PLEASE CHECK THIS HOW TO SOLVE Consider a second-price sealed bid auction of a single object with pure common values inwhich the common value v is either 0 or 1, with equal probability. Each bidder receives asignal in {e, h} about the value withPr(h|v=1)For each v, bidders' signals are statistically independent.(a) Suppose there are two bidders. Find a symmetric Nash equilibrium.Solution: There is an equilibrium in which, for each signal realization 8₁, each bidder ibids b;(8₁) = E[v | 8₁, 8; = 8;]. In this case, these bids areandandE[v | 8; = 8j = h] =Pr(l | v= 0) = 3/4.E[v | $i = 8j = l] =(b) Suppose there are three bidders. Find a symmetric Nash equilibrium.Solution: There is an equilibrium in which, for each signal realization s₁, each bidder ibids b;(8₁) = E[v | 8₁, 8; = maxji 8j]. In this case, these bids are==(1/4)²(1/4)² + (3/4)²110E[v | 8₁ = 8j = $k = l] =(3/4)²(1/4)² + (3/4)²91052E[v|s; = h and at least one s; = h for j ‡ i]=(1/4)³(1/4)³ +(3/4)³128(3/4)³ +2 (3/4)²-1/4(1/4)³ +2 · (1/4)² · 3/4+ (3/4)³ + 2 · (3/4)².1/445

Symmetric Nash equilibrium describes a situation in which all players in a strategic game use the…

Consider a second-price sealed bid auction of a single object with pure common values in which the common value is either 0 or 1, with equal probability. Each bidder receives a signal in {e, h} about the value with Pr(h|v= 1) = Pr(e|v=0)=3/4. For each v, bidders' signals are statistically independent. (a) Suppose there are two bidders. Find a symmetric Nash equilibrium.

Consider a second-price sealed bid auction of a single object with pure common values in which the common value is either 0 or 1, with equal probability. Each bidder receives a signal in {e, h} about the value with Pr(h|v= 1) = Pr(e|v=0)=3/4. For each v, bidders' signals are statistically independent. (a) Suppose there are two bidders. Find a symmetric Nash equilibrium.

Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)

14th Edition

ISBN:9781305506381

Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Publisher:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Chapter15A: Auction Design And Information Economics

Section: Chapter Questions

Problem 10E

See similar textbooks

Similar questions

(a) Design a two-player game with the property that "the payoff for Player1 is Playerl plays a strategy Sl and Player2 plays a strategy S2 is the same as the payoff for Player2 if Player2 plays Sl and Playerl plays S2". Express your game in terms of payoff matrices. Show the above property holds in your given payoff matrices. Give a case to show that there is one Nash equilibrium and such an equilibrium is also a Pareto efficient (justify your answers) in your given payoff matrices. Prove or disprove the above case that is unique. Give another example to show that there are two Nash equilibria in your given payoff matrices.
Two firms bid for a contract to build a university building. Their construction costs are independent and uniformly drawn from [0,1]: Both bidders submit their bids si- multaneously. The winner is the bidder who submits a lowest bid. Tie-breaking rule is random. This kind of bidding game is called i'reverse auction' because the bidders bid for the right to provide a service and the winner is paid for the service. The FCC in 2017 has adopted similar auctions designed to repurpose spectrum for new uses. (a) In the first auction, the winner gets paid the loser's bid. For example, if the winner's cost is 0.5, his bid is 0.56, and the loserís bid is 0.6, then the winner gets the contract and the university pays the winner 0.6. The winner's net profit from the contract is 0.6-0.5 = 0.1. Solve for a Bayesian Nash equilibrium of this auction. What is the equilibrium bid of a firm bid if his cost is actually 0.5? (b) In the second auction, the winner gets paid the his/her own winning…
Consider the following two-player game with three options for each player. (Payouts are listed for the row player first, then the column player.) player Y layery 3,3 A 1,5 4,4 6,2 K 8,1 3,7 5,2 0,6 1,1 Find a mixed Nash equilibrium for this game. Solution suggestion: Use two variables per player. If p and are the probabilities of selecting the first two strategies, then 1-p-q is the probability of selecting the third strategy. You will need to solve a system of equations.
For the friend-foe game, recall that there were 3 Nash equilibria possible, but the equilibria set didn't include the cooperative outcome, for which both players would win. Friend Foe Friend 500,500 0,1000 Foe 1000,0 0.0 a) If the game is played répeatedly. propose a play strategy that will enforce cooperation. For what valucs of o (discount factor) the equilibrium will be (Friend. Friend)?
2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.
Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
The payoff matrix below shows the payoffs for Firm A and Firm B, each of whom can either "cooperate" or "cheat." The numbers in parentheses are (payoff for A, payoff for B). Firm B Cooperate Cooperate (30, 30) (x, 10) Cheat (10, X) (20, 20) Firm A Chent If x = 40, the Nash equilibrium is. O a. (Firm A: cheat, Firm B: cheat) O b. (Firm A: cooperate, Firm B: cheat) O c. (Firm A: cheat, Firm B: cooperate) O d. (Firm A: cooperate, Firm B: cooperate) O e. There is no Nash equilibrium for this value of x.
Problem 2. Consider the partnership-game we discussed in Lecture 3 (pages 81-87 of the textbook). Now change the setup of the game so that player 1 chooses x = [0, 4], and after observing the choice of x, player 2 chooses y ≤ [0, 4]. The payoffs are the same as before. (a) Find all SPNE (subgame perfect Nash equilibria) in pure strategies. (b) Can you find a Nash equilibrium, with player 1 choosing x = 1, that is not subgame perfect? Explain.
Use the mixed method (Nash Equilibrium) to determine the following: What percentage of time should the maximizer play strategy H? 1. H6 17 ! 12 10 18.8% 84.6% O 15.4% 11.8%
Two firms bid for a contract to build a university building. Their construction costs are independent and uniformly drawn from [0, 1]. Both bidders submit their bids si- multaneously. The winner is the bidder who submits a lowest bid. Tie-breaking rule is random. This kind of bidding game is called "reverse auction" because the bidders bid for the right to provide a service and the winner is paid for the service. The FCC in 2017 has adopted similar auctions designed to repurpose spectrum for new uses. (a) In the first auction, the winner gets paid the loser's bid. For example, if the winner's cost is 0.5, his bid is 0.56, and the loser's bid is 0.6, then the winner gets the contract and the university pays the winner 0.6. The winner's net profit from the contract is 0.6-0.5 = 0.1. Solve for a Bayesian Nash equilibrium of this auction. What is the equilibrium bid of a firm bid if his cost is actually 0.5? (b) In the second auction, the winner gets paid the his/her own winning bid. For…
4. Consider a first-price, sealed-bid auction in which the bidders’ valuations are independently and uniformly distributed on [0,1]. Suppose that each bidder uses a strategy of b(vi) = avi. What is the symmetric Bayesian Nash equilibrium of this game when there are n bidders?