Econ 109 Game Teory Winter 2023 Problem Set 2

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Econ 109 – Game Teory – Winter 2023

Problem Set 2

1.  Consider the following normal-form game:

L       C      R

5, 10

5, 3

3, 4

1, 4

7, 2

7, 6

4, 2

8, 4

3, 8

2, 4

1, 3

8, 4

(a)  [1pt] Calculate the rationalizable set R.

(b)  [0.5pt] Find all Nash equilibria.

(c)  [1pt] Find all mixed-strategy Nash equilibria.

2.  Consider the Cournot oligopoly game in which frm i = 1, … , n is choosing a quantity gi . he market inverse demand is given by P = A−bO, where O = 〉i gi  is the total supply and each frm’s average cost is c : A. In other words, each frm’s proft is given by ui (g1 , … , gn ) = (A − c − b〉j(n)=1 gj )gi .

(a)  [1pt] Find the best response function BRi (g−i) of frm i.

(b)  [1pt] Find the symmetric Nash equilibrium.

3.  Consider a game in which simultaneously player 1 selects x ∈ s1  = [0, 6] and player 2 selects y ∈ s2  = [0, 6]. he payoTs are as follows:

u1 (x, y) =  − x2         u2 (x, y) =  − y2

(a)  [1pt] Calculate each player’s best response BRi (●) as a function of the opposing player’s pure strategy.

Hint: When you are maximizing a quadratic function with a negative quadratic coeacient over an interval, it is suacient to use the frst order condition as long as its solution is in the inverval.


(b)  [1pt] Find the set B1  of all best responses of player 1 to simple beliefs about player 2’s strategy.

(c)  [1pt] Find and report the Nash equilibrium of this game.


Hint: me roots of ax2 + bx + c = 0 are given by −b ± yb2 − 4ac

                                                                                 2a          .

(d)  [1pt] Find and report the rationalizable set R. You may rely on a graphical solution to illustrate the iterated dominance procedure.


Hint: Your graph may be very stylized and does not have to be precise. It should aim to re7ect the increasing/decreasing nature of the curves and where they intersect each other and the axes. It’s a good idea to calculate a couple of points (e.g. endpoints) to plot them.


4. here are two players. hey take stones from the pile of 6 stones. Player 1 can take only 2 or 3 stones. Player 2 can take only 2 or 4 stones. Players take turns, observe all previous moves, and player 1 moves frst. A player loses if she cannot make a legal move, while another player is declared to be a winner. Let the payoT of winning equal to 1 and the payoT of losing equal to 0.

(a)  [1pt] Represent the game in extensive form. (Note: Depict only legal moves).

(b)  [0.5pt] Find all SPNE of this game and explain your answer. Who will win?


2023-02-07