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3 The Revenue Equivalence Principle
Problem 3.1 (War of attrition) Consider a two-bidder war of attrition in which
the bidder with the highest bid wins the object but both bidders pay the losing bid.
Bidders’values independently and identically distributed according to F.
a. Use the revenue equivalence principle to derive a symmetric equilibrium bidding
strategy in the war of attrition.
b. Directly compute the symmetric equilibrium bidding strategy and the sellers’
revenue when the bidders’values are uniformly distributed on [0;1].
Solution. Part a. Suppose that there is a symmetric, increasing equilibrium of the
war of attrition,, such that the expected payment of a bidder with value 0 is 0.
Since the assumptions of Proposition 3.1 are satis…ed, we must have that for all x,
the expected payment is
0
On the other hand, we also have
m(x) = E[(Y1)jY1< x]
=1
F(x)Zx
0
(y)f(y)dy
where Y1is the bid from the other bidder. Combining the two equations, we have
Zx
0
F(x)Zx
0
Differentiating both sides with respect to xand rearranging this, we get
0
Part b. Suppose that bidder 1 has valuation x: He chooses bto maximize his expected
payoff
F1(b)x1
F1(b)Z1(b)
0
(y)f(y)dy
where the …rst term is the product of his probability of winning and his valuation,
and the second term is his expected payment.
Because F(x) = xand f(x) = 1, the expected payoff becomes
1(b)Z1(b)
0
Maximizing with respect to byields the …rst-order condition:
0 = 1
01(b)x+1(b)1
0(1(b))1(b) + 1
0(1(b))R1(b)
0(y)dy
1(b)2
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In equilibrium, b=(x);and thus the …rst-order condition becomes
0
Differentiating both sides with respect to x, we obtain
Combining with the initial condition (0) = 0, we can solve the equilibrium bidding
strategy as (x) = 3
2x2:
Therefore the seller’s revenue is
3
Problem 3.2 (Losers-pay auction) Consider a N-bidder losers-pay auction in which
the bidder with the highest bid wins the object and pays nothing while all losing bidders
pays their own bids. Bidders’ valuations independently and identically distributed
according to F.
a. Use the revenue equivalence principle to derive a symmetric equilibrium bidding
strategy in the losers-pay auction.
b. Directly compute the symmetric equilibrium bidding strategy for the case when
the bidders’values are distributed according to F(x) = 1 eax over [0;1).
Solution. Part a. Suppose that there is a symmetric, increasing equilibrium of the
losers-pay auction,, such that the expected payment of a bidder with value 0 is 0.
Since the assumptions of Proposition 3.1 are satis…ed, we must have that for all x,
the expected payment is
0
On the other hand, we also have
which is the product of probability of losing the auction and his own bid. If we
combine the two equations, we have
Zx
0
and therefore
(x) = Rx
0yg (y)dy
1G(x)
0yF N2(y)f(y)dy
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Part b. Suppose that bidder 1 has valuation x, then he chooses bto maximize
his expected payoff
G1(b)x1G1(b)b
where the …rst term is the product of the probability of winning and his valuation,
and the second term is his expected payment.
Maximizing with respect to byields the …rst-order condition:
g1(b)
In equilibrium, b=(x), and thus the …rst-order condition becomes
0(x)g(x)
Using exp Rx
0
g(y)
1G(y)dyas the integrating factor and the initial condition (0) =
0, it is easily seen that
g(z)
where the last equality comes from the fact that
exp Zx
0
g(y)
1G(y)dy= exp Zx
0
1
1G(y)dG (y)
1
Note that (1) is the same as the equilibrium strategy derived in Part a.
When the bidders’values are distributed according to F(x) = 1 eax, we have
G(x) = 1eaxN1
Therefore the equilibrium bidding strategy in (1) becomes
(x) = Rx
0g(y)ydy
1G(x)
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