課程名稱︰賽局論
課程性質︰經濟系選修
課程教師︰黃景沂
開課學院：社科院
開課系所︰經濟系
考試日期（年月日）︰109.11.09
考試時限（分鐘）：180
試題 :
註：部分數學符號與式子以LaTeX語法表示。
1. Two roommates each need to choose to clean their apartment, and each can ch-
oose an amount of time t_i \in [0, 10] to clean. If their choice are t_i and
t_j, then player i's payoff is (10 - t_j)t_i - t_i^2. (This payoff function im-
plies that the more one roommate cleans, the less valuable is cleaning for the
other roommate.) Consider only pure strategies for the following questions.
(a) (10%) What is the best response correspondence of each player i?
(b) (10%) Which choices survive one round of IESDS?
(c) (10%) Which choices survive IESDS?
(d) (5%) Which choices survive the first round elimination of rationalizability
2. Consider the following Bertrand game. There are two firms producing an iden-
tical product. Each firm posts a price for the product simultaneously. Suppose
that there is a minimal unit of $0.1 for the prices. Therefore, p_1, p_2 \in {
0, 0.1, 0.2, ...}. Assume that demand is given by q = 100 - p. The cost functi-
on is c_1(q_1) = 10q_1 for firm 1 and c_2(q_2) = 12q_2 for firm 2. All buyers
buy from the firm whose price is the lowest, p = \min {p_1, p_2}. When p_1 =
p_2, the market is splits equally between the two firms. Assume that both firms
only use pure strategies.
(a) (10%) What is firm 1's best response correspondence BR_1(p_2)?
(b) (5%) What is firm 2's best response correspondence BR_2(p_1)?
(c) (10%) What are the Nash equilibria in this game?
3. Two players are playing the rock-paper-scissors game. We use R (rock), P (p-
aper), and S (scissors) to denote the three possible actions. Their payoffs can
be described by the following matrix.
https://imgur.com/QBrTwxO
(a) (5%) Please find out all pure-strategy Nash equilibria of this game.
(b) (10%) Please find out all mixed-strategy Nash equilibria of this game.
4. Consider the following game of cards. Player 1 and 2 put a dollar each in a
pot, and player 1 pulls a card out of a deck of kings and aces, with an equal
probability of getting a king (K) or an ace (A). Player 1 observes his card and
then decides whether not to play the game (N), and forfeit his dollar to player
2, or proceed with the game (Y). If player 1 proceeds with the game, then with-
out knowing which card player 1 drew player 2 can fold (F) and forfeit his dol-
lar to player 1 or call (C), in which case each player must add another dollar
to the pot. After this, if player 1 has a king then player 2 wins the pot, whi-
le if player 1 has an ace then player 1 wins the pot.
As we mentioned in the class, we can express this game as the following game t-
ree.
https://imgur.com/undefined
(a) (10%) Please present this dynamic game in a game matrix.
(b) (10%) What are the Nash equilibria of this game (including both pure and m-
ixed strategies)?
(c) (5%) What are the subgame perfect Nash equilibria of this game (including
both pure and mixed strategies)?