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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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.
Someone at a party pulls out a $100 bill and announces that he is going to auction it off. There are n=10 people at the partywho are potential bidders. The owner of the $100 bill puts forth the following procedure: All bidders simultaneously submit a written bid. Only the highest bidders pay their bid (assuming that the highest bid is positive). If m people submit the highest bid, then each receives 1/m of the $100. Each person’s strategy set is {0,1,2,...,1000}{0,1,2,...,1000} so bidding can go as high as $1,000. The payoff of a player bidding bi is:0 if bi < max{b1,b2,…,bn},and 100/m − bi if bi = max {b1,b2,…,bn}where,m is the number of bidders whose bid equals max{b1,...,bn}. How many pure-strategy Nash equilibria does this game have? 1) 0 2) 1 3) 4 4) More than 4.
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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 $120 and it is submitted by players 6, 12 and 15, then the winner is Player 6). The losers don't get anything and don't pay anything. The winner gets the object and pays the third highest bid, which is defined as follows. Let be the winner and fix a Player j such that b₁ = max({b,,...,b,}\ {b}) [if GAME THEORY - Giacomo Bonanno 39 max({b,...,b}\{b;}) contains more than one element, then we pick one of them]. Then the third price is defined as max({b₁,...,b}\{b,,b;}).…
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B) A firm has the option to decide between two new technologies T1 and T2. The monetary payoffs and probabilities for each technology is given below X2 Probability under state X; Income ($) with TI under state X Income (S) with T2 under state Xi 0.3 0.4 0.3 8 18 6 16 The utility function for the firm is given by u = E(T) – 0 Var(1), where E(x) represents average income, Var(n) represents the variance of income, and 020 1) Show that the firms risk preferences exhibit constant absolute risk aversion (CARA) 2) When 0- 0.1 how does the firms risk premium change under the two technologies TI and T2?
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