A Common Value Auction with Bidder Solicitation Stephan Lauermanny
Asher Wolinskyz
August 2013
We study a common-value, …rst-price auction in which the number of bidders is endogenous: the seller (auctioneer) knows the value and solicits bidders at a cost. The number of bidders, which is unobservable to them, may thus depend on the true value, giving rise to a solicitation e¤ ect – being solicited already conveys information. The solicitation e¤ect is a key di¤erence from standard common value auctions. In contrast to standard auctions, the equilibrium bid distribution may exhibit atoms. We also discuss information aggregation in the case of small bidder solicitation cost. We show that there is a type of equilibrium that aggregates information well when the most favorable signals are informative. However, there may also be an equilibrium that fails to aggregate any information.
This paper considers a single good, common values, …rst-price auction in which the number of bidders is endogenous: the seller (auctioneer) knows the value and solicits bidders at a constant cost per sampled bidder. The bidders do not know the value and do not observe the number of solicited bidders. Each bidder only learns that he was summoned to the auction and observes a private noisy signal of the true value. In equilibrium, bidders bid optimally given their signals, the bidding strategies of others, and the solicitation strategy of the seller. The seller chooses optimally how many bidders to solicit given the true value and the bidders’strategies. The novel feature is the endogenous solicitation of the bidders by the informed seller. It implies that the number of bidders may vary across the di¤erent value states, which gives rise to a solicitation e¤ ect: The mere fact of being summoned The authors gratefully acknowledge support from the National Science Foundation under Grants SES-1123595 and SES-1061831. We thank Qinggong Wu, Au Pak and Guannan Luo for excellent research assistance. y University of Michigan, Department of Economics,
[email protected]. z Northwestern University, Department of Economics,
[email protected].
1
by the seller to bid conveys information to the bidder. This e¤ect is a key di¤erence between the analysis of the present paper and that of standard common value auctions. The relationship between the underlying value and the number of bidders is not always in the same direction. It is possible to have equilibria in which a seller of a high value good samples more bidders (solicitation is “good news,” a solicitation blessing) and equilibria in which a seller of a low value good samples more bidders (solicitation is “bad news,” a solicitation curse). This analysis has two objectives. First, to develop an understanding of the new model introduced here and, in particular, the nature of its equilibria. As will be seen later, the endogenous solicitation produces some peculiarities in the structure of equilibrium. Second, an exploration of the question of information aggregation— the relation between the expected winning bid (price) and the true value— when the cost of soliciting bidders is small. The question of information aggregation by markets is of central importance in economics and probably does not require justi…cation. One of our main observations is that sometimes a market of the sort we consider may totally fail to aggregate the information, even under conditions that normally are viewed as conducive to successful aggregation (participation of many bidders and existence of highly informative signals). Although the model features an auction augmented by a preliminary solicitation stage, we think of it as a model of a market with adverse selection in which an informed agent contacts simultaneously potential partners rather than a model of a formal auction mechanism. In line with this, the analysis does not adopt the perspective of mechanism design. In particular, it does not endow the auctioneer with the power to commit to an optimal solicitation policy. It is possible of course to explore this side as well, inquiring about optimal solicitation by a seller who has full or partial commitment power, but we chose here to focus mainly on the other aspects. We did not construct the model with a speci…c market in mind, but many markets share the feature that an informed agent contacts a number of partially informed agents for a potential transaction. For example, a potential borrower who contacts several lenders in an attempt to obtain funding for a project on which she has private information. The lenders obtain noisy signals and o¤er terms, while being aware that the borrower might be applying to other lenders. The borrower is the counterpart of the seller in our model (selling a bond) and the lenders are the bidders.1 Of course, in actual markets of this sort the contact is sometimes indeed simultaneous, 1 Broecker (1990) and Riordan (1993) have modeled this situation as an ordinary common value auction. However, accounting for endogenous (and unobservable) solicitation of terms by the borrower is natural in this environment and might produce new insights.
2
sometimes sequential and sometimes a combination of the two. We have explored the sequential scenario in a separate paper (Lauermann and Wolinsky (2012)) and focus here on the simultaneous case. There are two states: in state h the value of the good is high for all bidders; in state l it is low for all bidders. The seller knows the true value and selects the desired number of bidders randomly from a large population of potential bidders. Solicited bidders obtain conditionally independent signals and participate in a …rstprice auction. We characterize symmetric equilibria in which all bidders use the same bidding strategies. The analysis yields the following insights. The equilibrium bid distributions may exhibit atoms–bidders with di¤erent signals submit the same bid. This observation contrasts with the standard intuition that atoms induce bidders to overbid them slightly. This intuition fails when winning in the atom is pro…table only in the high value state, but on average more bids fall in the atom in the low value state. The atom then protects bidders in the low value state: Overbidding it would increase the probability of winning more signi…cantly when winning involves a loss and hence is unpro…table. One may think that a similar consideration could give rise to atoms in an ordinary common value auction, in which the number of bidders is constant across states. However, it turns out that this is not the case. In an ordinary common value auction, an atom may arise only under special circumstances (only at the bottom of the bid distribution and only if there is a mass of the lowest signals that share the same information content). When the solicitation cost is small, there are at most two kinds of distributions of the winning bid that may arise in equilibrium: a nearly atomless distribution and a distribution that places nearly all the mass on a common price below the ex-ante expected value. The nearly atomless outcome qualitatively resembles the equilibrium outcome of an ordinary common value auction. It is partially revealing in the sense that the expected winning bid di¤ers across the states. If the most favorable signal is very informative, it aggregates information well, in the sense that the expected winning bid is close to the true value. The outcome that exhibits the atom–the “pooling” outcome–fails to aggregate information. It is interesting to note that this outcome may arise even when the sampling cost is small and the most favorable signals are very informative. Moreover, since the atom occurs at a price that is strictly below the ex-ante expected value, the price not only fails to approach the value in each state but also the expected revenue is strictly below the expected ex-ante value despite the fact that many bidders participate and signals may be very informative. 3
Existence of equilibrium is established for the case in which the set of feasible bids is any su¢ ciently …ne grid. We show that a partially revealing equilibrium always exists. The pooling equilibrium exists under additional assumptions on the distribution that generates the signal. We do not know whether a pooling equilibrium exists for all signal distributions. As a by-product of the analysis, we derive explicitly the distribution of the winning bid of the ordinary auction when the number of bidders goes to in…nity. We are not familiar with such derivation in the relevant literature. Three strands of related literature should be mentioned. First, there is an obvious relation to the literature on common value auctions.2 This paper complements that literature by adding the endogenous solicitation of bidders, which may change the nature of the equilibrium in a signi…cant way. In particular, it complements the discussion within that literature of information aggregation by the price when the number of bidders becomes exogenous large (Wilson (1977) and Milgrom (1979, 1981)). Our results imply that exogenously large auctions may aggregate information quite di¤erently from the endogenously large auctions studied here. Second, Lauermann and Wolinsky (2012) studies the sequential search counterpart of the present paper (the counterpart of the auctioneer samples the counterparts of the bidders sequentially rather than simultaneously). The results of the present paper on information aggregation fall between the results of the two literatures just mentioned. Under the informational assumptions of the present paper, in the common values auction the equilibrium is partially revealing and, with a large number of bidders, aggregates the information well when there are highly informative signals (i.e., the likelihood ratio associated with the most favorable signal is large). In contrast, still under the same conditions on the informativeness of signals, in the sequential search environment information aggregation always completely fails. The present paper exhibits results of these two types in di¤erent equilibria. Finally, the present model can be interpreted as a simultaneous (“batch-”)search model like Burdett-Judd (1983), with the added feature of adverse selection. Section 9 explains further this connection.
1
Model
Basics.— This is a single-good, common value, …rst-price auction environment with two underlying states, h and l. There are N potential bidders (buyers). The common values of the good for all potential bidders in the two states are vl and vh , with 0
vl < vh . The seller’s cost is zero. 2
This literature is too voluminous to mention an arbitrary selection of names.
4
Nature draws a state w 2 fl; hg with prior probabilities
l
+
nw
h
l
> 0 and
h
> 0,
= 1. The seller learns the realization of the state w and invites nw bidders,
N . If nw < N , the seller selects the invitees randomly with equal probability.
We use n to denote the vector (nl ; nh ). The seller incurs a solicitation cost s > 0 for each invited bidder. We assume that N
vh s .
Therefore, N does not constrain the seller.
Each invited bidder observes a private signal x 2 [x; x] and submits a bid b from
a set of feasible bids P . Conditional on the state, signals are independently and identically distributed according to a cumulative distribution Gw , w 2 fh; lg. A
bidder does not observe w nor how many other bidders are invited to bid.
The invited bidders bid simultaneously: The highest bid wins and ties are broken randomly with equal probabilities. If in state w 2 fh; lg the winning bid is p, then the payo¤s are vw
winning bidder and zero for all others. The seller’s payo¤ is p
p for the
nw s.
Further Details.— The set of feasible bids P may either be the full interval [0; vh ] or a grid P = Here,
(
[0; vh ] [0; vl ] [ fvl +
; vl + 2 ;
; vh
if
= 0;
; vh g if
> 0.
is the step size of the grid. Notice that we leave the continuum of prices
on [0; vl ] even when
> 0. This avoids some irrelevant distinctions between the
case in which the bottom equilibrium bid is vl and the case in which it is vl Much of the following analysis holds for both of the cases
> 0 and
.
= 0. We
will mention it explicitly when the discussion focuses on just one of these cases. The signal distributions Gw , w 2 fl; hg, have identical supports, [x; x]
strictly positive densities gw . The likelihood ratio (x) limx!x gghl (x) ,
and
gh (x) gl (x)
=
(x) limx!x gghl (x) .
gh (x) gl (x)
is non-decreasing,
R, and gh (x) gl (x)
=
Thus, larger values of x indicate a higher
likelihood of the higher value. The signals are not trivial and boundedly informative, 0<
gh (x) gh (x) <1< < 1. gl (x) gl (x)
The boundedness of the likelihood ratios implies that Gh and Gl are mutually absolutely continuous, i.e., letting Gw (A) denote the measure of a set A Gh (A) = 0 , Gl (A) = 0.
[x; x],
The prior likelihood ratio and the likelihood ratio at the most favorable signal x
5
appear often in the analysis. We therefore dedicate to them special symbols, =
h
gh (x) . gl (x)
g=
and
l
Expected Payo¤s and Equilibrium.— The posterior probability of state w 2 fl; hg in the eyes of a bidder conditional on being solicited and receiving signal x is Pr[wjx; n] =
nw w gw (x) N nl nh l gl (x) N + h gh (x) N
=
w gw (x) nw . g (x) nl + h gh (x) nh l l
The terms gw (x) re‡ect the information contained in the signal, the terms
nw N
re‡ect
the information that is conveyed to the bidder by being invited, and the
w
re‡ect
the prior information. Since the signals accrue only to bidders who were sampled, we do not need a separate piece of notation for the information that this bidder was sampled. Notice that N cancels out and hence does not play any role in the analysis. The posterior likelihood ratio, Pr[hjx; n] = Pr[ljx; n]
h l
gh (x) nh , gl (x) nl
is thus a product of three likelihood ratios: The prior likelihood ratio likelihood ratio
gh (x) gl (x) ,
and the sampling likelihood ratio
We study pure and symmetric bidding strategies
h l
, the signal
nh nl .
: [x; x] ! P
that are mea-
surable. When there are n bidders who employ a bidding strategy , the cumulative distribution of the winning bid in state w is n
Fw (pj ; n) = Gw (fxj (x)
bg) =
"Z
gw (x) dx
xj (x) p
#n
.
The expected winning bid with n bidders in state w is Ew [pj ; n] = Let strategy
w
Z
x
pdFw (pj ; n) .
x
(bj ; n) be the probability of winning with bid b, given state w, bidding employed by the other bidders, and n bidders. The expected payo¤ to a
bidder who bids b, conditional on being solicited and observing the signal x, given the bidding strategy U (bjx; ; n) =
l gl
and the solicitation strategy n = (nl ; nh ), is (x) nl
l
(bj ; nl ) (vl b) + h gh (x) nh l gl (x) nl + h gh (x) nh
6
h (bj
; nh ) (vh
b)
. (1)
Denote by
0 (N; n; P
) the bidding game when the auctioneer is known to invite
n = (nl ; nh ) bidders and the set of possible bids is P . The ordinary common value auction is a special case of the bidding game with nl = nh . A bidding equilibrium of
0 (N; n; P
) is a strategy
such that, for all x, b =
(x) maximizes U (bjx; ; n) over P . Denote by is N =
vh s
(s; P ) the overall game in which the potential number of bidders
, the smallest natural number larger than
A pure equilibrium of
vh s .
(s; P ) consists of a bidding strategy
strategy n = (nl ; nh ) such that (i)
is a bidding equilibrium of
and a solicitation 0
vh s
; n; P
,
and (ii) the solicitation strategy is optimal for the seller, nw 2 arg max Ew [pj ; n] n2f1;2;
ns
for
;N g
w 2 fl; hg .
Since a pure equilibrium might not exist, we allow for mixed solicitation strategies. Let
= ( l;
h)
denote a mixed solicitation strategy, where
w (n)
is the
probability with which n = 1; ; N bidders are invited in state w. Let nw ( w ) = PN PN n=1 w (n) n w (bj ; n) =nw . These are the exn=1 n w (n) and w [bj ; ] = pected number of bidders and the weighted average probability of winning in state w and are analogous to nw and
w [bj
; n] in the deterministic solicitation case. To
make the expressions less dense we omit here and later the argument of nw (
w)
and
write just nw instead. The expected payo¤ to a bidder who bids b, conditional on being solicited and observing the signal x, given the common bidding strategy and the solicitation strategy U (bjx; ; ) =
l gl
= ( l;
h)
is
(x) nl l [bj ; l ](vl b) + h gh (x) nh h [bj ; l gl (x) nl + h gh (x) nh
h ] (vh
In a complete analogy to the above de…nitions (for pure strategies), is the bidding game given A bidding equilibrium of
= ( l; 0 (N;
h)
and
b)
0 (N;
.
(2) ;P )
(s; P ) is the full game.
; P ) is a strategy
such that, for all x, b =
(x) maximizes U (bjx; ; ) over P . The strategy pro…le ( ; ) is an equilibrium of (s; P ) if (i)
is a bidding equilibrium of
0
vh s
; ;P
and (ii) the solicitation
strategy is optimal, w
(n) > 0 ) n 2 arg max Ew [pj ; n]
ns.
n2f1;2;::::g
Before proceeding let us note that the paper contains the proofs of the formally stated results. Some proofs are in the body of the paper and the rest are relegated 7
to the appendix. We will not repeat it each time. If the proof of a formally stated claim is not in its immediate vicinity, then it is in the appendix.
2
Bidding Equilibrium: Single Crossing, Bertrand, and Monotonicity of Bids
This section derives some properties of a bidding equilibrium strategy . The main property is the monotonicity of the bidding equilibrium are invited in each of the states. If the likelihood ratio where, a bidding equilibrium
gh gl
when at least two bidders is strictly increasing every-
is necessarily non-decreasing. If the likelihood ratio
is constant over some interval, all signals in this interval contain the same information and, if
is not constant over this interval, the bids need not be monotonic.
Nevertheless, there is an equivalent bidding equilibrium that is monotonic and that is obtained by reordering the bids over such intervals. A bidding equilibrium e is said to be equivalent to a bidding equilibrium
if
the implied joint distributions over bids and states are identical.
Proposition 1 (Monotonicity of Bidding Equilibrium) Suppose that
l (1)
=
h (1)
= 0, and
is a bidding equilibrium.
1. If x0 > x, then U ( (x0 ) jx0 ; ; ) and only if
gh (x0 ) gl (x0 )
>
is such
gh (x) gl (x) .
U ( (x) jx; ; ). The inequality is strict if
2. There exists an equivalent bidding equilibrium e , such that e is non-decreasing on [x; x] and coincides with
over intervals over which
gh gl
is strictly increasing.
The proof of Proposition 1 relies on two lemmas. Lemma 1 (Single-Crossing) Given any bidding strategy , any solicitation strategy
and any bids b0 > b
1. If
0 w [b j
;
w]
vl .
> 0 for some w 2 fl; hg then, for all x0 > x,
U (b0 jx; ; )
U (bjx; ; ) ) U (b0 jx0 ; ; )
where the second inequality is strict if 0 w [b j U (b0 jx;
2. If
;
w]
gh (x0 ) gl (x0 )
>
= 0 for some w 2 fl; hg, then
; ) = U (bjx; ; ) = 0 for all x.
8
U (bjx0 ; ; );
gh (x) gl (x) . w [bj
;
w]
= 0 for both w, and
The following lemma collects a number of additional properties of a bidding equilibrium . One of them is a straightforward Bertrand property: when the seller solicits two or more bids in both states, then (x)
vl , for all x.
Lemma 2 (Bertrand and Other Properties) Suppose
l (1)
=
h (1)
= 0 and
is a bidding equilibrium. (x) j ;
w]
> 0 if
gh (x) gl (x)
w[
2.
(x) 2 [vl ; vh ) for almost all x.
3. U ( (x0 ) jx0 ; ; ) only if
gh (x0 ) gl (x0 )
>
>
gh (x) gl (x) .
1.
U ( (x) jx; ; ) if x0 > x. The inequality is strict if and
gh (x) gl (x) .
4. If P = [0; vh ], then
(x) 2 (vl ; vh ) for all x >x for which
gh (x) gl (x)
>
gh (x) gl (x) .
The proof of the lemma utilizes that the set of feasible bids is dense below vl . If the price grid is …nite below vl as well, equilibrium may involve bids just below vl — just like in the usual Bertrand pricing game with price grid— but such equilibria would not add anything important. The single crossing condition in Lemma 1 does not require that other bidders use a monotone (non-decreasing) bidding strategy. Furthermore, the proof also holds if other bidders use mixed strategies. It follows that, when there are at least two bidders in each state, every symmetric equilibrium is in monotone strategies and hence the restriction to pure bidding strategies is without loss of generality. In contrast, some existing single crossing conditions for auctions, such as the condition in Athey (2001), require monotonicity of the strategy of other bidders. The proof of the single crossing condition avoids assuming monotonicity by using the two state assumption: The condition that b
vl implies that (i) the low state
is unambiguously bad (pro…t is negative because the bid is higher than the value) and that (ii) the higher bid must be worse in the low state (because the increased probability of winning decreases pro…ts in the low state). With more than two states, such a strong result may not hold and single crossing may require stronger assumptions (such as monotonicity) on the strategies of other bidders. Observe that, when the number of solicited bidders depends on the state, monotonicity is not immediately obvious. Signals inform bidders not only about the expected value but also about the number of competitors. If fewer buyers are solicited when w = h, a higher signal implies both a higher value and less competition. The following example illustrates this consideration. It also clari…es why the assumption that at least two bidders are solicited in both states is needed for establishing monotonicity. 9
Example of a Non-Monotone Bidding Equilibrium: Let [x; x] = [0; 1], with gh (x) = 2x and gl (x) = 2
2x. Thus, the signals x = 1 and x = 0 reveal the state.3
Suppose that vl > 0, nh = 1 and nl = 100. It follows that Hence,
(1) = 0 in every bidding equilibrium. So, if
h [bj
; 1] = 1 for all b
0.
were weakly increasing, then
(x) = 0 for all x. However, this strategy cannot be an equilibrium. At x = 0 the expected payo¤ from bidding b = 0 is b0 = " is vl
1 100 vl
while the expected payo¤ from bidding
". Because vl > 0, a deviation to b0 is pro…table for small ".4 Thus, in
this example no bidding equilibrium strategy is weakly increasing. In light of Proposition 1, from now on, whenever
w (1)
= 0, attention will
be con…ned only to monotone bidding equilibria (whether or not
gh (x) gl (x)
is strictly
increasing).
3
Bidding Equilibrium: Atoms
One signi…cant consequence of the endogenous solicitation of bidders is the emergence of atoms in the bidding equilibrium. In auctions with private values, a standard argument involving slight overbidding (or undercutting) precludes atoms in which bidders get positive payo¤s. This argument does not apply directly to common value auctions, since overbidding the atom may have di¤erent consequences in di¤erent underlying states owing to possibly di¤erent frequency of bids that are tied in the atom in the di¤erent states. Still, as is shown below, a somewhat more subtle argument still precludes atoms in an ordinary common value auction (nl = nh = n), except at the lowest equilibrium bid. However, when nl > nh atoms may arise in a bidding equilibrium. Example of an Atom in a Bidding Equilibrium.— Suppose that vl = 0 and vh = 1, with uniform prior gl (x) = 1:2 4 [ 31 ; 10 ].
0:4x. Thus,
h = gh (x) gl (x)
l
= 21 . Let [x; x] = [0; 1], gh (x) = 0:8 + 0:4x and
is increasing as required. Let b be any number in
Claim: Suppose nl = 6 and nh = 2. There is a bidding equilibrium in which (x) = b
8x 2 [x; x] .
3
The example violates the assumption that likelihood ratios are bounded. This simpli…es the argument but it is easily possible to change the example so that signals are boundedly informative while the equilibrium bids are still decreasing. 4 In fact, one can show that must be strictly decreasing on [0; 1], using arguments analogous to the proof of Proposition 1.
10
Proof: The expected value conditional on x and winning with bid b is
E[vjx, win at b; , n] =
h
1 1+
h l
gh (x) nh gl (x) nl
h [bj l [bj
v + ] l
;nh ;nl ]
l
1+
gh (x) nh h [bj ;nh ] gl (x) nl l [bj ;nl ] vh . h gh (x) nh h [bj ;nh ] l [bj ;nl ] l gl (x) nl
At the atom, the assumption that ties are broken randomly implies that 1 nh
=
1 2
and
l
bj ; nl =
1 nl
=
1 6.
Further,
1+ gh (x) gl (x)
gh (0) gl (0)
l
=
2 3
bj ; nh =
= 1, vl = 0, vh = 1. Thus,
1 gh (x) 2 1 gl (x) 6
E[vjx, win at b; , n] =
The inequality is from
h
h
1 2 1 6
1 gh (x) 2 1 gl (x) 6
1 2 1 6
= 1
gh (x) gl (x) (x) + gghl (x)
4 . 10 4 10 ,
for all x. Because b
when bidding
b, almost all buyers expect strictly positive payo¤s whereas undercutting b yields zero payo¤. Consequently, for almost all signals (except possibly x = x if b = 0:4), buyers strictly prefer bidding b to any b < b. There is also no incentive for any bidder to overbid b. The expected value conditional on winning when overbidding b is
E[vjx, win at b > b; , n] =
The inequality is from because b
1 3,
gh (x) gl (x)
3 2
1
1 gh (x) 2 1 1 gl (x) 6 1 (x) 2 1 + 11 gghl (x) 61
1 . 3
for all x. Any bid above b is sure to win. Hence,
bidding b > b yields strictly negative payo¤s. However, when
bidding b, expected payo¤s are positive. Therefore, for all signals, buyers strictly prefer bidding b to any b > b. The key to the atom’s immunity to deviations is the fact that nl > nh . Slightly overbidding the atom would result in a discontinuous increase in payo¤ in state h, but an even more signi…cant decrease in state l. In other words, given the uniform tiebreaking rule, bidding in an atom provides insurance against winning too frequently (“hiding in the crowd”) in state l where the payo¤ is negative.5 A later result (Lemma 9) implies that, if nl = 3nh and nh is su¢ ciently large, there exists no equilibrium in strictly increasing strategies. Thus, atoms may be “unavoidable” if the number of bidders depends on the state. Finally, observe that bidding equilibria discussed here are not full equilibria. The 5 Atakan and Ekmekci (2012) …nd that atoms may occur when after the auction the winning bidders have to take an action whose payo¤ depends on the unknown state. Winning at the atom may inform the bidders about the state owing to the di¤erential probability of this event across states. Consequently, bidders may be reluctant to overbid if the value of information for the subsequent decision problem is su¢ ciently high.
11
seller’s solicitation strategy is obviously not optimal. Optimal solicitation in a face of a single atom would be nl = nh = 1. We return in Section 8.1 to the existence of a full equilibrium with an atom similar to the previous example. Winning Probability at Atoms.— To continue the discussion of atoms, the following lemma derives an expression for the winning probability in the case of a tie. De…ne the generalized inverse of
by
x (p) = inf fx 2 [x; x] j (x)
pg ,
x+ (p) = sup fx 2 [x; x] j (x)
pg ,
with x = sup ; and x = inf ;. When there is no danger of confusion we will omit
the arguments and write x and x+ . Lemma 3 Suppose
is non-decreasing and, for some b, x (b) < x+ (b). Then, w
(bj ; n) =
Gw (x+ )n Gw (x )n . n (Gw (x+ ) Gw (x ))
Building on this result, we can obtain useful bounds on the likelihood ratio h (bj l (bj
;nh ) ;nl )
that play an important role in a few points in the subsequent analysis
including the next proposition. Lemma 4 Suppose
is non-decreasing. If nh
nl
2, then
Gh (x)nh Gl (x)nl
is weakly
increasing and Gh (x )nh Gl (x )nl
1 1
; nh ] l [bj ; nl ]
The inequalities are strict unless nl = nh and The Case of nh
Gh (x+ )nh Gl (x+ )nl
h [bj
gh (x+ ) gl (x+ )
=
1 1
.
gh (x) gl (x) .
nl .— In this case the bidding equilibrium is essentially free of
atoms. Atoms may arise only if nh = nl and
gh gl
is constant at the bottom of the
signal distributions on some interval [x; x ^], and then only at the lowest possible bid. Thus, if either nh > nl or the case of nh
gh gl
is strictly increasing, then the bidding equilibrium in
nl cannot have an atom at all. The case of nh
nl includes of
course the ordinary common value auction nl = nh = n as a special case.6 For the standard common value auction (nl = nh = n), the absence of atoms when gghl is strictly increasing is well known. The second part of the proposition is related to results from Rodriguez (2000), which imply that when nl = nh = 2 and gghl is not strictly increasing, then atoms may occur only at the bottom of the bid distribution. In fact, one can easily show if gghl is constant on some interval [x; x0 ], then is constant on that interval as well; see the remark at the end of the proof. 6
12
Proposition 2 (No Atoms if nh given n, with nh 1.
(x) =
0.
is a bidding equilibrium
2, and P = [0; vh ].
is strictly increasing if
2. If
4
nl
nl ) Suppose that
gh (x0 ) gl (x0 )
>
gh (x) gl (x)
(x0 ) for some x 6= x0 , then
for all x0 > x and/or nh > nl .
gh (x0 ) gl (x0 )
=
gh (x) gl (x) ,
nh = nl , and U ( (x0 ) jx0 ; ; n) =
Optimal Solicitation: Characterization
The seller’s payo¤ when sampling n bidders who use bidding strategy
is Ew [pj ; n]
ns. This expression is strictly concave in n whenever the bidding strategy is not constant. Consequently, either there is a unique optimal number of sampled bidders or the optimum is attained at two adjacent integers. Lemma 5 Optimal Solicitation Given any symmetric bidding function , there is a number nw such that fnw ; nw + 1g
arg max Ew [pj ; n]
n2f1;2;
ns.
;N g
The lemma is an immediate consequence of the concavity of the expectation of the …rst-order statistic in the number of trials. Proof of Lemma 5: The probability that the winning bid is below p is Gw where
1
([0; p]) = fx :
1
([0; p])
(x) 2 [0; p]g. Recall that the expected value of a positive
random variable can be expressed by the integral of its decumulative distribution function, Ew [pj ; n] =
Z
vh
1
1
Gw
([0; t])
n
dt.
0
The incremental bene…t of soliciting one more bidder is therefore Ew [pj ; n + 1]
Ew [pj ; n] =
Z
vh
1
Gw
[0; t]
n
1
Gw
1
[0; t]
dt. (3)
0
The lemma is immediate whenever
is degenerate: If all buyers bid the same,
the uniquely optimal number is n = 1. If
is not degenerate, then inspection of the
incremental bene…t of soliciting one more bidder shows that it is strictly decreasing in n. Thus, the objective function is strictly concave, which implies the lemma. Given the lemma, we will restrict attention in the following to mixed strategies that have support on at most two adjacent integers. In addition, can represent any such mixed strategy
w
by nw 2 f1; :::; N g and 13
w
2 (0; 1], where
w
=
w
(nw ) > 0
n
,
and 1
w
=
w
(nw + 1)
0. A solicitation strategy is pure if
here on, when we talk about nw in the context of a strategy of the support of
w.
w,
w
= 1. Thus, from
we mean the bottom
In fact, since our characterization results pertain to the case
of small sampling costs and many bidders, they are not a¤ected by whether or not the equilibrium strategies are actually pure or mixed. Mixed solicitation strategies matter only for the existence arguments. Relative Solicitation Incentives: To understand how the incentive to solicit bidders depends on the seller’s type, observe that for a non-decreasing that x+ (p) = sup fxj (x) Ew [pj ; n + 1]
(and recalling
pg),
Ew [pj ; n] =
Z
vh
(Gw (x+ (p)))n (1
(Gw (x+ (p)))) dp.
0
Thus, the incremental bene…t of sampling another bidder depends on two terms, (Gw (x+ (p)))n — the probability that all n buyers bid below p— and (1
(Gw (x+ (p))))—
the probability that the additional buyer bids higher. The monotone likelihood ratio property implies that (Gl (x+ (p)))n 1
(Gh (x+ (p)))n while 1
(Gl (x+ (p)))
(Gh (x+ (p))). Intuitively, when w = l, the highest bid of the already sampled n
sellers is likely to be lower than when w = h, making further sampling more desirable. However, at w = l, there is a lower probability that an additional bid is high, rendering further sampling less bene…cial. The incentive to solicit bidders depends on the relative magnitudes of these two countervailing terms. As we demonstrate in examples, for given signal distribution, there can simultaneously be an equilibrium in which the high type samples more bidders than the low type (solicitation blessing) and an equilibrium in which the low type samples more bidders (solicitation curse).
5
Large Numbers: Basic Results
Our main characterization results and the associated insights into the question of information aggregation are derived for an environment in which the solicitation cost is small and the numbers of solicited bidders are large. This section obtains basic results that are used in the subsequent characterization of equilibria with large numbers of bidders. Recall the shorthand g =
14
gh (x) gl (x) .
Lemma 6 (Utilizing Poisson approximation for Binomial Distribution) x k ; nk
Consider some sequence (0; 1). If limk!1 Gl xk
nkl
with min nkl ; nkh ! 1 and limk!1
nkh nkl
=r2
= q then Gh (xk )
lim
k!1
nkh
= q gr .
For a given w and nkw , the number of signals above any cuto¤ xk is binomially distributed, with nkw independent trials and success probabilities 1
Gw (xk ). As is
well-known, when the number of trials is large and probability of success is proportionately small, the binomial distribution is approximated by a Poisson distribution. Speci…cally, if lim nkw [1 weakly above
xk )
Gw (xk )] = nkw
= Pr[m = 0] = e
be that xk ! x, and hence, nkl
Thus, if lim Gl xk
k!1
2 (0; 1), then the number m = #(signals
is Poisson distributed with parameter
Therefore, lim Gw xk
lim
w
(xk )
1 Gh 1 Gl (xk )
!
w
gh (x) gl (x) .
w
in the limit as nkw ! 1.
. Now, when Therefore,
h l
w
=
2 (0; 1), it must gh (x) gl (x)
nk
lim nhk = gr. l
= q, then
Gh (xk )
nkh
l
=e
h l
h l
nk
lim Gl l (xk )
=
k!1 k
Note that the lemma also implies that if
= q gr .
is non-decreasing and lim Fl pj
k
; nkl =
q, then lim Fh (p j
k
k!1
; nkh ) = q gr .
This is because for non-decreasing , Fw (pj
k
(4) k
; nkw ) = Gw (xk+ (p))nw .
As the number of bidders grows, the interim expected payo¤ for each bidder vanishes to zero. Lemma 7 (Zero Pro…t in the Limit) For every " there is an M (") such that, if nl > M (") and nh > M ("), then U ( (x)jx; ; ) < " for all x in every bidding equilibrium . k
Consider a sequence of solicitation strategies and a corresponding sequence any sequence of bids
bk
, if
k
such that min nkh ; nkl
!1
of bidding equilibria. Lemma 7 implies that for
lim bk
= b and lim
lim sup E[vjx, win at bk ; k!1
15
k
;
k
k
bk ;
l
]
b.
;
k l
> 0, then (5)
In addition, for any sequence of signals xk
k
xk = b, individual
b.
(6)
for which lim
rationality requires that lim inf E[vjxk , win at
k
k!1
Therefore, lim
k
l
k
xk ;
k l
;
(xk );
k
k
;
]
> 0 requires
lim E[vjxk , win at
k
k!1
(xk );
k
;
k
] = b.
(7)
Observations (4) and (7) together will imply a tight characterization (in Proposition 3 below) of the limiting distribution of the winning bid if there are no atoms in the limit. The last lemma provides a condition to be satis…ed in the limit by the seller’s optimal solicitation strategy. Lemma 8 (Total Solicitation Costs) Consider a sequence sk ! 0 and a sek
quence of bidding strategies given
k
. Suppose that
k w
is an optimal solicitation strategy
in state w and the winning bid distribution Fw j
k
;
k w
converges point-
wise. Then,
lim nkw sk =
k!1
Z
vh
lim Fw pj
k!1
0
k
;
k w
ln
lim Fw pj
k
k!1
;
k w
dp.
The lemma allows us to characterize the total solicitation costs in the limit as a function of the distribution of the winning bid. In particular, it immediately implies Corollary 1 If limk!1 Fw j
k
;
is non-degenerate, then lim nkw sk > 0.
k w
To understand the lemma intuitively, observe that the optimality of nk means that it is more pro…table for the seller to solicit nk than to solicit instead, for to Fw pj
k
6= 1. With
nk bidders
nk bidders, the distribution of the winning bid changes
; nk = Fw pj
k
; nk
, since Fw pj
k
; nk = (Gw (
k 1
([0; p])))
nk .
The expected payo¤ from soliciting nk bidders is therefore Z
vh
1
Fw pj
k
; nk
dp
nk s k .
0
Ignoring integer constraints, soliciting nk bidders is optimal if the derivative of the expected payo¤ with respect to Z
vh
Fw pj
k
; nk
vanishes at ln Fw pj
0
16
= 1, k
; nk
dp
nk sk = 0.
(8)
The lemma follows from the observation that when sk ! 0 either the number of solicited bidders is so large that the integer constraints can indeed be ignored or the
number of optimally solicited bidders is bounded. In the latter case, the Lemma is shown to hold trivially because the distribution of the winning bids must become degenerate.
6
Characterization of Equilibria with Small Sampling Costs
This section studies the nature of the equilibrium bid distribution when the sampling cost is small. In particular, it inquires about the extent of information aggregation by the equilibrium winning bid –whether the winning bid is near the true value when the sampling costs are small and many bidders may be sampled. Wilson (1977) and Milgrom (1979, 1981) studied the latter question in the context of ordinary common value auctions, without the solicitation element. Overall, the analysis implies that there are at most two kinds of equilibrium outcomes when the sampling cost is negligible: A partially revealing outcome that qualitatively resembles the equilibrium outcome of an ordinary common value auction and a degenerate “pooling” outcome that is qualitatively di¤erent. Consider a sequence of games and
k
! 0. (Recall that
price grid.) Let gies for
sk
sk ; P
k
k
and
k
=(
sk ; P
k
indexed by k where sk ! 0,
is the sampling cost and k; k ) l h
k
;
k w
approximation for Fw j
;
the step size of the
denotes the cumulative distribution
function of the winning bid. We study limk!1 Fw j k w
0
be equilibrium bidding and solicitation strate-
. Recall that Fw j k
k
k
when
sk
and
k
k
;
k w
thinking of it as an
are small.
From here on the term “limit” (and the operator lim) refers to a limit over a
subsequence such that all the magnitudes of interest are converging.7 We will not repeat this quali…cation each time, but it is always there. All the characterization results that we are about to report hold both for the case of a …nite price grid ( (
k
k
> 0,
k
! 0) and for the case of continuum of prices
0). To help the reading, we …rst present the results and prove them for the
continuum case and only later explain that they also hold when the limit is taken over a sequence of …nite grids. The following theorem characterizes the set of possible equilibrium outcomes when solicitation costs are negligible. It is perhaps the main result of this paper. 7
By Helly’s selection theorem, every sequence of cumulative distribution functions has a pointwise everywhere convergent subsequence. This is immediate from the monotonicity of Fw ; see Kolmogorov and Fomin (1970, p. 372).
17
Recall the shorthand notation
=
gh (x) gl (x) .
, g =
h l
parameter r > 0 (r 6= g), de…ne the functions
l
(pj ; g; r) =
and
8 1 > > < > > :
w
1 gr 1
1 p vl gr vh p
Observe that if gr > 1, then
p
if vl < p if p
j ; g; r) = (
l
> 0, g > 1 and a
( j ; g; r) by
if
0
h(
Given
vl + grvh 1+ gr ; vl + grvh 1+ gr ;
(9)
vl ,
( j ; g; r))gr :
( j ; g; r) is a cumulative distribution function.8
w
Theorem 2 (Equilibrium Characterization) Consider a sequence of games sk ; P0 , such that sk ! 0, and a corresponding sequence of equilibria r ( ; g) >
(i). There exists a unique number r
and min nkl ; nkh ! 1, then limk!1 lim Fw ( j
k
k!1
(ii). Otherwise, limk!1 Fw j
k
mass 1 on some number C
;
k w l vl
;
nkh nkl
1 g
k
;
such that if g limk!1
k nkh nkl
. >1
= r and
k w)
w
( j ; g; r ) .
(10)
is a degenerate distribution with probability +
h vh .
Since r g > 1, the functions limk!1 Fw j
k
;
k w
in (10) are indeed distribution
functions. They are partially revealing in the sense that they have the same support (hence, not perfectly revealing). However, limk!1 Fh j dominates limk!1 Fl j
k
;
k l
k
;
k h
stochastically
and, hence, gives rise to a higher expected price.
The theorem says that in the limit an equilibrium winning bid distribution takes
one of two forms. Either it is the unique partially revealing function described in Part (i), or it is a mass point below the ex-ante expected value. Furthermore, the partially revealing outcome arises if and only if the sampling behavior satis…es nk
min nkl ; nkh ! 1 and g lim nhk > 1. Uniqueness is not claimed, so that equilibria l
of both types may coexist, as veri…ed later.
The proof of Theorem 2 is split into three propositions that are presented over the following two subsections. The …rst subsection characterizes the limit of bidding equilibria given any sampling behavior (not necessarily optimal) such that 8
For the statement of the theorem, recall that by Lemma 5 we represent a mixed equilibrium strategy w by nw and w , where w = w (nw ) and 1 w = w (nw + 1).
18
min nkl ; nkh
! 1 and
nkh nkl
converges. The second subsection completes the char-
acterization for the case of an optimal sampling strategy. That subsection shows in particular how the number r is determined. Since r is uniquely determined by ( ; g), the partially revealing limit outcome depends only on
6.1
and g.
Bidding Equilibria
Consider …rst bidding equilibria alone, without the optimal solicitation requirement. Recall
=
h l
,g=
gh (x) gl (x)
and the functions
w
( j ; g; r) de…ned in (9).
Proposition 3 Consider a sequence of bidding games minfnkl ; nkh g k
equilibria
! 1 and limk!1
nkh nkl
0
N k;
k; P 0
such that
= r, and a corresponding sequence of bidding
.
(i). If rg > 1, then k
lim Fw ( j
k!1
(ii). If rg
1, then limk!1 Fw j
k
;
bility mass 1 on some number C
;
k w)
k w
=
w
( j ; g; r) .
is a degenerate distribution with probal vl
+
h vh .
The special case of r = 1 is of course the ordinary CV auction. Thus, the Proposition implies that
(pj ; g; 1) is the limiting winning bid distribution of the
w
ordinary auction as n ! 1.
More importantly, the proposition identi…es gr =
gh (x) gl (x)
nk
lim nhk as a key magnitude l
in the nature of the equilibrium distribution of the winning bid. When gr > 1, the limiting distribution is atomless. When gr
1, the limiting distribution is
degenerate. Notice that the relationship of gr to 1 determines whether being solicited and observing the most favorable signal x is “good news”or “bad news”for a bidder, in the sense of making the bidder more or less optimistic than the prior. If rg > 1, this is “good news”–for large enough k, the compound likelihood ratio
h l
gh (x) nkh gl (x) nkl
of
a bidder who observed the most favorable signal x is larger than the prior likelihood ratio
h l
. Conversely, if rg < 1, being solicited and observing x is “bad news.”
The formal proof of the proposition is in the appendix. Roughly speaking, it proceeds as follows. Part (i) is proved in three steps. First, it is established that if rg > 1, then limk!1 Fw pj
k
;
k w
has no atoms. This means that in the limit it
can be identi…ed with the probability of winning with bid p, lim
k!1
w [pj
k
;
k w]
= lim Fw pj k!1
19
k
;
k w
:
k
At a price p such that lim Fw pj
;
k w
> 0 the zero-pro…t plus IR condition (7) is
equivalent to gr
limk!1 limk!1
where we used the shorthands limk!1
w [pj
k
k] w
;
=
k
h [pj
; k l [pj ;
h l
k] h k] l
gh (x) gl (x) ,
,g=
p vl , vh p
=
r = limk!1
nkh . nkl
Substituting for
from the previous step and using the Poisson approximation
from Lemma 6 and Equation (4), we get two equations for limk!1 Fw pj gr
limk!1 Fh pj
k
limk!1 Fl pj
k
;
;
k h k l
=
k
;
k w
,
p vl , vh p
and
gr
lim Fh pj
k!1
k
;
k h
=
lim Fl pj
k!1
Solving the two equations yields limk!1 Fw pj
k
;
k w
k
k l
;
=
.
w
as stated in Part (i) of the proposition. The proof of Part (ii) observes that if limk!1 Fw pj
k
(pj ; g; r) for w 2 fl; hg, ;
k w
is strictly increas-
ing on any interval (p0 ; p00 ), then the argument of Part (i) applies, and over this interval limk!1 Fw pj rg < 1 then limk!1 Fw pj
w
k
;
k w
should coincide with
w
(pj ; g; r). However, when
(pj ; g; r) is decreasing— which implies a contradiction. Therefore, k
;
k w
should consist of atoms. The …nal argument rules out the
possibility of multiple atoms by pointing out a pro…table downward deviation from a prospective atom that is not the bottom atom. Proposition 3 does not tell us whether the atom that arises in the case of rg < 1 is a limit of atoms in winning bid distributions along the sequence or that whether the atom emerges only in the limit. The following lemma shows that it is the former case— signi…cant atoms are already present along the sequence. Lemma 9 (Tieing at the Top) Consider a sequence of bidding games such that minfnkl ; nkh g ! 1 and limk!1 bidding equilibria xk
k
nkh nkl
0
N k;
= r, and a corresponding sequence of
. Suppose that rg < 1. Then there is a sequence
xk ; xk+
xk+ such that: 1.
k
is constant on (xk ; xk+ ).
2. limk!1 nkw Gw xk+ 3. limk!1 Gw xk
nkw
Gw x k
= 1 for w 2 fl; hg.
= 0 and limk!1 Gw xk+
20
k; P 0
nkw
= 1 for w 2 fl; hg.
,
6.2
Full Equilibrium (Including Solicitation)
k
=(
k; k ) l h
are corresponding equilibrium bidding and
k
sk ; P0 , where
We return now to the full model: A sequence of games
solicitation9
and
strategies.
The …rst proposition states that with equilibrium solicitation if there exists a nk
partially revealing equilibrium of the sort described by Theorem 2-(i), then lim nhk is unique, which means that limk!1 Fw pj
k
l
;
k w
is unique as well by Proposition
3. sk ; P0 , such that sk ! 0. Suppose
Proposition 4 Consider a sequence of games k
that (
;
k)
is a corresponding sequence of equilibria such that min nkl ; nkh ! 1. nk
There exists a unique number r = r ( ; g) 2 ( g1 ; 1) such that if g lim nhk > 1 then nkh nkl
lim
l
=r .
The proof will use the following technical lemma. It is stated outside the proof, since it contains more material than needed for the proof and will be referenced in the subsequent analysis as well. De…ne the function J by J(r; ; g) =
Z
1
(x
0
ln x 1 gr1 1 )x dx. g (1 + x gr)2
(11)
Lemma 10 For the function de…ned in (11): > 0, g > 1, there is a unique number r = r ( ; g) 2 ( g1 ; 1) s.t.
(i). For any
J(r ; ; g) = 0.
(ii). J(r; ; g) < 0 for r 2 ( g1 ; r ) and J(r; ; g) > 0 for r 2 (r ; 1). nk
Proof of Proposition 4: Let r = lim nhk and suppose that r < 1 and gr > 1. By Proposition 3 limk!1 Fw pj
k
l
k w
;
=
8 imply that lim
k!1
nkw sk
=
Z
vl + grvh 1+ gr
vl
(
w
w
( j ; g; r). This together with Lemma
(pj ; g; r)) ln (
w
(pj ; g; r)) dp.
Since lim nkh sk = r lim nkl sk , it follows that 1 r
Z
vl + grvh 1+ gr
vl
(
h (pj
; g; r)) ln (
h (pj
; g; r)) dp =
Z
vl + grvh 1+ gr h
(
vl
l
(pj ; g; r)) ln (
Recall that by Lemma 5 we represent a mixed equilibrium strategy = w (nw ) and 1 w = w (nw + 1).
21
(pj ; g; r)) dp. (12)
9 w
l
w
by nw and
w,
where
Recall the function J from (11) above. Lemma 11 If r satis…es equation (12), then J (r; ; g) = 0. This lemma together with Lemma (10) imply that (12) is satis…ed by the unique r named r = r ( ; g) by Lemma (10). Therefore, for any sequence of equilibria nk
nk
nk
l
l
l
such that g lim nhk > 1 and lim nhk < 1, it must be that lim nhk = r ( ; g). nk
To complete the proof, it remains to show that indeed lim nhk < 1. Suppose to l
nk
the contrary that lim nhk = 1. Then Proposition 3 implies that lim Fw j
k
k w
;
l
is
a degenerate distribution with support vw . Lemma 8 implies that lim nkw sk = 0, so that seller type w’s equilibrium payo¤ converges to vw . By Lemma 6, if lim Fh pj if seller type l solicits
nkh
k
;
k h
= 0 then lim Fl pj
bidders, lim El pj
k
; nkh
k
;
k h
= 0. Therefore,
vh . Since lim nkh sk = 0, for
large k, seller type l’s payo¤ with this strategy is near vh which is larger than her equilibrium payo¤ near vl — contradiction. All of the above characterization results deal with the case of min nkl ; nkh = 1.
We show next that if min nkl ; nkh 9 1, then the limit distribution of the winning bid has probability mass 1 on some price C below the ex-ante expected value. Proposition 5 Consider a sequence of games that (
k
;
k)
sk ; P0 such that sk ! 0. Suppose
is a corresponding sequence of equilibria such that min nkl ; nkh 9 1.
Then limk!1 Fw pj
k
;
k w
has probability mass 1 on some C
l vl
+
h vh ,
for
both w = l and w = h. Propositions 3-5 complete the proof of Theorem 2.
6.3
Finite Grid
Theorem 2 and its proof were stated for the case of a continuum set of possible bids P0 . These results also hold for the case in which bids are restricted to …nite grids that become …ner along the sequence. It is important to know this for the subsequent discussion of existence. The required modi…cations are fairly small. Everywhere in Theorem 2 and sk ; P0 , such
Propositions 3-5 where it says “Consider a sequence of games that sk ! 0...” (or “...a sequence of bidding games say instead “Consider a sequence of games lim
sk ;
k
sk ; P
k
0
N k;
k; P 0
, such that
...”) it will k
0 and
= (0; 0)...”(and analogously for bidding games). The proofs go through
almost verbatim. The only changes are in the places where the proofs use a slight undercutting argument. There are two such places in the proof of Proposition 3. In
22
those places we have to make sure that, for a su¢ ciently …ne grid, there exist such undercutting bids that belong to the grid. This is done in the appendix. Each of these instances is followed by a remark that provides the needed argument for the case
7
k
k
> 0,
! 0.
Remarks on the Limit Equilibria
The Role of
gh (x) nkh gl (x) nkl .—
tween the inequalities
Theorem 2 and Proposition 3 expose the relationship be-
gh (x) nkh gl (x) nkl
? 1 and whether the equilibrium is partially re-
vealing. To understand this recall that competition drives bidders’payo¤s to zero when min nkl ; nkh ! 1 for some sequence of solicitation strategies nk . This could
happen either through convergence of bids to the expected value conditional on winning or through a vanishing probability of winning. That is— recalling that E vjx, x(1)
x,
k
, nk denotes the expected value for a bidder whose signal x
is the highest, with x(1) being the highest signal of the competitors— we have that, for a large k and a signal x such that Pr(winner’s signal cant, either
k
(x)
E vjx, x(1)
x,
k
,
nk
x) is signi…k
or there is an atom at
makes the probability of winning very small. Now, when E vjx, x(1)
(x) that
x;
k
, nk
is strictly increasing in x, it must be the former case. To see that an atom at bid p (with positive probability of winning) could not survive, consider a bidder with a signal x0 close to x+ (p) (the sup of x’s at that atom). This bidder would bene…t from slightly overbidding p since p is necessarily su¢ ciently below E vjx0 , x(1)
x+ (p);
k
, nk by virtue of being pro…table for signals at the bot-
tom of the atom’s range (i.e., p when E vjx, x(1) monotonicity of
x; k
k
E vjx (p), x(1)
k
, nk ).10 Conversely,
, nk is strictly decreasing, there is an atom, since the
rules out the former case.
Now, for su¢ ciently large x and k, E vjx, x(1) (x) (decreasing) in x if lim gghl (x)
is increasing i¤
x+ (p);
nkh nkl
x;
, nk is strictly increasing
> 1 (< 1). To see this recall that E vjx, x(1) k
h l
is increasing and notice that
k
gh (x) nkh Gh (x)nh gl (x) nkl Gl (x)nkl
Gh (x)nh
k
1
nk Gl (x) l
1
x;
k
, nk
1 1
is strictly increasing i¤
gh (x) nkh 1 gl (x) nkl 1
>
Gh (x) Gl (x) .
For large k and x near x— which are the only ones with signi…cant probability of winning when k is large— this is equivalent to 10
18.
The argument relies on the bounds on
nk 1 Gh (x) h k
n Gl (x) l
23
1
gh (x) nkh gl (x) nkl
> 1.
given by Lemma 4 and its extension in Lemma
A somewhat di¤erent way to present the above relationship is to note that gh (x) nkh gl (x) nkl
> 1 implies
h l
gh (x) nkh gl (x) nkl
>
h l
, which means that a sampled bidder who
observes the highest possible signal is more optimistic about h than he would be based on the prior alone. This implies that an atom is impossible in this case, since, for large k, the value conditional on winning at such atom would necessarily be smaller or equal to the ex-ante expected value,
l vl
+
h vh .
Hence, a bidder with
high enough signal would bene…t from slightly overbidding and winning with certainty, since in this case the expected value conditional on winning coincides with the expected value conditional on being sampled, which exceeds
l vl
+
h vh
by the
argument above. Seller’s Revenue–Consider a sequence of bidding games nkh nkl
lim
0
= r and a corresponding sequence of bidding equilibria
N k; k
k; P 0
such that
. As we know, when
gr > 1, the distribution of the winning bid converges to the partially revealing limit w
( j ; g 0 ; r) and, for large k and for any x that has a meaningful probability of win-
ning,
k
(x)
E vjx, x(1)
x,
k
,
k
. Letting y(1) denote the highest signal among
all sampled bidders, the winning bid is close to E vjx = y(1) , x(1)
x,
k
,
k
.
Therefore, the law of iterated expectations implies that the seller’s ex-ante expected revenue is approximately h h Ey(1) E vjy(1) , x(1)
y(1) ;
k
,
k
ii
=
l vl
+
h vh .
It follows that in the limit the seller extracts the whole ex-ante surplus. Inspection of h(
g
h
shows that for g 0 > g,
h(
j ; g 0 ; r) stochastically dominates
j ; g; r). Consequently, at w = h the expected revenue of the seller increases in gh (x) gl (x) —
the maximal signal likelihood ratio. Since the ex-ante expected revenue
equals the ex-ante expected value, this implies that the expected revenue of the seller decreases in g at w = l.11 When gr
1, the limit distribution of the winning bid has an atom with proba-
bility mass one on some price C
l vl
+
h vh .
The latter inequality may be strict.
This means that seller’s revenue may be strictly below the ex-ante value
l vl + h vh .
The bidders’interim expected payo¤s are still zero in the limit, since the probability of winning converges to zero. Total Solicitation Costs— The seller’s revenue discussed above is gross of the solicitation costs. Lemma 8 and Theorem 2 provide a complete characterization of the total solicitation costs nkw sk in the limit. Let (
k
;
k)
be a sequence of
11 See Figure 1 for an illustration. The straight black lines of the right panel show the expected revenue for each state as a function of g when = 1.
24
equilibria corresponding to sk ! 0. Either the limit distribution of the winning bid, limk!1 Fw pj
k
;
k w
, is partially revealing and the total solicitation costs converge
to lim
k!1
nkw sk
=
Z
0
vh
lim Fw pj
k
k!1
k w
;
ln
lim Fw pj
k!1
k
;
k w
dp,
or it is a mass point with probability 1 and the total solicitation costs vanish to zero, limk!1 nkw sk = 0. Thus, in the partially revealing limit, the seller does not enjoy the entire surplus extracted from the bidders since the total solicitation cost is positive. Because the total solicitation costs are positive in the partially revealing limit, the seller’s exante expected payo¤ may be higher in the pooling equilibrium than in the partially revealing equilibrium— especially if the atom is close to the ex-ante expected payo¤, meaning the seller can extract almost the entire surplus in the pooling equilibrium. The Large Ordinary Common Value Auction.— As a by-product, Proposition 3 also characterizes the limit distribution of the winning bid for the large ordinary common value auction. This distribution is given by
w
(pj ; g; r = 1), w = l; h. The
characterization shows in particular that the limit distribution is continuous in g. When g is large, then the winnings bids are close to the true values in probability: When g ! 1, inspection of
w
at r = 1 shows that the distribution becomes de-
generate with all its weight on vh and vl , respectively. To the best of our knowledge,
this complete characterization of the equilibrium outcome of the large common value auction is new to the literature.12 The key step towards this characterization is the Poisson approximation from Lemma 6. Comparison of Prices and Revenue.— Let us compare the outcome of the large ordinary common value auction with the outcome in the partially revealing equilibrium with bidder solicitation. Consider a sequence of bidding games such that lim k
nkh nkl
0
N k;
k; P 0
= r and gr > 1, and a corresponding sequence of bidding equilibria
. Thus, the distributions of the winning bid are given by
w
(pj ; g; r), w = l; h.
Let Ew [pj ; g; r] denote the limit of the seller’s expected revenue at w, Ew [pj ; g; r] =
Z
pd
w
(pj ; g; r) .
Of course, the limit revenue for the ordinary CV auction is obtained by plugging 12 We provide a closed form solution to the limit distribution of the winning bid. Kremer and Skrzypacz (2005) establish that the winning bid distribution is not degenerate for all k + 1-price auctions for k goods when g 2 (1; 1). Milgrom (1979) shows that the winning bid converges in probability to the true value if and only if g = 1.
25
1.5
1
1.4
0.9
1.3
0.8
1.2
0.7
1.1
0.6
1
0.5
0.9
0.4
0.8
0.3
0.7
0.2
0.6
0.1
0.5
0
5
10
15
20
25
30
35
40
0
0
5
10
15
20
25
30
35
Figure 1: Left Panel: The ratio of the number of sampled bidders, r (g; 1), as a function of g. Right Panel: Expected revenue as functions of g. Straight black lines are expected revenues with solicitation, Eh [pj1; g; r ] (top) and El [pj1; g; r ] (bottom); dashed grey lines are the expected revenues of the ordinary common value auction, Eh [pj1; g; 1] (top) and El [pj1; g; 1] (bottom). in r = 1, while the limit revenue for the equilibrium with endogenous solicitation is obtained with r = r ( ; g). Since
h (pj
; g; r) is decreasing in r, Eh [pj ; g; 1] >
Eh [pj ; g; r ], if r ( ; g) < 1 (i.e., when there is a solicitation curse), while the inequality is reversed if r ( ; g) > 1 (i.e., when there is a solicitation blessing). That is, when r ( ; g) < 1, there is less information revelation with bidder solicitation than in the ordinary auction; when r ( ; g) > 1, there is more information revelation with bidder solicitation. Figure 1 illustrates the shape of the ratio r (1; g) and compares the expected revenue of each type of seller with and without solicitation. As shown, when g is small, r (1; g) < 1 and when g is large, r (1; g) > 1.13
8
Existence of Equilibrium with Grid
This section studies the existence of non-trivial equilibria.14 Theorem 3 Consider a sequence of games lim
sk ;
k
sk ; P
k
, such that
k
> 0 and
= (0; 0).
13 We conjecture that one can …nd for all a cuto¤ g^ ( ) such that r ( ; g) ? 1 if g ? g^ ( ), but we have not been able to verify this conjecture analytically. 14 There is a trivial equilibrium where the auctioneer invites only one bidder and all summoned bidders bid 0.
26
40
(i). There always exists a sequence of equilibria that converges to the partially revealing outcome of Part (i) of Theorem 2. (ii). Under certain conditions on the distribution of signals, there also exists a sequence of nontrivial equilibria that converges to the pooling outcome of Part (ii) of Theorem 2. Observe that the grid of prices is …nite in every step (
k
> 0). This enables us
to adapt familiar techniques to prove existence. We comment on this point later. Part (ii) means that we are able to demonstrate the existence of pooling equilibria under certain conditions. However, it does not mean that such equilibria exist only under those circumstances. We do not know whether pooling equilibria exist for all speci…cations of the model. The Theorem is proved over the next two subsections. The …rst constructs a sequence of pooling equilibria. The second proves the existence of partially revealing equilibria.
8.1
Existence of Pooling Equilibria
We start with restrictions on the structure of signals that will be used in the construction of a sequence of nontrivial pooling equilibria. 1. Discrete Signals.
The range of the signal values [x; x] is divided into m
subintervals [x; ]; ( ; 2 ];
; (x
; x] .
The density functions gw are step functions that are constant over each of these intervals and jump upwards at the boundaries. 2. Strengthening MLRP. The likelihood ratios satisfy 1 Gl (x and
gh (x gl (x
)
<
) Gl (x ) Gh (x
gh (x) , gl (x) ) )
(13)
gh (x) . gl (x)
(14)
Condition 1 means that, as far as the information is concerned, this is a discrete signal structure with m values. Consequently, the likelihood ratio function as well, so there are at most m di¤erent likelihood
ratios.15
gh (x) gl (x)
is a step
The continuum
is kept only for puri…cation purposes. 15
The important assumption is the …niteness of the set of values that the likelihood ratio takes on. Density functions that are also step functions are consistent with that assumption but are not necessary.
27
Condition 2 can be thought of as a strengthening of the increasing likelihood ratio requirement at the top.16 The …rst part is naturally satis…ed if large, since then Gl (x Gw (x) =
x zw ,
is not too
> 1. It is satis…ed for example when
with zh > zl . Existence of Pooling Equilibrium Consider a sequence of
Proposition 6 games
) is near 1, while
gh (x) gl (x)
sk ; P
k
k
such that
> 0, lim sk ;
k
= (0; 0) and P
P
k
k0
, for
k < k 0 . Suppose that the signals are discrete and satisfy conditions (13) and (14) above. There exist bids b < b <
l vl
+
h vh
and a sequence of equilibria (
k
;
k)
such that min nkl ; nkh ! 1 and k
(x)
(
= b if x > x b if x
;
x
for su¢ ciently large k.
,
Thus, for this sequence of equilibria, the winning bid converges to b almost surely. The requirement that P
P
k
is needed only to assure that b 2 P
k0
k
, for all
su¢ ciently large k’s. However, essentially the same result can be proved without this assumption by looking at a sequence of bk ’s. Before turning to the formal proof, let us discuss some key steps of this construck
tion. First, it is immediate from the form of k
1. Next, given the strategies is approximately 1=nkw [1 pound likelihood ratio likelihood ratio
nkh nkl
h l
, the probability
Gw (x nkh gh (x) nkl gl (x)
h [bj l [bj
k
w [bj
; nkw ] of winning with bid b
)] when k is large. This implies that the comh [bj
k
l [bj
k
;nkh ] ;nkl ]
approaches
and signal likelihood ratio
winning likelihood ratio
that sk ! 0 implies min nkl ; nkh !
gh (x) gl (x)
h l
. That is, the sampling
exactly o¤set in the limit the
k
;nkh ] k k ;nl ]
. Therefore,
h lim E vjx, win at b;
k!1
k
i , nk =
l vl
+
h vh ,
independently of b and b. Thus, bidding b yields positive payo¤ to a bidder with signal x > x
. To verify
that bidding b is indeed optimal for such bidder, we have to consider all possible deviations. The deviation that requires a relatively more subtle argument is overbidding b by a bidder with signal x > x
. The payo¤ of such a bidder at b approaches
0 when k is large and slight overbidding assures a win. It turns out that optimality of the equilibrium sampling strategy assures that E vjx, win at b > b; 16
gh (x gl (x 1 Gh (x 1 Gl (x
In fact, for the existence proof we only need the implication that
This inequality is implied by 1 z is decreasing in z. ln z
gh (x gl (x
) Gl (x ) Gh (x
) )
gh (x) gl (x)
28
since
gh (x) gl (x)
=
k
, nk < b.
) Gl (x ) ln Gl (x ) 1. ) Gh (x ) ln Gh (x ) ") ln Gh (x ") < ln Gl (x ") because ")
To see the essence of this argument, suppose that the bidding strategy is simply k
(x) =
(
b if x > x
;
b if x
,
x
the optimal solicitation strategy is pure, and allow us to ignore integer problems. Ignoring integer constraints, optimal solicitation implies the equality of the marginal bene…t of an additional bidder to its cost in each state, k
(Gl (x
))nl (1
(Gh (x
))nh (1
k
Gl (x
)) b
b
= sk ,
Gh (x
)) b
b
= sk .
Substituting out sk , making a logarithmic transformation, rearranging and then taking limits we get nkh ln Gl (x = k k!1 n ln Gh (x l
) . )
lim
This ratio is smaller than one, so that being solicited is bad news. Moreover, gh (x) ln Gl (x gl (x) ln Gh (x
) )
< 1, which follows from
gh (x) gl (x)
in z. Hence, if solicitation is optimal given
= k
1 Gh (x 1 Gl (x
) )
and
1 z ln z
being decreasing
then
nk gh (x) lim hk < 1. gl (x) k!1 nl
(15)
Note that the limiting ratio of the number of solicited bidders is independent of the choice of b and b. Since a bid b > b wins with certainty, h
E vjx, win at b > b;
k
i
, nk =
vl +
h
1+
h
l
l
gh (x) nkh gl (x) nkl vh gh (x) nkh gl (x) nkl vh
.
Therefore, by (15) for large enough k, h E vjx, win at b > b; Choosing b su¢ ciently close to
l vl
+
k
i , nk <
h vh
l vl
+
h vh .
assures that this upward deviation is
unpro…table. The formal proof deals with the above deviation without the special simplifying assumptions, addresses the other potential deviations and shows how to choose b and b to assure immunity against all the deviations simultaneously. However, the more special argument that is tied to the endogenous sampling is contained in the 29
above discussion. Proof of Proposition 6: A
Auxiliary Game A: Let
s; P jb; b be an auxiliary game in which b < b and
the bidding strategies are constrained to satisfy (x)
(
= b if x > x b if x
x A
A strategy pro…le ( ; ) is an equilibrium of licitation strategy for the seller given
;
(16)
. s; P jb; b if
, and given
is an optimal so-
, the strategy
(x) is a best
response subject to (16). The heart of the proof consists of three lemmas on the equilibrium of the auxiliary game that are proved in the appendix. The …rst establishes existence when
> 0.
In this case, the auxiliary game is a …nite Bayesian game. Lemma 12 If
> 0,
A
s; P jb; b has an equilibrium.
The second lemma collects implications of the optimal sampling Lemma 13 Consider a sequence of auxiliary games 0. Let
k
k
satisfy (16) and
limk!1 nkw = 1; w 2 f`; hg.
2.
limk!1
3.
limk!1
=
sk ; P
k
jb; b such that sk !
be an optimal solicitation strategy given
1.
nkh nkl
A
ln Gl (x ln Gh (x
Gh (x
)nh
k
1
Gl (x
nk ) l
1
) )
k
, then:
< 1. (1 Gl (x (1 Gh (x
)) 1 )) Gh (x
).
The third lemma utilizes the previous lemma to calculate limiting expected values conditional on winning. Lemma 14 Consider a sequence of auxiliary games 0. Let
k
satisfy (16) and
k
A
sk ; P
k
jb; b such that sk !
be an optimal solicitation strategy given
there are numbers v1 , v2 , v3 independent of b, b such that h lim E vjx, win at b; k!1 h lim E vjx , win at b; k!1 h lim E vjx, win at b > b;
k!1
h lim E vjx
k!1
; win at b 2 b; b ; 30
k k k
, ,
k k
,
k
k
k
;
i
i
i
i
=
l vl
+
h vh ,
v1 <
l vl
+
h vh ,
v2 <
l vl
+
h vh ,
v3 <
l vl
+
h vh :
k
. Then
Select any b and b that satisfy max fv1 ; v2 ; v3 g < b < b < A
By Lemma 12, the auxiliary game (
k
;
k ).
k
We show next that (
k)
;
sk ; P
l vl
k
+
h vh . k
jb; b ,
(17)
> 0, has an equilibrium
is an equilibrium of the original game for sk
su¢ ciently small by proving that the constraints (16) do not bind if (17) holds. From Lemma 13, min nkl ; nkh ! 1. Step 1. Bidding
k
= b is optimal if x > x
.
(i) Bidding b > b is unpro…table. By the choice of b > v2 and Lemma (14), h lim E vjx, win at b > b;
k!1
k
k
,
i
< b.
(18)
Thus, there is some K1 such that bidding b > b is strictly unpro…table for all k
K1 .
(ii) Bidding b < b is unpro…table. First, by Lemma 14 and the choice of b, h
k
lim E vjx; win at b;
k!1
k
;
i
=
l vl
+
h vh
> b.
(19)
For any b < b, Lemma 3 implies
lim
k!1
w
bj
k
;
w
bj
k
;
k w k w
lim
1 1 nkw 1 Gw (x
k!1
1
)
(Gw (x
(Gw (x k
))nw
k
))nw = 1.
1
(20)
where the last equality follows from nkw ! 1. By (19), the payo¤ conditional on
winning at b is bounded away from 0. It now follows from (20) that there is some K2 such that for all k
K2 , the payo¤ from bidding b < b is an arbitrarily small fraction
of the payo¤ of bidding b, so that undercutting b is unpro…table for x > x Step 2. Bidding b > b is unpro…table for x
x
.
.
By Lemma 14, the choice of b > max fv1 ; v2 ; v3 g and MLRP, for all x
(i) Bidding b > b is unpro…table. For x h lim E vjx, win at b > b;
k
k!1
;
k
x
i
h lim E vjx, win at b > b;
k
k!1
,
x
k
i
. and all
K1 .
(ii) Bidding b is unpro…table, since for all x h lim E vjx, win at b;
k!1
k
;
k
i
x
h lim E vjx
k!1
31
:
,
Hence, (18) implies that bidding b > b is strictly unpro…table for x k
x
, , win at b;
k
;
k
i
v1 < b.
Thus, there is some K3 , such that bidding b is unpro…table for x k
x
when
K3 .
(iii) Bidding b 2 b; b is unpro…table, since for all x h lim E vjx, win at b 2 b; b ;
k!1
k
;
k
i
h lim E vjx
k!1
Thus, there is some K4 such that for all k for all x
x
.
Let K = max fK1 ;
x
, b 2 b; b ;
(
k)
;
k
;
k
i
v3 < b.
K4 bidding any b 2 b; b is unpro…table
; K4 g. Step 1 and Step 2 imply that the additional con-
straints of the auxiliary game do not bind when k k
,
is an equilibrium of the original game for k
any equilibrium. By construction,
k
K and (17) holds. Thus,
K. For k < K, we can pick
(x) = b for all x > x
and k
K.
Proposition 6 establishes Part (ii) of Theorem 3.
8.2
Existence of Partially Revealing Equilibria
Proposition 7 Consider a sequence of games sk ;
k
sk ; P
! 0. There exists a sequence of equilibria (
k
, sk > 0,
k
k)
;
k
> 0 and
that converges to the
partially revealing outcome of Part (i) of Theorem 2. Proof of Proposition 7:
Auxiliary Game B. We de…ne a second auxiliary game
B
(s; P jnl ; r) as follows:
(i) the two types are represented by separate players who choose nl and nh simultaneously; (ii) the chosen numbers (nl , nh ) determine the actual numbers of solicited bidders as n ^ l = max fnl ; nl g and n ^ h = max fr^ nl ; nh g; (iii) everything else is just as before. An equilibrium ( ; l ;
h)
of the auxiliary game is de…ned as usual:
bidding equilibrium given the distribution of solicited bidders implied by ( l ; given w
of
,
w
is a h );
maximizes seller w’s pro…t evaluated at the corresponding n ^ w ’s, i.e.,
(n) > 0 ) n 2 arg max Ew [pj ; n ^ w (n)]
n ^ w (n)s. Note that the sets of equilibria
n2f1;2;::::g
B
(s; P j1; 0) and
Lemma 15 If
> 0,
(s; P ) are identical since the constraints do not bind. B
(s; P jnl ; r) has an equilibrium.
The proof is analogous to the proof of Lemma 12 and omitted. Next, we show that for su¢ ciently large k, all the equilibria of a certain sequence B
s; P
k
jnkl ; r of auxiliary games are partially revealing.
32
B
Lemma 16 Consider a sequence of auxiliary games sk ;
k k
nkl
! (0; 0),
p1 sk
=
and r 2
( g1 ; r
;
Proof: Given the sequence of equilibria, let r = limk!1 k
Step 1. lim Fw pj
;n ^ kw =
w
k
jnkl ; r
such that
( ; g)). For any sequence of equilib-
k k k k ; k ) of B sk ; P k jnl ; r : nl > nl for large l h lim Fw pj k ; kw = w ( j ; g; r ), with w de…ned by (9).
ria (
s; P
nk
k, lim nhk = r ( ; g) and l
n ^ kh . n ^ kl
(pj ; g; r).
Proof of Step 1: The choice of nkl and r implies min n ^ kl ; n ^ kh ! 1 and g lim n ^ kh =^ nkl > k
1. Hence, Proposition 3 (in its extension to the case of lim Fw pj
k
;n ^ kw
=
w
0) implies that
(pj ; g; r), for all p and w = l; h.
Step 2. For k su¢ ciently large, nkl > nkl and nkh > rnkl Proof of Step 2: By choice of r and by the argument from Lemma 11 (in the proof of Proposition 4),
1 g
< limk!1
n ^ kh n ^ kl
< 1. Hence,
l
is not degenerate. Let mkw
denote an unconstrained optimal solicitation for type w given satis…es lim
k!1
Since nkl sk =
p
mkl sk
=
Z
k
. By Lemma 8, mkl
vh l
0
(pj ; g; r) ln (
l
(pj ; g; r)) dp > 0. nk
sk ! 0, lim mkl sk > 0 implies lim mlk = 0, so that mkl > nkl for
su¢ ciently large k. Thus, nkl = mkl > nkl , as claimed. Suppose to the contrary that mkh optimization implies n ^ kh = r^ nkl . By
l
r^ nkl . Then, the strict concavity of the seller’s 1 g
< r < r ( ; g) and Lemma 10, J(r; ; g) < 0.
From the proof of Lemma 11— especially Equation (55)— J(r; ; g) < 0 implies h lim n ^ kh Eh pj
k
k!1
i
h Eh pj
i
h Eh pj
;n ^ kh + 1
k
;n ^ kh
i
> lim n ^ kh sk . k!1
Hence, for su¢ ciently large k, h Eh pj
k
;n ^ kh + 1
k
i ;n ^ kh > sk .
That is, at n ^ kh sampling an additional bidder is strictly pro…table for type h. Therefore, nkh = mkh > r^ nkl , as claimed. Step 3. lim n ^ kh =^ nkl = lim nkh =nkl = r ( ; g). Proof of Step 3: By Step 2, n ^ kh and n ^ kl are both unconstrained optimal given Therefore, Lemma 11 requires that lim
n ^ kh n ^ kl
k
.
= r ( ; g).
Steps 1 and 3 together establish the lemma. Lemma 16 implies that, for suitably chosen nkl ; r and for su¢ ciently large k, all equilibria of the auxiliary game
B
sk ; P
33
k
jnkl ; r are also equilibria of
sk ; P
k
B
and are close to the partially revealing outcome. Lemma 15 implies that has an equilibrium when (
k
;
k)
for
sk ; P
k
k
sk ; P
k
jnkl ; r
> 0. Therefore, there exists a sequence of equilibria
that converges to the partially revealing outcome of Part (i)
of Theorem 2.
8.3
Existence without Grid
We use the …niteness of the relevant set of feasible bids to prove existence of equilibrium in the auxiliary games. This is the only place where we use the grid. The characterization results in Lemmas 14 and 16 hold also without the grid. Therefore, if we could prove existence of equilibrium for the auxiliary games without the grid, then we could also drop the requirement that
k
> 0 from Propositions 6 and 7.
The di¢ culty for showing existence without a grid is the presence of atoms in equilibrium, which implies that buyers’equilibrium payo¤s can be discontinuous in their bids. In particular, we cannot argue that the limit of a sequence of equilibria for a vanishingly small grid is an equilibrium of the continuum case. The reason is that there may be atoms in the limit that are absent in the sequence. To illustrate the problem, consider a sequence of games with grid P the sequence bidders bid either b or b +
k,
and suppose that along
k
depending on whether their signal is
below or a above some threshold x ^. The pointwise limit strategy as
k
! 0 would
be that all bidders bid the constant b. However, this bidding strategy would imply a strictly lower winning probability for buyers who bid b +
k
along the sequence and
a strictly higher winning probability for buyers bidding b. Thus, the limit strategy may not be an equilibrium of the game with a continuum of bids, even though the elements of the sequence may have been.17 A possible solution to the existence problem without a grid is to change the tiebreaking rule, as suggested by Jackson, Simon, Swinkels, and Zame (2002). Speci…cally, consider the following extension: Buyers submit two numbers, the …rst interpreted as a bid (just as before) and the second number interpreted as eagerness to trade. If there is a unique highest bid, the seller chooses to buy from that bidder. When several bids are tied, the seller may choose among the buyers based on their expressed eagerness. Extending our model in this way solves the existence problem, because the limit of a sequence of equilibrium strategies for a vanishingly small grid corresponds to an equilibrium of the extended game with a continuum of bids. For instance, in the example from the last paragraph, one may specify as the limit strategy of the extended game that buyers bid b for all signals. Buyers with signals above 17
There is no such problem for the seller’s strategy because of the continuity of the seller’s payo¤s in and . If k ; k converge pointwise to ( ; ), and if k is an optimal solicitation strategy given k , then is an optimal solicitation strategy given .
34
the threshold (who bid b +
k
along the sequence) all express the same eagerness,
say eh , and buyers with signals below the threshold (who bid b along the sequence) express a di¤erent eagerness, say el . If multiple bidders are tied at b, then the seller picks …rst among those bidders who express eh , choosing randomly if there are multiple such bidders; if no bidder expressed eh , the seller chooses randomly among bidders expressing el (and, …nally, choosing bidders who expressed anything else last). This limit strategy preserves the winning probabilities, and, hence, the payo¤s in a continuous way. Thus, if the elements of the described sequence of bidding strategies each constitute an equilibrium, so would the limit.
9
Discussion and Conclusion
9.1
Information Aggregation
For a common values auction environment, Wilson (1977) and Milgrom (1979) derived conditions on the informativeness of the signals under which the price aggregates information when the number of bidders becomes large. In their environment, the known number of bidders is exogenous and independent of the state of nature. They show that the winning bid approaches the true value when the number of bidders becomes large if and only if there are unboundedly informative, favorable signals, g
gh (x) gl (x)
= 1. If g < 1, then our results imply that the limit equilib-
rium of the standard common value auction is partially revealing, but it becomes continuously more revealing as g increases.18 In a related sequential search version of that model that di¤ers mainly in that the
seller searches sequentially for buyers, Lauermann and Wolinsky (2012) show that, when the search cost is negligible, nearly perfect information aggregation requires stronger conditions on the informativeness of the most favorable signals: Not only g = 1, but also the likelihood ratio
gh (x) gl (x)
has to increase at a su¢ ciently fast rate
when x approaches x. If g < 1, the equilibrium is complete pooling and both types
trade at a price equal to the ex-ante expected value.19
The present model combines elements from both of these environments. It is an auction in which the buyers compete directly in prices, but the endogenous state dependent solicitation of buyers is reminiscent of the search model. Indeed, in terms of information aggregation, the current model exhibits both patterns of information aggregation. The partially revealing equilibria resemble the equilibria of the standard auction. In particular, when g is large, the aggregation of information 18
See the remarks on large ordinary common value auctions in Section 7. In Lauermann and Wolinsky (2012), the roles of buyers and sellers are reversed, so that the buyer is the informed and the sellers are the uninformed agents. 19
35
is nearly perfect. To see this, recall that r ( ; g) is the solution to J(r; ; g) = 0. Lemma 17 limg!1 gr ( ; g) = 1. R1 Proof : Inspection of J(r; ; g) = 0 x
1 g
1
x gr
1
ln x dx (1+x gr)2
reveals that, if gr is
bounded, then J(r; ; g) < 0 for large g. Therefore, it must be that g ! 1 implies gr ( ; g) ! 1
Now, it can be observed from Equations (9) and (10) that when gr becomes large, the limiting distribution of the winning bid
w
( j ; g; r ) puts almost all its
weight on vw . Thus, large g implies nearly perfect information aggregation in the partially revealing equilibrium. In contrast, the pooling equilibrium of Section 8.1 aggregates no information— the winning bid is at or even strictly below the ex-ante expected value and such equilibria may exist independently of how large is g. In this sense it resembles the equilibrium of the corresponding sequential search model.
9.2
Information Aggregation and E¢ ciency
This paper devotes much attention to the question of information aggregation. The reader may wonder whether the aggregation of information is of importance in a common values environment. The answer is that it may have signi…cant e¢ ciency consequences. First, even in the model in its present form, the total solicitation cost is tied to the degree of information aggregation. It is negligible when the information is nearly perfectly revealed (in the partially revealing equilibrium in the case of large g) and when no information is revealed (in the pooling equilibrium). However, it is not negligible when the information is partially revealed. Second, for simplicity, we have assumed that the seller’s cost is zero and that the value is determined exogenously. Therefore, trade is always e¢ cient and the extent of information revelation has no e¢ ciency consequences in this respect. However, straightforward enrichments of the model will introduce such e¢ ciency consequences. For example, if the seller’s cost is c 2 (vl ; vh ), e¢ ciency requires that trade
takes place only in state h. In this case, failure of information aggregation implies allocative ine¢ ciencies. Alternatively, if the seller has an opportunity to invest in quality improvements prior to trade, failure of information aggregation could imply ine¢ ciently weak investment incentives.
9.3
Unbounded Likelihood Ratio
The boundedness of the signal likelihood ratio, g < 1, is important for our charac-
terization argument. Here, we report two additional results about limiting equilib36
rium outcomes when the signal likelihood ratio is unbounded and discuss directions for future research on this topic.20 First, the characterization result of Theorem 2 extends to unbounded likelihood ratios, that is, we can allow for g = 1 in the statement of the theorem in the following sense: for every sequence of equilibria for vanishing solicitation costs in
which at least two bidders are solicited, either the limit is perfectly revealing or there is a common atom at a price below the ex-ante expected value. This is a natural continuity implication of the theorem for g = 1, since we have already argued that
the theorem implies nearly fully revealing prices when g < 1 but su¢ ciently large in Section 9.1.
Second, for some distribution functions Gw ; w = l; h, with g = 1 and some
sequence sk ! 0, there exists both a sequence of equilibria along which prices converge to the true values (complete revelation) and a sequence of equilibria along which there remains an arbitrarily large atom at the top (pooling). We conclude that the characterization and the existence of limit outcomes with atoms do not depend on the assumption that the likelihood ratio is bounded. The identi…cation of general su¢ cient conditions for atoms to persist in the limit, however, is left for future research. It would likely require the use of di¤erent techniques than the one used in our analysis and is beyond the scope of the current paper.
9.4
Signaling: Observable Number of Bidders
If the number of solicited bidders is observable, it may signal the seller’s information.21 Consider a variation on our model in which the buyers observe the total number of solicited bidders before submitting their bids, while everything else remains unchanged. This variation has two types of pure strategy equilibria – separating and pooling. In the pooling equilibrium, both types of the seller solicit the same number of bidders. Multiple pooling equilibria can be supported by specifying that buyers believe that a seller who solicits an out-of-equilibrium number of bidders must be of the low type, consequently bidding at most vl . Bidding in the pooling equilibria is the same as in the standard common value auction, because the number of bidders is independent of the state. In the separating equilibrium nl = 2 20 We state these results informally without proofs because they would have further increased the length of the paper. Complete results and their proofs are available online in a supplementary note at www.sites.google.com/site/slauerma and is also included in this submission. 21 Our interest is in analyzing a speci…c trading environment in which the seller cannot veri…able communicate the number of solicited sellers. We discuss this variation as an exercise to provide further insight into the mechanism of the model. One may also be interested in the mechanism that is optimal for the seller. This mechanism likely resembles a full-rent extraction mechanism as in Cremer and McLean (1988) because the seller may utilize the correlation of buyers’signals, with the added di¢ culty that the seller has private information; see Severinov (2008).
37
and nh > 2. Bidders bid vl if two bidders are solicited and bid vh if nh bidders are solicited. To ensure incentive compatibility, it must be that vh
nh s = vl
2s.
Thus, in the separating equilibrium, the payo¤ of each type of the seller is vl
2s.
Therefore, in this equilibrium, if s is small, the seller’s revenue is lower than it is when the number of bidders is not observable, as in the model of this paper.22
9.5
Seller’s Commitment
Suppose that the seller can commit ex-ante to a solicitation strategy, with the rest of the game remaining unchanged. If the seller can commit ex-ante to a solicitation strategy, she can extract nearly the entire surplus when s is small: For example, the p auctioneer may commit to solicit 1= s bidders in both states. This would induce an ordinary auction. When s is small, the number of solicited bidders is large. Hence, the expected revenue is approximately equal to the ex-ante expected value while p the total solicitation cost is just s. The resulting pro…t is strictly higher than the seller’s pro…t in the partially revealing equilibrium of the original model without commitment where the total solicitation cost might be signi…cant. Thus, relative to the sellers’preferred number of bidders, in the absence of commitment the auction is “too large” in both states. Since the commitment described above is the same across the states, this argument also holds when the seller does not know the true state.
9.6
Uninformed Seller
Suppose now that the seller is uninformed about the state, again with the rest of the game remaining unchanged. Therefore, nl = nh = n(s). When s is small, n(s) is large and the distributions of the winning bid are close to
w
( j ; g; r = 1). Lemma
8 implies that n(s)s > 0. Thus, despite the fact that the ex-ante expected revenue would be near the ex-ante expected surplus, the uninformed seller’s expected pro…t would be smaller than that.23 It may be surprising that the seller incurs non-vanishing total solicitation costs even when uninformed. Intuitively, the seller is expected to solicit “too many” bidders which induces bidders to act more cautiously to mitigate the winner’s curse. 22
There are additional partially separating equilibria in mixed strategies. For example, the high type may randomly choose either 2 or nh bidders. If 2 bidders are chosen, bidders bid as in the corresponding common value auction in which the priors are adjusted appropriately for the seller’s strategy. 23 It is not obvious whether the expected pro…t of an uninformed seller is lower or higher than its counterpart in the partially revealing equilibrium of the informed seller’s case— the ex-ante expected revenue is equal but the total solicitation cost may di¤er.
38
This in turn induces the seller to indeed solicit a large number and incur signi…cant total cost even when the marginal solicitation cost is small.24 This teaches us that non-vanishing total solicitation costs with an informed seller are not only attributable to the interplay of the separating and pooling incentives of the two types, but that these costs are also due to a commitment problem that arises already when the seller is uninformed.
9.7
Simultaneous Search
Although this paper is couched in the terminology of auctions, it could be equivalently thought of as a simultaneous search model along the lines of Burdett and Judd (1983) with an added element of adverse selection. The seller in our model is the counterpart of the buyer in their model.25 The important di¤erence is in the private information that the sampling agent has in our model. The private information implies both additional substantive insights and some additional analytical challenges. Together, the current paper and Lauermann and Wolinsky (2012) span the two common modes of search, sequential and simultaneous.
9.8
About the Assumptions
The assumption of a binary state is used in the proof of the monotonicity of the bidders’best response. In this sense, the assumption plays an important role. Nevertheless, the assumption buys us more than we need— the best response to any strategy is monotone. Therefore, it may be possible to obtain similar results for monotone equilibria with more than two states. This seems to be an interesting extension that continued work on this subject may address. We have assumed that the seller is fully informed about the state. It is hard to see that anything substantial would change if the seller observed a noisy signal of the state instead. Of course, if the signal were not binary, then the model would be like a multi-state world.
24 As noted above, with commitment the uninformed seller’s pro…t can also be near the total surplus when s is small. 25 The roles of the seller and the buyers in our model can be reversed to make the models exactly parallel.
39
10
Appendix— For Online Publication
The proofs are grouped according to the section in which they appear in the main text.
10.1
Bidding Equilibrium Characterization
Proof of Lemma 1: b0 > b l (bj
; l ). These together with the hypothesis 0 l (b j
Hence, U (b0 jx; ; )
b0 ) < (vl
vl implies (vl
b0 <
; l ) vl
l
(b0 j
; l ) > 0 and
l (bj
; l ) (vl
h (bj
;
0 l (b j
b) and b0
>b
b) :
; l)
vl imply (21)
U (bjx; ; ) requires 0 h (b j
;
h)
b0 >
vh
h ) (vh
b) .
(22)
Rewriting U (b0 jx; ; ) yields l gl
(x) nl l gl (x) nl + h gh (x) nh
l (bj
; l )(vl
h gh (x) nh
b) +
l gl
h (bj
(x) nl
;
h ) (vh
b) . (23)
It follows from
U (b0 jx;
; )
h gh (x) nh l gl
(x) nl l (bj ; l )(vl Since x0 > x and
gh (x) gl (x)
0 h (b j
;
h)
vh
0 l (b j
b)
b0
; l )(vl
h (bj
;
h ) (vh
b)
h ) (vh
b)
b0 ) > 0:
is non-decreasing,
0 h gh (x ) nh 0 l gl (x ) nl l (bj
U (bjx; ; ) and (21) that
; l )(vl
0 h (b j
;
h) 0 l (b j
b)
vh
b0
; l )(vl
h (bj
;
b0 ) > 0.
(24)
which implies U (b0 jx0 ; ; ) 0 l gl (x ) nl = 0 0 l gl (x ) nl + h gh (x ) nh 0 l gl (x ) nl 0 0 l gl (x ) nl + h gh (x ) nh = U (bjx0 ; ; ): If
gh (x0 ) gl (x0 )
>
gh (x) gl (x) ,
0 l (b j l (bj
; l )(vl
b0 ) +
; l )(vl
b) +
0 h gh (x ) nh 0 h (b j ; h ) vh 0) n g (x l l l 0) n g (x h h h h (bj ; h ) (vh 0) n g (x l l l
then (24) and (25) hold with strict inequalities. 40
b0 b) (25)
The last part of the lemma is immediate because Gh and Gl are mutually absolutely continuous, so that Gh (fxj (x)
bg) = 0 , Gl (fxj (x)
bg) = 0.
Proof of Lemma 2: Step 0: If
w
(bj ; n) > 0 for some n
both w and any
2 and w = l or h, then
w
(bj ;
w)
> 0 for
w.
Proof of Step 0:
w
(bj ; n) > 0 for some n and w implies that Gw (fxj (x)
bg) >
0. Since Gh and Gl are mutually absolutely continuous, it follows that Gw0 (fxj (x) w0
0 also for
6= w. Therefore,
(x)
Step 1.
w
(bj ;
w)
> 0 for both w and any
w.
vl for almost all x.
Proof of Step 1: Let b It may not be that
inf fbj
(bj ; n) > 0 for some n and wg. Suppose b < vl . R has an atom at b (i.e., fx: (x)=bg gw (x)dx > 0) since by a w
standard Bertrand argument U (b + "jx; ; ) > U (bjx; ; ) for su¢ ciently small b). Therefore, there exists a sequence of xk such that
" 2 (0; vl w
U(
xk j ;
xk
w [bj
;
jxk ;
w]
! 0 (owing to
w
w (0)
=
w (1)
xk ! b and
= 0). Hence, equilibrium payo¤s
; ) ! 0. However, by the de…nition of b and monotonicity of
w,
is strictly positive for all b 2 (b; vl ). Thus, for all b 2 (b; vl ), the payo¤
U (bjx; ; ) > 0. This contradicts the optimality of
xk for su¢ ciently large k,
a standard Bertrand argument. Thus, b
w
implies that Gw (fxj (x) Step 2.
vl . Finally,
(bj ; n) = 0 for all b < vl
vl g) = 1, proving the step.
(x) < vh for all x.
Proof of Step 2: It clearly cannot be that Gw (fxj (x) > vh g) = 1 for any w,
since this would imply that bidders have strictly negative payo¤s in expectations. Suppose that implies [
(x0 )j
(x0 )
vh for some x0 . From Gl (fxj (x) > vh g) < 1,
; l ] > 0 and U (
(x0 )jx0 ;
(x0 )
vh
; ) < 0, a contradiction to optimality of
(x0 ). Step 3.
w
( (x)j ;
w)
> 0 for almost all x for w 2 fl; hg.
Proof of Step 3: Fix w 2 fl; hg. Let X = fxj
w
( (x)j ;
w)
= 0g. The probabil-
n n w (n)[Gw (X)] . Since in that n Pr[fWinning bidder n w (n)[Gw (X)]
ity that in state w all bidders are from that set is event some bidder has to win, we have R has signal x 2 Xgjw] nw x2X w ( (x)j ; Step 4. For any x0 > x, U ( (x0 ) jx0 ; ; ) strict if and only if
gh (x0 ) gl (x0 )
>
gh (x) gl (x) .
Thus,
gh (x0 ) gl (x0 )
is strictly positive.
w ) g (x) dx
>
= 0. Hence, Gw (X) = 0.
U ( (x) jx; ; ). The inequality is gh (x) gl (x)
implies that U ( (x0 ) jx0 ; ; )
Proof of Step 4: From (2) it follows (after dividing the numerator and denominator
41
bg) >
by gl (x)) that l nl l (bj
U (bjx; ; ) =
; l )(vl
b) + l nl
+
gh (x) h gl (x) nh h (bj gh (x) h gl (x) nh
;
h ) (vh
b) :
(26)
Therefore, for any x0 > x, U ( (x0 )jx0 ; ; )
U ( (x)jx0 ; ; )
U ( (x)jx; ; )
0;
(27)
where the …rst and last inequalities are equilibrium conditions; the second inequality owes to
gh (x0 ) gl (x0 )
gh (x) gl (x)
and
h(
(x)j ;
h ) (vh
(x))
0
l(
(x)j ; l )(vl
(x)),
which follows from Steps 1 and 2. Suppose
gh (x0 ) gl (x0 )
>
gh (x) gl (x) .
Now, either
0 and it follows from (26) and strict, or
w
( (x)j ;
w)
w ( (x)j ; w ) > 0, in which case h ( (x)j ; h ) (vh gh gh (x) gl (x0 ) > gl (x) that the second inequality in (27) is (x0 )
= 0 and hence U ( (x)jx; ; ) = 0. In the latter case,
by Step 3, there is some y 2 (x; x0 ) such that y such that h(
(y)j ;
fact that
gh (x0 ) gl (x0 )
>
h ) (vh
gh (y) gl (y)
(y))
(x) (recall that gghl (x) (x0 ) > 0. Since gghl (x 0)
w ) > 0. We can choose gh (x) limx!x gl (x) ). By Steps 1 and 2, gh (y) gl (y) , it follows from (26) and the
w
=
>
( (y)j ;
is a bidding equilibrium that
U ( (x0 )jx0 ; ; ) Conversely,
gh (x0 ) gl (x0 )
=
U ( (y)jx0 ; ; ) > U ( (y)jy; ; ) gh (x) gl (x)
0 = U ( (x)jx; ; ):
implies
U ( (x0 )jx0 ; ; ) = U ( (x0 )jx; ; )
U ( (x)jx; ; ) = U ( (x)jx0 ; ; )
U ( (x0 )jx0 ; ; );
where the inequalities are equilibrium conditions while the equalities owe to the fact that x and x0 contain the same information. Therefore, U ( (x0 )jx0 ; ; ) = U ( (x)jx; ; ). Step 5. The strict positivity of U ( (x)jx; ; ) implies immediately that 0 for any x for which
gh (x) gl (x)
>
gh (x) gl (x) .
w
( (x)j ;
(Step 3 established this only for almost all x).
This proves Part (1) of the Lemma. Step 6. If P = [0; vh ], then to just
(x) > vl for any x for which
gh (x) gl (x)
>
gh (x) gl (x)
(as opposed
established in Step 1).
Proof of Step 6: The same standard Bertrand argument used in the proof of Step 1 implies that there cannot be mass point at vl . Therefore, U (vl jx; ; ) = 0. Since Step 4 implies that U ( (x)jx; ; ) > 0 for all x >x, it must be that all x >x. This completes the proof of the lemma. 42
(x) > vl for
w)
>
(x)) >
Proof of Proposition 1: Part (1): Proved by Lemma 2. gh (x0 ) gl (x0 )
Part (2): Suppose that Since w[
>
gh (x) gl (x)
for some x; x0 2 (x; x], but
is a bidding equilibrium, U ( (x)jx; ; ) (x0 ) j
;
w]
(x0 )
> 0 and
U(
(x0 )jx;
(x0 ) <
(x).
; ). By Lemma 2,
vl . Therefore, by Lemma 1, U ( (x)jx0 ; ; ) >
U ( (x0 )jx0 ; ; ), contradicting the optimality of (x0 ) for x0 . Thus, the supposition (x0 ) < (x) is false. Hence, Next, suppose that
(x0 )
(x0 )
gh gl (x0 )
(x).
gh (x) gl (x)
=
for some x; x0 2 (x; x], but
Then there is some interval containing x and x0 over which
gh (x) gl (x)
(x0 ) <
(x).
is constant, say, C.
(x00 ) (x) 000 000 (x ) whenever x < x+ < x . De…ne e 1 (x) by
Let [x ; x+ ] be the closure of this interval. By the above argument, whenever x00 < x < x and
(x)
e (x) = inf fb : Gh (x) 1
Gh (ftj (t)
bg)g if x 2 [x ; x+ ]
Thus, on [x ; x+ ] the signals are essentially “reordered” to make e 1 (x) monotone. Outside [x ; x+ ], e 1 (x) coincides with (x). Note that ~ (x0 ) ~ (x) ~ (x00 ) for all x0 < x and x+ < x00 . With this de…nition, Gh (fxj e 1 (x)
bg) = Gh (fxj (x)
bg) ,
for all b. That is, the distribution of bids induced by e 1 is equal to the distribution of bids induced by in state h. It is also the same in state l because e 1 = outside
[x ; x+ ] and because the distributions Gl and Gh conditional on x 2 (x ; x+ ) are identical (owing to the constant
gh (x) gl (x) ).
The equality of the distributions of bids under e 1 and implies that, for any = [x ; x+ ] this follows immediately from x 2 = fx ; x+ g, ~ 1 (x) is optimal: for x 2 e (x) = (x); for x 2 (x ; x+ ) this follows from ~ (x) = (y) where y is some value 1
1
of the signal such that
gh (y) gl (y)
=
gh (x) gl (x) .
For x 2 fx ; x+ g, note that we can represent
the distribution of signals by an equivalent pair of densities that is equal to the original densities almost everywhere, so that the resulting equilibrium still corresponds to the same distributional strategy. Here, ~ can be rationalized at fx ; x+ g by chang1
ing the densities at the points x 2 fx ; x+ g. At x , if ~ 1 (x ) = ~ 1 (x + ") for some " (an atom), ~ 1 (x ) is rationalized by setting gw (x ) = lim"!0 gw (x + "). Otherwise, ~ (x ) is rationalized by setting gw (x ) = lim"!0 gw (x "). Simi1
larly for x+ . It follows that e 1 is monotone on [x ; x+ ] and that it is equivalent to .
Repeating this construction for all intervals over which
gh (x) gl (x)
is constant, we get
a sequence of bidding strategies (constructing the sequence by starting with the
43
(x) longest interval of signals on which gghl (x) is constant). Let ~ be the pointwise limit of this sequence on (x; x] and let ~ (x) = lim"!0 (x + "). Then, ~ is an equivalent
bidding equilibrium that is monotone on [x; x], as claimed.
10.2
Bidding Equilibrium: Atoms
Proof of Lemma 3: Since and Gw (fxj (x) > bg) = 1
=
=
= =
= = = =
w (bj n X1
1 n
1 i
(n (n
i=0 n X1 i=0
Gw (x+ (b)). We rewrite the winning probability at b:
; n)
n
i=0 n X1
is non-decreasing, Gw (fxj (x) < bg) = Gw (x (b))
(n
n 1X n (n k=1 Pn
1 Gw (x (b))n i+1
i 1
1 1)! Gw (x (b))n 1 i)!i! i + 1
[Gw (x+ (b)) i 1
n! 1 Gw (x (b))n 1 i)!i! i + 1 n! k)! (k
n! k=1 (n k)!k! Gw
1 Gw (x (b))n 1)! k (x (b))n
k
n [Gw (x+ (b))
Pn
n! k=0 (n k)!k! Gw
n k
(x (b))
Gw (x (b))]i
[Gw (x+ (b))
i 1
k
Gw (x (b))]i
Gw (x (b))]i
[Gw (x+ (b))
Gw (x (b))]k
[Gw (x+ (b))
1
Gw (x (b))]k
[Gw (x+ (b)) Gw (x (b))]
Gw (x (b))]k
[Gw (x+ (b))
Gw (x (b))n
n [Gw (x+ (b)) Gw (x (b))] (Gw (x (b)) + Gw (x+ (b)) Gw (x (b)))n Gw (x (b))n n [Gw (x+ (b)) Gw (x (b))] n Gw (x+ (b)) Gw (x (b))n . n [Gw (x+ (b)) Gw (x (b))]
The critical step is to apply the binomial theorem,
Pn
n! n k bk k=0 (n k)!k! a
= (a + b)n .
Proof of Lemma 4: Part (i) of Lemma 18 below appears in the text as Lemma 4. The proof of Lemma 18 follows its statement. The second and third parts of this lemma are used in later proofs. Lemma 18 (i) If nh
nl , then
Gh (x )nh Gl (x )nl
1 1
Gh (x)nh Gl (x)nl
is increasing and
h [bj
; nh ] [bj ; nl ] l
44
Gh (x+ )nh Gl (x+ )nl
1 1
,
with strict inequalities unless nl = nh and (ii) If
Gh (x)nh Gl (x)nl
0 @
1
1
nh 1
Gh (x ) Gl (x )nl
with strict inequalities if (iii) If 0 @
Gh (x)nh Gl (x)nl
1 1
1
gh (x+ ) gl (x+ ) A gh (x ) gl (x )
=
gh (x) gl (x) .
is increasing on [x ; x+ ], then
1
gh (x ) gl (x ) A gh (x+ ) gl (x+ )
gh (x+ ) gl (x+ )
Gh (x+ ) Gl (x+ )nl
h [bj
; nh ] l [bj ; nl ]
1
Gh (x)nh Gl (x)nl
nh 1
1 1
is strictly increasing or
1
0 @
gh (x+ ) gl (x+ )
1
gh (x+ ) gl (x+ ) A , gh (x ) gl (x )
>
gh (x ) gl (x ) .
is decreasing on [x ; x+ ], then nh 1
Gh (x ) Gl (x )nl
with strict inequalities if
h [bj
; nh ] l [bj ; nl ]
1
Gh (x)nh Gl (x)nl
nh 1
1 1
Gh (x+ ) Gl (x+ )nl
is strictly decreasing or
1
0 @
gh (x+ ) gl (x+ )
1
gh (x ) gl (x ) A , gh (x+ ) gl (x+ )
>
gh (x ) gl (x ) .
Proof of Lemma 18: Note that d dx
Gh (x)nh Gl (x)nl
=
nl nh nh Gghh(x) (x) Gh (x) Gl (x)
gh (x) Gh (x)
nl where the inequality is from nh gh (x) gl (x) Gh (x) Gl (x) Gh (x) Gl (x) = 0, which
MLRP implies and
gh (x) gl (x)
nl nh nl Ggll(x) (x) Gh (x) Gl (x)
(Gl (x)nl )2 gl (x) Gh (x)nh , Gl (x) Gl (x)nl
nl . Hence,
Gh (x)nh Gl (x)nl
is weakly increasing since the
0. In fact, it is strictly increasing unless both nh = nl requires
gh (x) gl (x)
=
gh (x) gl (x) .
Rewriting,
; nh ] nl Gh (x )nh = nh Gl (x )nl l [bj ; nl ]
h [bj
+ Gh (x )nh 1 + Gl (x )nl
1
45
Gh (x+ ) + ::: + Gh (x+ )nh 1 . 2 Gl (x+ ) + ::: + Gl (x+ )nl 1 2
Divide through by
h [bj
; nh ] [bj ; nl ] l
Gh (x+ )nh Gl (x+ )nl
1
to obtain
1
nh 1
=
Gh (x+ ) Gl (x+ )nl
1
!
1+
Gh (x ) Gh (x+ )
Gh (x ) Gh (x+ )
+
= 1+ 1+
Gl (x ) Gl (x+ )
Gh (x ) Gh (x+ ) Gl (x ) Gl (x+ )
1+
The …rst inequality follows from the fact that, since
+
Gl (x ) Gl (x+ )
+
Gh (x ) Gh (x+ )
+
Gl (x ) Gl (x+ )
Gh (x ) Gh (x+ )
2
2
2
2
+ ::: +
Gh (x ) Gh (x+ )
+ ::: +
Gl (x ) Gl (x+ )
+ ::: +
Gh (x ) Gh (x+ )
+ ::: +
Gl (x ) Gl (x+ )
nh 1
nl 1
nl 1
nl 1
=nh =nl =nl 1 =nl
< 1, the numerator after
the inequality is an average of the largest nl terms out of the nh terms that are averaged on the numerator before the inequality sign. The second inequality follows from
Gh (x ) Gh (x+ )
Gl (x ) Gl (x+ )
which in turn follows from
Gh (x ) Gl (x )
Gh (x+ ) Gl (x+ )
which holds by
MLRP. Analogously, dividing through by
h [bj
; nh ] [bj ; nl ] l
nh 1
=
Gh (x ) Gl (x )nl
1
!
Gh (x )nh Gl (x )nl
1+
1 1
,
Gh (x+ ) Gh (x )
Gh (x+ ) Gh (x )
+
= 1+ 1+ 1+
Gl (x+ ) Gl (x )
Gh (x+ ) Gh (x ) Gl (x+ ) Gl (x )
+
Gl (x+ ) Gl (x )
+
Gh (x+ ) Gh (x )
+
Gl (x+ ) Gl (x )
where the inequalities are explained by noting that
Gh (x+ ) Gh (x )
2
2
2
2
+ ::: +
Gh (x+ ) Gh (x )
+ ::: +
Gl (x+ ) Gl (x )
+ ::: +
Gh (x+ ) Gh (x )
+ ::: +
Gl (x+ ) Gl (x )
nh 1
nl 1
nl 1
nl 1
> 1 and reversing the
previous arguments. In both cases the two inequalities hold as equalities i¤ nh = nl and Gl (x ) Gl (x+ ) . However, the gh (x) gh (x) gl (x) = gl (x) for all x
last equality is equivalent to < x+ .
(ii) & (iii)
46
Gh (x ) Gl (x )
=
Gh (x+ ) Gl (x+ )
Gh (x ) Gh (x+ )
=
which holds i¤
=nh =nl =nl 1 =nl
Rx x
nl Gl (x+ ) nh Gh (x+ )
Gl (x ) Gh (x+ )nh Gh (x ) Gl (x+ )nl Rx x
+
=
Gh (x )nh = Gl (x )nl G (x)nh
+
Rx x
Gh (x)nh 1 gh (x)dx Rx + g (x)dx h x
+
1 g (x) h dx gl (x)
Gl (x)nl 1 gl (x) h n 1 Gl (x) l Rx + g (x) gh (x) dx l x gl (x) Rx + G (x)nl 1 g (x)dx l l x Rx + g (x)dx l x
Gl (x)nl 1 gl (x)dx Rx + g (x)dx l x
Now 0 @
1
gh (x ) gl (x ) A gh (x+ ) gl (x+ )
Rx x
+
G (x)nh
Gl (x)nl 1 gl (x) h n Gl (x) l Rx + g (x)dx l x Rx + G (x)nl 1 g (x)dx l l x Rx + g (x)dx l x
Rx
1 1
dx
x
0 @
The results for decreasing
Gh (x)nh Gl (x)nl
1 1
monotone . Assume that
G (x)nh
1 g (x) h dx gl (x)
Gl (x)nl 1 gl (x) h n 1 Gl (x) l Rx + g (x) gh (x) dx l x gl (x) Rx + G (x)nl 1 g (x)dx l l x Rx + g (x)dx l x
1
gh (x+ ) gl (x+ ) A gh (x ) gl (x )
Rx x
+
G (x)nh
Gl (x)nl 1 gl (x) h n Gl (x) l Rx + g (x)dx l x Rx + G (x)nl 1 g (x)dx l l x Rx + g (x)dx l x
1 1
dx
follow immediately.
Proof of Proposition 2: Suppose that nh rium given (nl ; nh ). Since nh
+
nl
nl
2 and
is a bidding equilib-
2, by Proposition 1, we restrict attention to
(x) = lim"!0 (x + "). The following claim implies the
proposition. A remark after the proof demonstrates that atoms must occur under the stated conditions. Claim 4 If for some p, x+ (p) > x (p) (an atom), then (i)
gh (x+ (p)) gl (x+ (p))
=
gh (x) gl (x) ,
(ii)
nh = nl , (iii) x (p) = x, and (iv) U (pjx; ; n) = 0. Proof of Claim: We prove the claim in a sequence of steps. First, by Lemma 2, vl
p < vh . Rewrite (1) slightly to get the expected payo¤ U (pjx; ; n) of a bidder with signal
x who bids at this atom, U (pjx; ; n) l gl (x) nl l (pj ; nl ) = (vl l gl (x) nl + h gh (x) nh
(28) p) +
47
h gh (x) nh l gl
(x) nl
h (pj
; nh ) (vh l (pj ; nl )
p) .
gh (x+ ) gh (x) gl (x+ ) = gl (x) . gh (x) gh (x) gl (x) = gl (x) for all
Gh (x+ )n Gl (x+ )n
;nh ) l (bj ;nl )
h (bj
Step 1: Recall from Lemma 18-(i) that
1 1
, with equality i¤
nl = nh and Step 2:
x
x+ , nh = nl , and U (pjx; ; n) = 0 for x 2 (x ; x+ ).
Proof of Step 2: The expected payo¤ of a bidder with signal x 2 (x ; x+ ) who bids “just above” p is approximately, lim U (p + "jx; ; n)
(29)
"!0 ">0
" nl 1 g (x) n G (x ) + l l l l [vl g (x) n + g l l l h h (x) nh
=
By optimality, U (pjx; ; n) l (pj
; nl ).
gh (x) nh Gh (x+ )nh p] + h nl l gl (x) nl Gl (x+ )
1 1
(vh
#
p) .
0 for all x 2 (x ; x+ ) and, by Lemma 3, Gl (x+ )nl
1
>
Therefore, it follows from (28), (29), the MLRP, and Step 1 that
lim"!0 U (p+"jx; ; n)
U (pjx; ; n) for all x. The inequality is strict if U (pjx; ; n) >
">0
0 or nh > nl or
gh (x+ ) gl (x+ )
>
gh (x) gl (x)
(or any combination). Therefore, optimality implies
the step. Step 3: x = x. Proof of Step 3: Suppose not and let x0 2 [x; x] be such that x < x0 < x (x0 )
< p. This and the monotonicity of
imply
w
(
fl; hg.
(x0 ) j
and
; nw ) > 0 for w 2
Observe that this and U (pjx; ; n) = 0 for all x 2 (x ; x+ ) implies
Since by Step 2, Gh (x+ )n Gl (x+ )n
1 1
h gh (x) nh
h (pj
; nh ) (vh l (pj ; nl )
(vl
p) +
gh (x) gl (x)
is constant on [x; x+ ), we have
l gl
(x) nl
; see Lemma 18. This together with (30) and
p) = 0.
(30)
(x0 )j ;n) h (pj ;n) = (x0 )j ;n) = l (pj ;n) (x0 ) < p imply that, for
h( l(
x 2 (x ; x+ ), (vl
0 h gh (x ) nh 0 l gl (x ) nl 0 h gh (x ) nh h ( 0 l( l gl (x ) nl
x0 ) +
> (vl
p) +
= (vl
p) +
However, this together with
h gh (x) nh l gl l
(x) nl
(x0 ) j ; n) vh 0 l ( (x ) j ; n) (x0 ) j ; n) (vh p) (x0 ) j ; n) h (pj ; nh ) (vh p) = 0. l (pj ; nl ) h(
x0
( (x0 ) j ; nl ) > 0 inferred above implies U ( (x0 ) jx0 ; ; n) >
0 contradicting Step 2. Therefore, x = x. Step 2 and Step 3 imply the Claim.
48
Remark (Existence of Equilibrium with Atoms): Suppose that nh = nl 2 and suppose that there is no atom at p0 =
(x0 ) for some x0 > x with
(x00 ) for some x00 2 (x; x0 ).
Let p =
Note that U ( (x) jx; ; n) = 0 in every bidding equilibrium. If
an atom at x, this is immediate from monotonicity of (x00 ) jx00 ;
at x, this follows from the claim. Thus, U (
and if
gh (x0 ) gl (x0 )
=
gh (x) gl (x) .
does not have
does have an atom
; n) = 0.
Now, 0 = (vl
p) +
h gh (x) nh
h (pj
(x) nl gh (x0 ) nh p) + h 0 l gl (x ) nl gh (x0 ) nh p0 ) + h 0 l gl (x ) nl
= (vl > (vl
l gl
; nh ) (vh p) l (pj ; nl ) 0 h (p j ; nh ) (vh p) 0 l (p j ; nl ) 0 h (p j ; nh ) vh p 0 , 0 l (p j ; nl )
where the second equality is from Lemma 18-(i) and the inequality from p0 > p. Thus, 0 > U (p0 jx0 ; ; n), a contradiction. Thus, it must be
10.3
(x0 ) =
(x), as claimed.
Large Numbers: Basic Results nkl
Proof of Lemma 6: Suppose lim Gl xk
= q 2 (0; 1). If q > 0, then lim xk = x
(for lim xk < x implies q = 0). The claim is immediate if xk = x for all k large nkl
enough since then q = 1 = Gl xk xk
< x for all k large enough but
lim xk
lim (1
lim
k!1
Gh x
k
Gl x k nkh
=
=
=
nkl = lim
k!1
lim
k!1
"
= e
nkh
for k large. So, suppose
= x. From lim Gl xk
nkl
= q,
Gl xk k nk nl ) l = q. nkl
1
k!1
This implies lim 1
= Gh x k
ln q. Therefore,
1
1
1
lim
k!1
ln q g lim
1
k
Gh x
Gl x k
1 + (ln q) g
1 nkl
nk h nk l
nk h nk l
=q
49
g lim
.
nkl
nkl
nkh
1 nkl 1 1
# nkl lim
xk
Gh 1 k Gl (x ) nkl nk h nk l
!nk nkhk l n
l
nkl
The lemma now follows analogously if lim Gl xk nkw
If lim xk < x, then lim Gw xk
= 0 for w 2 fl; hg.
= q 2 f0; 1g and lim xk = x.
Proof of Lemma 7: By contradiction. Suppose that there is some " > 0 and a sequence
k
= (nkl ,nkh ) such that min nkh ; nkl ! 1 and a corresponding sequences k
of bidding equilibria
and signals xk such that U ( (xk )jxk ; ;
By Lemma 2, U ( (x)jx; ;
k)
increases in x, it follows that U (
k)
k
> ", for all k. k
(x)jx;
k)
;
>"
for all k. Rewriting the expected payo¤s (analogously to (23)), observe that
U(
k
(x)jx;
k
k
;
By assumption, limx!x
)
gh (x) gl (x)
U(
=
k
(x)jx;
gh (x) gl (x) .
k
;
k
0
vl ) @ 1
) < (vh
Therefore, there exists x0 < x such that the
right side above is smaller than 2" . Hence, for all k, we have U(
k
k
(x0 )jx0 ;
k)
;
1
gh (x) gl (x) A . gh (x) gl (x)
" 2
< U(
k
(x)jx0 ;
k
where the last inequality follows from the optimality of
However, x0 < x implies that 1
k)
; k
(x0 ).
Gw (x0 ) > 0 and so the number of bidders with
signals above x0 goes to in…nity as k ! 1. Hence, we conclude that there are an unboundedly large number of bidders who each expect a payo¤ of 2" . However, the total available surplus for the bidders is bounded by vh ; a contradiction. Proof of Lemma 8: We prove the lemma for the case of a pure solicitation strategy. The proof extents directly to mixed strategies. Let X k (p) =
1
k
n ([0; p]) = xj
k
o p .
(x)
Case 1: nkw is bounded as sk ! 0. From sk ! 0, it must be that limk!1 Ew pj
Ew pj
k
k
; nkw + 1
; nkw = 0, that is, lim
Z
k!1 0
vh
Gw X k (p)
nkw
Gw X k (p)
1
dp = 0.
Thus, the integrand converges to zero almost everywhere. From the monotonicity of Gw , this requires that for some C 2 [0; vh ], k
lim Gw X (p) =
k!1
Hence, Fw pj
k
; nkw = Gw X k (p)
nkw
(
1
if p > C;
0
if p < C.
implies that
R vh 0
converges to zero, as claimed (recall lima!0 a ln a = 0).
50
Fw pj
k
; nkw
ln Fw pj
k
; nkw
dp
Case 2: nkw ! 1 as sk ! 0. From optimality of nkw nkw
Z
vh
0
nkw sk From lim Fw pj
nkw 1
Gw X k (p) 1 Gw X k (p) dp Z vh nkw k 1 Gw X k (p) Gw X k (p) nw
k
nkw
k!1
1
If lim Gw X k (p)
nkw
lemma holds.
10.4
(p)
nkw
it follows that
=
k
lim Fw pj
k!1
ln
Gw (X k (p)))nkw nkw
(1
2 (0; 1), then lim 1
; nkw
lim Fw pj
k!1
nkw
=
implies
lim
k!1
Gw X k (p)
1
Xk
In particular, if lim Gw lim(1 Gw (X k (p)))nkw
nkw
; nkw = lim Gw X k (p)
lim nkw Gw X k (p)
e
dp.
0
Gw X k (p)
nkw = ln
lim Fw pj
k!1
k
; nkw
.
2 f0; 1g, the claim follows from lima!0 a ln a = 0. Thus, the
Characterization: Partially Revealing Equilibrium
Proof of Proposition 3: Consider a sequence of bidding games such that min nkl ; nkh ! 1, limk!1
nkh nkl
= r and gr > 1. Let
k
0
N k ; nk ; P0
be a corresponding
sequence of bidding equilibria. By Proposition 1, we may assume that each bidding strategy
k
is monotone.
We prove the proposition for a sequence of pure solicitation strategies nk to shorten the algebra. However, the fact that min nkl ; nkh
! 1 implies that the
proof immediately extents to sequence of mixed solicitation strategies
k
that have
support on at most two adjacent integers. Part (i) is proved in three steps presented by the following three lemmas. The …rst establishes that there are no atoms in the equilibrium distribution of the winning bid in the limit. Recall that x+ (p) = sup fxj (x)
pg and that
w [pj
; nw ] is the
probability of winning.
Lemma 19 (No Atoms in Limit) Given rg > 1, for any sequence fxk g, and w 2 fl; hg,
lim
k!1
w[
k
xk j
k
k
; nkw ] = lim Gw (xk )nw k!1
51
1
k
; nkw
.
The proof of this lemma comes immediately after the end of the present proof. By de…nition of x+ (p), Fw (pj k
It follows that lim Fh pj
k
k
; nkw ) = Gw (xk+ (p))nw .
(31)
; nkh 2 (0; 1) implies that xk+ (p) ! x for otherwise the
RHS would go to 0. Recall that x(1) denotes the highest signal among the competitors of a …xed bidder. The next lemma uses the zero-pro…t plus IR condition (7) to obtain an equation holding for prices in the support of lim Fw pj Lemma 20 For every price for which lim Fw pj p = lim E[vjxk+ (p) , x(1) k!1
k
k
; nkw .
; nkw 2 (0; 1),
xk+ (p) ,
k
, nk ]:
Proof of Lemma 20: By the zero-pro…t plus IR condition (7), by lim Fh pj (0; 1) and by the de…nition of xk+ (p) lim E[vjxk+ (p) , x(1)
xk+ (p) ,
k!1
It may not be that limk!1 limk!1
k
k
k
, nk ] = lim
k
k!1
xk+ (p)
p
k
; nkh 2
(32)
xk+ (p) > p, since by Lemma 19 there is no atom at
xk+ (p) and hence far enough in the sequence a bidder with xk+ (p) has a
pro…table downward deviation from
k
xk+ (p) . Therefore, limk!1
k
xk+ (p) = p
and the result follows from (32). Finally, the last lemma combines the insights of the two previous lemmas to solve for lim Fw pj
k
; nkw for any p in its support.
Lemma 21 lim Fw pj
k
; nkw =
w
(pj ; g; r) for w 2 fl; hg and p 2 [0; vh ].
Proof of Lemma 21: Consider p such that lim Fw pj of Lemma 20 is equivalent to h l
gh xk+ (p) nkh limk!1 k k!1 gl xk + (p) nl limk!1 lim
k
; nkw 2 (0; 1). The result
k
; nkh ] p vl = k k vh p l [pj ; nl ]
h [pj
Using Lemma 19 and xk+ (p) ! x, it can be rewritten as k
h l
nk Gh xk+ (p )nh gh (x) p vl lim hk , k = n k gl (x) k!1 nl Gl x+ (p ) l vh p
The RHS is positive and …nite since lim Fw pj
52
k
(33)
; nkw 2 (0; 1) implies p 2 (vl ; vh ).
By Lemma 6 and (4), k
lim Fh pj
h
; nkl
= lim Fl pj
k
; nkh
i gh (x) lim gl (x)
nk h nk l
.
(34)
k
Using (31) and (34) to substitute for lim Gw xk+ (p )nw in (33) we get lim Fl pj
k!1 nk
where r = lim nhk , lim Fl pj
k
=
h l
l
; nkl =
l
k
; nkl
and g =
= gh (x) gl (x) .
.
Hence, if lim Fw pj
(pj ; g; r) and using again (34) lim Fh pj
Finally, the monotonicity of Fw and the de…nition of f0; 1g ,
1 gr 1
1 p vl gr vh p
w
k
k
; nkw > 0, then
; nkh =
h (pj
imply that lim Fw pj
(pj ; g; r) 2 f0; 1g.
w
Part (ii). Assume rg
1. Suppose to the contrary that limk!1 Fw pj
k
; g; r). k
; nkw 2
; nkw is
strictly increasing over some interval (p0 ; p00 ). Then the above analysis applies to p 2 (p0 ; p00 ): First, because there is no atom at p, the conclusion of Lemma 19 holds.
Second, given this conclusion, the proofs of Lemmas 20 and 21 also apply for rg k
Thus, if rg < 1, then by Lemma 21, lim Fw pj then by (34) lim Fh pj
k
; nkl
= lim Fl pj
k
; nkh
; nkl
=
w
(pj ; g; r); if rg = 1,
and hence by (33) p =
However, both of these observations lead to contradictions:
1.
w
l vl + h vh .
(pj ; g; r) is strictly
decreasing when rg < 1, and the constant p when rg = 1 clearly cannot hold for all p 2 (p0 ; p00 ). It follows that limk!1 Fw pj
to the contrary that limk!1 Fw pj
k
k
; nkw must consist of atoms. Suppose
; nkw has at least two di¤erent atoms. Let p1
and p2 , p1 < p2 , be two adjacent atoms. In what follows, we show that there may not be a sequence of equilibrium bid distributions Fw pj bidding games
0
N k ; nk ; P
k
k
; nkw (in the sequence of
) that converge to the postulated limit distribution.
First, observe that it may not be that far enough in the sequence Fw pi j
k
; nkw is
strictly increasing in the neighborhoods of pi for essentially the same argument that ruled out strict monotonicity of limk!1 Fw pj k, Fw pj k,
Gh (x)nh Gl (x)nl
at x i¤
k
k
; nkw . Therefore, for large enough
; nkw has an atom at pki ! pi . Observe that rg < 1 implies that, for large
is decreasing on [xk (pki ); xk+ (pki )]. This is because
gh (x) gl (x)
nkh nkl
<
Gh (x) Gl (x)
Gh (x)nh Gl (x)nl
is decreasing
and, since xk (pi ) ! x, it is implied for large k by rg < 1.
This together with Lemma 18-(III) imply that, for k large enough, n 1 Gh (pk2 ) h n 1 Gl xk (pk2 ) l
xk
1 (xk+ (pk2 )) k k B gl (xk+ (pk2 )) C Gh x (p2 ) @ g xk (pk ) A h( 2 ) Gl xk (pk2 ) gl (xk (pk2 )) 0g
h
53
nh 1 nl 1
>
h l
pk2 j ; nkh pk2 j ; nkl
It follows that, for a given " 2 (0; p2 x2
xk
(pk2 ); xk+ (pk2 )
p1 ) and a large k, the pro…t of a bidder with
from a downward deviation to pk2
" is bounded away from 0.
This is because the probability of winning after the deviation would still be bounded k
below by limk!1 Fw p1 j
; nkw and hence bounded away from 0, while the pro…t
conditional on winning would also be bounded away from 0, since h l
h l
"j k ; nkh ] "j k ; nkl ]
k h [p2 k l [p2
gh (x) nkh gl (x) nkl
h l
k k k h [p2 j ; nh ] k k k l [p2 j ; nl ]
gh (x) nkh gl (x) nkl
gh (x) nkh Gh xk (pk2 ) gl (x) nkl G xk (pk ) l 2
pk2 vh
nkh 1 nkl 1
>
(35)
vl pk2 " vl > pk2 vh (pk2 ")
Therefore, for a large k, a bidder with x 2 xk (pk2 ); xk+ (pk2 ) would pro…t from a downward deviation to pk2 that limk!1 Fw pi j
k
" in contradiction to equilibrium. Thus, the supposition
; nkw
has two (or more) atoms in its support is false, so there
must be a single atom. The proof for the case of rg = 1 is almost identical except that the middle
inequality in (35) should be replaced by “ ”.
Remark: Adapting above Proof of Proposition 3-(ii) to …nite price grid. k
With …nite grid
> 0 the only change to the above proof is that we have
to make sure that the undercutting argument of the second to last paragraph of the proof is compatible with the grid. That is, it is possible to …nd the required downward deviation pk2 k
k su¢ ciently large
" in the grid. However, this is obviously the case, since for
<< p2
p1 . k
Proof of Lemma 19 (No Atoms in Limit): Let xk = xk xk+
k
xk
so that xk = inf xj
Monotonicity of
k
k
implies Gw xk+
nkw
Suppose to the contrary that limk!1 k
This means that
w[
limit. That is, lim
Gw xk+
lim
k!1
(x) j
Gl xk+
k
; nkw ] k nw 1
k
(x)
xk
k
w w[
xk
j
xk j k
has an atom on
> lim Gw
nkl 1
and xk+ = sup xj
1 k
xk
= q > lim
k!1
xk
nkw
k
; nkw
; nkw ] 6= xk ; xk+ 1
k
and xk+ = (x)
Gw xk
limk!1 Gw
k nkw
xk 1
k (xk )nw 1 .
that persists to the
. So, let
G l xk
nkl 1
0
(36)
and hence, by Lemma 6,
lim
k!1
Gh
xk+
nkh 1
54
=q
gh (x) gl (x)
lim
nk h nk l
>0
(37)
.
.
nkw 1
Obviously, lim Gw xk+
> 0 requires that xk+ ! x.
Recall that individual rationality, (6), implies h E vjx, win at
k
k
(x) ;
i
;
k
k
; nkh
k
(x)
8k.
(38)
vl : (x)
(39)
xk .
(40)
which in turn requires gh (x) nkh gl (x) nkl
h l
k
h
k
l
(x) j
(x) j
k
; nkl
xk+ ;
k
;
k
(x) k
vh
Step 1. If (36), then h lim E vjxk ; x(1)
k!1
k
i
> lim
k
k!1
nkw
Proof of Step 1. Suppose that lim xk < x. This implies lim Gw xk nk h
hence
= 0 and
nk h
nk h
(Gh (xk+ )) (Gh (xk+ )) (Gh (xk )) 2 (0; 1). Therefore, Lemma 3 implies k ! k n n nk (G` (xk+ )) ` (G` (xk )) ` (G` (xk+ )) ` k
h
lim
k!1
k
l
xk j
(xk ) j
k k
; nkh
; nkl
nkh
G` xk ] Gh xk+ Gh x k ] G x k ` +
nk [1 = lim k` k!1 n [1 h
nk`
.
Substituting into (39) yields for x = xk ,
lim
h
k!1
l
gh xk [1 gl xk [1
G` xk ] Gh xk+ Gh xk ] G xk ` +
nkh nk`
vh
The monotone likelihood ratio property implies that nk
gh (xk+ )
l
gl (xk+ )
g lim nhk > 1 and
lim
k!1
h l
k
limk!1
xk
limk!1
k
vl . k (x )
gh (xk ) [1 G` (xk )] gl (xk ) [1 Gh (xk )]
1. Hence,
! g imply
gh xk+ nkh Gh xk+ gl xk+ nkl G xk ` +
nkh 1 nk` 1
>
limk!1 vh
k
xk
limk!1
k
vl . k (x )
(41)
which is equivalent to (40). Suppose that lim xk = x. nk
g lim nhk > 1 imply that l
Observe that lim
k Gh (x)nh 1 nk 1 G` (x) `
gh (xk+ ) gl (xk+ )
= lim
gh (xk ) gl (xk )
= g and
is strictly increasing on [xk ; xk+ ]. Therefore,
55
from Lemma 18-(ii) k
h
k
l
k
xk j
k
(xk ) j
; nkh
<
; nkl
Gh xk+
nkh 1
Gl xk+
nkl 1
nk h
and since by (36) and Lemma 6 limk!1
1
Gh (xk+ ) Gl (xk+ )
nk l
=q
1
,
g` (x) gh (x)
1
nk l nk h
lim
> 0 = limk!1
nk h
1
nk l
1
Gh (xk ) Gl (xk )
,
the strict inequality persists in the limit and k
h
lim
k!1
k
l
k
xk j
k
(xk ) j
; nkh
< lim
; nkl
k!1
Gh xk+
nkh 1
Gl xk+
nkl 1
,
which implies the claim via (39). k
Step 2. For k large enough there is a bid bk > U (bk jxk ;
k
; nk ) > U (
xk such that
xk jxk ;
k
k
;n
k
):
Proof of Step 2: Rearranging (1) shows that k
k
U (b jx ;
k
k
;n ) =
k
l
(x) j
k
; nkl
1+
h l
k
vh
"
(x)
gh (x) nkh gl (x) nkl
h l
gh (x) nkh gl (x) nkl
h l
k k
(x) j
(x) j k
Observe that, for large enough k, it is possible to choose a bid bk > k
is arbitrarily close to
xk and such that
w
k
is because there is a small neighborhood above (If limk!1 k
xk+
k
xk+
; if limk!1
k
= limk!1 k
xk+
xk
bk j
k
; nkw
xk
k k
k
; nkl
vh
(x)
# vl . k (x)
xk
i
xk that
nkw 1
Gw xk+
; nkh
. This
at which there is no atom
, there is such a neighborhood just above
> limk!1
k
k
xk , then
xk ;
k
xk+
is such a
neighborhood). Therefore, for large enough k, U (bk jxk ;
k
; nk ) =
l
h
bk j
Gl xk+
By (36) and (37),
k
i ; nkl +
nkl 1
h
h bk j
+ Gh xk+
h
nk 1 limk!1 Gl xk+ l
+
k
i h h E vjxk ; x(1) 1 h h E vjxk ; x(1)
; nkh
nkh
nk 1 Gh xk+ h
1, there is an " > 0 such that h lim E vjxk ; x(1)
k!1
xk+ ;
k
;
56
k
i
> lim
k!1
i
q+q
k
gh (x) gl (x)
xk + ".
i
bk ;
k
;
k
xk+ ;
k
k
lim
nk h nk l
;
i
. By Step
(42)
bk
i
k
Therefore, for su¢ ciently large k, U (bk jxk ; Lemma 7, limk!1 U (
k
k
Step 2 contradicts
k
xk jxk ;
k
; nk ) is bounded away from 0, and, by
; nk ) = 0, which establishes Step 2.
being an equilibrium, buyers with signals su¢ ciently close
to x can ensure strictly positive, nonvanishing pro…ts in the limit. This contradicts the zero-pro…t condition from Lemma 7. Hence, in the limit there cannot be any atoms in the distribution of the winning bid.
Remark: Adapting above Proof of Lemma 19 to …nite price grid k
With …nite grid
> 0 the only change to the above proof is that we have to
make sure that it is possible to select a bid bk , as described in Step 2, that also belongs to the grid. Let " > 0 satisfy (42). If limk!1 k
there is no problem since, for su¢ ciently large k,
k
xk+ > limk!1
k
xk , then
< min "; limk!1
k
xk ; limk!1
k
and therefore there exists a bk in the grid such that bk 2 limk!1 k
and bk < limk!1 k
limk!1
xk
xk + ", as needed for the proof.
xk ; limk!1
k
k
=
If limk!1
xk+
k
xk+
xk+
, let " > 0 be de…ned by (42). By Step 1, we can choose " > 0
such that
h lim E vjxk ; x(1)
xk+ ;
k!1
Observe that, for all k with
k
such that
k
;
k
i
xk + ".
k
> lim
k!1
k
< ", there exists some bk 2 nkl
Gl xk+ bk
Gl x k
nkl
bk
xk ;
k
xk + "
k
2 "
.
since otherwise,
Gl
xk+
k
x
k
+"
nkl
b"=
k
Xc
1
Gl xk+
k
xk + i
k
nkl
Gl x k
k
xk + i
i=0
Thus, there is no atom at bk in the limit, hence lim
k!1
w
h bk j
k
i ; nkw = lim Gw xk+ bk
nkw
lim Gw xk+
k!1
k!1
nkw
.
(43)
which is what is required for the proof. Proof of Lemma 9 (Unavoidable Ties): We consider a sequence of bidding games
0
N k;
k; P 0
such that minfnkl ; nkh g ! 1, limk!1
and a corresponding sequence of bidding equilibria
k
= r with rg < 1,
.
Step 1. Suppose that for some sequence xk , (i) xk
57
nkh nkl
xk
k
(x) for all k
k
nkl
> 1.
nkl
and (ii) lim Gl xk
2 (0; 1), then nkh
Gh x k
lim
k
Gl (xk )nl
k!1
k
h
> lim
k!1
Proof: Let
nkl
lim Gl xk
k
l
(x) j
(x) j
k k
; nkh
; nkl
.
(44)
= q^:
Then, Lemma 6, gr < 1, and q^ 2 (0; 1) implies lim
nkh
Gh x k
= q^gr
k
Gl (xk )nl
k!1
1
> 1.
(45)
Abbreviate xk = xk nkl
and let lim Gl xk
k
(x)
= q. nkl
Case 1. Suppose that lim Gl xk k
h
lim
k!1
k
l
= q = 1. Therefore, Lemma 6 implies
(x) j
(x) j
k k
; nkh
; nkl
q gr = 1. q
=
This and (45) implies (44). nkl
Case 2. Suppose that lim Gl xk 0 and
xk
= q < 1. By the hypothesis that lim Gl xk
xk nkl
lim Gl xk
=q
q^ > 0:
From Lemma 3,
lim
w
k!1
h
k
k
(x) j
i
; nkw = lim
k!1
1 nkw
nk
Gw x k w . 1 G w xk
From Lemma 6 and its proof, lim
k!1
l
h
k
(x) j
k
i 1 ; nkl =
q and lim k!1 ln q
Hence, lim
k!1
h l
k k
(x) j
(x) j
k k
; nkh
; nkl
=
58
h
h
1 q rg rg ln q 1 q ln q
k
=
(x) j
k
i 1 ; nkh =
1 q gr . gr (1 q)
q gr . ln q gr
(46)
nkl
>
Since 0 < gr < 1 and 0 < q < 1, straightforward manipulation shows that q gr nkl
From lim Gl xk q^gr
q gr
1
1,
1
>
1 q gr . gr (1 q)
q and gr < 1, q^gr
= q^
(47) q gr
1
1.
The equality from (45),
the inequality (47), and the equality (46) imply
lim
Gh x k
k!1
nkh
> lim
k
Gl (xk )nl
k!1
k
h
k
l
(x) j
(x) j
k k
; nkh
; nkl
,
as claimed. nkh
Step 2. Consider any sequence xk for which lim Gh xk xk = xk
and xk+ = xk+
xk
k
xk
k
2 (0; 1). Let
.
Then, nkh
lim Gh xk
nkh
= 0 and lim Gh xk+
= 1.
Proof: It is su¢ cient for the claim to prove that lim Gh xk+ lim Gh If
xk+
xk
nkh
xk
2 (0; 1), because one can choose lim Gh
nkh
(48)
nkh
= 1 whenever
arbitrarily small.
= x for su¢ ciently large k, we are done. So, suppose that xk+ < x for all k
large. Therefore, one can …nd a bid bk such that k
xk < bk <
k
(x)
(49)
and there is no atom at bk , xk By (49) and monotonicity of lim
k!1
k h [b j
k
; nkh ] = lim
k!1
k
bk = xk+ bk .
, xk bk
Gh xk
(50)
xk . Therefore, bk
nkh
lim
k!1
Gh x k
nkh
> 0.
(51)
Thus, the zero pro…t condition (5) requires that limk!1 gr limk!1
k k h [b j ; k k l [b j ;
k] h k] l
= gr
limk!1 Gh xk bk
nkh
limk!1 Gl xk (bk )
nkl
59
bk k!1 vh lim
vl , bk
(52)
while individual rationality of h
gr lim
k!1
k
Since, bk
k
(x) requires that
k
(x) j
k
l
(x) j
k k
; nkh
k
lim
; nkl
k!1
h
lim
k
vh
vl . (x)
(53)
k
l
; nkh
limk!1 Gh xk bk
nkh
; nkl
limk!1 Gl xk (bk )
nkl
k
(x) j
k
(x) j
From Step 1, (54) requires limk!1 Gl xk bk nkl
0 by (51), limk!1 Gl xk bk choices of
k
(x) for all k, (52) and (53) imply
k!1
bk
(x)
nkl
.
(54)
= 1 follows. Because this must be true for all
for which (49) and (50) hold, it must be true that lim Gh xk+
k
xk
1. As outlined at the start, this implies (48). Step 2 implies Claim 3 of Lemma 9. Claim 3 is su¢ cient for Claims 1 and 2.
10.5
Existence and Uniqueness of r ( ; g)
Proof of Lemma 10: Recall that J(r; ; g) =
R1 0
x
1 g
1
x gr
1
ln x dx. (1+x gr)2
Claim 1: For each g > 1, there exists an r0 (close to g
1)
such that J(r0 ; ; g) < 0.
Proof: Write J(r; ; g) =
Z
1 g
0
1 g
x Z
x 1 g
The term in the brackets f
are positive. Let
=
1 gr 1 .
( ln (x)) (x)
1
nkl
2 f0; 1g. Since limk!1 Gl xk bk
1 g
1 gr 1
(1 + grx)
( ln (x)) (x)
1 gr 1
2
dx
(1 + grx)
2
dx:
g is always nonnegative and therefore both integrals
The …rst integral is
60
nkh
=
>
Z
1 g
0
1 g
Z
=
x
1 g
1 g
0
1 g
Z
1 g
(1 + grx) +1
1+
( ln (x)) x
2 2
x
dx )
dx
( ln (x)) x dx
0 +2
1 g
=
x
1 gr 1
( ln (x)) (x) (
1 +1
1
ln g
:
Thus, the …rst integral vanishes to zero at a rate of at least g proaches 1 (or equivalently, r ! g
1
as
ap-
1 ).
The second integral is
Z
1
1 g
x 1 g
=
Z
1
x 1 g
1+
ln (x) (x) 1 g
(
+1
2Z 1
x
2
1 g
1 ( + 2)2
+ ( + 1)
(1 + grx) +1
1+
( ln (x)) x
1 g
=
1 gr 1
2
dx 2
x
)
dx
( ln (x)) x dx ln g 1 g 1 ( + 2) ( + 1)
g 1 + ( + 1)2
+2
+1
1 g 1 + g ( + 1)2
Thus, either the second integral stays positive or it vanishes at a rate of at most 2
as
approaches 1 (or equivalently, r ! g
To sum up, J(r; ; g) < 0 for r ! g
1 ).
1.
Claim 2: For su¢ ciently large r, J(r; ; g) > 0. Proof: We show that limr!1 r2 J(r; ; g) = 1. Let (x; r) denote the integrand of r2 J(r; ; g). That is,
(x; r)
x
1 g
1
ln (x) x gr
1
r 1 + grx
2
:
Observe that (x; r) is non-decreasing in r on the domain x 2 0; g increasing in r on the domain x 2 g
1; 1
1
, and is non-
. Therefore, by the monotone convergence
theorem,
61
g
1
+2
( + 2)2
!
:
Z
Z
1
Z
1
1 lim (x; r) dx = (x; r) dx = lim r J(r; ; g) lim r!1 r!1 r!1 0 ( g)2 0 "Z 1 Z 1 g 1 1 2 1 = x x ln (x) dx + x g ( g)2 0 g 1 2
1
Now, letting a 2 0; g Z
1
g
x
1
ln g
1
1 x g
0
= lim
a!0
1 2 R1
while
g
x
1
2
x
1 x g
1
0
1
1 x g
2
2
ln (x) dx #
ln (x) dx .
,
2
ln (x) dx
(ln (a))2 +
1 2 gx
1
1
Z
lim
g
1
x
a!0 a
1 g ln g g
1
1 x g
1
+1
a
1
2
ln (x) dx
(1 + ln (a))
= 1,
ln (x) dx is obviously bounded. Therefore, limr!1 r2 J(r; ; g) =
1 hence J(r; ; g) > 0 for large enough r. Claims 1 and 2 together with the continuity of J(r; ; g) in r establish the existence of r > 1=g such that J(r; ; g) = 0. Claim 3: Fix a g > 1. For r > g
1,
if J(r; ; g) = 0, then Jr (r; ; g) > 0.
Proof: Recall that Z
J(r; ; g)
1
1 g
x
0
1
Since x gr
1
ln x (1+ grx)2
1
x gr
1
ln x dx = 0. (1 + grx)2
< 0 for all x 2 (0; 1), the integrand is positive for all x 2 (0; g1 )
and is negative for all x 2 ( g1 ; 1). Therefore, at any r > g
1
that satis…es J(r; ; g) =
0,
Z
0
1 g
x
1 g
1
x gr
1
ln x dx = (1 + grx)2
Z
1
x 1 g
1 g
1
x gr
1
ln x dx > 0. (1 + grx)2
Consider the function r2 J(r; ; g) and observe that dr2 J(r; ; g) =r dr
Z
1
0
x
1 g
1
x gr
1
ln x gr ln x 2 2 2 + (1 + grx) dx. (1 + grx) (gr 1)
The integrand is equal to the integrand of J(r; ; g) times the term
h
gr ln x (gr 1)2
+
2 (1+ grx)
which is non-negative and decreasing in x. Therefore, at r such that J(r; ; g) = 0, the positive part over (0; g1 ) is weighted more heavily than the negative part over ( g1 ; 1) implying sgn
dr2 J(r; ;g) dr
dr2 J(r; ;g) dr
> 0. Now, at r such that J(r; ; g) = 0, sgn (Jr (r; ; g)) =
. Therefore, Jr (r; ; g) > 0 as required. 62
i
Claim 3 concludes the proof of the Lemma, since Jr (r; ; g) > 0 at any r such that J(r; ; g) = 0, there can be only one such r.
10.6
Proof of Lemma 11
Proof of Lemma 11: From lim nkh sk = r lim nkl sk , rewriting as in Lemma 8, h i h i lim nkh Eh pj k ; nkh + 1 Eh pj k ; nkh k!1 Z p Z p = h (pj ; g; r) ln h (pj ; g; r) dp + r vl
vl
Using (9) to spell out
w
lim nkh sk
k!1 l
(pj ; g; r) ln
l
(pj ; g; r) dp.
(pj ; g; r), rearranging and dividing through by gr=(gr
1); we get
=
i i h h gr 1 lim nkh sk ) Eh pj k ; nkh (lim nkh Eh pj k ; nkh + 1 gr 1 Z p 1 p vl 1 1 p vl gr 1 1 p vl dp. ln gr vh p g gr vh p gr vh p vl
Changing the integration variable by substituting for p the function (x) =
vl +x grvh 1+x gr
we get
= = = =
i h i h gr 1 lim nkh s Eh pj k ; nkh lim nkh Eh pj k ; nkh + 1 gr 1 Z p 1 p vl 1 1 p vl gr 1 1 p vl dp ln gr v p g gr v p gr vh p h h vl 1 Z 1 (p) 1 (x) vl 1 1 (x) vl gr 1 1 (x) vl ln 1 gr vh (x) g gr vh (x) gr vh (x) (vl ) Z 1 1 1 rg (vh vl ) x x gr 1 ln (x) dx g (1 + rxg )2 0 rg (vh vl ) J(r; ; g):
Lemma 8 requires that h lim nkh Eh pj
k
; nkh + 1
i
h Eh pj
k
; nkh
i
Hence, from gr > 1, optimality requires that J(r; ; g) = J(lim
63
nkh ; ; g) = 0: nkl
lim nkh s = 0:
(55)
0
(x)dx
10.7
Bounded Number of Bidders
Proof of Proposition 5: From Lemma 8, whenever lim Fw j degenerate,
lim nkw sk
> 0. Thus, if
nkw
k
; nkw
is non-
is bounded, the distribution of the win-
ning bid must become degenerate with support on some number C in state w, lim Fw C + "j
k
; nkw
Fw C
"j
k
; nkw = 1 for all ". We argue that the distrib-
ution of the winning bid must be degenerate on C in the other state as well. Suppose that limk!1 nkh = m < 1 (the case where nkl is bounded is argued
below). If nkl is bounded as well, we are done: In both states, the distribution of the winning bid is degenerate in the limit on some numbers and these numbers must be the same because the boundedness of the likelihood ratio implies that lim Fh j
k
; nkh and lim Fl j
k
; nkl are mutually absolutely continuous if nkh and
nkl are both bounded.
So, suppose nkl ! 1. Now, by nkh ! m < 1 and the bounded likelihood ratio,
whenever lim Fh pj
k
; nkh = 0 then lim Fl pj
the support of lim Fl j
k
k
; nkl = 0, i.e., the lower bound on
; nkl is weakly above C. If C
vl , then buyers’individual
rationality and the law of iterated expectations rule out that lim Fl pj for any p > C
vl . Hence, if C
k
; nkl < 1
vl , then the distribution of the winning bid
is degenerate on C in the low state as well. If C
vl , then nkl ! 1 rules out a
non-degenerate distribution of the winning bid below vl in the low state (Bertrand competition). Thus, the distribution of the winning bid becomes degenerate in the
low state with mass one on vl . However, when the distribution of the winning bid is degenerate in the low state, Lemma 8 implies that nkl sk ! 0. Hence, it must be
that C = vl : The boundedness of the likelihood ratio implies that when the high type samples nkl bidders, the winning bid is close to vl as well by Lemma 6, while total solicitation costs are close to zero. Thus, in both state, the distribution of the winning bid must become degenerate if nkh remains bounded as sk ! 1.
Now, if nkl ! m < 1, a similar argument applies. As before, the distribution of
the winning bid must become degenerate on some number C in the low state. Also
as before, if nkh is bounded as well, we are done. So, suppose nkh ! 1. In this case,
nkh =nkl ! 1, so that the interim expected value converges to vh for all signals. If there is some price p with C
p such that lim Fh pj
k
curse at p is bounded and, hence, lim E[vjx;win at p + ";
; nkh > 0, then the winner’s k
;
k]
= vh for all ". Thus,
the interim expected payo¤s are bounded from below by lim Fh pj
k
; nkh (p
almost all types. Feasibility requires therefore that p = vh whenever lim Fh pj
vh ) for k
; nkh >
0. Thus, it must be that the distribution of the winning bid becomes degenerate with mass one on vh in the high state. By buyers’individual rationality, this requires C
vl . Since the distribution of the winning bid is degenerate in the high state, 64
Lemma 8 implies that nkh sk ! 0. If the low type solicits nkn bidders, the low type would be sure to trade at vh as well by Lemma 6, at almost no solicitation costs, vl . Thus, if nkl is bounded, nkh must be bounded as well.
contradicting C
Therefore, in all case, the distribution of the winning bid must become degenerate on some number C. Of course, the number C must be below
l vl + h vh
by individual
rationality of buyers and the law of iterated expectations. This …nishes the proof of the proposition.
10.8
Existence of a Pooling Equilibrium
Proof of Lemma 12: The proof relies closely on Athey (2001). The existence of a bidding equilibrium (given the constraints on
) for a given
is an immediate
corollary of Athey’s Theorem 1 and our Proposition 1. We have to establish that an equilibrium exists also when Recall that P
is part of the equilibrium.
= [0; vl ] [ fvl +
; vl + 2 ; :::; vh
; vh g. Let B
; vl + 2 ; :::; b; b and
set of monotone bidding functions using bids from vl ; vl + let m =k fvl ; vl +
denote the
; vl + 2 ; :::; bg k. Using Athey’s idea,
is a set of vectors of
dimension m + 1 whose coordinates belong to [x; x] =f =( where
0;
1 ; ::: m )
]m+1 j x
2 [x; x
determines a bidding strategy
i = 0; :::; m
by
We say a bidding strategy
8x>x
arg max
De…ne the correspondence
0
;
0
) =
2(
0
;
0
) =
Both
if x 2 [ i ;
0
) =
(
2
U (bjx;
1(
0
;
0
)
x
,
; ).
from
D
over f1; :::; N g.
0
D into itself. For any
is best response against ( 0 ;
j
= ( l;
i+1 ),
.
D,
1(
g;
b2fvl ;vl + ;vl +2 ;:::;b;bg
Let D denote the set of probability distributions
( 0;
(x) = vl + i
x
m
is a best response against ( ; ) if for all x
(x) 2
2D
:::
1
1. We set (x) = b
0
0
h)
j if 2(
w 0
;
0
) for x
(n) > 0, then n 2 arg max Ew [pj 0
n2f1;::;N g
2
x
and
, 0
; n]
)
ns ,
):
and D are closed convex sets.
2
is convex valued and continuous by virtue
65
of being the set of maximizers of a concave problem on a convex set (see Lemma 5). That
1
is convex valued and upper hemi-continuous is established by Athey
(2001). To be precise, the convex valuedness is established directly by Lemma 2 of that paper, while Lemma 3 establishes the upper hemicontinuity of with respect to
0
(since
1(
0;
0)
only
is exogenously …xed in Athey’s model). Nevertheless,
since the bidder’s payo¤ function in our model, U ( jx; ; ), is continuous in Athey’s original argument establishes that
(
0;
0)
1(
0;
0)
,
is continuous with respect to
as well.
It follows that
=
Kakutani’s Theorem,
1
is convex valued and upper hemicontinous. By
2
has a …xed point. Since the strategies in
to use only prices from vl ; vl +
are constrained
; vl + 2 ; :::; b; b , the bidding strategy determined
by the …xed point satis…es (
(x)
= b if x > x b if x
In order to claim that a …xed point of
;
x
.
is indeed an equilibrium of the auxiliary
game A, it only remains to argue that there is no pro…table deviation to a price in [0; vl ), which is also in P and which is allowed by the constraint
(x)
b— but this
k
satis…es (16)
obviously true and was also argued in Lemma 2. Proof of Lemma 13. We want to show that whenever sk ! 0, and
k
k
is an optimal solicitation strategy given
1.
limk!1 nkw = 1; w 2 f`; hg.
2.
limk!1
3.
(1 Gl (x (1 Gh (x
nkh nkl
=
ln Gl (x ln Gh (x
)) 1 )) Gh (x
) )
< 1.
limk!1
)
, then:
Gh (x
)nh
k
1
Gl (x
nk ) l
1
.
Given convexity of the seller’s objective function, optimality requires that h Ew pj
k
; nkw
i
h Ew pj
k
; nkw
1
i
h Ew pj
sk
k
; nkw + 1
i
h Ew pj
k
i ; nkw . (56)
In particular, this implies h i h Eh pj k ; nkh Eh pj h i h El pj El pj k ; nk`
k
; nkh
k
; nk`
i 1 i 1
h i h i El pj k ; nk` + 1 El pj k ; nk` ; (57) h i h i Eh pj k ; nkh + 1 Eh pj k ; nkh : (58)
66
Observe that h Ew pj
k
where Ew (pjp
i ; n = (1 k
b;
)n ) b + Gw (x
Gw (x
)n Ew (pjp
k
b;
; n).
; n) is the expected winning bid conditional on p
b. It follows
that
)n
Gw (x
h Ew pj 1
(1
+Gw (x
i
)) [b
)n [Ew (pjp
b;
Gw (x
)n (1
k
;n + 1
Gw (x
h Ew pj
k
i ;n =
Ew (pjp
k
; n)
(59) k
b;
;n
Ew (pjp
1)]
b;
k
;n
1)]:
Hence, h Ew pj
k
Step 1:
;n + 1 nkw
i
h Ew pj
k
i ;n
Gw (x
)) [b Ew (pjp
b;
k
; n)].
(60)
! 1 for w 2 fl; hg.
Proof of Step 1: By (56) and (60),26 sk
h Ew pj
k
; nkw + 1
i
h Ew pj
k
; nkw
It follows from sk ! 0 that Gw (x Part (1) of the Lemma.
i
k
)nw (1
Gw (x
Gw (x
)) [b Ew (pjp
k
)nw ! 0 and, hence, nkw ! 1. This establishes
Step 2: G` (x
)
1 1
G` (x Gh (x lim
k!1
1 Gh (x
1 )1
G` (x Gh (x
[b ) lim ) k!1 [b
El (pjp Eh (pjp
Gh (x
)n`
k
1
G` (x
)nh
k
1
[b ) lim ) k!1 [b
El (pjp Eh (pjp
b; b;
k
b; b;
k
; nk` )] k ; nkh )]
(61)
; nk` )] . k ; nkh )]
Proof of Step 2: Using (59) for w = h and (60) for w = l to rewrite (57) and 26
The derivation of (60) does not assume monotonicity of
67
k
for x
x
.
b;
k
; nkw )].
rearranging we get, k
)nh
1
k
1
(
Gh (x )) [b Eh (pjp b; k ; nkh 1)] + k (1 G` (x )) [b El (pjp b; k ; nk` )] G` (x )n` ) Eh (pjp b; k ; nkh ) Eh (pjp b; k ; nkh 1) 1. (1 G` (x )) [b El (pjp b; k ; nk` )]
Gh (x
(1
(62)
Similarly, )n`
(
(1 G` (x )) [b El (pjp b; k ; nk` 1)] + k (1 Gh (x )) [b Eh (pjp b; k ; nkh )] Gh (x )nh ) El (pjp b; k ; nk` ) El (pjp b; k ; nk` 1) 1. (1 Gh (x )) [b Eh (pjp b; k ; nkh )]
G` (x
(63)
Now, nkw ! 1 implies lim Ew (pjp
k!1
k
b;
; nkw +1) = lim Ew (pjp
b;
k!1
k
; nkw ) = lim Ew (pjp k!1
b;
k
; nkw 1).
Hence, Ew (pjp b; k ; nkw ) Ew (pjp b; k ; nkw 1) = 0. k!1 (1 Gw (x )) [b Ew (pjp b; k ; nkw )] lim
Therefore, taking limits in (62) and (63) and combining them to a single chain of inequalities we get (61). Step 3: limk!1
nkh nkl
=
ln Gl (x ln Gh (x
) )
< 1.
Proof of Step 3:From (61), lim Gh (x Gl (x
0
B Gh (x lim B k!1 @ Gl (x
k
)nh nk ) l
)
2 (0; 1). Therefore,
nk h nk l
1nk l
C C ) A
2 (0; 1) ,
Since nkl ! 1, this requires that
lim
k!1
Gh (x Gl (x
68
)
nk h nk l
)
= 1.
nk
Therefore, ln Gh (x
) lim nhk = ln Gl (x
), hence,
l
nkh ln Gl (x = k k!1 n ln Gh (x l
) < 1; )
lim
where the last inequality is a consequence of Gl (x
) > Gh (x
). This estab-
lishes Part (2) of the lemma. Step 4: limk!1 El (pjp
b;
k
; nk` )
Proof of Step 4: Let Fw (pjp
limk!1 Eh (pjp k
b;
b;
k
; nkw ) denote the distribution of the winning
b. Recall (xk (p) = inffx j
bid conditional on being p
; nkh ) k
(x)
pg. Consider
a subsequence over which the following limits exist and let qw = lim Fw (pjp k
b;
; nkw ), w = `; h.
qw
lim Fw (pjp Gw (x
lim
k
b;
k!1
)
Gw (x
lim @1
k!1
; nkw ) = lim
k!1
k!1
0
[Gw (x Gw (x
Gw (
k
)
!nkw
Gw (xk (p))]
) ) 1
=
!nkw
k
nw ) Gw (xk (p)) k n w Gw (x ) A nkw
Observe that, for p such that lim sup xk (p) < x (xk
Gw (xk (p)) Gw (x )
:
, we have lim inf[Gw (x
)
(p)] > 0 and hence qw = 0; for p such that from some point in the sequence
1 (p)
=x
, we have qw = 1; for p such that xk (p) < x
we have lim
Gw (x
k!1
) Gw (xk (p)) k nw = Gw (x )
and lim xk (p) = x
,
ln qw :
Therefore, [Gh (x ln qh = lim k!1 [G` (x ln q` Now, since lim xk (p) = x k large enough, Gw (x observation and lim
nkh nkl
) nkh : ) nk`
and since gw are step functions, it follows that, for )
=
Gh (xk (p))] G` (x G` (xk (p))] Gh (x
) )
Gw (xk (p)) = gw (x
ln Gl (x ln Gh (x
xk (p)]. Using this
) [x
) )
ln qh gh (x = lim ln q` g` (x
) G` (x ) Gh (x
) ln Gl (x ) ln Gh (x
) : )
By condition (14), the right side is smaller than one, and so 69
ln qh ln q`
1 (see the
footnote). Therefore, lim Fl (pjp
Thus, lim Fl (pjp lim El (pjp
b;
k
b; k
k
b;
k!1
; nk` ) = q`
qh = lim Fh (pjp
b;
k!1
k
; nk` ) stochastically dominates lim Fh (pjp
; nk` )
lim Eh (pjp
k
b;
; nkh ):
b;
k
; nkh ). Hence,
; nkh ).
Step 5: Gh (x
lim
k!1
G` (x
k
)nh )
1
nk` 1
1 Gh (x
1 )1
G` (x Gh (x
[b El (pjp b; Eh (pjp b;
Proof of Step 5: Step 4 implies that lim [b with (61) implies the desired inequality.
k
) : )
;nk` )] ;nkh )]
1, which together
k
This proves Part (3) and concludes the proof of Lemma 13. Proof of Lemma 14. We want to show that there are numbers v1 , v2 ,v3 independent of b; b such that
h
h lim E vjx, win at b; k!1 h lim E vjx , win at b; k!1 h lim E vjx, win at b > b;
k k
k!1
lim E vjx
k!1
k
k
,
k
,
k
; win at b 2 b; b ;
k
,
;
k
i
l vl
=
+
h vh ,
i
v1 <
l vl
+
h vh ,
v2 <
l vl
+
h vh ,
i
v3 <
l vl
+
h vh :
i
Observe that h Pr hjx, win at b,
k
,
k
i
=
= = !
k
Pr h, x, k
Pr x,
, win at b
(64)
, win at b
n nh 1 1 (Gh (x )) h h gh (x) N nh 1 Gh (x ) n n nh 1 1 (Gh (x )) h nl 1 1 (Gl (x )) l h gh (x) N nh 1 Gh (x ) + l gl (x) N nl 1 Gl (x ) (Gh (x ))nh ) h (1 (Gh (x ))nh ) + l (1 (Gl (x ))nl ) h (1 k!1 h .
nw k and that by Lemma w gw (x) N = Pr w, x, n k 1 1 (Gw (x )) w k w bj ; nw = nw 1 Gw (x ) . The third equal-
For the second equality, note that 3, Pr win at bjw, ity follows from 1 convergence to
h
k
,
k
=
Gw (x
) = gw (x) and cancellation of terms. Finally, the
follows from nkw ! 1 by Lemma 13. Therefore h lim E vjx, win at b,
k!1
70
k
,
k
i
=
l vl
+
h vh ,
that is, the …rst equation holds. Replacing gw (x) by gw (x for Pr hjx out 1 h
Pr hjx
, win at b,
Gh (x
k
) in (64), we obtain the corresponding expressions k
,
. Then using gh (x) = 1
Gh (x
) to substitute
) and again canceling terms we get
, win at b,
k
,
k
i
gh (x ) h gh (x)
=
gh (x ) h gh (x)
(1
!
k!1
gh (x gl (x
h
) )
) )
gh (x) gl (x)
=
gl (x ) gl (x) l
gh (x) gl (x)
=
)nh )
Gh (x
)nh ) +
Gh (x
gh (x gl (x
h
(1
<
+
(1
Gl (x
h.
l
The expression following the convergence sign is obtained by dividing through by gl (x ) gl (x)
)nw ! 0 since nkw ! 1 by Lemma 13. The last
and noting that Gw (x
inequality owes to the increasing likelihood ratio
l
v1 = h
gh (x gl (x
) )
=
h
vl +
gh (x) gl (x)
+
l
gh (x gl (x
) )
<
gh (x gl (x
) )
gh (x gl (x
h
) )
gh (x) gl (x) .
Let
gh (x) gl (x)
=
=
gh (x) gl (x)
l vl
+
vh .
+
l
The second equation holds with h lim E vjx
k
, win at b;
k!1
k
,
i
= v1 <
h vh .
The winner at b > b does not learn anything from winning. Hence, h Pr hjx, win at b > b;
k
k
,
i
Pr x, win at b > b; k , nh h gh (x) N nh nl h gh (x) N + l gl (x) N
=
where the limit follows from
v2 =
nkh nkl
!
l ln Gl (x g h ln Gh (x
) )
ln Gl (^ x) ln Gh (^ x)
+
71
k
l
nkh h g nk l
+
l
ln Gl (x ) h g ln Gh (x ) k!1 ln Gl (x ) h g ln Gh (x ) +
; l
established in Lemma 13. Let
vl + l
k
,
nkh h g nk
=
!
k
Pr h, x, win at b > b;
=
ln Gl (x ) h g ln Gh (x ) ln Gl (x ) h g ln Gh (x ) +
vh . l
)nl )
From function
gh (x) gl (x) 1 z ln z is
1 Gh (x 1 Gl (x
=
) gh (x) ln Gl (x ) , gl (x) ln Gh (x
) )
1 Gh (x 1 Gl (x
=
) ln Gl (x ) ln Gh (x
) ).
strictly increasing in z 2 (0; 1) and because Gl (x 1 1
Gh (x Gl (x k
Therefore, Pr hjx, win at b > b,
) ln Gl (x ) ln Gh (x ,
k
<
h lim E vjx, win at b > b;
k!1
k
,
) > Gh (x
),
) < 1. )
and
h
k
Because the
i
= v2 <
l vl
+
h vh .
De…ne v3 by 1 Gl (x ) ) 1 Gh (x ) 1 Gl (x ) ) 1 Gh (x ) +
l
1 Gl (x ) 1 Gh (x
) )
1 h Gl (x
v3
1 h Gl (x
Since by condition (13),
1 Gl (x
h Pr vh jx w=h;` w
=
nk
lim
gh (x h gl (x
k!1
h
lim
k!1
k
h
+
G` (x
)
nk h nk ) `
)
Now,
k]
;
k
)nh
1 nk
l gl (x
k
)n`
) N`k G` (x
1
k
)nh l Gl
k
)nl
(x
k
)nh
G` (x
1
l vl + h vh .
k]
;
k
(65)
l
(66)
gh (x ) Gl (x ) ln Gl (x ) h gl (x ) Gh (x ) ln Gh (x ) Gh (x k ) Gl (x ) ln Gl (x ) )nh + ) Gh (x ) ln Gh (x ) Gh (x
Gh (x
Gh (x
k
vl :
+
i
) Nhk Gh (x )nh
) Nhk Gh (x
k
;
nk
h gh (x
h gh (x
=
k
, b 2 bk ; b jw;
Pr[x
) )
< 1, it follows that v3 <
, win at b 2 bk ; b jh;
P
l 1 Gl (x ) 1 Gh (x
1 h Gl (x
, win at b 2 bk ; b ;
h Pr[x
=
vh +
+
1 Gl (x ) ) 1 Gh (x ) 1 Gl (x ) ) 1 Gh (x ) +
1 h Gl (x
nk `
l
1 h Gl (x
<
h:
l
The expression after the 2nd equality sign is explained by Pr[win at b 2 bk ; b jw; k
;
k]
k
= Gh (^ x)nh
1
dividing through by
. The expression following the convergence sign is obtained by gh (x )nkh Gl (x ) and
nk
ln Gl (x ln Gh (x ) ln Gl (x ) ) ln Gh (x )
noting that lim nhk =
inequality follows from the assumption
l
gh (x gl (x
) Gl (x ) Gh (x
) ).
The following
1. The next in-
equality follows from Lemma 13. The …nal inequality follows from
x) 1 1 Gl (^ Gl (^ x) 1 Gh (^ x)
<1
which is implied by condition (13). The de…nition of v3 together with (66) imply h lim Pr vh jx
k!1
, win at b 2 b; b ;
72
k
;
k
i
v3 <
l vl
+
h vh .
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