signal in {e, h} about the value with Pr(h | v= 1) = Pr(l | v= 0) = 3/4. For each v, bidders' signals are statistically independent. (a) Suppose there are two bidders. Find a symmetric Nash equilibrium.
signal in {e, h} about the value with Pr(h | v= 1) = Pr(l | v= 0) = 3/4. For each v, bidders' signals are statistically independent. (a) Suppose there are two bidders. Find a symmetric Nash equilibrium.
Chapter1: Making Economics Decisions
Section: Chapter Questions
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0542aa5b-cdbd-4921-99ff-0dadb1c30f45%2Fe53e0e87-c9cb-4fe2-afab-2682ff0a07cf%2Fgu3mc7t_processed.png&w=3840&q=75)
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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