Q:1 Jhon have server and many clients or bidders bidders who are bidding with their resources (datasize,bandwidth) and price they want to sell their resources. Jhon is using first price sealed bid auction here. Jhon need to select winners. Please apply Nash equilibrium strategy for clients to maximize the expected profit and expected utility theory to give guidance to the aggregator
Q:1 Jhon have server and many clients or bidders bidders who are bidding with their resources (datasize,bandwidth) and price they want to sell their resources. Jhon is using first price sealed bid auction here. Jhon need to select winners. Please apply Nash equilibrium strategy for clients to maximize the expected profit and expected utility theory to give guidance to the aggregator
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.9P
Related questions
Question
Q:1 Jhon have server and many clients or bidders bidders who are bidding with their resources (datasize,bandwidth) and price they want to sell their resources. Jhon is using first price sealed bid auction here. Jhon need to select winners. Please apply Nash equilibrium strategy for clients to maximize the expected profit and expected utility theory to give guidance to the aggregator to obtain the expected resources from clients.
Note: related material is attached in pictures.
![1.3
Sealed Bid (First-Price) Auction
In a sealed bid, or first price, auction, bidders submit sealed bids b1, .., bn.
The bidders who submits the highest bid is awarded the object, and pays
his bid.
Under these rules, it should be clear that bidders will not want to bid
their true values. By doing so, they would ensure a zero profit. By bidding
somewhat below their values, they can potentially make a profit some of the
time. We now consider two approaches to solving for symmetric equlibrium
bidding strategies.
A. The "First Order Conditions" Approach
We will look for an equilibrium where each bidder uses a bid strategy
that is a strictly increasing, continuous, and differentiable function of his
value. To do this, suppose that bidders j # i use identical bidding strategies
b; = b(s;) with these properties and consider the problem facing bidder i.
Bidder i's expected payoff, as a function of his bid b; and signal s; is:
U (b;, s;) = (s; – b;) · Pr [b; = b(S;) < bi, Vj + i]
Thus, bidder i chooses bto solve:
max (s; – b;) F"-1
bị
' (b-" (b:)).
The first order condition is:
1
(8i – b;) (n – 1)F"-2
(6- (b;)) f (6-1(b:))
b (b-'(b;))
- F"-1 (b-1(b;)) = 0
At a symmetric equilibrium, b; = b(s;), so the first order condition reduces
to a differential equation (here I'll drop the i subscript):
f(s)
b'(s) = (8 – b(s)) (n – 1)
F(8)"
This can be solved, using the boundary condition that b(s) = s, to obtain:
Si F"-1(5)dš
Fn-1(s)
b(s) = s
'In fact, it is possible to prove that in any symmetric equilibrium each bidder must use
a continuous and strictly increasing strategy. To prove this, one shows that in equilbrium
there cannot be a "gap" in the range of bids offered in equilibrium (because then it would
be sub-optimal to offer the bid just above the gap) and there cannot be an “atom" in the
equilibrium distribution of bids (because then no bidder would make an offer just below
the atom, leading to a gap). I'll skip the details though.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48521d8e-d55b-4faa-b995-e6b5d82f052a%2Fb034a968-b5f9-49f8-b0d5-224acde07a35%2Fzc6xekb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.3
Sealed Bid (First-Price) Auction
In a sealed bid, or first price, auction, bidders submit sealed bids b1, .., bn.
The bidders who submits the highest bid is awarded the object, and pays
his bid.
Under these rules, it should be clear that bidders will not want to bid
their true values. By doing so, they would ensure a zero profit. By bidding
somewhat below their values, they can potentially make a profit some of the
time. We now consider two approaches to solving for symmetric equlibrium
bidding strategies.
A. The "First Order Conditions" Approach
We will look for an equilibrium where each bidder uses a bid strategy
that is a strictly increasing, continuous, and differentiable function of his
value. To do this, suppose that bidders j # i use identical bidding strategies
b; = b(s;) with these properties and consider the problem facing bidder i.
Bidder i's expected payoff, as a function of his bid b; and signal s; is:
U (b;, s;) = (s; – b;) · Pr [b; = b(S;) < bi, Vj + i]
Thus, bidder i chooses bto solve:
max (s; – b;) F"-1
bị
' (b-" (b:)).
The first order condition is:
1
(8i – b;) (n – 1)F"-2
(6- (b;)) f (6-1(b:))
b (b-'(b;))
- F"-1 (b-1(b;)) = 0
At a symmetric equilibrium, b; = b(s;), so the first order condition reduces
to a differential equation (here I'll drop the i subscript):
f(s)
b'(s) = (8 – b(s)) (n – 1)
F(8)"
This can be solved, using the boundary condition that b(s) = s, to obtain:
Si F"-1(5)dš
Fn-1(s)
b(s) = s
'In fact, it is possible to prove that in any symmetric equilibrium each bidder must use
a continuous and strictly increasing strategy. To prove this, one shows that in equilbrium
there cannot be a "gap" in the range of bids offered in equilibrium (because then it would
be sub-optimal to offer the bid just above the gap) and there cannot be an “atom" in the
equilibrium distribution of bids (because then no bidder would make an offer just below
the atom, leading to a gap). I'll skip the details though.
![It is easy to check that b(s) is increasing and differentiable. So any symmetric
equilibrium with these properties must involve bidders using the strategy
b(s).
B. The "Envelope Theorem" Approach
A closely related, and often convenient, approach to identify necessary
conditions for a symmetric equilibrium is to exploit the envelope theorem.
To this end, suppose b(s) is a symmetric equilibrium in increasing dif-
ferentiable strategies. Then i's equilibrium payoff given signal si is
U (s.) = (8 - b(s)) F"-(s).
(1)
Alternatively, because i is playing a best-response in equilibrium:
U(s,) = max (s - b,) F"-(6-(b,)).
Applying the envelope theorem (Milgrom and Segal, 2002), we have:
= F"-(6-'(b(s,)) = F"-(s,)
and also,
U (8,) = U(s) +
(2)
As b(s) is increasing, a bidder with signal s will never win the auction -
therefore, U(s) = 0.
Combining (1) and (2), we solve for the equilibrium strategy (again drop-
ping the i subscript):
b(s) = a
F-(s)
Again, we have showed necessary conditions for an equilibrium (i.e. any
increasing differentiable symmetric equilibrium must involve the strategy
b(s)). To check sufficiency (that b(s) actually is an equilibrium), we can
exploit the fact that b(s) is increasing and satisfies the envelope formula
to show that it must be a selection from i's best response given the other
bidder's use the strategy b(s). (For details, see Milgrom 2004, Theorems 4.2
amd 4.6).
Remark 1 In most auction models, both the first order conditions and the
envelope approach can be used to characterize an equilibrium. The trick is
to figure out which is more convenient.
What is the revenue from the first price auction? It is the expected
winning bid, or the expected bid of the bidder with the highest signal,
E (b(Sln]. To sharpen this, define G(s) = F"-1(s). Then G is the proba-
bility that if you take n - 1 draws from F, all will be below s (i.e. it is the
edf of Sln-1), Then,
b(s) = 8
ädF"-(3) = E [Sln-1|gln-1 < s] .
Fn-1(s)
Fn-1(s)
That is, if a bidder has signal s, he sets his bid equal to the expectation of
the highest of the other n-1 values, conditional on all those values being
less than his own.
Using this fact, the expected revenue is:
E (b (sm)] = E [s!m-|slm-1 < glm) = E [s2m],
equal to the expectation of the second highest value. We have shown:
Proposition 2 The first and second price auction yield the same revenue
in erpectation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48521d8e-d55b-4faa-b995-e6b5d82f052a%2Fb034a968-b5f9-49f8-b0d5-224acde07a35%2Fvqvry3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:It is easy to check that b(s) is increasing and differentiable. So any symmetric
equilibrium with these properties must involve bidders using the strategy
b(s).
B. The "Envelope Theorem" Approach
A closely related, and often convenient, approach to identify necessary
conditions for a symmetric equilibrium is to exploit the envelope theorem.
To this end, suppose b(s) is a symmetric equilibrium in increasing dif-
ferentiable strategies. Then i's equilibrium payoff given signal si is
U (s.) = (8 - b(s)) F"-(s).
(1)
Alternatively, because i is playing a best-response in equilibrium:
U(s,) = max (s - b,) F"-(6-(b,)).
Applying the envelope theorem (Milgrom and Segal, 2002), we have:
= F"-(6-'(b(s,)) = F"-(s,)
and also,
U (8,) = U(s) +
(2)
As b(s) is increasing, a bidder with signal s will never win the auction -
therefore, U(s) = 0.
Combining (1) and (2), we solve for the equilibrium strategy (again drop-
ping the i subscript):
b(s) = a
F-(s)
Again, we have showed necessary conditions for an equilibrium (i.e. any
increasing differentiable symmetric equilibrium must involve the strategy
b(s)). To check sufficiency (that b(s) actually is an equilibrium), we can
exploit the fact that b(s) is increasing and satisfies the envelope formula
to show that it must be a selection from i's best response given the other
bidder's use the strategy b(s). (For details, see Milgrom 2004, Theorems 4.2
amd 4.6).
Remark 1 In most auction models, both the first order conditions and the
envelope approach can be used to characterize an equilibrium. The trick is
to figure out which is more convenient.
What is the revenue from the first price auction? It is the expected
winning bid, or the expected bid of the bidder with the highest signal,
E (b(Sln]. To sharpen this, define G(s) = F"-1(s). Then G is the proba-
bility that if you take n - 1 draws from F, all will be below s (i.e. it is the
edf of Sln-1), Then,
b(s) = 8
ädF"-(3) = E [Sln-1|gln-1 < s] .
Fn-1(s)
Fn-1(s)
That is, if a bidder has signal s, he sets his bid equal to the expectation of
the highest of the other n-1 values, conditional on all those values being
less than his own.
Using this fact, the expected revenue is:
E (b (sm)] = E [s!m-|slm-1 < glm) = E [s2m],
equal to the expectation of the second highest value. We have shown:
Proposition 2 The first and second price auction yield the same revenue
in erpectation.
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