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, 2 nd -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(x 0,  )| X=x 0, Y=y] x0x0 E[v(x 0,  )| X=x 0, Y=x 0 ] Want to winWant to lose y b(y) = E[v(y,  )| X=y, Y=y] Suppose signal is x 0. Is optimal bid E[v(x 0,  )| X=x 0, Y=x 0 ]?

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 2 nd -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+1 st -price auction Equilibrium: b(x) = E[v(x,  )| X=x, Y=x] (X is owner’s signal, Y is m th -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? one two ; double-auction; n buyers, m sellers (But asymmetry introduced through endowments!)

Pr(max(b,s) = p 2 | x 3 )  Pr(max(b,s) = p 1 | x 3 ) NOT  in x 3 p1p1 p2p2 p3p3 p0p0 x 1, x 2 b(x1)b(x1) s(x2)s(x2) 1/21 High x 3 : x 1, x 2 ~ U(0,1) Low x 3 : x 1, x 2 ~ 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 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: v 1 (x,  ) and v 2 (x,  ) v 1 (x,  ) > v 2 (x,  ) for low and high, but not medium, values of  (and some range of signals x) It can then happen that v 1 (x,  ) > P(  ) for low and high, but not medium, values of  (and some range of signals x) What effect can this have?

Suppose v 1 (x,  ) > P(  ) for low and high, but not medium, values of  (and some range of signals x) P(  )  v1(x1,)v1(x1,) 10 b(x1)b(x1) 11

Suppose v 1 (x,  ) > P(  ) for low and high, but not medium, values of  (and some range of signals x) P(  )  v1(x2,)v1(x2,) 10 b(x2)b(x2) 22 b(x2)b(x2) A B 3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)