There are N ≥ 2 buyers who engage in the auction of a house. The buyers havea common valuation of the house, denoted by v, which is known to all. All buyersmake a simultaneous bid. Let pr denote the bid by buyer n = 1,, N. The onewith the highest bid wins the house. The winner receives payoff v – på. The other(s)receive zero payoff. If more than one buyer makes the same highest bid, then theyhave an equal chance of winning the house. Prove that:(a)(b)(c)-It is not a Nash Equilibrium (NE) if the highest bid is v and only onebuyer bids this price.this price.It is not a NE if the highest bid is less than v.It is a NE that the highest bid is v and more than one buyer bids

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Answered: There are N ≥ 2 buyers who engage in…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.3P

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Suppose two bidders compete for a single indivisible item (e.g., a used car, a piece of art, etc.). We assume that bidder 1 values the item at $v1, and bidder 2 values the item at $v2. We assume that v1 > v2. In this problem we study a second price auction, which proceeds as follows. Each player i = 1, 2 simultaneously chooses a bid bi ≥ 0. The higher of the two bidders wins, and pays the second highest bid (in this case, the other player’s bid). In case of a tie, suppose the item goes to bidder 1. If a bidder does not win, their payoff is zero; if the bidder wins, their payoff is their value minus the second highest bid. a) Now suppose that player 1 bids b1 = v2 and player 2 bids b2 = v1, i.e., they both bid the value of the other player. (Note that in this case, player 2 is bidding above their value!) Show that this is a pure NE of the second price auction. (Note that in this pure NE the player with the lower value wins, while in the weak dominant strategy equilibrium where both…
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David wants to auction a painting, and there are two potential buyers. The value for eachbuyer is either 0 or 10, each value equally likely. Suppose he offers to sell the object for $6, and the two buyers simultaneously accept or reject. If exactly one buyer accepts, the object sold to that person for $6. If both accept, the object is allocated randomly to the buyers, also for $6. If neither accepts, the object is allocated randomly to the bidders for $0. (a) Identify the type space and strategy space for each buyer. (b) Show that there is an equilibrium in which buyers with value 10 always accept. (c) Show that there is an equilibrium in which buyers with value 10 always reject.
There are N>=2 collectors who engage in the auction of an antique. The collectorshave a common valuation of the antique, denoted by v, which is known to all. Thecollectors make a simultaneous bid. Let pn denote the bid by collector n = 1,....,N. The one with the highest bid wins the antique. The winner receives payoff v-pi.The other(s) receive zero payoff. If more than one collectors make the same highestbid, then they have an equal chance of winning the item. Prove that: A) It is not a Nash Equilibrium (NE) if the highest bid is v and onlyone collector bids this price.(b) It is not a NE if the highest bid is less than v.(c) It is a NE that the highest bid is v and more than one collector bidsthis price
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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;}).…
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.
Suppose there are two players playing a game with east or west and south and nerth ways. Find the expected Nash equilibrium by using the concept of probabilities. Player X Left[L) Right|R) Player Y Up(U) (5,6) (0,8) (4,6) Down[D) (0,9)
Suppose there are two bidders for a single object. Each bidder has a value for the object, v1 and v2, which is randomly drawn uniformly from 0 to 80. (Note that this means the probability that a value is 25 or lower is 25/80, and more generally, the probability that a value is k or lower is k/80). Bidders see their own values but not the values of their rival bidders. Consider a first price sealed bid auction where the winner of the object is the bidder who submits the highest bid and pays the price that he bid. Suppose that you are bidder 1 and you believe that your rival always bids a constant fraction α of his value. (For example, if bidder 2’ value is 15, he will bid 15α.). 1. What is your expected winning probability of the auction from any given bid b1? (You win the auction when your bid amount b1 is greater than your opponent’s bid). 2. What is your expected payoff from any given bid, b1? (Write answer as a function of v1, b1, and α). 3. Compute your best response bid given your…
Suppose there are two bidders for a single object. Each bidder has a value for the object, v1 and v2, which is randomly drawn uniformly from 0 to 80. (Note that this means the probability that a value is 25 or lower is 25/80, and more generally, the probability that a value is k or lower is k/80). Bidders see their own values but not the values of their rival bidders. Consider a first price sealed bid auction where the winner of the object is the bidder who submits the highest bid and pays the price that he bid. Suppose that you are bidder 1 and you believe that your rival always bids a constant fraction α of his value. (For example, if bidder 2’ value is 15, he will bid 15α.). 1. What is your expected winning probability of the auction from any given bid b1? (Hint, you win the auction when your bid amount b1 is greater than your opponent’s bid). 2. What is your expected payoff from any given bid, b1? (You need to write your answer as a function of v1, b1, and α). 3. Compute your best…
What amount does each bidder bid in the Bayesian Nash equilibrium of a 2nd price auction? O The expected value of the second highest bidder. One half of the expected value of the second highest bidder. Their own value. One half of their own value.
A crime is witnessed by 3 citizens. Every citizen would like the police to be informed about the crime, but prefers that someone else reports it (filing a report is a hassle). Each of the three citizens chooses simultaneously (and independently) whether to call the police or not. When no one makes a call, every citizen receives payoff 0. If at least one citizen calls the police, citizens who call get payoff 5, and those who don't call get payoff 9. (a) Find all Nash equilibria in pure strategies. (b) Compute a symmetric Nash equilibrium in mixed strategies (i.e., an equilibrium in which every citizen calls with the same probability p E (0, 1)). (c) Does the game have a Nash equilibrium in pure or mixed strategies different from those you identified in (a) and (b)? If yes, construct one. If not, argue why not.

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