2. Consider a symmetric independent private values setting with two buyers. The seller owns an indivisible object which she is commonly known to value at zero. Suppose that v/E{1,2,3} for i = 1,2, and that each realization i equally likely (i.e., occurs with the probability 1/3). Suppose also that S,= (NO, 1, 2, 3} for fori= 1,2. That is a bidder may choose not to participate or he may bid one of his three possible valuations. Finally, suppose that ties in bidding are broken randomly without bias. a. Suppose the seller announces a second-price sealed bid auction with reserve price of 1. Find a symmetric Bayesian Nash Equilibrium (BNE), b (v), in which the bidders play weakly dominant strategies. How much expected revenue is generated in this BNE? b. Suppose the seller announces a second-price sealed bid auction with reserve price of 1. Find an asymmetric BNE, (b₁(v₁), b₂(v₂)). c. Suppose the seller announces a first-price sealed-bid auction with reserve price of 1. Find the symmetric BNE, b (v) which generates the highest expected revenue. How much revenue is generated in the BNE? Suppose the seller announces an all-pay auction with a reserve price of 1. Find a symmetric BNE, b (v). How much expected revenue is generated in this BNE? d.
2. Consider a symmetric independent private values setting with two buyers. The seller owns an indivisible object which she is commonly known to value at zero. Suppose that v/E{1,2,3} for i = 1,2, and that each realization i equally likely (i.e., occurs with the probability 1/3). Suppose also that S,= (NO, 1, 2, 3} for fori= 1,2. That is a bidder may choose not to participate or he may bid one of his three possible valuations. Finally, suppose that ties in bidding are broken randomly without bias. a. Suppose the seller announces a second-price sealed bid auction with reserve price of 1. Find a symmetric Bayesian Nash Equilibrium (BNE), b (v), in which the bidders play weakly dominant strategies. How much expected revenue is generated in this BNE? b. Suppose the seller announces a second-price sealed bid auction with reserve price of 1. Find an asymmetric BNE, (b₁(v₁), b₂(v₂)). c. Suppose the seller announces a first-price sealed-bid auction with reserve price of 1. Find the symmetric BNE, b (v) which generates the highest expected revenue. How much revenue is generated in the BNE? Suppose the seller announces an all-pay auction with a reserve price of 1. Find a symmetric BNE, b (v). How much expected revenue is generated in this BNE? d.
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.5P
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Transcribed Image Text:2. Consider a symmetric independent private values setting with two buyers. The seller owns an indivisible object
which she is commonly known to value at zero. Suppose that v,E{1,2,3} for i = 1,2, and that each realization is
equally likely (i.e., occurs with the probability 1/3). Suppose also that S,= {NO, 1, 2, 3} for fori = 1,2. That is
a bidder may choose not to participate or he may bid one of his three possible valuations. Finally, suppose that
ties in bidding are broken randomly without bias.
a. Suppose the seller announces a second-price sealed bid auction with reserve price of 1. Find a symmetric
Bayesian Nash Equilibrium (BNE), b´(v), in which the bidders play weakly dominant strategies. How
much expected revenue is generated in this BNE?
b. Suppose the seller announces a second-price sealed bid auction with reserve price of 1. Find an
asymmetric BNE, (b(v), b(v2)).
c. Suppose the seller announces a first-price sealed-bid auction with reserve price of 1. Find the symmetric
BNE, 6"(v) which generates the highest expected revenue. How much revenue is generated in the BNE?
d. Suppose the seller announces an all-pay auction with a reserve price of 1. Find a symmetric BNE, b (
How much expected revenue is generated in this BNE?
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v).
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