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;}). For example, if n = 3 and the bids are b₁ = 30, b₂ = 40 and b₁ = 40 then the winner is Player 2 and she pays $30; if b₁ = b₁₂ = b₁ = 50 then the winner is Player 1 and she pays $50. For simplicity, let us restrict attention to the case where n = 3 and v₁ > V₂ > V3 >0. Does Player 1 have a weakly dominant strategy in this auction?
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;}). For example, if n = 3 and the bids are b₁ = 30, b₂ = 40 and b₁ = 40 then the winner is Player 2 and she pays $30; if b₁ = b₁₂ = b₁ = 50 then the winner is Player 1 and she pays $50. For simplicity, let us restrict attention to the case where n = 3 and v₁ > V₂ > V3 >0. Does Player 1 have a weakly dominant strategy in this auction?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
![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;}). For example, if n = 3
and the bids are b₁ = 30, b₂ = 40 and b₁ = 40 then the winner is Player 2 and she
pays $30; if b₁ = b₁₂ = b₁ = 50 then the winner is Player 1 and she pays $50.
For simplicity, let us restrict attention to the case where n = 3 and v₁ > V₂ > V3 >0.
Does Player 1 have a weakly dominant strategy in this auction?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833227fa-6456-4698-8430-712200a97c6f%2Fd480b7e2-bfb5-42cb-b1d7-1bf009617af4%2Fwwuzdbj_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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;}). For example, if n = 3
and the bids are b₁ = 30, b₂ = 40 and b₁ = 40 then the winner is Player 2 and she
pays $30; if b₁ = b₁₂ = b₁ = 50 then the winner is Player 1 and she pays $50.
For simplicity, let us restrict attention to the case where n = 3 and v₁ > V₂ > V3 >0.
Does Player 1 have a weakly dominant strategy in this auction?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education