Consider a first-price sealed-bid auction with a reserve price. A single indivisible object is to beallocated. The reserve price r> 0 is a number set by the auctioneer in advance and known by all theparticipants. Each participant submits a bid inside a sealed envelope. The bids are submittedsimultaneously and independently. The auctioneer collects all bids and selects the highest one. If thehighest bid is no smaller than the reserve price, then the person with the highest bid pays their bidto the auctioneer and gets the object. Otherwise, if all the bids are smaller than the reserve price,then the auctioneer keeps the object and none of the participants pays anything.(a) Write down a game representing the sealed-bid first-price auction. You can assume that peoplehave quasilinear utility over money.(b) Suppose that r = 0, is overbidding weakly dominated? Justify your answer.(c) Suppose that r = 0, is underbidding weakly dominated by truthful bidding? Justify your answer.(d) How would your answer to (b) and (c) change if r> 0?

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Consider a first-price sealed-bid auction with a reserve price. A single indivisible object is to be allocated. The reserve price r2 0 is a number set by the auctioneer in advance and known by participants. Each participant submits a bid inside a sealed envelope. The bids are submitted simultaneously and independently. The auctioneer collects all bids and selects the highest one highest bid is no smaller than the reserve price, then the person with the highest bid pays the to the auctioneer and gets the object. Otherwise, if all the bids are smaller than the reserve pr then the auctioneer keeps the object and none of the participants pays anything. (a) Write down a game representing the sealed-bid first-price auction. You can assume that pe have quasilinear utility over money. (b) Suppose that r= 0, is overbidding weakly dominated? Justify your answer. (c) Suppose that r = 0, is underbidding weakly dominated by truthful bidding? Justify your ans (d) How would your answer to (b) and (c) change if r> 0?

Consider a first-price sealed-bid auction with a reserve price. A single indivisible object is to be allocated. The reserve price r2 0 is a number set by the auctioneer in advance and known by participants. Each participant submits a bid inside a sealed envelope. The bids are submitted simultaneously and independently. The auctioneer collects all bids and selects the highest one highest bid is no smaller than the reserve price, then the person with the highest bid pays the to the auctioneer and gets the object. Otherwise, if all the bids are smaller than the reserve pr then the auctioneer keeps the object and none of the participants pays anything. (a) Write down a game representing the sealed-bid first-price auction. You can assume that pe have quasilinear utility over money. (b) Suppose that r= 0, is overbidding weakly dominated? Justify your answer. (c) Suppose that r = 0, is underbidding weakly dominated by truthful bidding? Justify your ans (d) How would your answer to (b) and (c) change if r> 0?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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A seller has an indivisible asset to sell. Her reservation value for the asset is s, which she knows privately. A potential buyer thinks that the assetís value to him is b, which he privately knows. Assume that s and b are independently and uniformly drawn from [0, 1]. If the seller sells the asset to the buyer for a price of p, the seller's payoff is p - s and the buyer's payoffis b - p. Suppose the buyer makes a take-it-or-leave-it offer p to the seller. What's the optimal offer if the buyer's value is b = 1/2?
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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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Exercise 1.8. Consider the following "third-price" auction. There are n≥ 3 bidders. A single object is auctioned and Player / values the object $v,, with V, > 0. The bids are simultaneous and secret. The utility of Player / is: 0 if she does not win and (v-p) if she wins and pays $p. Every non-negative number is an admissible bid. Let b, denote the bid of Player i. The winner is the highest bidder. In case of ties the bidder with the lowest index among those who submitted the highest bid wins (e.g. if the highest bid is $120 and it is submitted by players 6, 12 and 15, then the winner is Player 6). The losers don't get anything and don't pay anything. The winner gets the object and pays the third highest bid, which is defined as follows. Let be the winner and fix a Player j such that b₁ = max({b,,...,b,}\ {b}) [if GAME THEORY - Giacomo Bonanno 39 max({b,...,b}\{b;}) contains more than one element, then we pick one of them]. Then the third price is defined as max({b₁,...,b}\{b,,b;}).…
Why do sellers generally prefer a Vickrey auction to a regular sealed bid if sellers don’t receive the highest bid in the Vickrey auction? Sellers only have to sell their item if the bid is the highest-price bid. The second-highest bid in a Vickrey auction is generally higher than the highest bid in a regular sealed-bid auction. The second-highest bid is about the same in both auctions. Sellers prefer the final price is not revealed to all bidders. Sellers would never prefer Vickrey auctions.
A seller has an indivisible asset to sell. Her reservation value for the asset is s, which she knows privately. A potential buyer thinks that the assetís value to him is b, which he privately knows. Assume that s and b are independently and uniformly drawn from [0, 1]. If the seller sells the asset to the buyer for a price of p, the seller's payoff is p-s and the buyer's payoffis b-p. Suppose simultaneously the buyer makes an offer p1 and the seller makes an offer p2. A transaction occurs if p1>=p2, and the transaction price is 1/2 (p1 + p2). Is the following strategy profile a Bayesian Nash equilibrium: the buyer chooses p1 = 1/2 if b>=1/2 and he chooses p1=0 if b<1/2; the seller chooses p2=1/2 if s<1/2 and she chooses p2=1 if s>1/2. Why or why not? Can there be a Bayesian Nash Equilibrium in which the transaction price is .9 whenever there is a transaction? Why or why not?
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