dynamic. Perhaps surprisingly, however, the results are similar. The bidders know the information
revelation is harmful but cannot avoid it.

To put more structure on this discussion, suppose that, following the analysis in Section 2, firm
A merges with seller 1. The structural change induced by the merger clearly affects the bidding
behavior of both A and B. Assuming that the merged firm wishes to maximize its joint profits, firm
A wants firm B to win the object whenever B’s bid is greater than vy. As a result, A simply bids
va in auction one, which has two effects. First, it causes B to change its behavior in auction one.
Second, it causes both firms to change their bidding behavior in auction two.

Vertical information flows may cause additional effects, and the behavioral changes described
above may be more pronounced if firm A (either secretly or publicly) sees B’s bid from the first
auction. The next two sections examine behavior in the presence or absence of an information

firewall within the vertically integrated firm.!9

Integration with a Firewall

Suppose A can credibly commit not to receive B’s bid. It could do this by having the bids in
the first auction evaluated by a third party or through fear of the government’s reaction if the
violation is detected. I continue to assume that type vz players submit no bid in either auction.
Furthermore, a type vy player B will not submit a positive bid in auction one, because B cannot
get the item if player A is type vy. This framework and its consequent behavior generate a lot of
information about the participants. If B wins auction one, then A knows that B is type vy and B
knows that A is type vy, and this is common knowledge. B then bids zero in auction two.20 If A
wins auction one, then B knows that A is type uy, and this is also common knowledge. However,
in this case A does not know B’s type, and in fact has learned nothing about B’s type. B’s payoff

in this situation is therefore

1
~Olvn) = 4 [orton 10,0] +5 [ow a (v4 | #)|
. Serene memmewemncna
Against vp, Against vy

49T am of course begging the question of which types of buyers merge. Clearly the type vz bidders will not merge if
there is a cost to doing so, and if the information gain is the only reason for the merger. However, if this information
acquisition aspect is only a byproduct of the merger, then for now I can ignore the reasons for merging. Of course,
the real reason for the merger, be it efficiency gains or increasing market power, must be considered when evaluating
the results of this model.

201f a type uz player A should happen to deviate and bid zero in auction one, the only effect is that A may win
and cause B to believe A is type vz. This will cause B to bid higher in auction two, and will not affect A’s payoff in
any way. Thus, assuming a type vz player A submits no bid is still a Nash equilibrium.

16


Proof: Because bidders use a mixed strategy in auction one, each bid in the bid distribution yields
the same total expected payoff over the entire auction. B’s expected payoff from bidding zero in
auction one is

v, UV; v;
7(0 [vy) = + + SF In(a) > vn.

This shows that the repeated auction offers the bidders higher expected payoffs than does a sequence
of two independent auctions, which yields an expected payoff to a type uy bidder of vy. O

There are two main implications of this softening of bidding behavior relative to a series of
independent auctions. First, if there is only one seller (that is, if seller 1 and seller 2 are actually
the same firm), then selling the items simultaneously is strictly preferred to selling them sequentially.
For example, a company hiring services through a procurement auction should prefer signing long
term contracts to short term contracts. This preference is strictly due to information transmission,
and it ignores other reasons for wanting long term contracts, such as negotiation costs and fears
about collusion.

Second, if the sellers are different, then they would like to differentiate their products. This
preference for differentiation is not because bidders will substitute one seller’s product for the
other’s; bidders want to acquire both items. The sellers wish to differentiate their products because
information acquisition leads to softer bidding, something absent from the series of independent
auctions in Proposition 1.

This analysis is complicated by the fact that bids are not revealed, so the updated beliefs about
one’s rivals are unknown. A simpler method is presented in the next section, which examines how
changing the auction framework to require bid revelation changes bidder strategies and auction
outcomes.

3 The Effect of Price Announcements

Most public sector auctions, such as those for construction contracts, reveal the bids submitted by
all participants. This ensures honesty by preventing transfers from bidders to the auctioncer, a
government representative, from altering auction outcomes.

While this concern has merit, it is well known in the industrial organization literature that
revealing private information or reducing the noise of unobservables enhances collusion. Stigler
[1964] notes that “collusion will always be more effective against buyers who report correctly and
fully the prices tendered to them.” Moreover, Green and Porter [1984] show that the ability of firms

to collude decreases and the incidence of price wars increases because firms cannot observe their

10


world mergers. If A merges with seller 2 before the repeated auction process begin, then in auction
two behavior is completely determined: a type vq player A bids vy, while a type vy player B bids
zero. Because first period behavior does not affect second period behavior, bidding in auction one
follows the predictions in Proposition 1. Thus, the sellers still prefer the vertical merger take place
if A merges with seller 2 rather than seller 1, though now. the firewall has no effect on the sellers’
payoffs.22

There are three realistic extensions of this merger question which are beyond the scope of this
paper. First, there might be T > 2 auctioneers. If A merges with an auctioneer besides the first
or the last (the two cases just analyzed), then there may be consequences for bidder behavior in
early auctions. Second, the vertical merger might take place after some auctions have taken place.
In this case, potential downstream merger partners may use the outcomes from early rounds to
decide with which firm to merge. This follows because the sellers in this model prefer merging
with type vy players rather than type vz. Third, there is the distinct possibility that the bidders’
characteristics are not fixed from period to period but simply have some systematic relationship.
This paper presents a first step in considering that question.

6 Conclusion

This paper models a small population of bidders whose members face each other repeatedly over
time. By assuming bidders desire multiple objects, and that the values of those items to a bidder
are correlated, I address strategic problems that do not arise either in a sequential auction in which
bidders desire only a single item, or in a series of independent single item auctions. With the
discrete model I use, I discover the following about three structural information policies and the
resulting strategic information transmission.

Without price announcements, bidding competition is softer in both the first and second auc-
tions than in a series of two independent auctions. Competition is softer in auction one as firms try
to acquire information about their rivals. It is softer in auction two because the winner of auction
one now more firmly believes he faces an inefficient opponent. The equilibrium outcome implies
that a single seller of the two items prefers a simultaneous sale. If the auctioneer is a firm acquiring
inputs, then this suggests that long term contracts are preferred to short term contracts to elimi-
nate information acquisition and soft bidding by input suppliers. Also, if the sellers are different,

then they prefer the auctions to be independent. In a sense, the sellers prefer to differentiate their

?More precisely, both sellers have expected revenues of “#, which is better than their payoffs premerger and is
exactly the same as if player A and seller 1 merge with no firewall.

20


1 Introduction

There is an ongoing public policy debate regarding vertical integration and its concomitant in-
formation flows. Of particular concern is that the information derived by an auctioneer (such
as a distributor) will be shared with its integrated bidder (such as a manufacturer), leading to
a reduction in competition between the bidders. Similar competitive concerns arise regarding bid
revelation policies, such as those used in public sector procurement. Modeling such situations as the
repeated auctions introduced in Thomas [1996a], this paper examines the transmission of private
information via the auction outcomes, and shows how that transmission is affected by the changes
in market structure described above. These concerns are not present in the existing auction liter-
ature, because such information transmission is irrelevant both in series of independent auctions
and in sequential auctions in which participants desire only a single item. However, when bidders
desire multiple items, and when the values of those items to a bidder are correlated, the incentive
to learn about opponents’ values and to obscure one’s own drive equilibria to depart systematically
from those in standard models.

It is certainly true in auction markets that firms often face the same competitors at several
points in time. For example, a few large firms compete for government defense contracts, while local
construction firms frequently compete for jobs in both the public and private sector. Additionally,
private firms submit bids to hire goods and services, as with logging companies bidding for timber
from the U.S. Forest Service, and with manufacturers bidding for the services of distributors in the
pharmaceutical industry.

In all of these situations there is a small population of bidders whose members face each other
repeatedly over time. Each interaction provides information about privately known attributes of
the participating firms, the form and precision of which depends upon the specifics of the auc-
tion framework used. For example, in municipal construction contracting, firms’ bids are publicly
revealed after the contract is awarded. In contrast, in many private sector auctions, bids are
considered proprietary information.

The value of the information gained by repeated interaction depends on the scrial persistence
of firms’ attributes. If the bidder’s valuations for items are independent across auctions, then no
information is gained from previous interaction. However, when a bidder’s valuations are correlated
across auctions, information is transmitted simply by winning or losing an auction. That is, bidders
learn about one another even if no bid information is revealed. For example, suppose that two

bidders with continuously distributed valuations for two identical objects bid on the first object


[23] Vickrey, W. “Counterspeculation, Auctions, and Competitive Sealed Tenders.” Journal of
Finance, Vol. 16 (1961), pp. 8-37.

[24] Vickrey, W. “Auctions and Bidding Games,” in Recent Advances in Game Theory, (1967), pp.
15-27.

(25] Vives, X. “Duopoly Information Equilibrium: Cournot and Bertrand.” Journal of Economic

Theory, Vol. 34 (1984), pp.71-94.

[26] Weber, R. “Multiple Object Auctions,” in Auctions, Bidding and Contracting: Uses and The-
ory, New York: New York University Press, 1983.

31


=i Lon) _ Sun
= glen ten] +5 (1) = 47

B may choose to submit no bid in auction one, but then auction two is exactly the same as the

single item symmetric auction. B’s payoff over the two auctions is

fen! (oy 22) 42 owed (om) 22
plore (e155) ] +5 [oe (#15.3)|

Against vz, Against vy

n® (“no bid” | vz)

UH

2

B does better submitting a bid in auction one than not, because of the chance that player A is type
vz, which would allow B the opportunity of both acquiring the first object and getting the second
for a low price. Therefore, B participates in auction one.

The payoff for a type vy player A in auction one is og 2 Because A learns nothing about B’s
type yet is known to be type vy, A’s payoff in auction two is

1 Vv,
ah (lon) = (% | e) = >

The following proposition summarizes the sellers’ payoffs.

Proposition 9 Ex ante expected profits of seller 1 and seller 2 when there is no bid transmission

between the upstream and downstream players are

L1\_ vn
Ra (5.5) = 1

and

Proof: See the Appendix.

The bidding strategies of A and B imply that seller 1 gets “# if A is type vy and zero otherwise.
If player A is type vz (which occurs with probability 3), then seller 2 gets nothing; either B is also
type vz or B is type vq yet knows that A is type vz. If player A is type vq, then seller 2 gets
either the payoff from two type vy players (if B is also type vy ) or only one type vy player.

?1For comparison purposes, I assume that the two merged firms split the rents.

17


Using B’s indifference relation, (2),
vy —b=(1—a)uq.

This implies b = avy.
2,3) Let F'4(b) and F'g(b) be the equilibrium bid distributions used by type vy players A and B, respectively.
To determine F'4(b) and Fg(b) I employ the typical mixed strategy approach: Use an indifference condition
on player 4 to solve for player j’s distribution. A type vq player A must be indifferent over all bids in [0,5],
so
(1 — 8)(vy — 6) + BF p(b)(vy — b) = (1 — B)uy + BFa(O)un. (1)
Against vy, Against uy

That is, A’s expected payoff from bidding b must be identical to the expected payoff from bidding zero.

Similarly, for a type vy player B

(1 ~ a)(vy — 6) + aF 4(b)(uy — b) = (1 — a)uy + aF 4(0)uy. (2)

Against vz, Against uy

Also note that this indifference relation must hold at 6, so for A

vy ~b= (1-6) + BFa(O)Jon (3)
while for B
vy —6=[(1—a) + aFa(0)Jun. (4)
Equating the two right hand sides of (1) and (2) gives
1~- 6+ 6Fp(0) =1—a+aF,4(0),
or

Fp(0) ~ aF4(0) = Ba. (5)

If a < @, then F4(0) = 0 and Fp(0) = 1 — 4- The information about F'4(0) and Fg (0) can be used in
conjunction with the indifference relations to solve for the equilibrium bid distributions.
4) Use the indifference relations (1) and (2) and the definition of b.

5,6) The expected payment of a type vy player A, when beliefs that A and B are type vz are a and ,

23


costs them nothing or is at least cheaper than the litigation costs. Moreover, because B bids the
same way with or without the internal wall, A has no incentive to secretly learn B’s bid. That is,

the firewall is trivially enforced.

Proposition 10 Ez ante expected profits of seller 1 and seller 2 when there is bid transmission

between the upstream and downstream players are

11 UH
Rs (5:3) <7

and

Proof: See the Appendix.

Seller 1’s payoff is computed exactly as it was when there was a firewall. Seller 2’s payoff
changes, however. Seller 2 still gets zero if A is type v,. However, seller 2 now receives zero if A is
type vy and B is type vz, and receives vy if both A and B are type vy. Thus, the knowledge that
information will be transmitted leads seller 2 to conclude ex ante that competition in auction two
will be more vigorous than if no information were transmitted.

These analyses of expected seller revenues can be combined for a striking conclusion.

Theorem 4 Seller 1 prefers merging with bidder A to not merging, regardless of the existence of
a firewall. Seller 2 prefers having seller 1 and bidder A merge and transmit information both to

having no merger take place and to having a firewall imposed as a condition for merger.

Proof: If seller 1 and bidder A merge, seller 1 gets *#. In the pre-merger situation in Section 2,
seller 1 gets strictly less than %. A similar argument holds for seller 2. 0

Both sellers do better with the merger and information transmission than in the pre-merger
situation. The driving force behind this is B’s knowledge that A knows B’s type if there is no
firewall. This leads to more aggressive bidding in auction two when both players are type vy. In
the context of a manufacturer/distributor relationship, this result implies that distributors (the
receivers of bids) expect to pay less. Lower bids to distributors translate into lower prices for final
consumers, suggesting that some of the concern about vertical integration involving proprietary
information issues is misplaced.

The above analysis assumes that firm A merges with seller 1. If the welfare predictions change if

A merges with seller 2, for example, then care must be taken when using this work to evaluate real

19


Figure 1

bid A

b(v)

b(v2)

b(v))



in the award process. Bid revelation greatly simplifies the analysis compared to that of the two
period auction in Section 2. The difficulty there was that the winning firm’s belief about the loser’s
type was not common knowledge. Bid revelation makes the winner’s beliefs about the loser’s type

common knowledge, which permits the use of the asymmetric auction template developed in Section

2.

Proposition 8 The symmetric equilibrium bid distribution in auction one with only the winner’s

bid revealed, FI"A(), is the solution to

FYAQ) = 5 + (

UH WA
wot H :) Inf + F”A(0)).

UH

Proof: See the Appendix.

Type vy bidders do not pool at zero when only the winner’s bid is revealed. The reason players
submit zero bids when all bids are announced is to disguise their type when they lose. With only
the winner’s bid revealed there is no need to worry about disguising one’s type after losing, so
there is no incentive to pool at zero. This also explains why there are no pooling or semi-pooling
equilibria.

Although revealing all bids leads to some pooling at zero, it is straightforward to show that the

auctioneers do better when all bids are announced.

Theorem 3 The sum of the sellers’ expected profits are higher when all bids are announced than

when only the winner’s bid is announced and when no bids are announced.

Proof: See the Appendix.

The intuition for this result is fairly simple. Although there is some pooling at zero when
all bids are revealed, this is mitigated somewhat because the bidders’ mixed strategy distribution
includes bids strictly higher than those employed in the distribution when only the winner’s bid is
announced, Bidders potentially bid much higher because, if they are going to risk revealing they
are type vy by bidding a positive amount, then they have a strong incentive to win the first item.
Thus, the net effect on revenues for the first seller are unclear. However, the environment following
auction one tends to be more competitive when all auction one bids are announced. This is so
because of the likelihood that, if both types are revealed to be vy, then in auction two they will

completely bid away any surplus they might hope to obtain ex ante.

14


yields Nash equilibrium outcomes. Further, I assume that the seller reveals a “no bid” as a zero
bid when auction policy requires bid revelation. This causes bidders to assess positive probability
that their opponent is type vy, which induces higher bidding than if bidders knew they faced a
type vy opponent.

Both type vy players have the same expected profit in equilibrium. This is so because each
type vy player has the same high bid in equilibrium. The high bid wins with probability one and
yields payoff vy — b. Because each bid in a mixed strategy equilibrium gives the same expected
payoff both A and B have the same expected payoff.

In the asymmetric case, a type vy player A’s expected payment is higher than a type vy
player B’s. This appears trivial, because parts 2 and 3 show that A’s equilibrium bid distribution
stochastically dominates B’s. It also appears strange because both players have the same expected
payoff. However, when determining expected payments one must take into account not simply the
expected bid made, but also the probability that that bid is successful. Thus, a given bid by player
A may be less likely to win than the same bid by player B, because A more likely faces a competitor
bidding a positive amount than does B.

Proposition 1 allows easy computation of continuation values for auctions with asymmetric
beliefs. In the repeated auctions I analyze, players begin with symmetric beliefs, but their actions
today determine their updated beliefs about their opponents tomorrow. By using the template
above, bidders can calculate the payoff they expect to receive in the future for given actions of
theirs in the current auction. To do so, they simply define a and @ by the appropriately updated
beliefs based on their current actions.

A second reason for constructing this benchmark is that the repeated auction involves bidding
behavior that is used to learn about the valuations of one’s competitors. This benchmark, by
construction, does not exhibit such behavior. By comparing the two, it is possible to show how
the desire to elicit information about one’s rivals leads to changes in bidding behavior and auction
outcomes. This comparison also shows whether a seller of T identical items should sell them
simultaneously or sequentially."

The Two Period Bidding Model

I now solve for the symmetric equilibrium of the two period repeated auction to see how bidding

1° Although I informally examine what happens when this method is not used.

This last part is equivalent to having a buyer’s option, in that one should see the single shot bids. Bidders will
not post the low bids that arise in the repeated setting because once a bidder has won he will buy all of the items at
the current bid. This induces bidders to bid more aggressively than without the buyer’s option.


when only the winner’s bid is announced or when no bids are announced is 1.34657359v;;. Therefore, the
auctioneers do better when all bids are announced.

Proof of Proposition 9: Seller 1 earns aa half the time (when merged with a type vq player) and zero
half the time. So ex ante expected profits for seller 1 are “Hi. Ex ante expected profits of seller 2 when there

is no bid transmission between the upstream and downstream players are

Pl _ ft 1 2) [pa (1 1) [pa (1 ae
tea) = (q+ (a) (3) fo" Goa)] + (@) fe" Ga) eP (2
vn
3°
Proof of Proposition 10: As in Proposition 9, seller 1 earns oH, Ex ante expected profits of seller two

when there is bid transmission between the upstream and downstream players are

ro($2) = a ()or (2) on

UH

4

References

{1] Bikhchandani, S. “Reputation in Repeated Second-Price Auctions.” Journal of Economic The-

ory, Vol. 46 (1988), pp. 97-119.

(2

Froeb, L. “Auctions and Antitrust.” Economic Analysis Group Discussion Paper 88-8 (1988).

3

Froeb, L., and McAfee, P. “Deterring Bid Rigging in Forest Service Timber Auctions.” Eco-
nomic Analysis Group Discussion Paper 88-5 (1988).

[4] Gal-Or, E. “Information Transmission-Cournot and Bertrand Equilibria.” Review of Economic

Studies, Vol. 53 (1986), pp. 85-92.

[5] Green, E., and Porter, R. “Noncooperative Collusion Under Imperfect Price Information.”

Econometrica, Vol. 52 (1984), pp. 87-100.

(6] Hausch, D. “A Model of Sequential Auctions.” Economics Letters, Vol. 26 (1988), pp. 227-233.

(7]

Laffont, J.J, and Tirole, J. A Theory of Incentives in Procurement and Regulation. Cambridge,
MA: MIT Press, 1993.

29


b =e. Substituting ¢ for b in the above inequality and rearranging implies

v, 1
0= > ~ 5F (vn - 9.
Taking ¢ arbitrarily small ensures that the right hand side of this inequality will be strictly positive, because
F,(b) goes to zero as b goes to zero. This leads to the contradiction 0 > 0, so it was incorrect to assume
x3 (b | uz) = 73(0 | vz). Thus, B will deviate by bidding zero and there cannot exist a pure separating
equilibrium.

Proof of Proposition 6: Suppose Fj(b) is the first period bid distribution for the semi-pooling equilibrium.
By definition, F,(0) > 0. If A follows F,(b), then B’s expected payoff from bidding b > 0 is

1 F,(0)
B - 2 _ H i
m(blvn) = 5 [om b) +7 (en | T+ AO! +

Against vy

50) [Cou — 0.4" (vu | EO a) +

1+ F4(0)
Beating vq bidding zero
1
5) ~ FiO) — b) + 27 (ow | 1,1] +

Beating vy bidding positive

pl — Fil + *4(vn 11,0),

ee,
Losing to vq

which when simplified yields

ble) = [tow bh) tnt (i BO 0) s1)] +

1+ F,(0)’

Fu} (var | Ee 1) + FF Olen ~d)

bole nie

The payoff to a type vy player B from bidding zero is

(0 |vy) = glow tat (on | EO 1] +

5h) F {on + 0H (ve | reno) pt {ota (v# | eo")}] +

5h FO) [own (e no")

In equilibrium, 73 (b | vz) = 78(0 | viz), which permits computation of F,(b). To conserve space, let
_Fi(0) Fi(0)
Qeart 1 H 1 :
(ol cera)" (ol Ro

26


consumer prices, and consequently the vertical merger is procompetitive.

The issues raised above are intriguing. The reason they have not been previously analyzed is
that, despite the recent growth of interest in sequential auctions, no work? examines the simple
model I propose here. Other papers dealing with repeated auctions do not examine interrelation-
ships between auctions and how they drive the asymmetric nature of learning, with its consequent
effect on behavior. These issues are critical to understanding how decision makers act in dynamic
situations with incomplete information.

The rest of the paper is organized as follows. First, I briefly describe papers from the auction
literature which are related to this topic. Section 2 introduces a repeated auction which is a
simplification of the model in Thomas [1996a]. Section 3 examines the outcomes which arise under
a public price announcement scheme. Section 4 discusses the implications of a merger between a
buyer and a seller, relative to the pre-integration equilibrium, while Section 5 briefly concludes.
Background Literature
The typical model of a multi-period auction has either a series of independent single item auctions in
which participants receive new draws from the distribution of types each period or a series of single
item auctions in which each participant desires only one of the items. Thomas [1996a] introduces the
idea of repeated auctions, in which participants would like to acquire all of the items and in which
a participant’s values for all the items are identical. Other papers examine related topics in multi-
period auctions, but do not discuss the serial linkage of auctions. Because no other work examines
such a model, the question of vertical integration and bid revelation are largely irrelevant and are
certainly different from the questions posed here. If the auctions are independent, then both bid
revelation and vertical integration will not change behavior, beyond the vertical integration causing
a change in behavior in the auction involving the merged entity. In a traditional sequential auction,
vertical integration may have some effect, while Milgrom and Weber [1982b] show that revealing
the winner’s bid has no effect on behavior. The papers by Hausch [1988] and Bikhchandani [1988]
use bid revelation, but do not make comparisons of different revelation policies.

2 A Two Period Repeated Auction
This section presents a model in which two players® bid for two identical items. Each player has a

valuation v for each object, which is constant regardless of the number of items acquired.® I call

With the exception of this paper’s predecessor, Thomas [1996a].

“Other issues regarding the repeated auction process in general are discussed more fully in Thomas [1996a].

*I will use the terms player, bidder, and firm interchangeably.

“I find this extreme to be more plausible in an industrial organization context than the assumption that the utility


in a first price auction using the bid function b(v) shown in Figure 1. By losing, player 1 with
valuation v; knows v2 > v:. Similarly, by winning, player 2 knows v2 > v1. Consequently, dynamic
interaction and the resulting information transmission mechanism suggest that, if there is a second
auction, then the bidding behavior in the first auction differs from that in a single item auction.
The desire to learn about one’s opponents and to conceal information about oneself provides strong
incentives for strategic behavior.!

In this paper I investigate two policies which affect structural information transmission, the
information revealed through the basic market structure. I concurrently examine strategic infor-
mation transmission, which arises as an optimal response to given structural policies. Following
the model introduced in Thomas [1996a], I describe a situation in which bidders do not see rival
bids and learn only the identity of the winner, ex post. This will be the benchmark against which
I compare the following two changes in market structure.

First, I examine the impact of bid revelation policies on bidder behavior and auction outcomes.
Comparing a full revelation policy with the benchmark suggests that current policies in public
sector procurement lead to stronger competition than a policy which reveals no bid information.
This result may be used to counter arguments that current bid revelation policies stifle competition
in the public sector procurement process.? Moreover, taking as given that some bid revelation
policy is required, I show that revealing all bids leads to greater revenues for auctioncers than does
revealing only the winner’s bid. Finally, as is also true when bids are not revealed, I find that a
single seller bound by bid revelation rules prefers to bundle items for sale. This suggests that long
term contracts should be preferred by firms accepting bids from input suppliers, for example.

Second, in light of recent government interest regarding information flows between vertically
related firms, I examine the impact on firm behavior and consumer welfare of a vertical merger
between a buyer and a seller. This is relevant if manufacturers compete for the services of distrib-
utors, for example, and a manufacturer and a distributor consider merging. I find that the merged
manufacturer and distributor are willing to commit to a policy which prevents the transfer of infor-
mation about opponents’ bids, a remedy which has been applied in recent decisions. However, I also
show that if bid information is transmitted vertically, then expected payments by all distributors

(the receivers of bids) are lower than if there were no merger. These lower bids translate into lower

'See Thomas [1996c] for a more general discussion of this phenomenon.

?Of course, other factors may determine the efficacy of bid revelation policies, such as concerns about collusive
behavior.


This section illustrates how bidding behavior changes when the auction framework changes by
revealing bids. Most importantly, it shows how a single auctioneer restricted to short term contracts
is better off revealing all bids rather than only the winner’s bid or revealing no bid information.
This helps counter arguments that current policies in public sector procurement stifle competitive
behavior.

4 Vertical Mergers

Understanding vertical integration requires examination of several difficult and often conflicting
concepts. Beyond the typical efficiency enhancing and competition dampening claims about vertical
mergers, there are issues concerning the flow of information between vertically related entities. If
upstream horizontally related firms compete repeatedly for the services of downstream firms, and
if the resulting vertical relationships entail the transfer of proprietary information, then upstream
firms outside a particular merger may fear that information they transmit will flow directly to their
horizontal competitor via the competitor's link to the downstream firm.!8 Not only might this
transfer of information be anticompetitive by making it easier for the upstream firms to monitor
its rivals and hence coordinate their actions, these mergers may induce upstream outsiders not
to submit bids to integrated downstream firms, a consequence which could adversely affect social
welfare.

One proposed remedy in these situations is the imposition of a firewall between the merging
parties, a device which prohibits information gained by one party from being shared with the
other. Such a device, if it can be successfully implemented, should alleviate concerns about allegedly
harmful information flows. However, a larger and unanswered question is whether these information
flows actually harm competition.

The literature on information sharing in oligopoly examines this question in a related context
by modeling the incentive to share private cost and demand information by both price setting and
quantity setting firms. Vives [1984] shows that quantity setting firms with private information
about a common linear demand curve do not wish to share their private information, though
price setting firms do. Shapiro [1986] shows that quantity setting firms with privately known
linear marginal costs do want to share their private information. Most similar to this repeated
auction context is Gal-Or [1986], which shows that price setting firms do not wish to share private

cost information. The main difference is that the Gal-Or model is static while my formulation is

**Such concerns have arisen in the satellite (Martin Marietta/General Dynamics merger), pharmaceutical
(Lilly/PCS merger), and data processing (First Data/Western Union merger) industries, among others.

15


auctions has no effect on beliefs and hence on behavior in the current auction.?

Let F'4(b) and Fg(b) be the equilibrium bid distributions used by type vy players A and B in
each period. Let r*(v | a,@) and 7% (v | a, 8) denote the equilibrium expected profits of player
A and player B, respectively, whose type is v when the probabilities that A and B are type vy
are @ and 8. The L and H refer to A’s more likely being a low valued player than B. Similarly,
let P*(v | a,8) and P¥(v | a, 8) denote the equilibrium expected payments of player A and B,
respectively, when their type is v and the probabilities that A and B are type vy are @ and £.
Clearly P“(0 | a, 8) = P#(0 | a,8) = 0. More interesting are the expected payments of the
high valuation players. Finally, let R(a, 3) denote the seller’s expected revenue when player A and
player B are type vy with probability a and .

Proposition 1 The equilibrium in the asymmetric T item simultaneous auction is in mized strate-

gies and has the following properties:

1) Type vy players mix over (0, avy).

2) Fal) = (252) (35) — (458)

9) Fat) = (58) (sits) - C34)

4) (up | a, B) = 0" (vy | 0,8) = (1—a)onT
5) P* (vq | @,B) = (234) T

6) PH (vn | a,8) = (Sgt) 7

1) Re, B) = (o?vy)T

Proof: See the Appendix.

From part 3, one sees that if B is more likely to be vy than is A, then B will bid zero (vz) with
positive probability. The careful reader will notice that if Fp(0) is positive, then B faces the risk of
tying with a type vy player A. This implies that B should bid slightly above zero, thus unraveling
the purported equilibrium. This problem is minor and may be resolved in two ways. First, one
may assume that in equilibrium type vz players do not submit a bid. Second, one may assume the
existence of an infinitesimal bid € > 0 on which the probability mass actually rests. These methods
are employed in Maskin and Riley [1994], and they are typically equivalent. However, in Sections
3 and 4 I examine situations in which bids are revealed, and thus firms may truly wish to bid zero

and hide the fact that they are type vy. For this reason I employ the first method, which still

°Concerns about collusion are trivially dismissed in the simultaneous sale. In the sequential sale, collusion unravels
via backward induction arguments.


products.

With price announcements, the sellers do better having all bids announced rather than having
only the winner’s bid announced. This follows because revealing all bids generates a great deal of
public information which leads to vigorous competition. For similar reasons, the sellers do better
having all bids announced rather than revealing no bid information. These two results may be used
to counter arguments that current bid revelation policies stifle competition in public sector pro-
curement. A single seller, such as the government, prefers simultaneous sales rather than sequential
sales, for the same reasons as in the repeated auction without price announcements. In Froeb and
McAfee [1988], simultaneous sales are preferred because they reduce the possibility of collusion by
making orders more lumpy. Here the preference arises because simultaneous sales prevent firms
from gaining information about their rivals and eliminate the resulting soft competition.

With mergers, I find that bidders outside the merger would like to know that there is no
transmission of bid data between the upstream and downstream members of the merged entity.
Surprisingly, I find that the inside bidder is indifferent between knowing and not knowing the
bids of rival bidders. Also, because the outsider’s action in the first auction is unchanged by the
presence of an internal wall, the insider has no incentive to secretly see the bid of the outsider.
Finally, the sellers do better with the merger and no firewall than in the pre-merger equilibrium.
In a manufacturer/distributor relationship, this result implies that distributors pay less on average,
which leads to lower prices for final consumers and to the conclusion that the vertical merger
is procompetitive. Therefore, some of the concern about informationally related anticompetitive
effects of vertical mergers may be misplaced.

Though these results are from a two period, two bidder model, I believe the character of the
results remains as the number of auctions or bidders increases. For example, in the benchmark with
no bid revelation, bids should start out lower than usual, then go up over time, albeit more slowly
than in the two period case. The reason they will rise more slowly is that, say in the second of three
auctions, while the environment has grown more competitive following the first auction, there is
still reason to try to elicit information for use in the third auction. Of perhaps greater importance
is the robustness of these results to a change in the distribution of types. In particular, using
a continuum of types may generate, following bid revelation or vertical integration, the complex
pooling that arises in models exhibiting the ratchet effect.23

Appendix

?8See Laffont and Tirole [1993].

21


Market Structure and the Flow of Information
in

Repeated Auctions

Charles J. Thomas!

This revision: December 2, 1996

1Federal Trade Commission. First draft on May 23, 1996. I am indebted to Igal Hendel for some helpful
discussions and to Robert Willig for his valuable guidance. This paper is based in part on Chapter 1 of my
Ph.D. dissertation from Princeton University, and was supported in part by a National Science Foundation
Graduate Fellowship. Send correspondence to Federal Trade Commission-Room 5022, 601 Pennsylvania
Ave., NW, Washington, DC 20580, or to cjthomas@ftc.gov. This work represents the opinions of the au-
thor and is not intended to represent the position of the Federal Trade Commission or any individual
Commissioner.


which is independent of A’s bid a < FEA Gan

Firm B’s Problem

Suppose that for a given unsuccessful bid by in auction one, A bids according to the strictly
increasing function L(b4)) = pean) in auction two. A is willing to make this bid according to
the preceding analysis. B wins with bid 6 if and only if

vi F(ba1) 1 b
o> ba) ob > bar.
Thus, B’s expected payoff is
—1(_b
7B (b | bai) = 1 ( Fi(bai) ) A (Ai (ats) fox —) = 2 _,
1+ F,(ba1) 1+ Fi(bgi) F,(bp1) 1+ F,(bp1)

which is independent of b. Thus, B is willing to bid according to Fg(blbgi) = TRG (#5).
By examining the maximization problems of firms A and B, it is clear that for a given mixed

strategy distribution F\(b) in auction one, the equilibrium in auction two is given by the pure

strategy L(b) = yea for the loser of auction one and the mixed strategy distribution Fg (b|b1) =

TRG) (5) for the winner of auction one.

Characterization of Equilibrium

Let m(b) denote the equilibrium expected payoff in auction two of a type vy player who won

auction one with bid 6. Define 7j(b) analogously for a type vy loser of auction one. Let 7 be the

expected profit for a type vy player in a symmetric single shot auction. From Proposition 1, with

a=B= }, t= h.
Theorem 1 For all first period bids b € (0,61), m4,(b) > % and 17,(b) > 7.

Proof: From the derivation above, for all first period bids b € [0,1),

Tv) = TFA > > yA

and

i (b) = {in(2) — Inf + Fi(b)]] > a =

i 1-A®
This is the desired result. 0

Given the characterization of bidding in the two possible contingencies of auction two, having


in the two periods differs. I also compare bidding in the repeated auction to that in a series of
independent auctions. As is usual in this sort of dynamic game, I begin in auction two, taking as
given some bid distribution from auction one.!2

Auction Two

Suppose that type vy players A and B used the bid distribution F\(b) on (0, &)] in auction one, and
that player B won with a bid bgi, while player A lost with bid b41. Initial beliefs put probability
3 that either player is type vy. Following auction one, it is now common knowledge that B is type
vn, while player B’s updated belief that player A is type vy is Te. Notice that, conditional
on winning, B learns more about A’s type the lower was B’s winning bid. Similarly, a type vy
player A learns more about B’s first period bid the higher was A’s losing bid.

Though the type vy players used mixed strategies in auction one, I can characterize an equilib-
rium in auction two in which the loser of auction one uses a pure strategy and the winner of auction
one uses a mixed strategy. In this equilibrium, players condition their bid or bid distribution in
auction two on their bid in auction one and whether the bid was successful or not.

Firm A’s Problem
Suppose that for a given successful bid bg1, B bids according to

_ 1 UH
Fabe.) = tay (S = 7) .

This is the mixed strategy distribution B would use if B were known to be type vy and A were

type uy with probability TR Ga" A maximizes its expected profits for a bid a, given bay:

BAL

1— Fy(ba1)

Sb, Fo(als)Fi(s)ds

mH (a | bai) = ‘vy — al.

One concern with this formulation of A’s expected payoff is that there may exist pairs of (a, s) such
that Fg(als) > 1, when for those pairs it should be noted F (als) = 1. However, if A is restricted

to bids a < ¥8Albay | the maximum bid if a = Albay then Fp(als <1 for all s € [b41, 4].
THF (bai TAGs) ?
Thus, A’s expected payoff may be re-written as

by ”
Un Fi(s) Un
ds = In(2) — Inf. + Fyar))),
T=Fi@ad) Jog 1 AG) = TFG) N@) ~ M+ Aiea]

m$(a | bai) =

“Note that there will be no out of equilibrium beliefs, because players will not see the action of their opponents,
only auction outcomes.


seller in auction two are

Ra (5-5) = (3) w+ (2) eea.ay 4

(3) [R(Q,1)} + (3) [1 — Fy(0))2oe + (3) (1=[1 - F(0)}2) RO, 1).
Straightforward calculation yields

11)\_ (49-11Vi7
nea (58) ° (Stace)

Proof of Proposition 8: Suppose a type vy player A is following F(b), the first period symmetric Nash

equilibrium bid distribution. A type vy player B’s payoff from bidding b > 0 is

wB(bl un) = ; [ow —b) +04 (vs | Fi(b) 1)| +

3 T+’
Against vy,
5h) [ow —b) 4a" (% | ey) +

Beating vy

Sint (on | BG5.1) wide

1~ F\(b)

A — F,(b)]

Losing to vy

This may be simplified as

By} vy) = yee) Alen ,
7? (b| vn) = [1 + Fi(®)] (vu ~ 8) + 7 +3 ih, ithe)

In equilibrium, 79(b | vj7) and 78(0 | vj) must be equal. Equating the two implies

1 1 Y, 1
50+ AON en ~ 8) + = + 5[ln@2) — Ind + A))] = a + o + 5/2).

Minor algebra yields

Fi(b) = —2 + (25) ma + AO.

vn b
which is the desired result. 0
Proof of Theorem 3: When all bids are announced, a bidder’s expected payoff is the expected payolt

fromm bidding zero in auction one. Straightforward calculation using the profit function shows that a type

vp bidder expects to earn 1.280776406v,; when all bids are announced. A type vj bidder's expected payoll

28


Simplifying 7 (0 | viz) gives

7? (0 | vq) = 8 + On 9,
2 4
Equating (0 | vz) and 75(b | vz) yields
F,(b)(vy — bt) =Q+ 64 Aen ~ Fy(0)Q.
Substituting zero for b and simplifying gives
r= Vs

which can then be used to show

wera (ats) + (ats) (22),

Proof of Proposition 7: The expected payment of a type vy player, when beliefs that A and B are type

vy are 3 and 4, respectively, is
H 2 a Ys

P (v | 3 ) = [> [; + 50] Fi(B)db.

I solve this integral directly, not using the indifference relation on the integrand as in the proof of Proposition

1. The result of this tedious integration” is

2
Plot |«,8) = (5%) vn.

Expected revenues for the seller in auction one are
11 1 1 1 1 11
fo (3) = (3)(2)+(3) (3) [P(t 53)]+
1 1 1 1 1 1. > lil
(3) (3) [P (vm ba) ]* (3) G) [P (e155) + (om #3)

(=)

Let
__F(0) _ Vi7-3
~ 1+ RQ) Vi7-1’

the probability that a player is type vy given he bid zero in auction one. Ex ante expected revenues for the

*4Which is available upon request.

27


this a “repeated auction” to contrast it with a “sequential auction” in which players desire only a
single item. For simplicity I assume that players can be one of two types.

First, I characterize the equilibrium of a series of T independent single item asymmetric auctions.
There can be no learning in such an environment, so it will be the benchmark showing how behavior
changes once players have incentives to acquire information about their rivals. Second, I characterize
the equilibrium of the two period repeated auction in which players learn only the identity of the
winner. I show that bidding competition in auction one is softer than if there were no later auction.
Also, bidding in auction two is softer than if there were no previous auction. Thus, the bidders
earn higher payoffs in the repeated auction than if the two auctions were independent or were held
simultaneously.

To avoid any confusion, recall that the analyses of buying auctions and selling auctions are
conceptually identical. The model I use is of a buying auction in which bidders are buyers and
auctioneers are sellers. However, some of the examples I use have bidders as sellers. When I offer
insights about such situations I adjust my explanations accordingly.

A General Asymmetric Auction

Consider an asymmetric first price auction in which two players, A and B, can be one of two types,
uz and vy, with vy, < vg. For simplicity, assume vz = 0.7, Player A is type vy with probability
a, while player B is type vy with probability @. Without loss of generality, assume that a < 3.8
Suppose there are T items to be sold, and that a buyer has an identical valuation for each. If the
items are sold simultaneously, then it is as if there is a series of T independent auctions. Therefore,
the equilibrium is a T times repeated version of the equilibrium from a single item auction. A
buyer may submit the same bid for each item, or may draw each bid from the mixed strategy
distribution of bids. This indifference arises because the bidders are assumed to be risk neutral and
both methods of bidding offer the same expected payoff.

Alternatively, suppose the T items are sold sequentially but are only stochastically equivalent
to the bidders. That is, a bidder receives a new draw from the distribution of valuations for each

item, In this case, each auction is equivalent to a single item auction. Information from prior

of a second object is zero, as would be the case in a standard sequential auction. Of course, more generally it would
be nice to consider positive and negative synergies. My extreme assumption makes the analysis easier, while still
adding the correlation of values across auctions. .
This assumption will not affect the qualitative nature of the results. Suppose 0 < vz < uy. Type vz players earn
zero in equilibirum and bid vz. Methods similar to those described following Proposition 1 may be used to correct
for any problems associated with the possibility of ties.
®This model follows and extends example 2 in Maskin and Riley [1995]


Integration without a Firewall

Suppose A cannot credibly commit not to see B’s bid, even seeing a “no bid” as such. As is the case
when there is an internal wall, B will not bid a positive amount in auction one. Again, if B wins
auction one, then A knows B is type vq and B knows A is type uz, and this is common knowledge.
Also, if B loses auction one, then B knows A is type vy..The difference that arises when A sees
B’s bid is that if B loses and has submitted a bid, then A knows that B is type vy. B’s payoff from

bidding zero in auction one is

(Olen) = Slow +a (vn [0,1] + 510+ 24(vn | 1,0)

Against uz Against uy

1
= gles + vx] = v4.

If B submits no bid in auction one, the payoff is
1 11 1 11
Bi&no bid” m= o L at 2 L as
a? (“no bid” | uz) = [or (vw 15.5) | +3 [ore («7 15.5)]

Against vp Against uy

Again, B participates in auction one. This is because there is the chance that player A is type
uz, and B does not wish to lose the opportunity of both acquiring the first object and getting the
second for a low price. Regardless of the firewall, B does equally well in auction one. Examining
the payoffs in the second auction, it is clear that B is harmed in auction two by the revelation of
bid information. Thus, B definitely prefers there be a wall within the merged firm; B gets ree
versus vy. This result corresponds to the potential concerns of outside firms.

A’s payoff is not changed by the transmission of bid information. The payoff for a type ug
player A in auction one is 4. Because A learns B’s type following auction one, A receives a
positive payoff in auction two if and only if B’s type is vy. Thus, A’s payoff in auction two is

me (-| on) = 50" (0,1) =
2 2
Surprisingly, A is indifferent about seeing B’s bid; A’s total payoff over the two auctions is vy in

either case. This corresponds to the acquiescence of firms to having a firewall imposed; it either

18


respectively, is

7
PX (vy |a,6)= [HQ — 8) + BFo(O) FAO).

Using the indifference relation (1), the integrand can be rewritten to yield

5
P*(oy 1,8) = [a ~B)vm + BFa(t)vn) F(ab
cy
= (a—f)vn + Bon [ Fp(b)F4(b)db.
0

Some straightforward integration shows

7
| Fa(0)Fa(0)db = 1 — 55,

au

P¥(oy |, 8) =

A similar argument establishes P# (vy | 0,8) = 954".

7) Expected revenues for the seller are

R(a, 8) (1— @)(1 — B)(0] + (1 — a) B[PTM (vp | a, B)] +

a(t — B)[P* (vi | 0, 8)] + aB[P* (vi | «,8) + P*(vx | a, 8)]

= avy.

Each term in the expression for R(q, 3) is the probability of a certain state of the world (combination of

player types) times the expected payment of players in that state of the world.
Proof of Proposition 4: Suppose player A of type vy submits a zero bid with probability one. The payoff
to a type vy player B from bidding zero is

780 lon) = 5 [ow +a (»13.2)| 45 [5 fon +a (v1 ja)} + Bota (15.4) }].

Against v;, Beating vy Losing to vy

The beliefs in the continuation game are computed as follows. If B wins with a zero bid, this is either because
he beat a type vz, player who submitted no bid, or because he won the tie break against a type vy player
A bidding zero. The probability that A is type vy given that B beat A with a zero bid is } Similarly, if
B lost to A, then B knows A will use the same thought process to determine new beliefs that B is type vy.
Simplification of 4(-) by using part 4 of Proposition 1 yields

l7uy

7m? (0| un) = 1

24




opponents’ output and are uncertain about demand. If firms could monitor each other’s output
choices, then they could collude and avoid price wars, even with low demand realizations,

A related yet often overlooked point when discussing collusion in auctions is that firms’ ability
to collude is reduced when they are uncertain about one another’s production costs.!3 This gives
another strong motive not to reveal bids in repeated auctions, becanse otherwise firms will form
better estimates about the costs of their rivals. From the point of view of a government department
soliciting bids from contractors, these two points suggest that bid revelation leads to higher expected
payments.

One issue not previously considered in the procurement context is the informational effect
on non-collusive bidding behavior which arises from bid revelation. In a single item auction, or
equivalently, when firms redraw their types for each auction in a repeated setting, the information
gained through seeing rivals’ bids is irrelevant.!4 However, once firms face each other repeatedly and
with unchanging valuations, such information can be crucial. If the objects sought have a common
value aspect, then the bids of other players reveal information about the common state variable.
Similarly, if the objects have private value aspects, then the bids of other players reveal information
about their type. Given the properties of equilibrium in Section 2, with no bid announcements,
a natural and compelling question to ask is how the price announcement affects bidding behavior
and expected revenues.

I use the model developed in Section 2 to examine the effect of bid revelation. In particular,
I study two different bid announcement policies seen in practice. The first entails revealing all
bids, while the second reveals only the bid of the winner. Because generating public information
about firms’ costs leads to more vigorous competition when firms set prices, revealing all bids
leads to greater revenues for the auctioneer than revealing only the winner’s bid, which helps
counter arguments that current policies in public sector procurement stifle competitive behavior.
Surprisingly, revealing no bids leads to the identical revenues for the auctioneer as revealing only
the winner’s bid.

Revealing All Bids

Suppose A and B participate in a two item repeated auction, but both players’ first period bids are

*SRefer to the information sharing literature, such as Vives [1984], Gal-Or [1986], and Shapiro [1986].

“Unless a firm’s distribution of types is unknown. Then knowing the draws a firm has had will afffect beliefs about
the true distribution of that firm’s type.

**Hausch [1988] presents a repeated auction model with bid revelation, but his model is inapplicable to the inde-
pendent private values framework I use. However, he does explain behavior in the context of the common and the
strictly mixed common and private value frameworks I mentioned above.

11


A winning player is revealed as type vy no matter what bid revelation scheme is used. The
reason for mimicking type vz players is to hide one’s type in case of a loss in auction one. This
response intuitively contrasts with the results in both the single item symmetric auction and the
two period auction without price announcements, both of which have no pooling of bids at zero.
Because of the mimicry by the type vy bidders in auction one, seller 1 does worse than if there
were no later auction. Also, because of the likelihood of the auction one winner thinking the loser

is type uz, seller 2 does worse than if there were no previous auction.

Proposition 7 In the semi-pooling equilibrium with all bids announced, the ex ante expected profits

R. 1W_ 5= vir 7
s1\ 55) = q H

The ex ante expected profits of seller 2 are

of seller 1 are

Roo (Gj ) _ 49 — Lvit
272 4(/17 — 1)?
Proof: See the Appendix.

Because both sellers do worse in this dynamic environment relative to the static one, if there
is a single seller who is legally required to release bid information, then she prefers selling them
simultaneously. A preference for simultaneous sales arises in Froeb and McAfee [1988], which
suggests making sales more lumpy to increase firms’ incentive to cheat on collusive agreements. In
the repeated auction model, the preference for simultaneous sales is because their use reduces the
flow of information to bidders.

Another interpretation of the result that sellers do better in static auctions, as with the result
without price announcements, is that the sellers do better when the auctions are independent. In
a sense, the sellers would like to differentiate their products. They are harmed by the fact that
they are perfect substitutes, though not because bidders are choosing one item instead of the other;
bidders wish to acquire both items. The sellers are harmed for two reasons. First, the bidders
use the similarity of the items to learn about each other’s valuation and thus reduce their bids in
auction two. Second, the learning is accomplished by bidding low in the first auction.

Revealing the Winning Bid
In some auctions only the winner’s bid is announced, a policy which should seemingly be preferred

by the sellers. It reveals fewer bids, which eases fears about collusion, and yet it prevents corruption

13


Abstract

There is an ongoing public policy debate regarding vertical integration and its concomitant in-
formation flows. Of particular concern is that the information derived by an auctioneer (such
as a distributor) will be shared with its integrated bidder (such as a manufacturer), leading to
a reduction in competition between the bidders. Similar competitive concerns arise regarding bid
revelation policies, such as those used in public sector procurement. Modeling such situations as the
repeated auctions introduced in Thomas [1996a], this paper examines the transmission of private
information via the auction outcomes, and shows how that transmission is affected by the changes
in market structure described above. These concerns are not present in the existing auction liter-
ature, because such information transmission is irrelevant both in series of independent auctions
and in sequential auctions in which participants desire only a single item. However, when bidders
desire multiple items, and when the values of those items to a bidder are correlated, the incentive
to learn about. opponents’ values and to obscure one’s own drive equilibria to depart systematically
from those in standard models. I examine structural information transmission, created through
various policies for conducting auctions, and its effect on strategic information transmission, which
arises as an optimal response to given structural policies. I initially model information acquisition
in situations where bidders do not see rival bids and learn only the identity of the winner. I then
extend this model to show how the desire to conceal information about oneself affects behavior
by examining repeated auctions with publicly announced bids. The auctioneers prefer a policy of
revealing all bids to revealing only the winner’s bid and to revealing no bid information. Finally,
I show that a vertical merger between a buyer and a seller can be procompetitive due to both

structural and strategic information transmission.


Proof of Proposition 1: 1) Suppose player A bids a when his type is vy, while Player B bids b. If b > a,
then B prefers bidding adh, which still wins with probability one but involves a lower payment. The reverse
holds when a > b. If a = 6, then B’s payoff is

1 lta
43 (b | v7) = a(vy — 6) + (1—a) (5) (vq — 6) = (vn — 6) (4) :
B’s payoff from bidding 6 + € is

mB (b+e| un) =vy —b-e.

For small ¢, 79(b + | vz) > 7(b | vz), so B cannot bid 6 in equilibrium. Thus, there is no pure strategy
equilibrium.

If both players do not mix over the same set of bids, then one player has an incentive not to use the
higher bids. Suppose the range of bids is not a connected interval. For example, suppose bids are over
(b;, 61) and (by, 62) with b; < by. Then a player bidding very near by could lower his bid to Fitba decreasing
his probability of winning by an arbitrarily small amount yet lowering his expected payment when he wins.
Thus, both players bid over an interval (6, 8].

Suppose player A bids bid b € (8, 8] with positive probability. B must bid b* € (b—€,8] by the previous
argument. If B instead bids b* +, his probability of winning increases discontinuously, while his expected
payment when he wins goes up by an arbitrarily small amount. This is a profitable deviation for B, who will
therefore not bid 6* € (b — ,b], a contradiction. Thus, there are no mass points in (b— e,}].

Suppose both A and B bid b with positive probability. If B deviates to bidding b + €, his probability of
winning increases discontinuously, while his expected payment when he wins goes up by an arbitrarily small
amount. This is a profitable deviation for B, who will therefore not bid b. Thus, at most one player bids b
with positive probability

Suppose b > 0 and that A bids according to F'4(b) .with F'4(b) > 0. The payoff to B from bidding b+

git + Fab+ oon ~2~6).

The payoff from bidding ¢ is
1 (
glen €).

Taking ¢ arbitrarily small ensures that bidding ¢ yields a higher expected payoff than bidding ) + €. Thus,

it must be that b= 0.

22




(8] Lebrun, B. “Existence of an Equilibrium in First Price Auctions.” to appear in Economic

Theory (1995a).

{9] Lebrun, B. “First Price Auctions in the Asymmetric N Bidder Case: Part I.” unpublished
manuscript (1995b).

(10) Lebrun, B. “First Price Auctions in the Asymmetric N Bidder Case: Part II.” unpublished

manuscript (1995c).
{11] Lebrun, B. “Reputation Differences in First Price Auctions.” unpublished manuscript (1995d).

{12] Maskin, E., and Riley, J. “Equilibrium in Sealed High Bid Auctions.” unpublished manuscript
(1994).

{13] Maskin, E., and Riley, J. “Asymmetric Auctions.” unpublished manuscript (1995).

{14] McAfee, P., and Vincent, D. “The Declining Price Anomaly.” Journal of Economic Theory,
Vol. 60 (1993), pp. 191-212.

(15) Milgrom, P. and Weber, R. “A Theory of Auctions and Competitive Bidding.” Econometrica,
Vol. 50 (1982a), pp. 1089-1122.

{16] Milgrom, P. and Weber, R. “A Theory of Auctions and Competitive Bidding, II.” unpublished

manuscript, Northwestern University (1982b).

[17] Ortega-Reichert, A. “Models for Competitive Bidding Under Uncertainty.” Ph.D. thesis, Stan-
ford University (1967).

[18] Shapiro, C. “Exchange of Cost Information in Oligopoly.” Review of Economic Studies, Vol.
53 (1986), pp. 433-446.

[19] Stigler, G. “A Theory of Oligopoly.” Journal of Political Economy, Vol. 72 (1964), pp. 44-61.

{20} Thomas, C. “Low Bidding as a Learning Device: Information Acquisition in Repeated Auc-

tions.” unpublished manuscript (1996a).
{21] Thomas, C. “On Bidding Functions in Sequential Auctions.” unpublished manuscript (1996b).

{22} Thomas, C. “Testing the Water as a Means of Information Acquisition.” unpublished manu-
script (1996c).

30


If B deviates and bids € > 0, then he will win for sure while revealing himself to be type vz. However,

he would also reveal himself to be type vy if he won auction one with a bid of zero. More importantly, B

learns nothing about A’s type. B's payoff from deviating is
1
= fom ~etant (v | 3)}

1 1
me | vn) = 3 [vw ~etat (v0 | 3] +5
Against vz, Against uy
which simplifies to
we lun) = Se ~e.

This is a profitable deviation for B, for suitable e. Thus, there cannot be a pooling equilibrium.
Proof of Proposition 5: Suppose there is a pure separating equilibrium with first period bid distribution
F,(»). By definition, F,(0) = 0. If A follows F, (6), then B’s expected payoff from bidding 6 > 0 when his

type is uy is
[(vir ~ 8) + (vy | 0,1)] +

m(bl vr) =

:

Against v;,
1 1
5Fi(0) [(ver — 8) + 2# (vy | 1,1)] +50- AO) (O+a*(vy |1,1)].
Losing to vz

Beating vy
After beating a type vz player A, B knows it. Similarly, when both bid positive amounts each knows the
other is type vq. Now 7¥ (uy | 1,1) = 2*(vq | 1,1) =0 and +¥ (vy | 0, 1) = vy, so

bo1
m1? (b| vn) = vn — 3 + 5Fil)(en — 6).
Suppose B deviates by bidding zero. B will still beat a type v,, player A, and will fool a type vy player A

into thinking that B is type vz. Thus, in auction two a type vy player A will bid zero and lose to player B,

who will bid « > 0. The expected payoff to B from this strategy is

1 1 3
7 (0 vn) = glen +2 (vir |0,1)] + 510+ (on — 0] = SE

‘Against vz Against vy

If r?(b| vy) = 1 3(0 | viz), then
bol Sun €
MH 9 + 5 Fi()(ru — 0) = = - oe

This inequality must hold for any in the support of F,(b), [0,5], for some b > 0. In fact, it must hold for

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either won or lost auction one, it is possible to determine optimal bidding behavior in the first
auction.

Auction One

The expected payoff to player B from bidding b in auction one, given optimal play in auction two

and assuming that a type vy player A is using F,(b), is

w3(bl| uy) = sll + F,(0)] [(ux — 6) + wel +5 stl — F,(b)] 7y(b)

= 95 t + F\(d)] (vy — 6) + 2% 4 SH F [in( 2) —In(1+ F,(b))]

Proposition 2 The equilibrium bid distribution in auction one, F,(b), is the solution to

n= (aa) *(Gs

Proof: Because bidders use a mixed strategy in auction one, each bid in the bid distribution yields

) ma 40).

the same total expected payoff over the entire auction. Thus, 1? (b | vi) = 73 (0 | v7), so
zt + FQ] Ow ~0) + + fing) - In. + HO))] = on + H In(2).

Minor algebra yields
_ b UH
F,(b) = Gs — :) + (5 — 3) In(1 + Fy(0)),

which is the desired result. 0
It is easy to show that equilibrium bids in auction one are lower on average if there is a later

auction. That is, bidders bid less aggressively if there exists a later auction.

Proposition 3 The equilibrium bid distribution from Proposition 1 with a = B = 4 first order
stochastically dominates F,(b).

Proof: Follows from a simple comparison of the two distributions. 0
It is straightforward to rank the profits from the repeated auction and the series of independent

auctions.

Theorem 2 Each bidder’s expected payoff is higher in the two period repeated auction than in a

series of two independent auctions.


revealed after auction one. As in Section 2, I assume that type vy players submit no bids in either
auction, so the only interest is in the behavior of the type vy players. It is easy to show that the
incentive for type vy players to deviate from either a pure pooling equilibrium or a pure separating
equilibrium is too great, and that the only equilibrium is semi-pooling.

Pooling equilibria entail bids of zero in auction one by. both types of players. More precisely,
type uy, bidders do not bid, while type vq bidders bid zero, with the seller revealing both as zero

bids.!© The winner, if there is one,!”

is revealed to be type vy, but no information is revealed
about the loser. Thus, the second auction will have the same expected outcome as the single item
symmetric auction in Proposition 1. The following proposition shows that type vy bidders have an
incentive to deviate from a pooling strategy. By bidding « > 0 a type vy bidder can win auction

one for sure without affecting his payoff in auction two.
Proposition 4 There does not exist a pooling equilibrium in this game.

Proof: See the Appendix.

In a pure separating equilibrium types vz and vy bid different amounts with probability one. A
type uy player has an incentive to bid zero and mimic a type vy, player, however. The bidder will
still beat a type vy rival and will convince a type vy rival that a bid of zero is sufficient to win

auction two. Thus, the bidder “lying in the grass” can obtain the second object for almost nothing.
Proposition 5 There does not exist a pure separating equilibrium in this game.

Proof: See the Appendix.

In a semi-pooling equilibrium, type vy players bid zero in auction one with positive probability.
Thus, seeing a zero bid by one’s opponent is not proof that he is type vy. However, positive bids
will still reveal that one is type vy. Proposition 6 describes bidding behavior in the first auction

of a semi-pooling equilibrium.

Proposition 6 The symmetric bid distribution in auction one with all bids announced, F/‘4(b),

is FAA(p) — —& vn (VI7-3)
is FY ()=so+ ab) .

Proof: See the Appendix.

+The seller has an incentive to do this. If “no bids” were revealed as such, then the seller will earn zero in the
second auction.

‘Note that both players may be type vz, in which case there will be no winner.

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