ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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Consider a second-price sealed bid auction of a single object with pure common values in which the common value v is either 0 or 1, with equal probability. Each bidder receives a signal in {e, h} about the value with Pr(h|v=1) For each v, bidders' signals are statistically independent. (a) Suppose there are two bidders. Find a symmetric Nash equilibrium. Solution: There is an equilibrium in which, for each signal realization 8₁, each bidder i bids b;(8₁) = E[v | 8₁, 8; = 8;]. In this case, these bids are and and E[v | 8; = 8j = h] = Pr(l | v= 0) = 3/4. E[v | $i = 8j = l] = (b) Suppose there are three bidders. Find a symmetric Nash equilibrium. Solution: There is an equilibrium in which, for each signal realization s₁, each bidder i bids b;(8₁) = E[v | 8₁, 8; = maxji 8j]. In this case, these bids are = = (1/4)² (1/4)² + (3/4)² 1 10 E[v | 8₁ = 8j = $k = l] = (3/4)² (1/4)² + (3/4)² 9 10 52 E[v|s; = h and at least one s; = h for j ‡ i] = (1/4)³ (1/4)³ +(3/4)³ 1 28 (3/4)³ +2 (3/4)²-1/4 (1/4)³ +2 · (1/4)² · 3/4+ (3/4)³ + 2 · (3/4)².1/4 45
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Transcribed Image Text:Consider a second-price sealed bid auction of a single object with pure common values in which the common value v is either 0 or 1, with equal probability. Each bidder receives a signal in {e, h} about the value with Pr(h|v=1) For each v, bidders' signals are statistically independent. (a) Suppose there are two bidders. Find a symmetric Nash equilibrium. Solution: There is an equilibrium in which, for each signal realization 8₁, each bidder i bids b;(8₁) = E[v | 8₁, 8; = 8;]. In this case, these bids are and and E[v | 8; = 8j = h] = Pr(l | v= 0) = 3/4. E[v | $i = 8j = l] = (b) Suppose there are three bidders. Find a symmetric Nash equilibrium. Solution: There is an equilibrium in which, for each signal realization s₁, each bidder i bids b;(8₁) = E[v | 8₁, 8; = maxji 8j]. In this case, these bids are = = (1/4)² (1/4)² + (3/4)² 1 10 E[v | 8₁ = 8j = $k = l] = (3/4)² (1/4)² + (3/4)² 9 10 52 E[v|s; = h and at least one s; = h for j ‡ i] = (1/4)³ (1/4)³ +(3/4)³ 1 28 (3/4)³ +2 (3/4)²-1/4 (1/4)³ +2 · (1/4)² · 3/4+ (3/4)³ + 2 · (3/4)².1/4 45
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