Information Aggregation in Auctions:

Recent Results (and some thoughts on pushing them further)

Philip J. Reny

University of Chicago

Information Aggregation

• n bidders, single indivisible good, 2nd-price auction

• Equilibrium: b(x) = E[v(x,)| X=x, Y=x] (X is owner’s signal, Y is highest signal of others)

(e.g., stocks, natural resources, fashion goods, quality goods)

• unit value, v(x,), nondecreasing (strict in x or )

• state of the commodity, ~ g(), drawn from [0,1]

• signals, x ~ f(x|), drawn indep. from [0,1], given

• f(x|) satisfies strict MLRP:

x > y f(x|)f(y|) strictly in

(Wilson, Restud (1977), Milgrom, Econometrica (1979, 1981))

Claim: b(x) = E[v(x,)| X=x, Y=x] is an equilibrium.

E[v(x0,)| X=x0, Y=y]

x0

E[v(x0,)| X=x0, Y=x0]

Want to win Want to lose y

b(y) = E[v(y,)| X=y, Y=y]

Suppose signal is x0. Is optimal bid E[v(x0,)| X=x0, Y=x0]?

• 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(Wilson, Restud (1977), Milgrom, Econometrica (1979, 1981))

• 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 .)

Information Aggregation(Pesendorfer and Swinkels, Econometrica (1997))

• -sided market

• ex-ante symmetry (values and signal distributions)

• single-unit demands

• single good/market

• one-dimensional signals; one-dimensional state

(Perry and Reny (2003))

What Next?

onetwo ; double-auction; n buyers, m sellers

(But asymmetry introduced through endowments!)

Pr(max(b,s) = p2 | x3 ) Pr(max(b,s) = p1 | x3) NOT in x3

p1

p2

p3

p0 x1, x2

b(x1)

s(x2)

1/2 1

High x3 : x1, x2 ~ U(0,1)

Low x3 : x1, x2 ~ U(0,1/2)

• -sided market

• ex-ante symmetry (values and signal distributions)

• single-unit demands

• single good/market

• -dimensional signals; one-dimensional state

What Next? cont’d

one

multione(Pesendorfer and Swinkels 2000)

• 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

0

2

1 2

y

x

b(x,y) = x + yx, y independent uniform;

Pr(b = 2|x = 0)Pr(b = 1|x = 0) = ∞

Pr(b = 2|x = 1)Pr(b = 1|x = 1) = 0

• existence of equilibrium (even monotone and pure) is restored when signals are continuous and bids discrete and sufficiently fine

• single-crossing holds despite affiliation failure

• 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

• -sided market

• ex-ante (values and signal distributions)

• single-unit demands

• single good/market

• one-dimensional signals; one-dimensional state

What Next? cont’d

one

asymmetrysymmetry

• 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)

P()

v1(x1,)

10

b(x1)

1

• Suppose v1(x,) > P() for low and high, but not medium, values of (and some range of signals x)

P()

v1(x2,)

10

b(x2)

2

b(x2)

A

B

3

• 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)

• -sided market

• ex-ante (values and signal distributions)

• -unit demands

• single good/market

• one-dimensional signals; -dimensional state

What Next? cont’d

one

multi

symmetry

single

onemulti

• many goods/markets

(dynamics?)

• 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)

• nature chooses α ≤ uniformly from [0,1]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)