Search with Adverse Selection Stephan Lauermann, University of Michigan Asher Wolinsky, Northwestern University

Presented at the 2011 Econometric Society Winter Meeting

Search with Adverse Selection .... and Noisy Signals

Search with Adverse Selection .... and Noisy Signals

Price: $2000

Search with Adverse Selection and Noisy Signals I

For a common values auction environment, Wilson and Milgrom derived conditions on the informativeness of the signals under which the price aggregates information when the number of bidders is large.

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We consider a search version of this environment I I I

The seller samples bidders sequentially and makes TOL-o¤ers Bargaining is private and unobserved by other buyers The counterpart of many bidders is small sampling cost

Search with Adverse Selection and Noisy Signals I

For a common values auction environment, Wilson and Milgrom derived conditions on the informativeness of the signals under which the price aggregates information when the number of bidders is large.

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We consider a search version of this environment I I I

The seller samples bidders sequentially and makes TOL-o¤ers Bargaining is private and unobserved by other buyers The counterpart of many bidders is small sampling cost

Main Result: Characterization of the limit equilibrium price as a function of properties of the signal distribution. I

Compared to auctions, with search stronger conditions on the signal distribution are needed for information aggregation

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Reason: A special winner’s curse with search

Model (1): The Adverse Selection Problem I

A single seller and a continuum of buyers

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Seller has no use value for object (cost c = 0)

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Seller’s type (quality) w 2 fL, H g; prior probabilities gL and gH Buyers have common values, depending on quality vw 2 fvL , vH g

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The quality is private information of the seller Endogenous opportunity cost of trading

Model (2): Noisy Signals I

Each buyer observes signal x 2 [x, x¯ ] I

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conditional on the quality w , signals are i.i.d., with atomless c.d.f. Fw continuous density fw that is strictly positive on (x , x¯ ).

Likelihood ratio is continuously di¤erentiable and increasing, d dx

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fH (x ) fL (x )

> 0.

The most informative signal towards H is x, ¯ fH (x¯ ) fL (x¯ )

lim

x !x¯

fH (x ) . fL (x )

Model (3): Search In every period t 2 f1, 2, 3, ...g (t is not observed by buyers) 1. Seller draws one buyer at cost s

2. Buyer receives a signal x, observed by seller as well 3. Seller o¤ers a price 4. Buyer accepts or rejects the price 5. If price accepted: trade happens and game is over; payo¤s Buyer: vw 6. If price o¤er rejected: Seller samples another buyer.

p

Seller: p

c

ts

Equilibrium (prep): Payo¤s and Beliefs I

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Equilibrium objects: Markovian pure o¤er strategy Pw (x ), acceptance probability A (p, x ), belief β (p, x ). (Continuation) payo¤ of the seller Vw (P, A) = E [p jP, A, w ]

c

s E [# of buyers sampledjP, A, w ] .

Equilibrium (prep): Payo¤s and Beliefs I I

Expected number of sampled buyers nw = E [# of buyers sampledjP, A, w ] Interim belief of a buyer conditional on signal x and conditional on being sampled (before observing the o¤er)

βI (x, P, A)

g H f H (x ) n H g L f L (x ) n L . g H f H (x ) n H g L f L (x ) n L + 1

Equilibrium: De…nition An equilibrium is a Markovian (pure) o¤er strategy P = (PL , PH ), an acceptance strategy A (p, x ) and beliefs β(p, x ) s.t. 1. Pw (x ) 2 arg maxp 0 fA(p 0 , x ) (p 0 c ) + (1 A(p 0 , x ))Vw (P, A)g. 2. p 7 β (p, x ) vH + (1 β (p, x )) vL implies A(p, x ) = 1 and 0. 3. β(Pw (x ) , x ) is Bayesian updating of βI (x, P, A).

“Undefeated” Equilibrium

“Undefeated” Equilibrium Interim expected value: EI [v jx, P, A] = βI (x, P, A) vH + (1

βI (x, P, A)) vL .

De…nition: An “Undefeated” Equilibrium is such that the high quality seller receives at least the pooling payo¤ in every bargaining game given x, A(PH (x ), x ) (PH (x ) maxfVH , EI [v jx, P, A]

c ) + (1 cg

A(PH (x ), x )) VH

Result 1: Boundedly Informative Signals

Result 1: Boundedly Informative Signals I

To what extent is information aggregated into equilibrium prices when s is small? I I

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Maximal: price received by H is close to vH ; Minimal: both types receive the same price(s).

Sampling cost sk , sk ! 0; equilibria (Pk , Ak ); prices E [p jPk , Ak , H ]

Result 1: Boundedly Informative Signals I

To what extent is information aggregated into equilibrium prices when s is small? I I

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Maximal: price received by H is close to vH ; Minimal: both types receive the same price(s).

Sampling cost sk , sk ! 0; equilibria (Pk , Ak ); prices E [p jPk , Ak , H ] f (x¯ )

Proposition. Suppose the most informative signal fH (x¯ ) < ∞. Consider a L sequence sk ! 0 and a sequence (Pk , Ak ) of corresponding undefeated equilibria. The limiting expected price is lim E [p jPk , Ak , H ] = gH vH + gL vL .

s k !0

Sketch of Proof Step 1: When sk small, equilibrium characterized by cuto¤ xk . Both types o¤er common (pooling) price whenever x above xk : PL (x ) = PH (x ) = EI [v jx, Pk , Ak ]

A (EI [v jx, Pk , Ak ] , x ) = 1;

and sellers o¤er unacceptable price if x is below xk : A (PL (x ) , x ) = A (PH (x ) , x ) = 0.

Sketch of Proof Step 1: When sk small, equilibrium characterized by cuto¤ xk . Both types o¤er common (pooling) price whenever x above xk : PL (x ) = PH (x ) = EI [v jx, Pk , Ak ]

A (EI [v jx, Pk , Ak ] , x ) = 1;

and sellers o¤er unacceptable price if x is below xk : A (PL (x ) , x ) = A (PH (x ) , x ) = 0.

Step 2 The cuto¤ xk converges to x¯ (both types search for the most optimistic buyers).

Sketch of Proof: Winner’s Curse with Search Limit likelihood ratio of being sampled 1

1 1 F H (x k ) n lim kH = lim = lim 1 xk !x¯ xk !x¯ 1 k !∞ nkL 1 F L (x k )

FL xk FH xk

=

fL (x¯ ) . fH (x¯ )

Sketch of Proof: Winner’s Curse with Search Limit likelihood ratio of being sampled 1

1 1 F H (x k ) n lim kH = lim = lim 1 xk !x¯ xk !x¯ 1 k !∞ nkL 1 F L (x k )

FL xk FH xk

=

fL (x¯ ) . fH (x¯ )

Interim belief of a sampled buyer with xk 2 xk , x¯ for all k: lim βI (xk ) =

k !∞

g H fH (xk ) n kH g L fL (xk ) n kL lim g k !∞ H fH (xk ) n kH + 1 g L fL (xk ) n kL

=

g H fH (x¯ ) fL (x¯ ) g L fL (x¯ ) fH (x¯ ) g H fH (x¯ ) fL (x¯ ) g L fL (x¯ ) fH (x¯ ) + 1

=

gH gL

gH gL

+1

= gH .

Sketch of Proof: Winner’s Curse with Search Limit likelihood ratio of being sampled 1

1 1 F H (x k ) n lim kH = lim = lim 1 xk !x¯ xk !x¯ 1 k !∞ nkL 1 F L (x k )

FL xk FH xk

=

fL (x¯ ) . fH (x¯ )

Interim belief of a sampled buyer with xk 2 xk , x¯ for all k: lim βI (xk ) =

k !∞

g H fH (xk ) n kH g L fL (xk ) n kL lim g k !∞ H fH (xk ) n kH + 1 g L fL (xk ) n kL

=

g H fH (x¯ ) fL (x¯ ) g L fL (x¯ ) fH (x¯ ) g H fH (x¯ ) fL (x¯ ) g L fL (x¯ ) fH (x¯ ) + 1

Interim expected value (=price) lim EI [v jxk , Pk , Ak ] = gL vL + gH vH

k !∞

=

gH gL

gH gL

+1

= gH .

The Winner’s Curse in Search and in Auctions

The Winner’s Curse in Search and in Auctions I

A …rst price auction with N bidders and signal distributions FH and FL

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E [p jN, H, FL , FH ]: expected equilibrium winning bid

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Consider sequence of equilibria for N ! ∞ (Milgrom 1979)

The Winner’s Curse in Search and in Auctions I

A …rst price auction with N bidders and signal distributions FH and FL

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E [p jN, H, FL , FH ]: expected equilibrium winning bid

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Consider sequence of equilibria for N ! ∞ (Milgrom 1979)

Proposition. "Auction" For every ε > 0, there is some R such that f (x¯ ) whenever FL and FH are such that fH (x¯ ) > R, L

lim E [p jN, H, FL , FH ]

N !∞

vH

ε.

The Winner’s Curse in Search and in Auctions I

A …rst price auction with N bidders and signal distributions FH and FL

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E [p jN, H, FL , FH ]: expected equilibrium winning bid

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Consider sequence of equilibria for N ! ∞ (Milgrom 1979)

Proposition. "Auction" For every ε > 0, there is some R such that f (x¯ ) whenever FL and FH are such that fH (x¯ ) > R, L

lim E [p jN, H, FL , FH ]

N !∞

vH

ε.

The “winner’s curse” in an auction is di¤erent from the “winner’s curse” with search.

Result 2: Arbitrarily Informative Signals Proposition. Suppose the signal distribution is as follows: (i) the support contains arbitrarily informative signals, f (x ) lim H x !x¯ fL (x )

=

fH (x¯ ) = ∞; fL (x¯ )

(ii) there is some λ 2 [0, ∞] such that λ=

f H (x ) 1 F H (x ) lim f H (x ) x !x¯ d dx f L (x )

.

Consider a sequence sk ! 0 and a sequence (Pk , Ak ) of corresponding undefeated equilibria. The limiting expected price is p¯ H =

λvL + (1 λ)vH gL vL + gH vH

if if

λ 2 [0, gL ], λ gL .

Interpretation The parameter λ is the hazard rate of the distribution of the likelihood ratios. If λ = 0, then lim

z !∞

n

f (x )

z +y x j fH (x ) o n L f (x ) z FH x j fH (x ) L

FH

o

= 1.

Interpretation The parameter λ is the hazard rate of the distribution of the likelihood ratios. If λ = 0, then lim

z !∞

n

f (x )

z +y x j fH (x ) o n L f (x ) z FH x j fH (x ) L

FH

o

= 1.

Corollary: The limit price equals the true value if the distribution of likelihood ratios in the high state (i) has an unbounded support with (ii) a long tail.

Conclusion I

When lemons search, signals are less valuable: Search by bad sellers leads to a severe winner’s curse

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The limit price depends on the support and the tail properties of the distribution of likelihood ratios I

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If the support is bounded or if the tail is too thin (λ is high): Complete pooling in the limit If the support is unbounded and if the tail is su¢ ciently long (λ is low): Almost complete separation and trading at true values

Conclusion I

When lemons search, signals are less valuable: Search by bad sellers leads to a severe winner’s curse

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The limit price depends on the support and the tail properties of the distribution of likelihood ratios I

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In the paper: I

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If the support is bounded or if the tail is too thin (λ is high): Complete pooling in the limit If the support is unbounded and if the tail is su¢ ciently long (λ is low): Almost complete separation and trading at true values

welfare implications and other extensions

Open problems: I I

Auctions with endogenous (unobservable) number of bidders Observable Duration (friends and strangers)