7. In a first price sealed bid auction with 2 bidders, let v be the valuation ofbidder i and b; be her bid. The bids are made simultaneously and the payofffunction of player i is given as follows:if b; > bj,(v; – b:)/2 if b; =b;,if b; < b;.u:(b1, 62; V1, V2) =0.The optimization problem of player i is given by(a) maxb, (vi – b;)(b) mat, (v; – b:)Prob{b; > b;(v;)}(c) maxs, (vi – bi)Prob{b, > b;(v;)} + }(v; – b;) Prob{b; = b;(v;)}(d) mar, (v; – bị)Prob{b; > b;(v;)} +}(v; – 6:)Prob{b, < b;(v;)}

In the first price sealed bid auction, person i will win and pay his bid i.e. bi Now, the…

7. In a first price sealed bid auction with 2 bidders, let v; be the valuation of bidder i and b; be her bid. The bids are made simultaneously and the payoff function of player i is given as follows: Vi - - bi ui (b1,b2; V1, V2) = (v₁ - bi)/2 0 if bi > bj, if b₁ = bj, if bi < bj. The optimization problem of player i is given by (а) тахь; (v; — b;) (b) maxь; (vi — b₁) Prob{b; > b;(vj)} - (c) maxь, (vi — b₁) Prob{b; > bj(vj)} + ½ (vi − b;) Prob{b; = b;(vj)} (d) maxь; (vi — b₁) Prob{b; > b;(v;)} + ½ (v₁ − b;) Prob{b; < b;(vj)}

7. In a first price sealed bid auction with 2 bidders, let v; be the valuation of bidder i and b; be her bid. The bids are made simultaneously and the payoff function of player i is given as follows: Vi - - bi ui (b1,b2; V1, V2) = (v₁ - bi)/2 0 if bi > bj, if b₁ = bj, if bi < bj. The optimization problem of player i is given by (а) тахь; (v; — b;) (b) maxь; (vi — b₁) Prob{b; > b;(vj)} - (c) maxь, (vi — b₁) Prob{b; > bj(vj)} + ½ (vi − b;) Prob{b; = b;(vj)} (d) maxь; (vi — b₁) Prob{b; > b;(v;)} + ½ (v₁ − b;) Prob{b; < b;(vj)}

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter18: Asymmetric Information

Section: Chapter Questions

Problem 18.8P

See similar textbooks

Similar questions

Q56 A Nash equilibrium is an outcome... a. Achieved by cooperation between players in the game. b. That is achieved by collusion where no party has an incentive to change their behaviour. c. Where each player's strategy depends on the behaviour of its opponents. d. That is achieved when players in the game have jointly maximized profits and divided those profits according to market share of each player. e. Where each player's best strategy is to maintain its present behaviour given the present behaviour of the other players.
Consider the following game 1\2 Y Z A 10,3 3,9 B 8,5 6,1 Suppose Player 2 holds the following belief about Player 1: θ1 (A,B) = (9/10,1/10) What is the expected payoff from playing ‘Y’ ? What is the expected payoff from playing ‘Z’ ? Based on these beliefs, player 2 should respond by playing _____
5 Consider a first-price sealed-bid auction in which bidders valuations are independently and identically distributed according to the Uniform distribution on the interval [0, 1]. Explain what the rules of the First Price Sealed bid auction are. Set it up as a Bayesian game. Compute a symmetric Bayesian Nash equilibrium for the two bidder case.
Consider a Common Value auction with two bidders who both receive a signal X that is uniformly distributed between 0 and 1. The (common) value V of the good the players are bidding for is the average of the two signals, i.e. V = (X1+X2)/2. Compute the symmetric Nash equilibrium bidding strategy for the second-price sealed-bid auction assuming that players are risk-neutral and have standard selfish preferences. Furthermore, you may assume that the other bidder is following a linear bidding strategy. Make sure to explain your notation and the steps you take to derive the result.
Suppose that 5 risk neutral competitors participate in a rent seeking game with a fixed prize of $100. Each player may invest as much as he wishes in the political contest, although those investments have an opportunity cost equal 1. The probability of winning is directly proportional to the candidate’s share of the total rent-seeking investment. What is the profit-maximizing investment by player 1 as a function of the investment by all the others? What is a Nash equilibrium investment by each player in a symmetric game?
Consider the charity auction. In many charity auctions, altruistic celebrities auction objects with special value for their fans to raise funds for charity. Madonna, for example, held an auction to sell clothing worn during her career and raised about 3.2 million dollars. In the charity auction the winner of the lot is the highest bidder. The difference with the standard auction is that all bidders are required to pay an amount equal to what they bid. Suppose there are two bidders and assume bidders have valuations randomly drawn from the interval [2, 4] according to the uniform distribution. 1. Derive the equilibrium bidding function. Hint: After getting the differential equation given by the FOC, propose a non-linear bidding function b(v) = α + βv2 as solution. Your task is to find α and β. 2. Derive the revenue of the seller in the charity auction. 3. Would the seller obtain higher profits if she organized a first-price sealed bid auction instead? A. Yes, higher revenue B. No, lower…
Consider a sealed-bid auction in which the seller draws one of the N bids at random. The buyer whose bid was drawn wins the auction and pays the amount bid. Assume that buyer valuations follow a uniform(0,1) distribution. 1. What is the symmetric equilibrium bidding strategy b(v)?2. What is the seller’s expected revenue?3. Why doesn’t this auction pay the seller the same revenue as the four standard auctions? That is, why doesn’t the revenue equivalence theorem apply here?
The mixed stratergy nash equalibrium consists of : the probability of firm A selecting October is 0.692 and probability of firm A selecting December is 0.309. The probability of firm B selecting October is 0.5 and probability of firm selecting December is 0.5. In the equilibrium you calculated above, what is the probability that both consoles are released in October? In December? What are the expected payoffs of firm A and of firm B in equilibrium?
Consider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs: Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00 a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they exist
Two players bargain over 1 unit of a divisible object. Bargaining starts with an offer of player 1, which player 2 either accepts or rejects. If player 2 rejects, then player 1 makes another offer; if player 2 rejects once more, then player 2 makes an offer. If player 1 rejects the offer of player 2, then once more it is the turn of player 1 where he makes two consecutive offers. As long as an agreement has not been reached this procedure continues. For example, suppose that agreement is reached at period 5, it follows that player 1 makes offers in period 1,2 then player 2 makes an o er in period 3, then player 1 makes offers in 4,5. Negotiations can continue indefinitely, agreement in period 't' with a division (x, 1- x) leads to payoffs ( , (1-x)).(The difference from Rubinstein's alternating offer bargaining is that player one makes two consecutive offers, whereas player 2 makes a single offer in her turn.) a. Show that there is a subgame perfect equilibrium in which player 2's…
Consider a Common Value auction with two bidders who both receive a signal X that is uniformly distributed between 0 and 1. The (common) value V of the good the players are bidding for is the average of the two signals, i.e. V = (X1+X2)/2. the symmetric Nash equilibrium bidding strategy for the second-price sealed-bid auction assuming that players are risk-neutral and have standard selfish preferences. Furthermore, you may assume that the other bidder is following a linear bidding strategy. Make sure to explain your notation and the steps you take to derive the result.
Q2.1 In the second round with two buyers remaining, the probability that a buyer with valuation v wins is vN-1, where N is the number of buyers in the first round. Use the revenue equivalence theorem to derive the symmetric equilibrium bidding function b(v) for the buyers in stage two. Show your work. Q2.2 At the end of the auction what is the value of the actual (not expected) revenue that the seller receives? Round your answer to at least three decimal spaces.