sep 25
th Wednesday
SHOW ALL YOUR WORK and REASONING in DETAIL !!!
Please submit back on blackboard assignments as a SINGLE LEGIBLE PDF file.
[1] Assume rationality is common knowledge in the two player game on the right. What is game theory’s prediction for this game?(IESDS!)
[2] (Social Unrest) Suppose there is a continuum [0,1] of people in a country, each person is indexed with i ∈ [0,1]. Each person can independently choose to either stay home (H) or protest (P) the government. Staying home gives 0 utility to each person. The utility of protesting is : u(x , i) = 4x – 2 + αi where x is the proportion of the population that is protesting, and is a given parameter. Hence u implicitly depends on what others are doing (home or at the protest). Is this a game of strategic substitutes or complements? For each of α=1 and α=3,
a) Find the set of pure NE,
b) Find the set of rationalizable (IESDS) outcomes.
[3] (Hotelling Competition) Consumers are uniformly distributed along a boardwalk that is 1 mile long. They all like ice cream the same and dislike walking the same. Prices are regulated and equal for every vendor. The cost of producing ice creams is zero. If more than one vendor is at the same location, they split the business evenly (similarly, if two vendors are at the same distance, the consumer goes to each of them with the same probability). Assume that at the regulated prices the maximum distance that a consumer is willing to walk is 1 mile.
a) Consider a game in which two ice-cream vendors pick their locations simultaneously. Write down the utility function of each vendor. Find the pure strategy NE of the game.
b) Find the pure strategy NE of the game when three vendors choose locations simultaneously and the maximum distance that a consumer is willing to walk is 1/2 mile.
c) (will not be graded) There exists a maximum distance x that consumers are willing to walk such that a pure strategy Nash Equilibrium with 3 vendors exists. Find it. What happens when the maximum distance is greater than x?
[4] Give an example for the following scenarios, or show that it is impossible (a two player 2x2 or 2x3 game should be sufficient to find an example if it is possible).
a) (Is NE payoffs monotonic in game payoffs?) Assume game G has a unique pure NE. We increase player 1’s payoffs strictly for each strategy profile (for each outcome box in the game) to arrive at another game G’, again with a unique pure NE (possibly different than the NE of game G); but such that player 1 is worse off in the new NE (of the new game) compared to the old game.
b) (Is NE payoffs monotonic in available strategies?) Assume game G has a unique pure NE. We make some of player 1’s
strategies unavailable, to arrive at another game G’, again with a unique pure NE (possibly different than the NE of
game G); but such that player 1 is better off in the new NE (of the new game) compared to the old game’s unique NE.
[5] (Coordination) Suppose there are 100 drivers on the road, each simultaneously and independently choosing
driving speeds v [70 , 100]. Each driver wants to drive as fast as possible, but does not want to get a speeding ticket
at any speed (you don’t need this, but assume their net utility equals their net speed minus $100 if they get a ticket).
The cops will ticket anyone who is driving strictly faster than n other drivers where n ∈ {1 , 2, 3,. .,99}, where n is a
parameter of the game. For each n, find the set of pure NE.
[6] Suppose we are constructing a 2 player game where each player has 2 actions, by “filling in “ any payoff with an
i.i.d. draw from U[0,1] , the uniform distribution.
a) What is the probability that this game has a strictly dominant equilibrium?
b) Now solve the problem for N players with each having “a” many strategies (a∈N)