Short Answer/Essay Questions
Problem b1-1
The following three problems are about the mini-ultimatum game in which the proposer receives 1000 yen as endowment and Y yen as a possible offer amount. Here, let XXX be the last three digits of your student ID, and Y=XXX yen. However, in case your XXX is 000 or 001, then let Y=521.
Answer the problems, assuming that each player is a selfish economic man for the analysis based on game theory (Do not consider mixed strategies as this course only treats pure strategies).
A) Consider a mini-ultimatum game in which the proposer can offer either 1 yen or Y
yen to the responder. Assume that each player shows his own strategy at the
beginning of the game, and the responder makes a commitment to his own strategy
even if the mood changes. Write a payoff table for this game and find all the Nash
equilibria. Explain why each of these strategy pairs is a Nash equilibrium, while the
others are not.
B) Suppose that the mini-ultimatum game in the previous problem is played as a
sequential game. Draw a game tree, illustrating that offering 1 cent is a subgame
perfect equilibrium and that offering Y yen is not subgame perfect equilibrium.
Write the payoffs of the proposer and the responder as two numbers in parentheses
in the game tree diagram.
C) Consider another mini-ultimatum game in which the proposer can offer can either 0 yen , 1 yen, or Y yen to the responder. Draw two game trees, illustrating that there
are two subgame perfect equilibria, and that there is no other subgame perfect
equilibrium. Draw one tree for a subgame perfect equilibrium and show the best response in each node by making the appropriate branch a bold line. Write the
payoffs of the proposer and the responder as two numbers in parentheses in each of
the game tree diagrams.
Problem b1-2
In a market experiment with an imaginary good, assume a case where five buyers have values of 10, 20, 30, 40, 50 yen, respectively, and five sellers have costs of 10, 20, 30, 40, 50 yen, respectively. Assume that each buyer can buy only one unit and that each seller can sell only one unit. Draw a diagram with demand and supply curves and find the market equilibrium price and quantity. Explain your diagram.
Problem b1-3
In a market experiment with an imaginary good, suppose the following market
experiment: all buyers have the same value, Y yen (let Y=300 + XX where XX is the
last two digits of your student ID) and all sellers have the same cost (180 yen). Assume
that each buyer can buy only one unit and that each seller can sell only one unit.
Answer the following two questions:
A) Find the market equilibrium price and quantity when there are 20 buyers and 30
sellers, by drawing a diagram with demand and supply curves. Explain your diagram.
B) Find the market equilibrium price and quantity when there are 30 buyers and 20
sellers, by drawing a diagram with demand and supply curves. Explain your diagram.
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