Information Revelation in Auctions with Common and
Private Values∗
Xu Tan†
January, 2014
Abstract
Incentives to reveal private information are explored when there are both common
and private components to agents’ valuations and private information is held on both
dimensions. When agents only observe one signal, they do not have incentives to
reveal it, because revealing the only signal makes them fully lose their information
advantage. However, when agents observe multiple signals, they may have incentives
to reveal some signals in order to earn higher profits from other signals. This paper
shows that there exists an equilibrium with revelation of common-value signals and
concealment of private-value signals in standard auctions. This equilibrium achieves full
efficiency and may give the seller a higher profit. Experiments confirm these theoretical
predictions. For example, subjects tend to hide private information in a pure common
value environment, and choose to reveal the common-value signal in the presence of an
additional private-value signal.
Keywords: information revelation, auctions, two-dimensional values, efficiency
JEL Classification Codes: D44, D82, L13
∗
I am very grateful to my advisors Matthew Jackson, Kyle Bagwell and Muriel Niederle for the valuable
suggestions and support. I also thank Douglas Bernheim, Yan Chen, Matthew Elloit, Ben Gulob, John
Hatfield, Han Hong, Yuichiro Kamada, Fuhito Kojima, Eric Mayefsky, Paul Milgrom, Tomas RodriguezBarraquer, Ilya Segal, Alex Wolitzky, and audience from Caltech, University of Washington, MEDS Kellogg,
Johns Hopkins University, Cornell and UBC for their helpful comments. Financial support from the B.F.
Haley and E.S. Shaw Fellowship for Economics through SIEPR is gratefully acknowledged.
†
Department of Economics, University of Washington, Seattle, Washington 98105, USA. Emails:
tanxu@uw.edu.
1
1
Introduction
Communication plays an important role in many games with private information, such as
deliberation among voters and information sharing in oligopoly competitions. Private information is prevalent in auctions, however, communication does not get much attention.
Among a few papers studying the value of information in auctions with one signal (like
Milgrom and Weber (1982b) and Kovenock, Morath and Munster (2010)), the common conclusion is that information is valuable and no incentive to share it. But when we extend to
a general setting of auctions with multiple signals, communication becomes important.
Valuations in many auctions include both common and private components. For instance,
in markets with clear common values, private values exist when players are heterogeneous
in some aspects. One example is the oil tract auction, where firms care about not only the
common quantity of the oil, but also their private costs of extracting from the tract.1 On
the other hand, in markets with clear private values, common values exist if the item will be
resold in the future. For instance in the housing market, buyers have private preferences for
a particular house, and they also care about the future price (common value) of the house.
This paper explores the incentives of verifiable information revelation2 in auctions, where
private information includes both common-value and private-value signals. Consider the
following example: The government sells an oil tract in an auction. There are two bidders:
one “insider” who knows the quantity of the oil in the tract and one “outsider” who does
not know the quantity information. Both bidders have their own costs of extracting from
the tract, which are private information. We ask the following questions: Do they want to
share the quantity and/or cost information (via verifiable evidence) with each other? And
how does the social welfare and the seller’s profit change with information revelation?
The first contribution of this paper is to illustrate the differences in information revelation between auctions with one signal and multiple signals. Following Milgrom and Weber
(1982b), the idea that agents earn a positive rent from private information is generally accepted, and thus there is no incentive to share information. After all, if there is only one
signal, sharing the signal makes agents fully lose their information advantage, and hence is
not profitable. However, this implication may not hold with two-dimensional signals. Take
the oil tract example: Suppose the insider has an extremely low cost of extracting from the
tract and observes the quantity of the oil is small. If the insider can still make a positive
profit because of the extremely low cost, I claim that the insider may have incentives to
reveal the quantity. This is because if the insider reveals the small quantity, the outsider
1
Another example is the sale of timber-harvesting contracts studied in Pesendorfer and Swinkels (2000).
This paper focuses on revelation with verifiable evidence: Agents need to provide evidence when revealing
a signal, such that they can truthfully reveal a signal or hide it, but cannot lie about the signal.
2
2
would submit a lower bid as the value of the tract is lower. As a result, the insider can win
the tract with a higher probability and/or with a lower price. Therefore, revealing the small
common quantity increases the insider’s payoff from the low private cost. To generalize this
intuition: when there are two signals, agents do have incentives to reveal the signal that
could lower other competitors’ bids, and earn a higher profit from the other signal.
The second contribution of this paper is to explain the fact that considering information
revelation makes the two-dimensional signal model more tractable and yields different results.
Recent studies point out some issues in auctions with two-dimensional signals, such as
losing full efficiency as discussed in Pesendorfer and Swinkels (2000) and losing existence of
equilibrium as shown in Jackson (2009). When observing two signals, agents can submit
only one bid. Unlike the case of one signal, where the equilibrium bidding strategy is a
one-to-one monotone mapping,3 the bidding strategy now is a two-to-one mapping where
monotonicity cannot be naturally defined. For instance, a realization of a low private value
and a high common value cannot be compared to another realization of a high private value
and a low common value. Thus, the equilibrium turns out to be a harder (even impossible)
problem to solve. We also lose full efficiency due to the loss of monotonicity, since there is
no guarantee that the winner is the agent with the highest value.
If information revelation is considered, the predictions are different. First, there generally
exists an equilibrium where agents fully reveal their common-value signals. The auction is
then reduced to a pure private-value setting, and agents use the monotone bidding strategy. Second, there is no efficiency loss in this monotone equilibrium, because the winner is
guaranteed to be the agent with the highest value. In addition, the seller may also prefer
information revelation, since it increases the seller’s profit by reducing agents’ information
rents.
The third contribution of this paper is to validate the theoretical predictions on information revelation by experiments. I confirm that subjects do not want to reveal information
when there is only one signal, while they are incentivized to reveal some information in the
case of multiple signals. For instance, with just one low common value, only 25% of subjects
chose to reveal at least once, and none of them chose to reveal with a probability higher than
50%. While with two-dimensional signals where the common value is low, the revelation is
much higher: 83% of subjects chose to reveal at least once, and 58% chose to reveal with a
probability higher than 50%. This difference in revelation between pure common value and
two-dimensional values is significant.
In summary, agents’ incentives to reveal information are different between auctions with
one signal and multiple signals. In contrast to the fact that agents do not want to reveal the
only signal, they choose partial revelation when observing multiple signals. Common-value
3
Monotonicity generally means bids are monotone in signals.
3
signals are revealed to lower other agents’ bids and to earn higher profits from private-value
signals. The reduction of information asymmetry from revelation also leads to a improved
social welfare and a possible higher profit for the seller.
In addition, this paper also studies the incentive to acquire the common-value signals.
Milgrom and Weber (1982b) predicts that agents want to overtly collect information in
order to show off their information advantage. However, if communication is considered, the
prediction is the opposite. Common-value information would be fully revealed after being
overtly collected, such that agents don’t have incentives to do so if it is costly. On the other
hand, agents do want to covertly acquire some information in order to get an information
advantage.
1.1
Related Literature
This paper is mainly related to two strings of literature: strategic communication and twodimensional valuations.
First, in strategic communication, one related paper is Okuno-Fujiwara, Postlewaite and
Suzumura (1990), where they also work with information revelation of multiple signals. They
provide sufficient conditions for full revelation, which require differentiable payoffs.4 However, auctions do not have differentiable payoffs, because the winner takes all. A recent paper
by Hagenbach, Koessler and Perez-Richet (2013), nesting Okuno-Fujiwara, Postlewaite and
Suzumura (1990), doesn’t require continuity. They show that the full revelation equilibrium
exists if we can find a worst-case type, such that if a hidden signal is believed to be this
worst-case type, agents have incentives to reveal the signal. We share a similar spirit when
constructing the incentives of revelation. This paper examines and emphasizes the communication in two-dimensional auctions, which is a well-acknowledged important setting but
hard to analyze without considering revelation. Moreover, if we consider revelation of both
common and private signals, this paper doesn’t predict full revelation.
Analysis here makes a strong use of the communication models where agents can conceal
information but cannot overtly lie. This fits many applications where the signals are in the
form of verifiable evidence (hard information), such as the inspection reports in the housing
market. There is a large literature that distinguishes soft information (Crawford and Sobel,
1982) from hard information (Milgrom, 1981 and Grossman, 1981). The “persuasion games,”
as introduced by Milgrom (1981) and Grossman (1981)5 , are cases with hard information.
The unraveling argument from this literature is used in this paper.
4
See Assumption 4 in Okuno-Fujiwara, Postlewaite and Suzumura (1990) for details.
Subsequent investigations include Milgrom and Roberts (1986), Shin (1994) and Lipman and Seppi
(1995), among others (see Milgrom (2008) for some recent discussion).
5
4
Very few studies have been done on information revelation in auctions, mainly because
bidders rarely have the incentive to share information in the symmetric one-dimensional
valuation setting. For example, Milgrom and Weber (1982b) established the idea that private information is valuable by showing “the bidder with only public information make no
profit at equilibrium and the bidder with private information generally makes positive profits
in common-value auctions.” Moreover, Kovenock, Morath and Munster (2010) considered
information revelation in all-pay auctions, and they concluded that in both pure commonvalue and pure private-value auctions, agents do not have incentives to share information
individually.
The closest related literature on information revelation in auctions is Benoit and Dubra
(2006), where they show that bidders have incentives to share information when their values
are asymmetric. Incentive of revealing information in their paper is similar. Agents want to
reveal information in order to lower other agents’ bids, when they can earn positive payoffs
after the revelation. The positive payoffs may come from asymmetric values as in Benoit
and Dubra (2006) or from another dimension of private values as in this paper. Instead of
the one-dimensional signal setting in Benoit and Dubra (2006)6 , this paper considers a more
general setting of two-dimensional signals, and shows that information revelation happens
in this very standard setting and on the other hand it helps to make the model tractable.7
Another remotely related literature is information revelation in an oligopoly, which mainly
focuses on ex ante choices (trade association) with one-dimensional signals, and the results
are mixed8 . The ex ante problem is very different and does not answer the interim questions,
such as after observing a change in the cost, does one firm has incentive to reveal or conceal
it? Some oligopoly (such as Bertrand) is similar to auctions, so some results can be adapted.
Second, recent studies of two-dimensional valuations in auctions show that many standard results are not robust when changing from one signal to multiple signals. Pesendorfer
and Swinkels (2000) show that inefficiency appears with a positive probability in auctions
with two-dimensional valuations, but the asymptotic efficiency could be achieved in a large
society. Goeree and Offerman (2003) solve the equilibrium when the common value has a
specific (linear) form, and further confirm the positive efficiency loss. Jackson (2009) provides a simple discrete example in a standard second-price auction that equilibria can fail
6
In their paper, they consider adding private-value components to a common-value auction. But the
private-value components are common knowledge, so agent still observe one signal.
7
Another difference: in their general theorem 1 and 2, they impose restrictions on the equilibrium payoffs
in auctions, and these restrictions simplify the analysis because equilibrium payoffs from asymmetric auctions
could be very complicated. In this paper, I don’t impose such restrictions but instead focus on three popular
types of auctions, first-price, second-price and all-pay auctions.
8
Vives (1990) provides a good survey, and the main papers in the literature include Clarke (1983), Vives
(1984), Gal-Or (1985) and (1986), Li (1985), Shapiro (1986), among others.
5
to exist with two-dimensional valuations. Similarly, I show that incentives of sharing information are also different between cases with one signal and multiple signals. Considering
communication yields different predictions: the existence of equilibrium and full efficiency
hold with information revelation.
Another related paper is McLean and Postlewaite (2004), where they study a general
multi-dimensional signal setting, and show that a modification of a Vickrey auction with
side payments from the seller to the bidders could lead to an efficient outcome. Even though
I don’t focus on the mechanism design question, it can be addressed by the main results of
this paper in the setting of two-dimensional signals. With communication among bidders, a
standard second-price auction could achieve full efficiency and a higher profit for the seller
than that in the mechanism designed in McLean and Postlewaite (2004), because there are
no side payments to the bidders.
Finally, Jehiel and Moldovanu (2001) study the efficient mechanism with multi-dimensional
signals, and show that it only exists in nongeneric cases. The communication method in Jehiel and Moldovanu (2001) is cheap talking, while this paper considers communication with
verifiable evidence. Thus, the efficiency result in this paper doesn’t imply that the model is
nongeneric. In contrast, it covers most two-dimensional valuation models in the literature
such as Pesendorfer and Swinkels (2000) and Goeree and Offerman (2003).
The reminder of the paper is organized as follows. Section 2 introduces the basic intuition
of information revelation in the oil-drilling example. Agents have incentives to reveal the
common-value signal in the presence of an additional private-value signal, while they don’t
have such incentives without the private-value signal. Section 3 describes the setup of the
model, and Section 4 characterizes the equilibrium of the general model, the social welfare
and the seller’s profit. Section 5 provides two extensions of the model, including extensions
on communication and information acquisition. Section 6 presents the design and results of
the experiments, and confirms the theoretical predictions. Section 7 concludes.
2
Oil-Drilling Example
This section uses an oil-drilling example to explain the following results:
1. When agents only observe one common-value signal, there is no incentive to reveal
that signal (intuition we know from the literature).
2. When agents observe both the common-value and the private-value signals, there is
incentive to reveal the common-value signal.
Suppose there is one oil tract for sale in a second-price auction. The quantity of the oil in
the tract is q ∈ {qH , qM , qL } (qH > qM > qL > 0), where q = qi with a probability pi and
6
pH + pM + pL = 1. There are two bidders: the “insider” knows the exact quantity, while the
“outsider” only knows its prior probability distribution.9
2.1
Pure Common Value
The value of the oil tract for both bidders equals the quantity of the oil times the difference
between the price of the oil and the cost of extraction:
u = (p − c)q
The price of the oil (p) and the cost of extraction (c) are exogenous and common knowledge
in the pure common-value setting.
The timeline is as follows: At time 0, the insider privately observes the true quantity
q; at time 1, the insider decides whether to truthfully (via verifiable evidence) reveal q or
hide it; the outsider observes the insider’s revelation decision and the true quantity q if it is
revealed; at time 2, both bid in a second-price auction.
I focus on the perfect Bayesian equilibrium surviving iterative elimination of weakly
dominated strategies.10 The refinement requires:
• The bidding strategy is undominated in a second-price auction.
• The revelation strategy is undominated given bidders use undominated bidding strategies, and so on.11
Considering undominated strategies is helpful to eliminate some degenerate equilibria. It
is well known that there exists an asymmetric equilibrium in second-price auctions. For
example, let the outsider unconditionally bid the highest possible value (p − c)qH , and let
the insider bid 0. This is an equilibrium regardless of the true quantity. In order to eliminate
this sort of degenerate equilibrium, the bidding strategy is required to be undominated. In
a second-price auction, the undominated strategy is bidding the true value. Given bidders
use undominated bidding strategies in the auction, some revelation strategies are weakly
dominated as follows.
Consider the insider’s incentive of revealing the quantity q: if the insider reveals q, the
outsider knows the true value is u = (p − c)q, and his/her unique undominated strategy
is bidding the true value in a second-price auction. Such bidding strategy of the outsider
9
These settings, such as asymmetric bidders, discrete quantities and second-price auctions, simplify the
example. The general model doesn’t require these settings.
10
This refinement is useful to predict the unique equilibrium in this example. The main existence results
for the general model don’t need it.
11
The proofs use 2-3 iterations of eliminating weakly dominated strategies, and the order of elimination
doesn’t matter unless clearly stated as in Example 1.
7
makes the insider earn a payoff of 0. This is because even if the insider wins the auction,
the price is the outsider’s bid, which equals the true value u = (p − c)q. On the other hand,
if the insider doesn’t reveal the quantity (assuming q 6= qL ), depending on the outsider’s
bidding strategy there are cases where the insider could earn a positive payoff. So revealing
the quantity is weakly dominated given bidders use undominated bidding strategies.
Claim 1 Consider the oil-drilling auction with two bidders (one insider and one outsider)
and pure common value (quantity); in all iteratively undominated perfect Bayesian equilibria,
all quantities except the lowest one (q > qL ) are concealed.
Proof of Claim 1: First eliminate all weakly dominated bidding strategies in the secondprice auction. The undominated strategy for the insider is bidding the true value u = (p−c)q,
and for the outsider is as follows: bidding the true value u = (p − c)q if q is revealed, and
bidding within the lowest and highest possible values otherwise, bO ∈ [(p − c)qL , (p − c)qH ].
Then consider the insider’s revelation strategy when q > qL : If the insider reveals q, the
outsider bids the true value u = q(p − c) and the payoff for the insider is 0; if the insider
doesn’t reveal q, the outsider bids in an interval of [(p − c)qL , (p − c)qH ], and the payoff for
the insider is either positive or 0. Thus, revealing the quantity is weakly dominated, and is
eliminated. When q = qL , the payoff for the insider is 0 regardless of revealing or concealing
the quantity, so the insider is indifferent.
So, in all iteratively undominated perfect Bayesian equilibria, only the lowest signal qL
might be revealed, and all other signals must be concealed.
Revealing the only signal is not profitable, because it completely eliminates an agent’s
information advantage and reduces his/her the payoff from the auction to zero. The result
holds in a more general setting where q takes values from any finite or infinite set.12 The
result could also be extended to a setting where bidders are symmetric: each bidder observes
one common-value signal, and the true quantity is determined by both signals. In this
symmetric setting, concealing information is a perfect Bayesian equilibrium. Kovenock,
Florian and Munster (2010) obtain similar, but ex ante, results in all-pay auctions. They
show that agents don’t prefer to reveal private signals about their common or private values
individually.13
These results ensure that we can usually skip discussing information revelation before
auctions and proceed to analyze the bidding strategies assuming information remains private.
12
Let q ∈ Q where Q is a finite or infinite subset of R+ , and q is the infimum of Q. Claim 1 could
be generalized as follows: Consider the oil-drilling auction with two bidders (one insider and one outsider)
and pure common value (quantity); in all iteratively undominated perfect Bayesian equilibria, all quantities
except the infimum value (q > q) are concealed.
13
Besides no revelation on the individual level, they also show that an industry-wide agreement to share
information can arise in equilibria with private values.
8
However, the next part shows that this intuition doesn’t hold when agents observe multiple
signals.
I remark that the refinement of iterative elimination of weakly dominated strategies
removes some equilibria where q is revealed. For example, there exists a perfect Bayesian
Nash equilibrium: the insider fully reveals q and bids the true value; the outsiders believes any
hidden signal is qH and bids the value according to the belief. This equilibrium involves the
insider using weakly dominated revelation strategy when bidders use undominated bidding
strategies, so it is eliminated by the refinement.
Lastly, if revelation is costly, all quantity signals (including qL ) are predicted to be concealed in the pure common-value setting. The cost of revelation is not introduced to the
theory part to keep it simple, but it is introduced to the experiments to eliminate the indifference between revealing and concealing the low common-value signal and thus give a
unique prediction of the equilibrium revelation strategy.
2.2
Values with both Common Quantity and Private Costs
Bidder i’s cost of extraction (ci ) is now assumed to be a private-value variable, which is
identically independently distributed in [cL , cH ] with a strictly increasing and atomless distribution (density f (c)). Assume cH < p, such that the price of the oil is always higher than
the costs, and therefore the value of the oil tract to bidder i, ui = (p−ci )q, is always positive.
The updated timeline is as follows. At time 0, the insider privately observes the quantity
q and the cost cI , and the outsider privately observes the cost cO ; at time 1, the insider
decides whether to truthfully reveal q or conceal it; and at time 2, they bid in a second-price
auction. To make the example simple, bidders’ incentives of revealing their costs are not
considered for now. They are discussed later in Example 1.
Let’s say an equilibrium is common-revealing equivalent if, with probability one, the
outcome is the same as in an equilibrium where all common-value signals (the quantity) are
revealed.
Proposition 1 Consider the oil-drilling auction with two bidders (one insider and one
outsider) and values with both common-value quantity and private-value costs; all iteratively
undominated perfect Bayesian equilibria are common-revealing equivalent.
Proofs are provided in the Appendix.
Suppose the insider observes a low quantity, but still wants to win the auction because of
a low private-value cost (cI < cH ). If the insider reveals the low quantity, the outsider would
generally bid lower, and hence the insider can win with a higher probability and pay a lower
9
price. So by revealing the low quantity, the insider earns a higher profit from the privatevalue cost.14 Given the low quantity is fully revealed, when the quantity is not revealed, the
outsider can infer it must be the medium or high quantity. Following the same argument, the
insider has the incentive to reveal the medium quantity. Thus it unravels to full revelation
of all quantities, and this is the unraveling argument from Milgrom (1981) and Grossman
(1981). This result holds in other settings, such as a different form of the utility ui = pq − ci .
Minimal incentives of information revelation in the pure common-value setting are not
robust to a perturbation in agents’ information structure. With only one common value
(quantity), the insider generally doesn’t want to reveal the value in order to keep his/her
information advantage. With an additional private-value cost, the insider wants to reveal
the quantity signal to earn a higher profit from the private-value cost. Such incentive of information revelation would also happen when costs are common-knowledge but asymmetric.
For example, if the insider has a lower cost, he/she has incentives to reveal the low quantity;
but if the insider’s cost is higher, there is no strict incentive to reveal. Benoit and Dubra
(2006) provide more details on this part.
When bidders can also truthfully reveal their cost information, the common-revealing
equilibrium in Proposition 1 remains to be one equilibrium, and under some orders of eliminating weakly dominated strategies it is unique. However, under some other orders it does
not survive.
Example 1 Order of iterative elimination of weakly dominated strategies matters when
agents can reveal both signals.
Consider the oil-drilling example with two modifications: (i) ci ∈ [cL , cH )15 ; (ii) At time
1, the insider can truthfully reveal both q and cI , only reveal q or cI , or conceal both; and
the outsider can truthfully reveal cO or conceal it.
The common-revealing equilibrium in Proposition 1 is equivalently unique under some
orders of eliminating weakly dominated strategies:
1. Agents do not play weakly dominated bids in the second stage conditional on the
information is given to them: The outsider bids in an interval ([(p − cO )qL , (p − cO )qH ])
if seeing nothing, bids the true value if seeing q, and the insider bids the true value.
2. Conditional on 1, eliminate the insider’s weakly dominated revelation strategy which
is hiding qL .
14
Recall that when there are no endogenous costs, lowering the outsider’s bid doesn’t make the insider
earn a higher profit because the payoff of the insider is always 0 when q = qL .
15
Assuming the interval is open at cH makes this example simple, so that I can focus on illustrating the
effect of the order on eliminating weakly dominated strategies. The proof of Proposition 1 in the Appendix
considers the closed interval, and shows why it makes the proof more complicated.
10
3. Conditional on 2, eliminate the outsider’s weakly dominated strategy when nothing is
revealed. So the outsider bids in an interval of [(p − cO )qM , (p − cO )qH ].
4. Conditional on 3, eliminate the insider’s weakly dominated revelation strategy which
is hiding qM .
So the outcome is equivalent to full revelation of quantities under this order, where qL
and qM are fully revealed and the concealment of qH can be fully inferred. The revelation of
private costs is irrelative, as agents have a unique undominated bidding strategy in secondprice auctions.
However, there are some other orders of eliminating weakly dominated strategies, which
lead to different equilibrium behaviors:
1. Eliminate the insider’s weakly dominated bidding strategy, so the insider bids the true
value.
2. Conditional on 1, eliminate the outsider’s weakly dominated bidding strategy conditional on the information: the outsider bids in [(p − cO )qL , (p − cO )qH ] when seeing
nothing, bids the true value when seeing q, bids (p − cO )qL when only seeing cI < cO ,
and bids (p − cO )qH when only seeing cI > cO .
When only cI < cO is revealed and q is not revealed, the insider bids uI = (p − cI )q
and the outsider bids as low as possible because the profit is always negative when
winning; on the other hand when only cI > cO is revealed, the outsider bids as high as
possible because the profit is always positive when winning.
3. Conditional on 2, eliminate insider’s weakly dominated revelation strategy: when q =
qM or q = qH , q is concealed and cI is revealed; when q = qL , revealing q or cI is
indifferent.
Revealing only cI dominates revealing q when q = qM or q = qH . When q is revealed,
the outsider bids uO = (p − cO )q, and the insider wins only when cI < cO and pays a
price of (p − cO )q. On the other hand, when only cI is revealed, the insider still wins
only when cI < cO , but pays a lower price of (p − cO )qL .
So the equilibrium surviving includes full revelation of the insider’s cost and full concealment
of the quantity.16 Two lessons can be learned from this example.
First, the argument that there is no communication in auctions because of “information
rent” highly depends on the one-dimensional information setting. When there are multiple
16
Special thanks to John Hatfield for pointing out this possible equilibrium.
11
pieces of information, agents usually want to reveal some information, common-value or
private-value signals. Therefore, communication should play an important role in auctions.
Second, the communication with both signals could be very complicated. Even the refinements as iterative elimination of weakly dominated strategies cannot give a clear prediction.
In most applications, the verifiable evidence usually exists for the common-value signals,
while it rarely exists for private values because they are very personal. Thus in the main
model focuses on communication of the common-value signals, and the revelation of both
signals is discussed in the extension section.
3
General Model
A group of agents, N = {1, ..., n}, bid for one item in an auction.17
3.1
Valuation and Revelation
There are two components related to agent i’s valuation of the item:
u(vi , q) : V × Q → R+
a private-value component vi ∈ [vL , vH ] = V , which only affects agent i’s valuation, and a
common-value component q ∈ [qL , qH ] = Q, which affects all agents’ valuations. We assume
the valuation u(vi , q) is increasing in vi and q. The valuation is called linear if there exists
some λ ∈ (0, 1) such that u(vi , q) = λvi + (1 − λ)q.
Agent i’s private-value component, vi , is drawn independently according to a continuous
and strictly positive density f (·). Agent i doesn’t observe the true common-value component,
q, but instead observes a realization of a random signal si ∈ [sL , sH ] = S. Signals are drawn
according to a continuous and strictly positive joint density g(s1 , ..., sn ). For simplicity,
we assume all private values vi are independent from all common-value signals si .18 For
example, the information about the quantity of the oil is independent of the private-value
costs of extraction, which depend on firms’ own technologies. The common-value component
is fully determined by all signals,19
q = q(s1 , ..., sn )
17
The results can be generalized to auctions with multiple identical items, for instance the setting in
Pesendorfer and Swinkels (2000).
18
When vi and si are correlated, the results in this paper remain true in second-price auctions. In other
auctions (e.g. first-price auctions), this correlation leads to asymmetric private-value bidders (because they
observe different common-value signals), which worths a whole paper to study (see Maskin and Riley (2000),
and Reny and Zamir (2004)).
19
When q is not fully determined by all signals, considering the expected common-value component is
usually sufficient for risk-neutral agents.
12
where q is (weakly) increasing in each si . It is possible that si has no impact on q, then
agent i is an outsider since his/her common-value information is irrelevant. Thus the general
model covers the example except expanding the set of common-value signals from a finite
set to an interval.
I remark that this two-dimensional valuation setting is different from the affiliated signals
discussed in Milgrom and Weber (1982a). In their paper, each bidder observes only one
signal, which may relate to both common and private values, but the bidding strategy is
still a one-to-one mapping. Here each bidder observes two signals, where one relates to the
common value and the other relates to the private value. The equilibrium strategy is a more
complicated mapping. In reality, agents usually observe many signals. For instance, in the
oil tract auction, firms might know the quantity of the oil, the depth of the tract, the weather
condition nearby, the salary of the captain, the technology of the extracting equipments, and
etc. Some of them could be summarized to one common-value signal, and others could be
summarized to one private value.
Prior to the auction, agents could communicate their signals in the form of verifiable
evidence.20 I focus on the communication of common-value signals, because they are usually
supported by hard documents. Agents can choose to either reveal the signal or not, but they
cannot “lie”. This matches a variety of applications. For instance, in the housing market
the inspection reports could indicate the value of a house. As shown in Example 1, the
communication involving both private-value and common-value signals is less tractable, so I
defer the analysis to the extension section.
Formally, a revelation strategy for an agent i ∈ N is a function (including mixed strategies),
ri (vi , si ) ∈ 4{si , ∅}
r(v, s) = (r1 (v1 , s1 ), ..., rn (vn , sn )) represents all agents’ revelation strategies.
All of the settings are common knowledge except for the private information (vi , si ). All
agents are risk-neutral.
3.2
Game and Equilibrium
The game takes two stages. In the first stage, agents simultaneously choose their revelation
strategies, ri (vi , si ). All revealed signals are observed by all agents. I = I(r(v, s)) is the
belief of the signals after the revelation. In the second stage, agents simultaneously choose
their bids, bi (vi , si , I(r(v, s))), which is a function of agent i’s own signals and the belief I.
20
The main result holds when agents can partially reveal the evidence but not all of them, and it is
discussed in the remarks after the theorem.
13
An equilibrium is a list of strategies, ri (vi , si ) in the first stage and bi (vi , si , I(r(v, s))) in
the second stage for each agent i, and the belief system I, that form an interim undominated
perfect Bayesian equilibrium, where both the revelation and bidding strategies could be
mixed. Undomination is required to eliminate degenerate equilibria, such as the ones in
second-price auctions.
4
Equilibria and Welfare
This section starts by showing the set of PBE is non-empty, then describes the revelation
strategies in these equilibria, and concludes with efficiency analysis.
4.1
Equilibrium Existence
We conjecture that there exists at least one equilibrium where agents fully reveal their
common-value signals. This is based on two observations from the oil-drilling example: 1)
agents bid lower when the common value is lower; 2) agents observing low common-value
signals want to reveal their signals, and it unravels to full revelation.
The following lemma establishes the first observation: the monotone relationship between
agents’ equilibrium bidding strategies and the common value.
Lemma 1 When q is common knowledge, the equilibrium bidding strategy bi (vi , q) increases
in q, in any first-price and all-pay auction with linear values and all second-price auctions.
The lemma is proved by examining the equilibrium bidding strategies in these auctions.
In any second-price auction, the equilibrium is bidding the true valuation,
A
bSP
(vi , q) = u(vi , q)
i
thus the bidding strategy is an increasing function of q. With linear valuations u(vi , q) =
λvi +(1−λ)q, an increase to the common value, say q 0 = q +, leads to an increase of (1−λ)
to all valuations. In a first-price auction, the equilibrium bid is
bFi P A (vi , q 0 ) = bFi P A (vi , q) + (1 − λ)
this increase in valuations is fully absorbed by the increase in the bids. In an all-pay auction,
the equilibrium bid is
A
A
(vi , q) + (1 − λ)(F n−1 (vi ) − F n−1 (vL ))
bAP
(vi , q 0 ) = bAP
i
i
the increase in bids equals the increase in valuations times the probability of winning the
auction. Agents with low private values are less likely to win and enjoy the increase in q but
they need to pay their bids regardless, so the increase in their bids is smaller.
14
However, this monotone relationship between the bidding strategy and the common value
cannot be easily generalized to cover non-linear value functions. For example, it is possible
that a lower q also suggests a stronger competition when the valuation does not change
linearly, and this stronger competition may lead to higher bidding strategies. If the effect
of an increase in competition dominates the decrease in valuations, the equilibrium bidding
strategy may increase even if the common value decreases.
Example 2 In first-price auctions, equilibrium bidding strategies may increase when q decreases with a non-linear valuation.
Suppose there are 2 bidders in a first-price auction. Let q ∈ [0, 1], and vi is uniformly
distributed in [0, 1]. Bidder i’s value of the item equals:
vi ∈ [0, 1/2)
V aluei = vi ,
= 1/2 + (vi − 1/2)q, vi ∈ [1/2, 1]
When q = 1, V aluei is uniformly distributed in [0, 1] and the equilibrium bidding strategy
is bq=1 (v) = v/2. When q = 0, V aluei is uniformly distributed in [0, 1/2) with a probability
of 1/2, and V aluei = 1/2 with a probability of 1/2. The equilibrium bidding strategy is
v ∈ [0, 1/2)
bq=0 (v) = v/2,
= mixed in [1/4, 3/8] with a cdf G(b),
v ∈ [1/2, 1]
1
where G(b) = 2(1−2b)
− 1 is the distribution function of the mixed strategy.
When q decreases from 1 to 0, the equilibrium bidding strategy changes from bq=1 (v) to
bq=0 (v). This change is not monotone. For instance, if bidder 1 has v1 = 5/8, bq=1 (v1 ) = 5/16
and bq=0 (v1 ) ∈ [1/4, 3/8]. Thus, I focus on linear valuations for first-price and all-pay auctions when establishing
the existence result. An equilibrium is common-revealing if in the outcome all common-value
signals are revealed.
Theorem 1 There exists a common-revealing equilibrium in any first-price and all-pay auction with linear values and all second-price auctions.
Proof of Theorem 1: The common-revealing equilibrium includes revelation and bidding
strategies and a belief system. For the strategies, agents reveal their common-value signals
and bid according to the following equilibrium bidding strategies in these three auctions.
R vi
F (s)n−1 ds
v
A
, bSP
(vi , q) = u(vi , q)
bFi P A (vi , q) = vi + q − L
i
F (vi )n−1
15
A
(vi , q)
bAP
i
Z
vi
(s + q)dF (s)n−1
=
vL
For the belief system, if no signal is revealed, all other agents believe the signal is sH .
We need to verify this is a perfect Bayesian equilibrium. First, no signal being revealed
happens off equilibrium path, so the belief is valid. Also all the bidding strategies are the
equilibrium strategies in auctions with private values. Thus they are best responses given
the belief of q. The last part is to prove revealing si is a best response for agent i. Suppose
agent i hides si instead, other agents believe it is sH and thus believe the common value is
weakly higher. As a result, other agents bid higher and agent i’s payoff gets smaller. So
revealing common-value signals is a best response.
This equilibrium existence result remains true in some more general settings. I provide
two possible generalizations.
Remark 1 Consider the generalization based on the setting in Pesendorfer and Swinkels
(2000): 1) k(≥ 1) identical items is sold by (k + 1)th-price auction; 2) the private values
vi can be correlated with the common-value signals si . The equilibrium bidding strategy is
still bidding the true value, bi (vi , q) = u(vi , q), which increases in q. So the same perfect
Bayesian equilibrium constructed in the proof remains valid.
Remark 2 Consider a partial revelation of the common-value signals: agents can reveal
some evidence but not all of them, which narrows the set of the possible common-value
signals:
ri (vi , si ) ∈ 4{B : si ∈ B, B is closed}
The perfect Bayesian equilibrium still exists: the same strategies as in the proof and the
belief system is believing si = sup(B) when the revelation from agent i shows B.
4.2
Revelations in Equilibria
The theorem establishes the existence of the equilibrium where the common-value signals are
fully revealed. But the possibility of other equilibria with no revelation or partial revelation
remains.
First, other equilibria might appear because agents’ beliefs could be more complicated.
For instance, the following example shows no revelation of common-value signals could appear in an equilibrium. If one agent (agent 1) deviates from the common-hiding equilibrium
and reveals the common-value signal, agent 2 may not only update the belief of agent 1’s
signal, but also update the beliefs of others’ signals. For example, agent 2 might completely
change his/her belief of other agents’ revelation strategies from common-hiding to commonrevealing. Thus all others’ hidden signals are now believed to be the highest possible signal,
16
and agent 2 might submit an even higher bid based on this new belief. So agent 1 does not
have an incentive to reveal any common-value signal.
Example 3 Fully-hiding equilibrium in a second-price auction.
Suppose there are 4 agents in the society (n = 4), common-value signals are independent,
f and the marginal density gsi are uniform on [0,1]21 , the common value is the average of
the four signals q = (s1 + s2 + s3 + s4 )/4, and the value for each agent (agent i) is ui = vi + q.
Let ci = vi + si /4 be the surplus of agent i. From Goeree and Offerman (2003), without
information revelation the symmetric bidding strategy (for agent i) is an increasing function
of the surplus
b(ci ) = ci + E(s/4|c = ci ) + 2E(s/4|c < ci )
Thus b(ci ) < ci + 1/4 + 2(1/8).
Suppose in the equilibrium, all 4 agents won’t reveal any signal. If some agent (say agent
1) deviates and reveals a common-value signal s1 , all other agents change their beliefs to the
common-revealing equilibrium where the hidden signal is believed to be sH = 1. Then all
other agents bid their values under their beliefs b0 (ci ) = ci + s1 /4 + 2(1/4) in a second-price
auction. The new bid is higher than the original bid (b0 (ci ) ≥ b(ci )). Thus, revealing any
common-value signal increases other agents’ bids, which is not a profitable deviation. The
initial fully-hiding equilibrium is valid. The example above cannot be refined away by iterative elimination of weakly dominated
strategies nor intuitive criterion, but it can be eliminated by trembling-hand perfection.
Thus in the following part, I focus on trembling-hand perfect Bayesian equilibrium where
agents have independent beliefs on other agents’ revelation strategies (e.g. one change in
one agent’s revelation does not change beliefs about other agents’ revelations).
The second challenge is the non-existence and complication of equilibrium bidding strategies in auctions with multiple signals (Jackson (2009) and Pesendorfer and Swinkels (2000)).
When agents don’t fully reveal the common-value signals, they may have multiple private
signals when bidding. Goeree and Offerman (2003) is the only setting where the equilibrium
can be solved to the best of my knowledge, So I focus on their settings and show there
doesn’t exist an equilibrium where the common-value signals is fully hidden. This is because
agents have incentives to reveal low common-value signals.
(A1) (Goeree and Offerman (2003)) Common value signals si are independent, q = (s1 +
... + sn )/n, ui = vi + q and the densities f and gsi are logconcave.
21
This is enough to ensure that conditions in Goeree and Offerman (2003) are satisfied, such that the
equilibrium with no information revelation can be solved.
17
Let ci = si /n + vi be the surplus of agent i. Goeree and Offerman (2003) shows that the
symmetric equilibrium bidding strategy in a second-price auction is
b(c) = E(q + vi |ci = c, yi = c)
(1)
where yi is the highest surplus from all other agents.
Proposition 2 Consider the setup in (A1) and agents use symmetric monotone bidding
strategy as a function of their surplus c, the common-hiding equilibrium is not a PBE surviving trembling-hand perfection in second-price auctions
Proof of Proposition 2: In second-price auctions, agents bid the expected value when
they are the pivotal winner (tie with another agent) in equilibrium. The equilibrium bidding
strategy after player 1 observes and reveals s1 = sL is (b1 (v1 ), B(c2 ), ..., B(cn )).
b1 (v1 ) = E(q + v1 |s1 = sL , B(yi ) = b1 (v1 ))
B(c) < E(q + vi |ci = c, s1 = sL , y1,i = c)
where y1,i is the highest surplus from agents other than 1 and i (i ≥ 2). The last inequality
comes from the fact that with some probability P , 1 submits the highest bid among all other
bidders (except i), in which case the expected utility from being pivotal is E(q + vi |ci =
c, s1 = sL , y1,i < c). So to be precise,
B(c) = P E(q + vi |ci = c, s1 = sL , y1,i < c) + (1 − P )E(q + vi |ci = c, s1 = sL , y1,i = c)
where the first part of the sum is smaller than the second part. Since s1 = sL , B(c) < b(c)
in (1) unless c = sL + vL . Revealing sL can lower other bidders bids, so it is profitable to do
so.
I remark that the result is presented in a quite restricted setting because of the complication of the equilibrium bidding strategies. But the intuition of non existence of commonhiding equilibrium is quite general. As long as revealing some signals (usually bad news
of the common value) can lower others’ bids and increase one agent’s profit, agents have
incentives to communicate.
Lastly, the partial revelation is harder to rule out, because there are many possible partial
revelation strategies and each of them is associated with a complicated (if existing) bidding
strategy. A possible equilibrium with partial revelation is provided later in Example 5 in
the experiment setting. In general, proving the uniqueness of common-revealing equilibrium
must require more subtle refinements to remove all equilibria with partial or no revelation.
18
4.3
Welfare and Profit
This section proceeds to show that there is no efficiency loss in monotone equilibria with
revelation of common-value signals, in contrast to positive efficiency loss in equilibria with
no revelation.
It is efficient to select the winner to be the agent with the highest private value since
the common-value component is the same for all agents. However, in auctions without
information revelation, there is a positive expected efficiency loss22 . Because it is quite
possible that the winner observes a high enough common-value signal, but does not have
the highest private value. With information revelation, the story is different. If agents
fully reveal their common-value signals, the auction only involves private information of
private values and thus the winner is the agent with the highest private value in a monotone
equilibrium. Thus, there is no efficiency loss in auctions with two-dimensional valuations if
information revelation is considered.
Corollary 1 There is no efficiency loss in a monotone common-revealing equilibrium in
any first-price and all-pay auction with linear values and all second-price auctions.
Not only does the social planner support information revelation, the seller may also prefer
information to be shared. As before, in order to calculate and compare the seller’s profit
when information is not revealed, I focus on the settings in Goeree and Offerman (2003).
The following example shows that the expected payoffs of agents decrease with information
revelation, and thus the expected profit of the seller increases when n = 2.23
Example 4 The seller’s expected profit increases when common-value signals are revealed.
Suppose there are two agents in a second-price auction (n = 2), and (A1) holds except
q = s1 + s2 . Let ci = vi + si be the surplus of agent i.
With revelation of si , the auction is a standard private-value second-price auction, and
the bid equals the true value, b(v1 ) = c1 + s2 . The ex ante expected payoff of agent 1 is
Z v1
∗
π = Ev1
(v1 − v2 )f (v2 )dv2
vL
where agent 1 wins with a payoff of v1 − v2 when v1 > v2 .
From Goeree and Offerman (2003), without information revelation, the symmetric bidding strategy (for agent 1) is an increasing function of the surplus
b(c1 ) = c1 + E(s2 |c2 = c1 )
22
See Pesendorfer and Swinkels (2000) and Goeree and Offerman (2003).
When n > 2, the agents’ payoffs may increase or decrease with information revelation depending on the
distributions of signals, and thus the change in the seller’s profit is uncertain.
23
19
The ex ante expected payoff of agent 1 is
Z c1
∗∗
π = Ev1 ,s1
(u(c1 , c2 ) − b(c2 ))fc (c2 )dc2
cL
where agent 1 wins when c1 > c2 . Conditional on c2 , the value for agent 1 is u(c1 , c2 ) =
c1 + E(s2 |c2 ), and the price to pay is agent 2’s bid which is b(c2 ) = c2 + E(s1 |c1 = c2 ). Since
E(s2 |c2 ) = E(s1 |c1 = c2 ), we have
Z c1
∗∗
(c1 − c2 )fc (c2 )dc2
π = Ev1 ,s1
cL
In order to show π ∗∗ ≥ π ∗ , we need this claim:
Claim 2 If x and y are two independent variables with mean 0 and symmetric density h(x)
and k(y)24 on [−x∗ , x∗ ] and [−y ∗ , y ∗ ] , then
Z
Z x∗
(x + y)h(x)k(y)dxdy ≥
xh(x)dx
(2)
x+y>0
0
The proof is in the Appendix and Equation (2) can be re-written as
Z z∗
Z x∗
zhz (z)dz ≥
xh(x)dx
0
(3)
0
where z = x + y, z ∈ [−z ∗ , z ∗ ], and hz (z) is the density of z. Take x = v1 − v2 , y = s1 − s2
and z = x + y = c1 − c2 , π ∗∗ ≥ π ∗ is true by (3).
The expected payoffs of agents decrease with information revelation while the social
welfare increases as discussed in Corollary 1, thus the seller’s expected profit must increase.
Figure 1 shows the cumulative distribution functions of the changes in social welfare
and the seller’s profits with 100 simulations.25 Figure 1(Up) shows that with about 80%
probability, there is no change in social welfare, and with the remaining 20% probability,
there is an increase in social welfare with information revelation. Figure 1(Down) shows that
with about 20% probability, the seller’s profit decreases with information revelation, mainly
because low common-value signal is revealed, and with the remaining 80% probability, the
seller’s profit increases with information revelation, mainly because of the increase in social
welfare or the revelation of high common-value signal. Overall, the seller’s expected profit
increases with the revelation of common-value signals. The intuition of the decrease in agents’ payoffs and the increase in the seller’s profit is easy
to see in pure common-value auctions. If signals are common knowledge, all agents submit
24
Symmetry suggests h(x) = h(−x) and k(y) = k(−y).
When vi and si are uniformly distributed in [0, 1], the equilibrium bidding strategy with no information
revelation is b(ci ) = 3/2ci .
25
20
Cumula.ve Distribu.on Func.on of Welfare with Info Revela.on -­‐ Welfare without Info Revela.on 1 probability 0.8 0.6 0.4 0.2 0 -­‐0.4 -­‐0.3 -­‐0.2 -­‐0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Welfare with Info Revela=on -­‐ Welfare without Info Revela=on Cumula.ve Distribu.on Func.on of Seller's Profit with Info Revela.on -­‐ Profit without Info Revela.on 1 probability 0.8 0.6 0.4 0.2 0 -­‐0.4 -­‐0.3 -­‐0.2 -­‐0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Profit with Info Revela>on -­‐ Profit without Info Revela>on Figure 1: Cumulative distribution functions for the differences (values with information
revelation minus those without information revelation) for social welfare (up) and the seller’s
profits (down), based on 100 simulations. The settings follow Example 4 where all signals
are uniformly distributed in [0, 1].
21
the same bid, which is the true common value in the equilibrium. Thus, the expected payoff
is 0 for all agents. However, if signals are private information, agents generally have positive
expected payoffs. So with full information revelation, agents’ expected payoffs decrease and
the seller’s expected profit increases.
In reality, communication is restricted by rules, such as the anti-trust law. One potential
policy implication of this welfare analysis is that laws shouldn’t restrict certain kinds of
communication, for instance the public announcement with verifiable evidence about the
common value.
5
Extensions of Theory
In this section, two extensions are considered: (i) communication of private values; (ii)
incentives of acquiring the common-value signals.
5.1
Communication Including Private Values
In this part, agents can reveal their private values in addition to their common-value signals:
ri (vi , si ) ∈ 4{(vi , si ), (vi , ∅), (∅, si ), (∅, ∅)}
The set of PBE is still non-empty: there exists a common-revealing and private-hiding
equilibrium where the common-value signals are revealed and the private values are hidden.
Let’s start with examining agents’ incentives to reveal private values when q is common
knowledge.
Proposition 3 There exists a private-hiding equilibrium in first-price auctions with linear
values and any second-price auction, when q is common knowledge.
The proof is in the Appendix. The intuition is similar to most literature on information
revelation in auctions: no incentive to reveal the only signal, because otherwise agents completely lose their information advantage. Moreover, Kovenock, Morath and Munster (2010)
obtain a similar result for all-pay auctions. Combining Theorem 1 and Proposition 3, we
can get a full description of the revelation strategies in equilibrium.
Proposition 4 There exists a common-revealing and private-hiding equilibrium in firstprice auctions with linear values and any second-price auction.
Proof of Proposition 4: The common-revealing and private-hiding equilibrium includes
revelation and bidding strategies and a belief system. For strategies, agents reveal their
22
common-value signals, hide their private values, and bid according to the equilibrium bidding
strategies in these auctions. For the belief system, if no common-value signal is revealed, all
other agents believe the signal is sH .
The proof of Theorem 1 shows that the bidding strategies are best responses, the belief
system is consistent, and agents don’t have incentives to hide any common-value signal.
From Proposition 3, agents don’t have incentives to reveal any private value. So the only
thing that we haven’t checked is a double deviation, e.g. one agent revealing vi and hiding
si . Suppose agent i does this double deviation, other agents believe si = sH . I claim that
this double deviation cannot be profitable, because other agents bid higher under the belief
A
(vi , q) = u(vi , q). In firstsi = sH . In second-price auctions, agents bid their valuation bSP
i
0
FPA
FPA
price auctions, bi (vi , q ) = bi (vi , q) + (1 − λ) which is true even if agent i reveals vi .
Thus no deviation is profitable and the equilibrium is valid.
Enlighten by Example 1, we wonder whether there exists another equilibrium where
private values are revealed and common-value signals are hidden. I claim that in general,
this equilibrium doesn’t exist. Suppose this equilibrium happens; the revealed private values
are almost surely different among agents, so they are asymmetric. Benoit and Dubra (2006)
examine such asymmetric one-signal auctions and show agents have incentives to reveal
common-value signals, because low common-value signals can lower agents’ bids. Since
agents want to deviate from the revelation strategies, the private-revealing and commonhiding equilibrium doesn’t exist. In addition, if the common-value signals are fully revealed,
agents don’t have incentives to fully reveal their private values as shown in the next claim.
The following claim shows that there doesn’t exist a private-revealing equilibrium in
first-price auctions with linear values.26
Claim 3 There doesn’t exist a private-revealing equilibrium in first-price auctions with linear
values, when q is common knowledge.
Proof of Claim 3: Suppose there exists one private-revealing equilibrium, and let v(1) be
the highest private value and v(2) be the second highest one. I claim that in the first-price
auction, all possible equilibrium behaviors are equivalent to the following one: (i) If there are
multiple agents with v(1) , at least two of them must bid v(1) because of competition; (ii) If
there is only one agent with v(1) , this agent bids v(2) and wins with probability one. Part (ii)
is proved in 2 steps. First, the agent with v(1) won’t bid higher than v(2) because all others’
bids are at most that high by undomination. Suppose the agent with v(1) bids strictly lower
than v(2) with a positive probability, the agent with v(2) could bid close enough to v(2) and
earn a positive payoff. In that case, the agent with v(1) wants to deviate and bid higher than
26
In second-price auctions, private-revealing equilibrium is identical to private-hiding equilibrium.
23
the agent with v(2) . So the agent with v(1) must bid v(2) with probability 1, and the agent
with v(2) must use a mixed strategy in the equilibrium.
Now consider some agent i deviates from the private-revealing equilibrium by hiding the
i
be the highest signal from other agents.27 We claim the agent
signal vi > vL , and let v(1)
i
i
with probability 1 in the equilibrium after agent
must bid strictly lower than v(1)
with v(1)
i deviates. It is proven in two cases depending on the common belief of vi . First, if it is
i
with a positive probability, the agent
common belief that the hidden vi is lower than v(1)
i
i
. Second, if it is the
with v(1) earns a positive expected payoff by bidding lower than v(1)
i
common belief that the hidden vi is higher than v(1) with probability one28 , it is similar to
i
i
uses a
and the agent with v(1)
case (ii) above such that in the equilibrium, agent i bids v(1)
i
mixed strategy which is lower than v(1) with probability 1. Thus agent i is strictly better off
i
, agent i now have a chance of winning with a positive payoff while
by hiding vi : if vi ≤ v(1)
i
i
there are no such chance without the deviation; if vi > v(1)
, agent i could still bid v(1)
(or
even lower) which is identical to the case when vi is revealed. So, the deviation is profitable
which means the private-revealing equilibrium is not valid.
5.2
Incentives of Acquiring Common-Value Signals
In a lot of applications, agents need to pay some cost in order to get signals of the commonvalue component. For instance, firms need to hire experts to estimate the potential value of
a forest or an oil tract, or individuals need to pay for the inspection of a house.
In this subsection, incentives of acquiring information about the common-value component are explored. I focus on the oil-drilling example because it has a unique prediction of
the equilibrium. The timeline is as follows. At time 0, the insider first observes the privatevalue cost (cI ), and then makes a choice whether to acquire information or not.29 The insider
decides whether to pay a positive cost and acquire the quantity or not pay the cost and not
acquire the quantity. If the insider doesn’t acquire the quantity, both bidders hold the same
prior belief about q, which takes value from {qL , qM , qH } with probability pL , pM and pH .
Between time 0 and time 1, the insider observes q if choosing to acquire information. The
rest of the game proceeds as before.
There are two possibilities about the common knowledge after time 0: whether the
action of acquiring information is publicly observed by the outsider or not. Both of these
27
i
i
We only need to consider the case where there is only one agent has v(1)
and v(1)
> vL , because other
cases have zero probability.
28
i
There is another case where the common belief is vi = v(1)
. Since agent i chooses to hide vi before
i
knowing v(1) , the probability of this case is 0 and omitted.
29
I look at the ex post question where the insider observes cI before making acquisition decision. The
result of ex ante question, where the insider makes acquisition decision before observing cI , is the same.
24
possibilities fit some applications, and they lead to different results. On one hand, if the
action of acquiring information is observable, in the end of time 0 it is common knowledge
whether the insider observes the quantity or not. It fits the applications like offshore oildrilling auctions, where the action of investigation is easy to detect. On the other hand, it
is possible that agents could privately investigate the common-value component or purchase
information from consulting firms. The following proposition states the results separately
for these two possibilities:
Proposition 5 Consider the oil-drilling example:
• If the action of acquiring information is overt, no information is acquired in all iteratively undominated perfect Bayesian equilibria;
• If the action of acquiring information is covert and the cost of information is small
enough, some information must be acquired in all undominated perfect Bayesian equilibria.
Proof of Proposition 5: In the first case, the action of acquiring information is observable.
If the insider pays the cost and acquires the quantity information, by Proposition 1 all
iteratively undominated equilibria are common-revealing equivalent, and both bidders bid
the true value. The insider’s expected payoff is
Z cH
a
π (cI ) = Eq
(cO − cI )qf (cO )dcO
cI
On the other hand, if the insider chooses not to acquire the information, both agents only
know the prior distribution of the quantity, and bid the expected value because they are
risk-neutral. The insider’s expected payoff is
Z cH
na
π (cI ) =
(cO − cI )Eqf (cO )dcO
cI
Thus, in time 0, the insider’s expected payoffs are the same between acquiring the quantity
information or not. However, acquiring information requires a positive cost, such that it is
not profitable.
In the second case, the action of acquiring information is not observable. Supposing there
is an equilibrium where the insider doesn’t acquire the quantity information, we claim that
the insider has incentives to deviate when the cost of acquiring information is small enough.
In particular, there exists c∗I ∈ (cL , cH ) such that: if the insider has cI ∈ [c∗I , cH ], he/she is
strictly better when observing q = qH and weakly better otherwise; and if the insider has
cI ∈ [cL , c∗I ], he/she is strictly better when observing q = qL and weakly better otherwise.
25
When q = qH , the insider doesn’t have incentives to reveal this signal. The insider’s bid
when observing qH is (p − cI )qH , instead of (p − cI )Eq when not acquiring information. With
the higher bid, the insider wins with a higher probability.
(p − cI )Eq < (p − cO )Eq < (p − cI )qH
(4)
where the left inequality means the insider doesn’t win when not acquiring the information,
and the right inequality means the insider wins when observing qH (ignoring tie here). ConqH
−Eq
−Eq
dition (4) implies cO ∈ ( Eq
cI − qHEq
p, cI ), and the length of the interval is qHEq
(p − cI ).
qH −Eq
Let δ = min( Eq (p − cH ), cH − cL ) is the shortest length of all intervals, and define
c∗I = cH − δ/2. For all cI ∈ [c∗I , cH ], cO ∈ (cH − δ, c∗I ) guarantees condition (4). With the
additional probability of winning, the additional gain is at least:
Z c∗I
H
((p − cH )qH − (p − cO )Eq)f (cO )dcO
4π =
cH −δ
When q = qL , the insider reveals this signal. The outsider’s bid when observing qL is
(p − cO )qL , instead of (p − cO )Eq when not observing qL . With the outsider’s lower bid, the
insider (with cI ∈ [cL , c∗I ]) wins with the same probability but pays a lower price:
Z cH
L
(p − cO )(Eq − qL )f (cO )dcO
4π =
c∗I
where (p − cO )(Eq − qL ) is the change of the price.
The insider always can make the outsider submit the same bid by hiding the signal q,
such that the insider is always weakly better off. Let = min(4π H pH , 4π L pL ), the insider
earns a higher expected payoff by deviating and acquiring q if the cost is smaller than .
I remark that if the insider has a choice of whether acquire the information overtly or
covertly, he/she should choose to acquire the information covertly. Because if the information
is acquired overtly, it will be revealed publicly in the communication stage and yields no
benefit to the insider. On the other hand, if the information is acquired covertly, the agent
could hide positive information, show negative information, and earn a higher payoff. This
gives a different prediction against Milgrom and Weber (1982b).
Moreover, the social welfare could be ranked in the following way: It is optimal if no
information is acquired, such that no cost (assumed to be loss of the society) is spent to
acquire information and the winner is the agent with a lower cost of extraction; it is the
second best if information is acquired and revealed, such that the loss is only the cost of
information; it is the worst if information is acquired and concealed, where the loss includes
both the cost of information and the possibility of a misallocation. Thus, for a social planner,
it is best if acquiring information action is overt, then in equilibrium no information is
acquired.
26
6
Experiment Design and Results
The experiment is designed to be the same as the oil-drilling example, except adding a
positive cost to any revelation.
The positive cost has two benefits. First, it strengthens any observed revelation behavior
from the experiments: agents must believe revealing the signals is profitable enough when
choosing to reveal them. Without the positive cost, we cannot distinguish whether agents
truly find the revelation profitable or they just randomize since the revelation is free. If we
consider an application where revelation is free, if anything, more signals would be revealed.
Second, introducing the cost eliminates the multiplicity of equilibrium behaviors. For example, theory now predicts no revelation of the low common-value signal in pure common-value
auctions, instead of indifference between revealing or not when it is not costly. Adding the
cost requires some additional examinations of the theory, but it worths the price.
6.1
Theory
Consider a second-price auction with two bidders, labeled as one “insider” and one “outsider”. Bidder i observes his/her private value vi , and the insider observes the common value
q.
The private value vi is identically independently chosen from a finite set of positive
values V , where v j ∈ V (0 < v 1 < v 2 < ... < v m ) is chosen with a probability of pj such that
Pm
j=1 pj = 1. The common value q is randomly chosen from {qH , qM , qL } (qH > qM > qL > 0)
with probability pH , pM and pL such that pH + pM + pL = 1. Bidder i’s valuation is the sum
of his/her private value and the common value
ui = vi + q
There are 2 different treatments of values: (i) pure common value: private values are 0 for
both bidders; (ii) two-dimensional values: private values are randomly chosen from V .
Pure Common Valuation
The timeline of the game is as follows: At time 0, the insider privately observes the true
common value q; at time 1, the insider decides whether to truthfully reveal q to the outsider
with a cost > 0 or hide it with no cost; at time 2, the outsider observes the insider’s
revelation decision and the true common value q if it is revealed, and both bidders bid in a
second-price auction.
The positive cost of revelation eliminates the indifference between revealing and concealing the low common value in Claim 1, so all common values are now fully concealed.
27
Claim 4 Consider the second-price auction with two bidders (one insider and one outsider),
pure common value and a positive revelation cost, in all undominated perfect Bayesian equilibria, all common-value signals are concealed.
Proof of Claim 4: First eliminate all weakly dominated bidding strategies in the secondprice auction. The remaining strategy for the insider is bidding the true value u = q, and
for the outsider is as follows: bidding the true value q if it is revealed, and bidding in an
interval from the lowest to the highest possible value otherwise, bO ∈ [qL , qH ].
Then check the insider’s revelation strategy: If the insider reveals the common value
(by paying a cost ), the outsider bids q and the payoff for the insider is −; if the insider
doesn’t reveal the common value, the outsider bids in an interval of [qL , qH ], and the payoff
for the insider is either positive or 0. Thus, revealing the common value is strictly worse
than concealing it. In all undominated perfect Bayesian equilibria, all common-value signals
are concealed.
Two-Dimensional Valuations
When changing from the pure common value to two-dimensional valuations, incentives of
sharing the common value change from no revelation to positive revelation.
The updated timeline is as follows: At time 0, the insider privately observes the common
value q and own private value vI , and the outsider privately observes own private value vO ;
at time 1, the insider decides whether to truthfully reveal q to the outsider with a cost > 0
or hide it with no cost30 ; at time 2, the outsider observes the insider’s revelation decision and
the true common value q if it is revealed, and both bidders bid in a second-price auction.
Claim 5 Consider the second-price auction with two bidders (one insider and one outsider)
and valuations with both common and private values, when the revelation cost is small
enough, there exists an equilibrium where the insider reveals q when vI > v 1 and q < qH ,
conceals q otherwise, and the outsider bids as if q = qH when q is concealed.
Proof of Claim 5: Construct the equilibrium as follows: The insider reveals q when vI > v 1
and q < qH , conceals it otherwise, and always bids q + vI ; the outsider bids q + vO if q is
revealed, and bids qH + vO otherwise. In order to show it is an equilibrium, we need to prove
the insider doesn’t want to deviate from the revelation strategy and the outsider doesn’t
want to deviate from the bidding strategy.
Check the outsider’s bidding strategy first. When q is concealed, there are two possibilities: (a) vI = v 1 , and (b) q = qH . If (b) is true, bidding qH + vO is the outsider’s unique
30
Same as the oil-drilling example, subjects are only allowed to reveal common-value signals in the experiments.
28
undominated strategy. If (a) is true, the insider’s bid is q + v 1 , and the outsider’s value is
uO = q + vO , which is equal to or larger than the insider’s bid since vO ≥ v 1 . The outsider
earns a non-negative payoff when winning, so there is no incentive to deviate.
Then, check the insider’s revelation strategy. The insider doesn’t have an incentive to
reveal q when (a) or (b) above is true. If (a) is true, the insider earns 0 when revealing q but
needs to pay ; if (b) is true, the insider’s revelation of qH doesn’t change the outsider’s bid
but costs . We claim that the insider has incentives to reveal q when q < qH and vI > v 1 .
If the insider reveals q, the outsider bids q + vO , and if the insider conceals q, the outsider
bids qH + vO . If the insider wins in both cases, the price is lower by qH − q, and if the insider
wins only in one case (this must be the case with revelation), the winning payoff is vI − vO .
With probability p1 , vO = v 1 and at least one of the above is true. The increase in payoff by
revealing q is at least
4 = p1 min(vI − vO , qH − q)
= p1 min(v 2 − v 1 , qH − qM )
When < 4, the insider prefers to reveal q.
In the experiments, = 1, both v and q are randomly selected from {10, 30, 50} each
with a probability of 1/3. Thus,
4 =
1
20
min(30 − 10, 50 − 30) =
>1
3
3
The condition < 4 is satisfied.
However, adding a positive cost of revelation changes the nature of the game, and leads
more equilibria to appear. The following example describes two other possible equilibria,
which turn out to be close to the behaviors observed in the experiments.
Example 5 Two other equilibria under the setting in the experiments.
In the experiments, the cost of revelation = 1, both v and q are randomly selected
from {10, 30, 50} each with a probability of 1/3. It is a second-price auction with bids being
integers.31
• Equilibrium #1:
– The insider: reveal q only when (vI , q) = (50, 10) and (30, 10), and bid the true
value;
31
Requiring bids to be integers simplifies the strategies in the experiments, and the equilibria below hold
when withdrawing this assumption. For instance, in equilibrium #1 below, the bid 61 could be replaced by
any value in (60, 63], and in equilibrium #2 below, the bid 70 could be replaced by any value in [63, 80].
29
– The outsider: bid the true value when observing q, and otherwise bid (40, 61, 100)
when vO = (10, 30, 50).
• Equilibrium #232 :
– The insider: reveal q only when (vI , q) = (50, 10), (30, 10) and (50, 30), and bid
the true value;
– The outsider: bid the true value when observing q, and otherwise bid (40, 70, 100)
when vO = (10, 30, 50).
In both equilibria, low common value is revealed when the insider has a high or medium
private value. The medium common value is revealed when the insider has a high private
value only in equilibrium #2, where the outsider with a medium private value bids higher
than 63. This is because revealing q = 30 reduces such outsider’s bid to 60 (vO = 30 with a
probability of 1/3), and it only worths paying the cost of 1 when the reduction is at least 3.
6.2
Design of the Experiment
6 sessions, with 12 participants per session, have been conducted. 3 sessions played the
pure common value treatment, and the other 3 sessions played the two-dimensional value
treatment. Each session had 40 periods and was completely computerized33 . Subjects are
undergrad and graduate students recruited at Stanford University. Each session lasted 90
minutes and subjects on average earned $30.8. The show-up fee ($5) and the completion fee
($10) are high enough such that no subject went bankrupt.
Basic Setup
In each period, the common value (q) is randomly chosen from {10, 30, 50} experimental
dollars each with a probability of 1/3. In each period with two-dimensional valuations, the
private value (vi ) is randomly chosen from {10, 30, 50} experimental dollars each with a
probability of 1/3, and it is emphasized to all subjects that vI and vO are two independent
draws. Two different treatments of values are: (i) pure common value: values for both
bidders are q; (ii) two-dimensional values: value for bidder i is vi + q (i ∈ {I, O}).
32
This is also an equilibrium under the general model setting, e.g. no cost of revelation and no restriction
of integer bids.
33
The experiment was programmed and conducted with the experiment software z-Tree (Fischbacher,
2007).
30
Part I: Training Periods
In the very beginning of the study, there is a sequence of three quizzes to teach subjects how
the second-price auction mechanism works. In each quiz, there is a description of a certain
second-price auction with two specific bids, and subjects need to answer two questions: (q1)
what is the price the winner needs to pay, and (q2) what is one bidder’s payoff? On the
next page of each quiz, correct answers with explanations are provided. Their answers don’t
affect their payoffs from the experiment.
In the final quiz, 70 out of 72 got (q1) correct and 61 out of 72 got (q2) correct. So there
is some confidence that most subjects understand second-price auctions.
Part II: Real-Game Periods
The real game includes 40 periods. At the start of each period of the auction, each subject
is randomly paired with another subject, and only competes against that subject in that
period of the auction. In each pair, one is randomly selected to be the insider and the other
is the outsider. The timeline of each period is as follows:
• Observe Signals: insiders observe q, and in periods with two-dimensional valuations
insiders observe vI , and outsiders observe vO .
• Insiders’ Revelation: insiders choose to (truthfully) reveal q with a cost of 1 experimental dollar or conceal it with no cost, and then outsiders observe the insiders’ revelation
decision and q if it is revealed.
• Bidding: each subject submits one sealed bid (must be an integer between 0 and 100
inclusively) in the second-price auction, and the tie is broken randomly.
• Results: all subjects observe full information within their own pair, including q, vI ,
vO , both bids, who is the winner and their earnings in that period.
Each experimental dollar is equal to one real dollar. 2 times the average earning from all
40 periods (plus $5 show-up fee and $10 completion fee) is the final payoff to each subject.
For simplicity, I omit the unit as experimental dollars for the rest of the description of the
experiments.
6.3
Results and Analysis
This part focuses on two questions: (Q1) Is there a significant difference in revelation behaviors between settings with pure common value and those with two-dimensional values?
(Q2) How do the revelation and bidding strategies look, and are they optimal?
31
Revelation of Low Common Value
Following Example 5, revelation of the low common values is discussed first, because insiders
are consistently predicted to have incentives to reveal them. I compare revelation of low
common value in treatment (i) pure common value (“PCV” for short) and treatment (ii) twodimensional values (“TDV” for short). Theory suggests that subjects do not have incentives
to reveal any common value in treatment (i) because of the positive revelation cost, while
they have incentives to reveal the low common value when they have high or medium private
values in treatment (ii).
Frequency of Revelation
Never
Once+
50%+
(1) % of Subjects in PCV (q = 10)
(2) % of Subjects in TDV (q = 10)
Difference (% in TDV - % in PCV)
75%
17%
-58%***
25%
83%
58%***
0%
58%
58%***
(3) % of Subjects in TDV (q = 10, vI = 10)
(4) % of Subjects in TDV (q = 10, vI = 30)
(5) % of Subjects in TDV (q = 10, vI = 50)
56%
36%
28%
44%
64%
72%
28%
53%
72%
*** Significant difference (TDV - PCV) at 1 percent level
Table 1: The percentages of subjects who chose to reveal low common value (q = 10) never,
at least once, or with a probability of at least 50% in: (1) pure common value, and (2-5)
two-dimensional values aggregately and separately based on the insiders’ private values.
Table 1 first shows the summary of subjects’ revelation frequencies in the settings with
pure common value and two-dimensional values. With low pure common value (q = 10),
only 25% of subjects chose to reveal at least once, and none of them chose to reveal with a
probability higher than 50%. It is consistent with the theory suggesting no revelation. On
the other hand, with two-dimensional values (q = 10), the revelation is much higher: 83%
of subjects chose to reveal at least once, and 58% chose to reveal with a probability higher
than 50%. It is also consistent with the theory suggesting strong incentives to reveal.
There is a clear difference in subjects’ revelation behaviors, suggesting that an additional
private-value dimension indeed changes subjects’ incentives of revelation. The Fisher exact
test shows that the difference in revelation is significant at 1% level. For example, in PCV
27 subjects never revealed and 9 revealed at least once; in TDV, 6 never revealed and 30
revealed at least once. The p-value of a Fisher exact test for (27, 9) vs (6, 30) is 1.13 ∗ 10−6 .
Claim 5 and Example 5 also predict that the insider has different incentives to reveal
q = 10 depending on his/her private value: reveal when the private value is high or medium,
but not reveal with low private value.
32
The second part of Table 1 shows the summary of subjects’ revelation frequencies in the
setting with two-dimensional values (q = 10) and different private values. Clearly, insiders
with high private value have the strongest incentives to reveal q = 10, e.g. 72% of subjects
revealed with a probability higher than 50%; followed by insiders with medium common
value, e.g. 53% of subjects revealed with a probability higher than 50%; and the ones with
low common value have the least incentives to reveal, e.g. only 28% of subjects revealed
with a probability higher than 50%.
Revelation Strategy
The full picture of subjects’ revelation is presented in the following 2 tables: (i) Table 2
provides the revelation of each common value in settings with pure common value and twodimensional values aggregately; (ii) Table 3 provides the revelation of each common value in
the two-dimensional values setting separately with different private values.
Frequency of Revelation
Never
Once+
50%+
(1) % of Subjects in PCV (q = 10)
(2) % of Subjects in TDV (q = 10)
Difference (% in TDV - % in PCV)
75%
17%
-58%***
25%
83%
58%***
0%
58%
58%***
(3) % of Subjects in PCV (q = 30)
(4) % of Subjects in TDV (q = 30)
Difference (% in TDV - % in PCV)
86%
58%
-28%**
14%
42%
28%**
3%
3%
0%
(5) % of Subjects in PCV (q = 50)
(6) % of Subjects in TDV (q = 50)
Difference (% in TDV - % in PCV)
89%
78%
-11%
11%
22%
11%
0%
0%
0%
*** Significant difference at 1 percent level
** significant difference at 5 percent level
Table 2: The percentages of subjects who chose to reveal never, at least once or with a
probability of at least 50% with: (1) low pure common value; (3) medium pure common
value; (5) high pure common value; (2) two-dimensional values with low common value; (4)
two-dimensional values with medium common value and (6) two-dimensional values with
high common value.
Table 2 provides the complete picture of the revelation behavior: Most subjects chose to
never reveal in the pure common value setting, e.g. less than 3% of subjects revealed with
a probability higher than 50%, which is consistent with Claim 4. Also, most subjects chose
to never reveal the high common value in the two-dimensional value setting, e.g. no subject
revealed with a probability higher than 50%, which is consistent with Claim 5.
The revelation of q = 30 in TDV is less transparent, because different equilibria predict
33
different behaviors. The equilibrium in Claim 5 predicts revelation of q = 30 when vI = 30
or 50; one equilibrium in Example 5 predicts revelation only when vI = 50, and the other
predicts no revelation at all. In the data, we observe a significant difference (42% vs. 14%)
in the revelation of q = 30 when comparing the percentages of revealing at least once or not,
but no significant difference (3% vs. 3%) under the measure of revealing at least 50%. It
could be caused by the fact that subjects mis-cooperated on the equilibrium they play, when
multiple equilibria exist. Combining these three equilibria together, we get: some agents
revealed q = 30 when vI = 50, a few revealed when vI = 30, and no one revealed when
vI = 10. Thus, the difference shows up when measuring based on revealing at least once or
not where the revelation with vI = 50 plays an important role, and it disappears when using
stronger measures such that requiring a revelation with a probability higher than 50% where
the revelation with vI = 50 gets diluted.
We proceed to take a closer look at the revelation strategies in TDV when insiders have
different private values.
Revelation in TDV
% of Subjects (q = 10)
% of Subjects (q = 30)
% of Subjects (q = 50)
vI = 10
44%***
11%
6%
Once+
vI = 30
64%***
6%
0%
*** Significantly positive at 1 percent level
vI = 50
72%***
33%***
19%**
vI = 10
28%***
11%
0%
50%+
vI = 30
53%***
3%
0%
vI = 50
72%***
33%***
11%
** significantly positive at 5 percent level
Table 3: The percentages of subjects who chose to reveal with two-dimensional values at
least once or with a probability of at least 50% (breakdown by the insiders’ private values).
Table 3 suggests a significantly positive revelation of q = 30 when vI = 50, and revelation
of q = 30 when vI = 30 and vI = 10 is not significantly positive. I use the same Fisher
exact test to determine whether a revelation frequency is significantly positive or not. For
example, with q = 30 and vI = 50, 12 subject revealed at least once and 24 never revealed.
The p-value of a Fisher exact test for (24, 12) vs (36, 0) is 0.0001629. These results are not
far from the theory about revelation of q = 30: 2 out of 3 equilibria predict revelation when
vI = 50, while only 1 predicts revelation when vI = 30, and all predict no revelation when
vI = 0.
While there are multiple equilibria with inconclusive predictions, another way is to check
whether subjects best responded to the behaviors observed in the experiments. Consider a
new insider (different from all subjects in the experiments) playing the same auction with one
representative outsider of all subjects, we can estimate this new insider’s optimal revelation
strategy. Given the new insider’s optimal revelation strategy greatly depends on subjects’
34
bidding strategies, I provide a brief result below and will go back to explain the details after
examining the bidding strategies.
It turns out the empirically optimal revelation strategy is revealing only when q = 10
and vI ≥ 30 and concealing otherwise. From Table 3, a significantly positive revelation
is indeed observed when q = 10 and vI ≥ 30. The revelations when (q = 10, vI = 10)
and (q = 30, vI = 50) even though significantly positive, are not profitable and show lower
revelation frequencies than those when q = 10 and vI ≥ 30.
Bidding Strategy
The insiders’ bidding strategies are presented in Table 4, and the outsiders’ bidding strategies
are presented in Table 5.
Insiders’ Bids
q = 10
q = 30
q = 50
No vI
Bid Overbid
13.01
30%
34.75
16%
65.93
32%
vI = 10
Bid Overbid
25.60
28%
47.69
19%
68.06
13%
vI = 30
Bid Overbid
49.25
23%
64.87
8%
87.39
9%
vI = 50
Bid Overbid
66.20
10%
88.14
10%
96.37
-4%
Table 4: The insiders’ average bidding strategies and the percentages of overbidding based
on the true value.
Theoretically, the insiders know the true value and have a unique undominated strategy
in second-price auctions, which is bidding the true value. Table 4 presents the insiders’
average bidding strategies, unconditional on the insiders’ revelation strategies.34 The bids
exhibit a positive level of overbidding except the case where the true value is 100. Given
the highest bid is restricted to 100, it is impossible to have positive overbidding when the
true value is 100. It is not a surprising result since overbidding has been reported in many
previous experiments on auctions, such as Kagel and Levin (1993).
The outsiders’ bidding strategy depends on whether they know the common value or
not. If the common value is known, outsiders are in the same situation as insiders, e.g.
bidding the true value is the unique undominated strategy, and their bidding patterns in the
experiments also look very similar to the insiders’ patterns as shown in Table 4. A more
interesting question is the outsiders’ bidding strategies when the common value is unknown,
which depends on not only the prior belief of the distribution of the common values, but
also the belief of the insiders’ revelation strategies. The expected common value is 30 based
on the prior distribution, and it goes higher than 30, after taking into the consideration that
34
The bidding strategies show similar patterns when examining them conditional on the insiders revealing
or not revealing the common values.
35
most insiders reveal low common value signals and hide medium and high common value
signals.
Outsiders’ Bids
Bids
Bids - vO
Estimated Optimal Bids
Profit Increase
No vI
25.23
25.23
10
0.54
vI = 10
vI = 30
vI = 50
42.43
63.77
87.35
32.43
33.77
37.35
21
69
87
1.32
0.03
0.01
Table 5: The outsiders’ average bidding strategies, the differences between the bids and their
private values, their estimated optimal bidding strategies and the profit increases by using
the optimal bids (all conditional on the insiders conceal the common value).
Table 5 presents the outsiders’ average bidding strategy when the common value is unknown. The difference between their bids and their private values equals the expected
common value conditional upon winning, which is 25.23 in the pure common value setting
and in the range of 32.43 - 37.35 in the two-dimensional value setting. Same as in the
analysis of the revelation strategy, multiple equilibria don’t give conclusive predictions of
the bidding strategies. So Table 5 also provides the estimation of the empirically optimal
bidding strategies, and the potential gains by using the optimal bids.
Consider a new outsider (different from all subjects in the experiments) play an auction
uniformly randomly chosen from all auctions in the experiments where the insider hided the
common value, we can estimate this new outsider’s optimal bidding strategy. The results
are as follows:
• PCV: the new outsider would bid 10 instead of 25.23, because the insider’s overbidding
makes winning most likely lead to negative payoffs.
• TDV with vO = 10: the new outsider would bid 21 instead of 42.43, similarity because
of the insider’s overbidding.
• TDV with vO = 30: the new outsider would bid 69 instead of 63.77, but the potential
gain by changing the bid is very small.
• TDV with vO = 50: the new outsider would bid 87, which is very close to the empirical
average bid.
Overall, given the potential gain by using the optimal bids is relatively small, subjects might
have little incentive to figure them out.
Finally, we can estimate the optimal revelation strategy given the observed bidding strategies. Recall we consider a new insider (different from all subjects in the experiments) playing
36
the same auction with one representative outsider of all subjects. Assume this representative outsider use the average bidding strategy of all insiders if knowing the common value
in which case insiders and outsiders are in the same situation,35 and the average bidding
strategy of all outsiders if not knowing the common value. Then we can conclude that it
is optimal for this new insider to reveal q = 10 when vI ≥ 30, which would lower the representative outsider’s bid from 42.43 to 25.60 when vO = 10 and from 63.77 to 47.69 when
vO = 30. While it is optimal for this new insider to conceal in all other cases. For example
when q = 30 and vI = 50, revealing q could increase the representative outsider’s bid from
42.43 to 49.25 when vO = 10 and from 63.77 to 64.87 when vO = 30 (although the difference
is small). Thus, despite a significant positive revelation of q = 30 is observed when vI = 50,
it is not optimal but it is not far from the optimal outcome.
7
Conclusions and Discussions
This paper characterizes information revelation when agents observe multiple signals. The
basic intuition is that if there are two signals, agents have incentives to reveal the signal that
could lower others’ bids and earn a higher profit from the other signal.
Specifically, with only one signal, agents don’t have incentives to reveal the only information in order to retain their information advantage. While with multiple signals, agents have
incentives to reveal the signals which directly influence others’ valuations (e.g. commonvalue signals). Revelation of such signals can manipulate others’ bids, and let the agent earn
a higher profit from the remaining signals that only relate to his/her own valuation (e.g.
private-value signals). Considering information revelation makes the model more tractable:
there usually exists one equilibrium with full revelation of common-value signals. Such equilibrium leads to full efficiency and possibly a higher profit for the seller. So, one policy
implication is that public communication of information with verifiable evidence should be
encouraged.
This model could be extended in several directions.
The first direction is the setup of the signal structure. This paper focuses on the twodimensional signals including a signal related to one’s own private value and a signal related
to the common value. This is clearly important in many settings (e.g. housing markets,
timber-harvesting contracts, and oil tract auctions), but there are other types of signal
structure worth considering. For instance, Fang and Morris (2006) purpose a two-dimensional
signal setting, where each agent observes his/her own private value and a noisy signal of the
other bidder’s private value. They show that it also involves the difficulty of charactering the
35
The following result remains the same if assume the representative outsider use the average bidding
strategy of all outsiders if knowing the common value.
37
equilibrium bidding strategy as a two-to-one mapping. So it might be interesting to explore
incentives of revealing signals related to others’ private values, and even consider a broader
multi-dimensional signal structure.
The second direction is to consider the sale of multiple items. If these k items are
homogeneous and sold in a simple (k + 1)th-price auction, all of the main results of this
paper hold, for instance bidders have incentives to reveal common-value signals. However, if
these items are heterogeneous and agents only have one unit demand, incentives of revelation
might be different. For instance, if agent i observes a low quantity for tract A and a high
quantity for tract B, revealing the low quantity in A could reduce competitors’ bids for A
but may also increase competitors’ bids for B. In this case, the intuition of revealing low
quantity leading to lower bids is not clear. Auctions of multiple heterogeneous items are
complicated to design by themselves, and the incentives of information revelation in such
settings deserve further research.
Lastly, the observations from the experiments are also worth further study. For instance,
it would be interesting to see whether the results are robust if subjects play first-price auctions
or if subjects can reveal both signals. In addition, analyzing subjects’ learning behaviors
throughout the 40 periods would be useful.
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Appendix
Proof of Proposition 1: First eliminate all weakly dominated bidding strategies in the
second-price auction. The undominated strategy for the insider is bidding the true value
u = (p − cI )q, and for the outsider is as follows: bidding the true value u = (p − cO )q
if q is revealed, and bidding within the lowest and highest possible value otherwise, bO ∈
[(p − cO )qL , (p − cO )qH ].
Consider the case where the insider observes qL : if the insider reveals qL , the outsider bids
(p − cO )qL , and if the insider hides qL , the outsider bids in an interval [(p − cO )qL , (p − cO )qH ],
such that revealing qL reduces the outsider’s bid. When revealing qL , the insider wins if
cI < cO and the expected payoff for the insider is
Z cH
(cO − cI )qL f (cO )dcO
πI (cI ) =
cI
which is positive if cI < cH . Revealing qL weakly dominates concealing it when cI < cH ,
because revealing qL reduces the outsider’s bid and increases the insider’s expected payoff.
Thus qL is revealed with probability one (except cI = cH ) in the iteratively undominated
equilirbium.
Consider the case where the insider observes qH : if the insider reveals qH , the outsider bids
(p−cO )qH , and if the insider hides qH , the outsider bids in an interval [(p−cO )qL , (p−cO )qH ],
such that revealing qH increases the outsider’s bid. Thus concealing qH weakly dominates
revealing it, and all qH is concealed in the iteratively undominated equilibrium.
The remaining case is q = qM , which takes several steps to discuss. First, we claim
bidding (p − cO )qM weakly dominates bidding less than (p − cO )qM for the outsider, when q
is hidden and cO < cH . There are three possibilities when q is hidden: (i) q = qL and cI = cH ;
(ii) q = qH ; and (iii) q = qM . For (i) the outsider is indifferent among submitting any bids
in [(p − cO )qL , (p − cO )qH ], because the insider’s bid is fixed to be (p − cH )qL ; for (ii) bidding
(p − cO )qM is weakly better than bidding lower because q = qH ; for (iii) bidding (p − cO )qM
is the unique undominated bidding strategy. Thus, outsiders bid at least (p − cO )qM when
q is hidden and cO < cH in the iteratively undominated equilibrium.
Second, we claim that the insider observing qM and cI < cH hides qM in the equilibrium,
only if the outsider bids (p − cO )qM for all cO > cI except for a measure zero set of cO . Otherwise, the insider could lower the outsider’s bid by revealing qM and increase the expected
41
payoff. Let C denote the set of all cI with which insider observing qM hides the signal, and
let c be the infimum of C. Third, we claim the outsider bids (p − cO )qM for all cO > c except
for a measure zero set of cO , which follows directly from the second claim.
Then we claim: C is empty or C = {cH }, such that qM is revealed with measure one
(except cI = cH ). Suppose the contrary is true, which is c < cH . There exist some small
enough > 0 satisfying several conditions described as follows. Consider the outsider with
cO ∈ (c, c + ) (assuming c + < cH ), by the third claim above, the outsider should bid
(p − cO )qM almost surely. Let’s consider the loss and gain for such outsider when deviating
by bidding (p − cO )qH instead. The loss and gain appears only when q is hidden, which has
a probability of p(q = qH ) + p(q = qM )p(cI ∈ C). The loss appears when q = qM , the insider
hides the quantity (c ≤ cI ) and the outsider wins with a negative profit (cI < cO ≤ c + ),
T
which is bounded by the probability of q = qM and cI ∈ (c, c + ) C conditional on q is
hidden
T
p(q = qM )p(cI ∈ (c, c + ) C)
p(loss) ≤
p(q = qH ) + p(q = qM )p(cI ∈ C)
The value of the loss (when it appears) equals the value for the outsider minus the price to
pay which is the insider’s bid
|loss| = |(p − cO )qM − (p − cI )qM | = |(cI − cO )qM | ≤ qM
So the loss is at most
p(loss) × qM which is the probability of the loss times the value of the loss (when it appears). Similarly,
the gain appears when q = qH , the outsider wins with a positive profit (cI > cO ) but doesn’t
win with the original bid:
(p − cO )qM < (p − cI )qH
where the LHS is the outsider’s original bid and the RHS is the insider’s bid, and it implies
the bound for cI is
qH − q M
qM
cO < cI <
p+
cO
qH
qH
Thus, the gain appears for sure when cI ∈ (c + , c∗ ), where c∗ = min(cH , c0 ) and c0 =
qH −qM
p + qqM
c.36 The gain is at least
qH
H
p(q = qH )
p(q = qH ) + p(q = qM )p(cI ∈ C)
Z
c∗
(cI − cO )qH f (cI )dcI
c+
which is the outsider’s gain when winning with q = qH and cI ∈ (c + , c∗ ) conditional on q is
hidden. When converges to 0, the loss shrinks to 0 and the gain increases from a positive
36
Since p > cH , c0 > c and assuming is small enough that c0 > c + .
42
value such that we can find some where the gain is higher than the loss. Then the deviation
is profitable, which is a contradiction. Thus, our original claim is true: qM is revealed with
probability one (except cI = cH ).
Overall, in all iteratively undominated perfect Bayesian equilibria: in revelation, qL and
qM is revealed with probability one, all qH is concealed; in bidding, the insider bids bI =
(p − cI )q, and the outsider bids bO = (p − cO )q when q is revealed, and bO = (p − cO )qH
otherwise.
R x∗ R y ∗
Proof of Claim 2: The LHS of Equation (2) is higher than 0 max(−x,−y∗ ) (x+y)k(y)dyh(x)dx,
because the event {x < 0, x + y > 0} is dropped. We want to show D as defined below is
non-negative,
Z y∗
(x + y)k(y)dy − x
D =
max(−x,−y ∗ )
∗
Z
y∗
yk(y)dy − x
= xP (y > max(−x, −y )) +
max(−x,−y ∗ )
Z
y∗
yk(y)dy − xP (y ≤ max(−x, −y ∗ ))
=
max(−x,−y ∗ )
If P (y ≤ max(−x, −y ∗ )) = 0 (including the case max(−x, −y ∗ ) = −y ∗ ), D is surely nonnegative. Otherwise, because the density is symmetric, P (y ≤ −x) = P (y ≥ x),
Z
y∗
yk(y)dy − xP (y ≥ x)
D =
−x
Z y∗
Z
x
(y − x)k(y)dy +
=
Z
yk(y)dy
−x
x
y∗
(y − x)k(y)dy ≥ 0
=
x
where
Rx
−x
yk(y)dy = 0 because k(y) is symmetric.
Proof of Proposition 3: First, it is easy to show the existence of a private-hiding equilibrium in a second-price auction. Agents bid their true values, and the revelation of others’
signals doesn’t affect this equilibrium bidding strategy.
Now we prove the existence of a private-hiding equilibrium in a first-price auction when
q is common knowledge. Without loss of generality, we assume q = 0 in the following
proof, otherwise it is a constant adding to all agents’ values and all bids, but won’t affect
their payoffs. Without information revelation, there exists an equilibrium with a symmetric
43
and (strictly) monotone bidding strategy β(vi )37 , the profit of agent i in such a monotone
equilibrium is
π(vi , bi ) = (vi − bi )F n−1 (β −1 (bi ))
where all other bidders (j 6= i) bid β(vj ).
If some agent (say agent 1) deviates by revealing v1 , we claim this deviation is not
profitable. Assume b01 is agent 1’s bidding strategy after revealing v1 , which might be mixedstrategy. Define b1 as follows,
b1 = sup{b1 ∈ B1 : P (b01 < b1 ) = 0}
Intuitively b1 is the lower bound of b01 except for a measure zero set. This definition of
b1 implies that: P (b01 < b1 ) = 0 and for any δ > 0, P (b01 ∈ [b1 , b1 + δ)) > 0. Otherwise
if P (b01 ∈ [b1 , b1 + δ)) = 0 for some δ, P (b01 < b1 + δ/2) = 0, which is a contradiction
b1 ≥ b1 + δ/2.
We claim the profit agent 1 earns in the deviation of revealing vi > vL 38 is
π 0 (v1 , b1 ) = (v1 − b1 )F n−1 (b1 )
There are two cases: (i) b1 is in agent 1’s pure or mixed bidding strategy, or (ii) b1 is not in
agent 1’s bidding strategy. Since for any δ > 0, P (b01 ∈ (b1 , b1 + δ)) > 0, in case (ii) there
(m)
exists a decreasing sequence of bids (bi > b1 , m → ∞) converging to b1 and the sequence
belongs to agent 1’s mixed strategy. For other agents with v < b1 , it is undominated to
bid strictly lower than b1 . Also, other agents with v > b1 must bid strictly higher than b1 .
Because if agent 1 bids b1 with a positive probability, it is better to bid a bit higher than b1
to win over that positive probability; or if agent 1 bids strictly higher than b1 almost surely,
it is also better to bid higher than b1 to get a positive probability of winning39 instead of 0.
So the probability that agent 1 could win by submitting b1 is the probability that all
other agent’s values are lower than b1 :
P (b1 > max bj ) = F n−1 (b1 )
j6=i
(5)
Also, the probability that some other agent bidding b1 is 0 because all other agents who
might bid b1 must have a value of b1 :
P (b1 = max bj ) = 0
j6=i
37
(6)
Monotonicity means β(vi ) is (strictly) increasing in vi , and see Milgrom and Weber (1982a) for details.
There is no incentive to reveal vL . The equilibrium payoff is always 0 for the agent with vL , since other
agents’ values are higher than vL with probability 1.
39
The probability of winning is positive because for any δ > 0, P (b1 ∈ (b1 , b1 + δ)) > 0.
38
44
In case (i), the profit from bidding b1 is (v1 − b1 )F n−1 (b1 ), which is the difference of the
value minus the bid times the probability of winning, and it is consistent with the claim.
(m)
Let’s consider case (ii) where there exists a decreasing sequence of bids (b1 ) converging to
b1 . Agent 1 earns the same positive profit from all bids in the sequence because agent 1 is
indifferent among submitting all these bids. The probability of winning by submitting these
decreasing bids also decreases and should converge to a constant, say P , because probabilities
are non-negative. The limit P equals the probability that b1 is higher than or equal to the
highest bid from all other agents:
(m)
lim P (b1
m−>∞
(m)
because limm−>∞ b1
> max bj ) = P (b1 > max bj ) + P (b1 = max bj )
j6=i
j6=i
j6=i
= b1 . Put (5) and (6) into the equation,
(m)
lim P (b1
m−>∞
> max bj ) = F n−1 (b1 )
j6=i
(m)
(m)
The profit from submitting bids in the sequence is (v1 − b1 )P (b1
converges to and equals (v1 − b1 )F n−1 (b1 ).
We prove the profit of agent 1 by deviating is
> maxj6=i bj )40 , which
π 0 (v1 , b1 ) = (v1 − b1 )F n−1 (b1 )
Let b∗1 be the strategy to maximize the profit π 0 (v1 , b1 ), b∗1 in the original equilibrium (without
deviation) gives agent 1 a higher profit such that π(vi , b∗1 ) ≥ π 0 (v1 , b∗1 ). This follows from
undomination (β(v) ≤ v), which implies β −1 (b∗1 ) ≥ b∗1 . Thus, the deviation of revealing v1 is
not profitable and the private-hiding equilibrium is valid.
40
Tie is ignored here, because when m is large enough, the probability of tie must converge to 0.
45