Nash equilibrium.
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- Consider the following normal form of the game with two players. What is the Nash Equilibrium of this game?What is the pure strategy Nash equilibrium outcome if there is one Is this a socially optimal outcome? If not, which outcome is preferredThe Chinese government has created a fund worth more than 20 trillion won to foster the semiconductorindustry. Although there is a large technological difference between memory semiconductors, systemsemiconductors can be developed in a short period of time. The number of companies producinghomogeneous quality products has increased.■Question (a) In the Cournot game, when the number of firms increases from 2 to n, compare the output,total output, and profit of each firm with N.E. in the Cournot model. ■Question (b) If the number of companies participating in the semiconductor market increases to infinity,that is, in a perfectly competitive market, what will be the equilibrium point?
- can we find nash equilibrium through mixed strategy as well?Consider a first-price sealed bid auction of a single object with two bidders j = 1,2 and no reserved price. Bidder 1's valuation is v1 = 2. and bidder 2's valuation is v2= 5. Both v1 and v2 are known to both bidders. Bids must be in whole dollar amounts. In the event of a tie, the object is awarded by a flip of a fair coin. Is there a Nash equilibrium? What is it? Is it unique? Is it efficient?Consider an economy with two states of nature and one time period. An option pays $3 in state 1 and $0 in state 2, and is being traded for the price of $1. A forward contract pays $3 in state 1 and -$2 in state 2. Using this information, compute the risk-free rate and the risk-neutral probabilities for each state of nature.
- Little Kona is a small coffee company that is considering entering a market dominated by Big Brew. Each company’s profit depends on whether Little Kona enters and whether Big Brew sets a high price or a low price: a) Does either player in this game have a dominant strategy? b) Does your answer to part (a) help you figure out what the other player should do? What is the Nash equilibrium? Is there only one? c) Big Brew threatens Little Kona by saying, “If you enter, we’re going to set a low price, so you had better stay out.” Do you think Little Kona should believe the threat? Why or why not?A seller would like to sell a painting using second price (Vickrey) auction. The seller knows that there are three buyers and each buyer is equally likely to have High (H), Medium (M), or Low (L) valuation for the painting. The valuations of the buyers are independently distributed. Assuming that the buyers will play the Nash equilibrium in the second price auction, how much revenue will the seller make?Two players are engaged in a game of Chicken. There are two possible strategies, Swerve and Drive Straight. A player who chooses to Swerve is called Chicken and gets a payoff of zero, regardless of what the other player does. A player who chooses to Drive Straight gets a payoff of 36 if the other player swerves and a payoff of −36 if the other player also chooses to Drive Straight. Find the Nash equilibrium in mixed strategies. What is the probability of a crash in this mixed strategy Nash equilibrium?
- Find the saddle point of the game given by the payoff matrices A below.Consider the following extensive form game: Find the subgame-perfect equilibrium for this game.Consider a symmetric game with 10 players. Each player chooses among three strategies: x, y, and z. Let nx denote the number of players who choose x, ny denote the number of players who choose y, and nz denote the number of players who choose z. (So, nz = 10−nx−ny.) The payoff to a player from choosing strategy x is 10−nx (note that nx includes this player as well), strategy y is 13−2ny (again ny includes this player as well), and strategy z is 3. (a) Show that a Nash equilibrium must have at least one person choosing x and at least one person choosing y. (Hint: In a Nash equilibrium, no player can do better by doing something different.) b) Find all Nash equilibria.