Exercise 5 In a first-price all-pay auction, the bidders simultaneously submit sealed bids. The highest bid wins the object and every bidder pays the seller the amount of his bid. Assume that there are two bidders and each bidder's value is drawn from the uniform distribution on [0, 1]. (a) Show that b; (vi) = for i = 1, 2 is a Bayesian Nash equilibrium. (b) Calculate the expected revenue for the seller.

Exercise 5 In a first-price all-pay auction, the bidders simultaneously submit sealed bids. The highest bid wins the object and every bidder pays the seller the amount of his bid. Assume that there are two bidders and each bidder's value is drawn from the uniform distribution on [0, 1]. (a) Show that b; (vi) = for i = 1, 2 is a Bayesian Nash equilibrium. (b) Calculate the expected revenue for the seller.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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(The All-Pay Auction). The seller has an item for sale. The valuations of the bidders are independently and identically distributed on R+ with a c.d.f. F. Find the symmetric equilibrium of an auction with two bidders in which both bidders pay their own bids but only the highest bidder wins the object. Show that each bidder’s expected payment is the same in this auction and in the first-price auction.
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Assume that two collectors, X and Y are in a first prize sealed bid auction for a batch of vintage comic books. X and Y have different valuations (V) for this batch of comic books e.g. VX And VB are between $2000 and $4000. Both collectors know their own V but does not know the V of the other collector. All they know is that the other collector’s V is a uniformly distributed number between $2000 and $4000. Assume risk neutrality for X and Y e.g. expected payoff for X is: (VX – bX)Pr(bX) and expected payoff for Y is (VY – bY)Pr(bY). These collectors will make their bids strategically. Show how X’s bidding strategy is bX = ½ Vx + 1 and Y’s is bY = ½ Vy +1 in a Nash equilibrium.
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P1| P2 Left Middle Right Left 4,2 3,3 1,2 Middle 3,3 5,5 2,6 Right 2,1 6,2 3,3 Consider the simultaneous move game represented by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted, or the discount factor is equal to 1. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. a) (Left, Left) is played in both periods. b) (Right, Right) is played in both periods. c) (Middle, Middle) is played in both periods. d) (Middle, Middle) is played in the first period, followed by (Left, Left). e) (Middle, Middle) is played in the first period, followed by (Right, Right).
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Exercise 11 Dominance and Best Responses (H) (a) Show that if an action is a best response (to some strategy of the other players), then it is undominated. (b) Show that every weakly dominant strategy is pure. (c) Now, let N = {1, 2} and si be a best response for player 1 to both t and t'. Show that si is also a best response to all mixtures t of t and t,, where for any pe [0, 1], t is defined by t = [p : t, (1 – p) : t].
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Faris Co. and Fahad Co. are rivalry in Bahrain's market. Suppose the payoff to each of four strategic interactions are as follows: Faris Co. Response Action Reduce Price Don't Reduce Price Gain = BD 30,000 Reduce Price Loss = BD 1,000 Don't Reduce Price Loss = BD 5,000 No loss or gain The probability of Faris Co. to reduce price is 95 % (otherwise, probability of Faris Co. for not reducing price is 5 %), assume that Fahad Co. choose "reduce price" strategy a. What is size of loss from "reduce price" for Fahad Co.? b. What is size of gain from "don't reduce price" for Fahad Co? c. What is the expected payoff of a "reduce price" for Fahad Co.? and give an example (application) of d. Explain the concept of 'game theory' game theory Fahad Co. Response
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NE 2). Consider the following extensive form game between two players. (1,10) u D 1 B X (a) List all pure strategies of player 2. (b) Represent this game in normal form. (c) Find all pure-strategy Nash equilibria of this game. (d) Find all SPNE (in pure strategies) of this game. (6,3) (4,2) (5,1)