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Question 2 Consider a standard auction setting with two bidders (i = 1,2). Let F₁(v) = √u where v € [0, 1] the cumulative distribution which bidder 1's valuation is drawn from (and f₁ = ½v- the corresponding density function). Let F₂(v) = ¼½v² where v € [0, 2] the cumulative distribution which bidder 2's valuation is drawn from (and f2 = /v the corresponding density function). Consider Second price sealed auction. (a) Is bidding one's own valuation a weakly dominant strategy for bidder 1? Explain why why not. (b) Is bidding one's own valuation a weakly dominant strategy for bidder 2? Explain why why not. (e) What is the expected payment of bidder 1? (d) What is the expected payment of bidder 2? Now, consider First price sealed auction. (e) Suppose each bidder has the lowest possible valuation. That is, v₁ = V₂ = 0. In equilibrium, b₁ (0) = b₂(0) = 0 because it would be dominated for a bidder to bid more than the value. Now suppose each bidder has the highest possible valuation. That is, V₁ = 1 and v₂ = 2. In equilibrium, b₁(1) = b₂(2). Explain why. (f) Let b = b₁(1) b₂(2) denote the common highest bid. In the following questions, assume b; < b for i = 1,2. Assume b₁(v) = ß₁v and b₂(v) = ß₂v where ß₁ and ß₂ are real numbers. What is the expected payoff of bidder 1? Express as a function of b₁, V1, and ₂. -

Question 2 Consider a standard auction setting with two bidders (i = 1,2). Let F₁(v) = √u where v € [0, 1] the cumulative distribution which bidder 1's valuation is drawn from (and f₁ = ½v- the corresponding density function). Let F₂(v) = ¼½v² where v € [0, 2] the cumulative distribution which bidder 2's valuation is drawn from (and f2 = /v the corresponding density function). Consider Second price sealed auction. (a) Is bidding one's own valuation a weakly dominant strategy for bidder 1? Explain why why not. (b) Is bidding one's own valuation a weakly dominant strategy for bidder 2? Explain why why not. (e) What is the expected payment of bidder 1? (d) What is the expected payment of bidder 2? Now, consider First price sealed auction. (e) Suppose each bidder has the lowest possible valuation. That is, v₁ = V₂ = 0. In equilibrium, b₁ (0) = b₂(0) = 0 because it would be dominated for a bidder to bid more than the value. Now suppose each bidder has the highest possible valuation. That is, V₁ = 1 and v₂ = 2. In equilibrium, b₁(1) = b₂(2). Explain why. (f) Let b = b₁(1) b₂(2) denote the common highest bid. In the following questions, assume b; < b for i = 1,2. Assume b₁(v) = ß₁v and b₂(v) = ß₂v where ß₁ and ß₂ are real numbers. What is the expected payoff of bidder 1? Express as a function of b₁, V1, and ₂. -

University Physics Volume 3
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter7: Quantum Mechanics
Section: Chapter Questions
Problem 7.1CYU: Check Your Understanding If a=3+4i , what is the product a* a?
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Question 2 Consider a standard auction setting with two bidders (i = 1,2). Let F₁(v) √ where v € [0, 1] the cumulative distribution which bidder 1's valuation is drawn from the corresponding density function). Let F2(v) v² where v € [0, 2] (and fi the cumulative distribution which bidder 2's valuation is drawn from (and f₂ corresponding density function). = V the = Consider Second price sealed auction. (a) Is bidding one's own valuation a weakly dominant strategy for bidder 1? Explain why why not. = (b) Is bidding one's own valuation a weakly dominant strategy for bidder 2? Explain why why not. (e) What is the expected payment of bidder 1? (d) What is the expected payment of bidder 2? Now, consider First price sealed auction. (e) Suppose each bidder has the lowest possible valuation. That is, v₁ V2 = 0. In equilibrium, b₁ (0) = b₂(0) = 0 because it would be dominated for a bidder to bid more than the value. Now suppose each bidder has the highest possible valuation. That is, = 2. In equilibrium, b₁(1) = b₂(2). Explain why. U1 1 and 2 = = (f) Let b = = b₁ (1) b₂(2) denote the common highest bid. In the following questions, assume b; < b for i = 1, 2. Assume b₁(v) = B₁v and b₂(v) = ß₂v where ₁ and ₂ are real numbers. What is the expected payoff of bidder 1? Express as a function of b₁, V₁, and ₂. = =
Transcribed Image Text:Question 2 Consider a standard auction setting with two bidders (i = 1,2). Let F₁(v) √ where v € [0, 1] the cumulative distribution which bidder 1's valuation is drawn from the corresponding density function). Let F2(v) v² where v € [0, 2] (and fi the cumulative distribution which bidder 2's valuation is drawn from (and f₂ corresponding density function). = V the = Consider Second price sealed auction. (a) Is bidding one's own valuation a weakly dominant strategy for bidder 1? Explain why why not. = (b) Is bidding one's own valuation a weakly dominant strategy for bidder 2? Explain why why not. (e) What is the expected payment of bidder 1? (d) What is the expected payment of bidder 2? Now, consider First price sealed auction. (e) Suppose each bidder has the lowest possible valuation. That is, v₁ V2 = 0. In equilibrium, b₁ (0) = b₂(0) = 0 because it would be dominated for a bidder to bid more than the value. Now suppose each bidder has the highest possible valuation. That is, = 2. In equilibrium, b₁(1) = b₂(2). Explain why. U1 1 and 2 = = (f) Let b = = b₁ (1) b₂(2) denote the common highest bid. In the following questions, assume b; < b for i = 1, 2. Assume b₁(v) = B₁v and b₂(v) = ß₂v where ₁ and ₂ are real numbers. What is the expected payoff of bidder 1? Express as a function of b₁, V₁, and ₂. = =
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