MATH4321 Game Theory (2023 Spring)
Assignment 2
Submission deadline of Assignment 1: 11:59p.m. of 4th Apr, 2023 (Tue)
Instruction: Please complete all required problems. Full details (including (i)
description of methods used and explanation, (ii) key formula and theorem used
and (iii) final answer) must be shown clearly to receive full credits. Marks can be
deducted for incomplete solution or unclear solution. You may earn extra score by
completing some bonus problems. Also, additional score will be given for well-
written assignment.
Please submit your completed work via the submission system in canvas before the
deadline. Late assignment will not be accepted.
Your submission must (1) be hand-written (typed assignment will not be accepted,
you may write on ipad if you wish), (2) in a single pdf. file (other file formats will not
be accepted) and (3) contain your full name and student ID on the first page of the
assignment.
Problem 1 (Required, 30 marks)
Three bidders are bidding an object. The bidders will take turn submitting their bids. The
bidder who submits the highest bid will get the object. If there are more than bidders
submitting the highest bid, the object will be randomly assigned to one of these bidders
with equal chance. The bidder who get the object has to pay for the object immediately.
It is given that
 Bidder 1 first submits the bid. After knowing the bidder 1’s bid, bidder 2 submits
the bid. After knowing the bids submitted by bidders 1 and 2, bidder 3 submits
his bid.
 Bidder 1 values the object at $120 and he has $100.
 Bidder 2 values the object at $100 and he has $90.
 Bidder 3 values the object at $95 and he has $120.
Assuming the bid amount submitted must be a non-negative integer, find the
sequentially rational Nash equilibrium for this bidding games. Provide full justification to
your calculation.
Problem 2 (Required, 35 marks)
We consider the following Battle of sexes games: A couple decides a place for dinner.
They can choose one of the followings: Chinese food (C), Western food (W) or Japanese
food (J). Now the boy can choose whether to let the girl to choose her preference first
(Lady first).
 If the boy chooses “lady first”, then the girl will choose her preference and let
the boy knows her preference. Then the boy then chooses his preference. The
payoff matrix is given as follows:
Girl (Player 2)
Boy
(Player 1)
(10,6) (−4,−4) (−4,−4)
(−4,−4) (7,8) (−4,−4)
(−4,−4) (−4,−4) (4,10)
 If the boy does not want “lady first”, then they will choose their preference
simultaneously (without knowing the preference of each other). The
corresponding payoff matrix is as follows:
Girl (Player 2)
Boy
(Player 1)
(9,5) (−3,−3) (−3,−3)
(−3,−3) (7,7) (−4,−4)
(−3,−3) (−3,−3) (5,9)
(a) Draw the game tree for this games.
(b) Hence, find all possible subgame perfect equilibrium for this games.
Problem 3 (Required, 35 marks)
Three companies are producing a product for the market. Each company will decide the
number of products produced. You are given that
 Given the quantites produced by three companies (denoted by 1, 2, 3
respectively), the market price of the product is = 300 − 1 − 2 − 3.
 Cost of producing one product is 20 (*Note: So the total cost for producing
units of product is 20.
(a) Suppose that three firms chooses the quantity produced simultaneously,
determine the Nash equilibrium of the games.
(b) Suppose that firm 1 is the industry leader and it will first choose the quantity
produced 1, other two firms will then choose the quantities produced
simultaneously after knowing the decision of the firm 1, determine all possible
subgame perfect equilibrium.
Provide full justification to your answer.
Bonus Problem (Optional, 30 marks)
We consider two-person multi-stage games which a game is played 3 times
sequentially. You are given that
 Two players will choose a single strategy (from , , ) simultaneously in each
game .
 The payoff matrix of the games is given by
Player 2
Player 1 (10, 10) (4, 12) (0, 13)
(12, 4) (, ) (0, 0)
(13, 0) (0, 0) (1, 1)
where 0 < < 10 is a constant.
Apparently, it is good for both players to choose (, ) in the game . However, it is not
the Nash equilibrium since the players have incentive to deviate and achieve a higher
payoff. In this problem, we would like to investigate the possibility that they may adopt
(, ) in some stage in this multi-stage games.
We let ∈ [0,1] be the discounting factor. Find all possible values of such that there
exists ∗ ∈ [0,1] such that for > ∗, there exists a subgame perfect equilibrium
which two players play (, ) in the first two stages.