Information Aggregation in Auctions:
Recent Results (and some thoughts on
pushing them further)
Philip J. Reny
University of Chicago
Information Aggregation
(Wilson, Restud (1977), Milgrom, Econometrica (1979, 1981))
• n bidders, single indivisible good, 2nd-price auction
(e.g., stocks, natural resources, fashion goods, quality goods)
• state of the commodity,  ~ g(), drawn from [0,1]
• signals, x ~ f(x|), drawn indep. from [0,1], given 
• unit value, v(x,), nondecreasing (strict in x or )
• f(x|) satisfies strict MLRP:
x>y 
f(x|)f(y|) strictly  in 
• Equilibrium: b(x) = E[v(x,)| X=x, Y=x]
(X is owner’s signal, Y is highest signal of others)
Claim: b(x) = E[v(x,)| X=x, Y=x] is an equilibrium.
Suppose signal is x0. Is optimal bid E[v(x0,)| X=x0, Y=x0]?
b(y) = E[v(y,)| X=y, Y=y]
E[v(x0,)| X=x0, Y=y]
E[v(x0,)| X=x0, Y=x0]
Want to win
x0 Want to lose
y
Information Aggregation
(Wilson, Restud (1977), Milgrom, Econometrica (1979, 1981))
• Equilibrium: b(x) = E[v(x,)| X=x, Y=x]
(X is owner’s signal, Y is highest signal of others)
• outcome efficient for all n
• Equilibrium Price: P = E[v(z,)| X=z, Y=z],
where z is the 2nd-highest signal.
• if  is U[0,1] and x is U[0,], then P
v(,)
the competitive limit, and information is aggregated.
(fails if conditional density is continuous and positive.)
Information Aggregation
(Pesendorfer and Swinkels, Econometrica (1997))
• m units for sale; m+1st-price auction
• Equilibrium: b(x) = E[v(x,)| X=x, Y=x]
(X is owner’s signal, Y is mth-highest signal of others)
• if n = 2m, then price set by bidder with median signal
• n = 2m
 implies P
v(x(),)) the competitive
limit, and information is aggregated.
(where x() is median signal in state .)
What Next?
• one
two-sided market; double-auction; n buyers, m sellers
(Perry and Reny (2003))
• one-dimensional signals; one-dimensional state
• ex-ante symmetry (values and signal distributions)
(But asymmetry introduced through endowments!)
• single-unit demands
• single good/market
Pr(max(b,s) = p2 | x3 ) Pr(max(b,s) = p1 | x3) NOT  in x3
p3
s(x2)
High x3 : x1, x2 ~ U(0,1)
p2
Low x3 : x1, x2 ~ U(0,1/2)
p1
b(x1)

p0
1/2
1
x1, x2
What Next? cont’d
• one-sided market
• multi
one -dimensional signals; one-dimensional state
(Pesendorfer and Swinkels 2000)
• ex-ante symmetry (values and signal distributions)
• single-unit demands
• single good/market
• Jackson (1999) provides an important counterexample to
existence with discrete multi-dimensional signals and
continuous bids; what is the root cause?
• affiliation, a key property for single-crossing, fails
x, y independent uniform; b(x,y) = x + y
y
Pr(b = 2|x = 0)
Pr(b = 1|x = 0) = ∞
2
Pr(b = 2|x = 1)
=0
Pr(b = 1|x = 1)
0
1
2
x
• single-crossing holds despite affiliation failure
• existence of equilibrium (even monotone and pure) is
restored when signals are continuous and bids discrete and
sufficiently fine
• this may lead to information aggregation in the limit; a
promising avenue for confirming the results proposed in PS
(AER, 2000), and for obtaining more general related results
What Next? cont’d
• one-sided market
• one-dimensional signals; one-dimensional state
• ex-ante asymmetry
symmetry (values and signal distributions)
• single-unit demands
• single good/market
• single-crossing holds despite asymmetries
• existence of equilibrium (monotone and pure) can be
established when bids are discrete and sufficiently fine
• convergence to a fully revealing REE appears promising
despite asymmetries
• even so, ex-post efficiency, and so information
aggregation, can fail as follows
• Consider a limit market with two types of bidders:
v1(x,) and v2(x,)
• v1(x,) > v2(x,) for low and high, but not medium,
values of  (and some range of signals x)
• It can then happen that v1(x,) > P() for low and
high, but not medium, values of  (and some range of
signals x)
• What effect can this have?
• Suppose v1(x,) > P() for low and high, but not
medium, values of  (and some range of signals x)
v1(x1,)
P()
b(x1)
0 1
1

• Suppose v1(x,) > P() for low and high, but not
medium, values of  (and some range of signals x)
v1(x2,)
B
b(x2)
P()
A
b(x2)
0
2
3
1

• price is fully revealing
• equilibrium bids are ex-ante, but not necessarily
ex-post, optimal; outcome can be ex-post inefficient
• despite fully-revealing price, information can have
strictly positive value (even when there is no private value
component)
• related to Dubey, Geanakoplos and Shubik (JME 1987)
What Next? cont’d
• one-sided market
• one-dimensional signals; multi
one -dimensional state (dynamics?)
• ex-ante symmetry (values and signal distributions)
• single
multi-unit demands
• many
single goods/markets
good/market
• nature chooses α ≤  uniformly from [0,1]2
• x is uniform on [α,]; v(x,α,) = x + 2α + 
• half as many goods as agents
• b(x) = x + E(2α +  | α +  = 2x) = 7x/2;
(x < 1/2)
• P(α,) = 7(α + )/4, reveals only α + ;
(α +  < 1)
• information has value
(even if pure common values)
• how general is existence of partially-revealing REE?
• dynamics and efficiency? (multiple price observations)