A Common Value Auction with Bidder Solicitation

Stephan Lauermann, University of MichiganAsher Wolinsky, Northwestern University

September 2012

Information Aggregation with Bidder Solicitation

I A single-good, first-price auction with common values.... novel feature: the number of bidders is endogenous:

I Seller (auctioneer) knows the value and solicits bidders at some costsI Buyers (bidders) privately observe noisy signals;number of solicited bidders unobservable

I The number of solicited bidders depends endogenously on the value,which leads to a “solicitation effect” (sometimes “solicitation curse”).The solicitation effect is a key difference to standard common valueauctions.

I Two objectives:I Some understanding of equilibrium in this environmentI Revisit information aggregation in large auction (Milgrom 1979)Auction is endogenously large when solicitation costs small

Two Benchmarks: Search and AuctionsCV Auctions

I Exogenous number of bidders, independent of the state of natureI Seller of higher value receives higher price in expectation. Wilson(1977) and Milgrom (1979) derived conditions on the informativenessof signals s.t. price equals value when number of bidders becomes large.

Sequential Search

I In a sequential search environment, Lauermann, Wolinsky (“Search withAdverse Selection”) show that sellers expected price is independent ofvalue if signals are boundedly informative. Information aggregationrequires stronger conditions on signal distribution.

I Number of contacted bidders endogenous, similar to “solicitation curse”

Simultaneous (“Noisy”) SearchOur model can be interpreted as a batch search model as in Burdett-Judd(1983), with the added feature of adverse selection.Example: Borrower with private information applying at banks for loan.

Model (1): Seller and Buyers

I A single seller and N̄ potential buyersI Seller’s cost c is commonly knownI Seller’s type w ∈ {L,H}; prior probabilities ρL and ρHI Buyers have common values,

vw ∈ {vL, vH } , c ≤ vL < vH

I w is private information of the seller

Model (2): Signal Distribution

I Each buyer observes signal x ∈ [x , x̄ ]I conditional on type w , signals are independent and identically distributedI atomless c.d.f. Gw admits a density gw that is strictly positive on [x , x̄ ]

I Likelihood ratio gH (x )gL(x )

is weakly increasing;

likelihood ratio is right-continuous at x and left-continuous on (x ,x̄ ]I Most favorable signal is x̄ . Signals boundedly informative,

0 <gH (x)gL (x)

< 1 <gH (x̄)gL (x̄)

< ∞

Model (3): First Price Auction with Bidder Solicitation

1. Seller knows w ; solicits n randomly drawn bidders at marginalcost s > 0, with n ∈ {1, ..., N̄}, N̄ ≥ vH

s .

2. Each solicited bidder privately observes a signal x ∼ Gww and n unobservable to buyers

3. Solicited bidders submit bids simultaneously.

4. Highest bidder wins; ties are broken randomly

Payoffs when p is the winning bid:Winning Bidder: vw − p; Other Buyers: 0; Seller: p − c − ns

Interim Beliefs and Solicitation Effect

I A pure solicitation strategy is n = (nL, nH ) ∈ {1, ...., N̄} × {1, ...., N̄}.

I Interim belief is

Pr [H | signal x , solicited;n] =ρHgH (x)

nHN̄

ρLgL (x)nLN̄ + ρHgH (x)

nHN̄

=

ρHρL

gH (x )gL(x )

nHnL

1+ ρHρL

gH (x )gL(x )

nHnL

I The ratio nHnLcaptures “solicitation effect”. Solicitation is bad news

("curse") if nHnL < 1.

Bidding Equilibrium

A symmetric and pure bidding equilibrium given solicitation strategyn = (nL, nH ) is a bidding strategy β : [x , x̄ ]→ R such that for all x ∈ [x , x̄ ],b = β (x) maximizes interim expected payoffs.

Nests equilibrium of standard common value auction with nH = nL = n

Bidding Strategies are Weakly Monotone

Lemma. Given any solicitation strategy n = (nL, nH ) such that each typesolicits at least two bidders, nw ≥ 2. In every bidding equilibrium,gH (x1)gL(x1)

> gH (x2)gL(x2)

implies β (x1) ≥ β (x2).

Counterexample: Suppose nH = 1 and nL ≥ 2. Then, in every biddingequilibrium, gH (x1)gL(x1)

> gH (x2)gL(x2)

implies β (x1) < β (x2).

Intuition: Consider gH (x )gL(x )= 0 and gH (x̄ )

gL(x̄ )= ∞

Example of a Bidding Equilibrium

I Values vL = 0 and vH = 1; Uniform prior, ρH = ρL =12

I Signals x ∈ [x , x̄ ] = [0, 1]I gH (x) = 0.8+ 0.4x and gL (x) = 1.2− 0.4x

Claim: Let N̄ = 10 and solicitation strategy nL = 6 and nH = 2. For allb̄ ∈ [1/3, 0.4], there is a bidding equilibrium in which

β (x) = b̄ ∀x ∈ [x , x̄ ] .

Verification I: No Incentives to Overbid AtomClaim: No incentive to overbid atom at b̄ ≥ 1/3.

I Expected value conditional on winning at b′ > b̄

E[ v | signal x , solicited;n] =1

1+ ρHρL

gH (x )gL(x )

nHnL

vL +

ρHρL

gH (x )gL(x )

nHnL

1+ ρHρL

gH (x )gL(x )

nHnL

vH

≤ 0+113226

1+ 113226(1) =

13

I Solicitation curse is overwhelming if

gH (x̄)gL (x̄)

nHnL< 1

and implies

E[ v | signal x̄ , solicited;n] < ρLvL + ρH vH .

Verification II: No Incentive to Undercut Atom

Claim: No incentive to undercut the atom at b̄ ≤ 0.4.

I Expected value conditional on winning at b̄

E[ v | signal x , solicited, win at b̄;n, β] =

ρHρL

gH (x )gL(x )

nHnLPr[Win|H ]Pr[Win|L]

1+ ρHρL

gH (x )gL(x )

nHnLPr[Win|H ]Pr[Win|L]

vH

≥2326

1216

1+ 2326

1216

1 = 0.4.

I Winning is "good news," Pr[Win|H ]Pr[Win|L] =

1nH1nL

=1216= 3.

I Bidding in Atoms provides insurance (“hiding in the crowd”) givenuniform tie-breaking rule.

Observations

I Whenever nHnL =13 and nH ≥ 2 there is a bidding equilibrium where all

bidders bid b̄ ∈ [1/3, 0.4].I [Result] If nL = 3nH and nH is suffi ciently large, there exists noequilibrium in strictly increasing strategies. Atoms are “unavoidable.”

I [Result] No atoms in standard common value auction (nH = nL) ifgHgL

is strictly increasing.

I Construction is not an equilibrium: Seller’s solicitation strategy notoptimal.

EquilibriumΓ (s,G ) is the solicitation game with N̄ =

⌈ vhs

⌉and signals G = (GL,GH )

A symmetric and pure strategy equilibrium of Γ (s,G ) consists of abidding strategy β : [x , x̄ ]→ R and a solicitation strategy (nL, nH ) such that

(i) β is a bidding equilibrium given solicitation strategy (nL, nH );(ii) solicitation is optimal,

nw ∈ arg maxn∈{1,....,N̄}

[E [p|n,w , β]− c − ns ] .

An equilibrium (without qualifier) allows for mixed solicitation strategy,denoted ηw ∈ ∆ {1, ..., N̄}.

Solicitation Strategy

Lemma. Given any bidding strategy β, the support of the optimalsolicitation strategy ηw contains at most two adjacent numbers, that is, forsome n∗,

{n∗, n∗ + 1} ⊇ arg maxn∈{1,....,N̄}

E [p|w ; β, n]− c − ns.

I Expected winning bid E [p|w , β, n] is a non-decreasing, concavefunction of n.

I When the number of bidders grows large, mixing can often be ignored.

Equilibrium with Small Solicitation Costs

I A sequence {sk}∞k=1 such that s

k → 0.

I A sequence of equilibria {βk , ηk} of the games {Γ(sk ,G )}I F ( · |w ; ηk , βk ) the distribution of the winning bidI What are the limits of n

kHnkLand F (p|w ; ηk , βk )?

Characterization of All Equilibria

PropositionThere are functions F̂L, F̂H , R̂, such that for {sk} with sk → 0, for allconvergent (sub-)sequences of equilibria {βk , ηk} of Γ(sk ,G ):

[Partial Separation]

If lim gH (x̄ )gL(x̄ )

nkHnkL> 1 and (nkL , n

kH )→ ∞, then lim nkH

nkL= R̂( gH (x̄ )gL(x̄ )

) and

limF (p|w ; ηk , βk ) = F̂w (p; gH (x̄ )gL(x̄ )) for all p, with

vH > limk→∞

E[p|H; ηk , βk ] > ρLvL + ρH vH > limk→∞

E[p|L; ηk , βk ] > vL.

[Pooling]Otherwise, for some C ≤ ρLvL + ρH vH and all ε > 0,

limk→∞

F [C + ε|w ; ηk , βk ] = 1 and limk→∞

F [C − ε|w ; ηk , βk ] = 0.

Overwhelming Solicitation Curse

Fix one bidder, denote highest signal of other bidders by y (n−1)1

I If gH (x̄ )gL(x̄ )nkHnkL> 1 then

E[ v | signal x , solicited, y (n−1)1 ≤ y ; ηk ]

is increasing in y , for y suffi ciently high (“winner’s curse”)

I If gH (x̄ )gL(x̄ )nkHnkL< 1 (overwhelming solicitation curse), then

E[ v | signal x , solicited, y (n−1)1 ≤ y ; ηk ]

is decreasing in y , for y suffi ciently high (“winning is good news”)

Existence of Equilibrium: Good News / Bad News

I Atoms on equilibrium path upset standard existence proofsI In paper, discretize bid setI Here: Good News / Bad News

gH (x)gL (x)

=

gH (x̄ )gL(x̄ )

> 1 if x > x̂ ,gH (x )gL(x )

< 1 if x ≤ x̂ .

Existence of “Pooling Equilibrium”I Good News/Bad News: gH (x) /gL (x) constant on [x , x̂ ], (x̂ , x̄ ]I Symmetric Signals: GL (x̂) = 1− GH (x̂)I Suffi ciently Informative: GH (x̂) ≤ 0.3

Proposition.Suppose signals are as described before. Then, for all {sk}∞

k=1 withlimk→∞ sk = 0, there exists a sequence of equilibria {βk , ηk} such that(nkL , n

kH )→ ∞ and for some b̄ and k large enough,

βk (x) = b̄ < ρLvL + ρH vH ∀x > x̂ .

I Auction does not become “competitive”I GH (x̂) can be arbitrarily small, i.e., signals can be arbitrarily informative

Construction of Pooling EquilibriumI Bidding is step function for sk small,

βk (x) ={b̄ if x > x̂ ,bk if x ≤ x̂ .

I With bids bk < b̄, and

limk→∞

bk < b̄ < ρLvL + ρH vH .

I When sk → 0, optimal solicitation strategy implies both typesI solicit unboundedly many biddersI end up trading almost surely at b̄

Pooling: SamplingI Ignoring integer problem, solicitation strategy (nkH , n

kL ) optimal if

(GL (x̂))nkL (1− GL (x̂))

(b̄− bk

)= sk

(GH (x̂))nkH (1− GH (x̂))

(b̄− bk

)= sk

I This implies

nkL lnGL (x̂) + ln (1− GL (x̂)) = nkH lnGH (x̂) + ln (1− GH (x̂))I Taking limits and re-ordering

limk→∞

nkHnkL=lnGL (x̂)lnGH (x̂)

< 1

I From gH (x̄ )gL(x̄ )

= 1−GH (x̂ )1−GL(x̂ ) ,

gH (x̄)gL (x̄)

limk→∞

nkHnkL=1− GH (x̂)1− GL (x̂)

lnGL (x̂)lnGH (x̂)

< 1.

Pooling: BiddingStep 1: No incentives to overbid b̄ if b̄ close to ρLvL + ρH vH

I Seller’s optimality implies overwhelming solicitation curse:

gH (x̄)gL (x̄)

limk→∞

nkHnkL< 1.

I Therefore

limk→∞

E[v |solicited, signal x̄ ] < ρLvL + ρH vH .

Step 2: No incentives to undercut b̄ if b̄ < ρLvL + ρH vH .

I Observe that

limk→∞

E[ v | solicited, signal x̄ , win at b̄ ] = ρLvL + ρH vH .

Existence of “Separating Equilibrium”

Proposition.Suppose signals are either good news or bad news. If sk → 0, then there

exists a sequence of equilibria {βk , ηk} of Γ(sk ,G

)such that

gH (x̄ )gL(x̄ )

limk→∞nkHnkL> 1,

(nkL , n

kH

)→ ∞ and

vH > limk→∞

E[p|H; ηk , βk ] > ρLvL + ρH vH > limk→∞

E[p|L; ηk , βk ] > vL.

Almost Full Separation.For every ε > 0, there is λS (ε) such that gH (x̄ )gL(x̄ )

≥ λS (ε) implies,

limk→∞

E[p|H; ηk , βk ] > vH − ε,

limk→∞

E[p|L; ηk , βk ] < vL + ε.

Benchmark 1: Separation in Standard CV Auctions

Proposition. Almost Separation in the Limit with many Bidders.

Given a sequence {nkL , nkH } with nkL = nkH = nk and limk→∞ nk = ∞. For

every ε > 0, there is λC (ε) such that whenever gH (x̄ )gL(x̄ )≥ λC (ε), for every

sequence of bidding equilibria {βk},

limk→∞

E[p|nk ,H, βk ] ≥ vH − ε,

limk→∞

E[p|nk , L, βk ] ≤ vL + ε.

Milgrom (1979): expected winning bid converges exactly to vw if and only if

limx→x̄gH (x )gL(x )

= ∞.

Benchmark 2: Pooling with Sequential SearchLauermann-Wolinsky "Search with Adverse Selection"

I Seller samples individual buyers sequentially at costs z per buyer andmakes price offers.

I Each buyer observes signal x ∼ Gw , but does not observe past historyand how many buyers have been sampled before.

Proposition. Always Pooling with Sequential Search.If gH (x̄ )gL(x̄ )

< ∞ and sampling costs zk → 0, all sequences of “undefeated”

equilibria σk are such that

limk→∞

E[p|σk ,H ] = limk→∞

E[p|σk , L] = ρLvL + ρH vH .

Conclusion

I Introduced a common values auction with bidder solicitationI Endogenous relationship between value and number of bidders:Identified “solicitation effect”(“curse”)

I Multiple limit outcomes (in a "two-signal" example):I Perfect poolingI (Partial) separation

I Auction with Solicitation “in between”Auction and Search:I Large Standard Auction: Always (Partial)SeparationI Sequential Search: Always Pooling

Outlook and Related Questions

I Relation of number of solicited bidders and type?Who solicits more? Can there be a “solicitation blessing”?

I What happens when number of solicited bidders observable? Incentiveto signal? Trade-offs?