Present an example of a game with a pure-strategy Nash equilibrium Answer

Part I. Present an example of a game with a pure-strategy Nash equilibrium (call it s*) in which each player chooses a weakly dominated strategy. (Two players with two strategies apiece will suffice).

Part II. Present an example of a game with two pure-strategy Nash equilibria (call them s* and s**). These two pure-strategy Nash equilibria should satisfy the following three properties: (a) in the Nash equilibrium s* , each player chooses a weakly dominated strategy, (b) in the Nash equilibrium s**, each player chooses a strategy that is not weakly dominated, and (c) both players receive a higher payoff in equilibrium s than from the alternative Nash equilibrium s** :

QUESTION 2. For the game “A War Between Two Generals” in Question 4 of Homework 1, apply “iterated elimination of strictly dominated strategies” to solve the game. Is this game “dominance solvable”?

General A General B
ab ac bc
ab (0,0) (1,-1) (1,-1)
ac (-1,1) (0,0) (1,-1)
bc (-1,1) (-1,1) (0,0)

QUESTION 3. Consider the following two-player game:
Column
North South East West
Earth 1, 3 3, 1 0, 2 1, 1
Row Water 1, 2 1, 2 2, 3 1, 1
Wind 3, 2 2, 1 1, 3 0, 3
Fire 2, 0 3, 0 1, 1 2, 2

(a) Find strictly dominated/dominant strategies for both players (Row and Column), if any.
(b) Use iterated elimination of strictly dominated strategies to reduce the game as much as possible.
(c) Is the game dominance solvable? Explain.
(d) Final all pure-strategy Nash equilibria of this game.

QUESTION 4. Consider a three-player game in which there is a prize worth $30. Three contestants, Alice, Bob and Charles are the three players. Each player can buy a ticket worth $15 or $30 or not buy a ticket at all (hence each player has three strategies). They make these choices simultaneously and independently. After the decisions are made, the game organizer observes the ticket-purchase decisions and awards the prize according to the following rule:

If no one has bought a ticket, the prize is not awarded.

If there is only one player who has purchased the ticket, the prize is awarded to that player, regardless of whether the player has purchase a $15-ticket or a $30-ticket.

If there are two or more players who have purchased the ticket, the organizer awards the prize to the player who has purchased the highest-cost ticket, and the prize is split equally between the players if there are ties among the highest-cost ticket buyers.

Answer the following questions with regard to the above game:

(a) Represent the above strategic interaction using a game table, using Alice as the row player, Bob as the column player, and Charles as the matrix player.
(b) Find all pure-strategy Nash equilibria of this game.

QUESTION 5.
(A Duopoly Model of Cournot Competition) Consider a duopoly model of Cournot competition where two firms compete by choosing quantities of an identical good. Suppose firm 1 has a marginal cost of 1, firm 2 has a marginal cost of 2. Both firms face a general linear demand curve represented by P = 4-Q.

(a) Write down the profit functions of the firms.
(b) Find the best-response “functions” of the two firms. (You should write down a complete best-response “function”, i.e., firms’ quantities cannot be negative.)
(c) Solve for the Nash equilibrium of this game. In addition, express the Nash equilibrium graphically, as in the lecture notes.
(d) What is each firm’s equilibrium quantity and equilibrium profit? In the equilibrium, which firm produces more and which firm’s profit is higher? Explain.
(e) Now suppose the two firms in the market forms a cartel. If the cartel maximizes total industry profits (that is, if the two firms act like a monopoly), what does this imply about the division of output between the firms? (Ignore here the issue of whether the cartel is stable or not).