Implementation by Gradual Revelation
Gorkem Celik
yJuly 29, 2014
Abstract
We investigate the feasibility of implementing an allocation rule with agradual revelation mech-anism in which agents reveal their private information over time (rather than all at once). With independently distributed types, private values, and transferable utilities satisfying a single-crossing
property, anex-post monotonicity condition is su¢ cient forbudget-balanced implementation of any incentive compatibleallocation rule with any gradual revelation scheme. When we extend the single-crossing property over the set ofrandomized allocations, a weaker monotonicity condition is both necessary and su¢ cient for budget-balanced implementation by gradual revelation.
KEYWORDS: Mechanism Design; Sequential Revelations; Budget-balanced Implementation
1
Introduction
In her celebrated novel,Sense and Sensibility, Jane Austen describes the lives of the Dashwood sisters in 18th century England. Elinor Dashwood falls in love with Edward Ferrars. The attraction is mutual, but
Edward is already secretly engaged to another woman who is in fact more interested in Edward’s family
fortune than in Edward himself. At the end of the novel, Edward loses his rights to the family money
but ends up marrying Elinor. Of course, not all love stories have such happy endings. In the same novel,
Elinor’s sister, Marianne, is romantically involved with John Willoughby. However, unlike Edward,
Willoughby prefers living a comfortable lifestyle to pursuing true love. Thus, Willoughby marries a
young lady with a large fortune rather than Marianne.
From a cold-blooded mechanism design perspective, it is simple to formulate a straightforward
in-centive scheme that would generate anallocation rule pursuant to which idealist Edward would lose his fortune but succeed in love and materialist Willoughby would secure a …nancially comfortable life but
Acknowledgements to be added.
fail to …nd romance. Adirect revelation mechanism would ask each gentleman whether he is more of an idealist or more of a materialist and then assign him the appropriate spouse.
Although the revelation principle is an essential tool for analyzing design problems, it does not provide a good formula for writing a literary masterpiece. In Sense and Sensibility, it takes three
volumes and 300 pages for Elinor and Edward to unite. First, Edward begins acting detached and
Elinor concludes that he has lost interest in her. Then, she learns of Edward’s prior engagement, which
he feels compelled to honor. However, Edward’s mother also …nds out about the engagement and
disinherits him in disapproval. Once Edward loses the connection to his wealthy family, Edward’s …ancé
breaks up with him and marries his brother, who is now the sole heir of the family estate. Elinor hears
about the marriage in Edward’s family and assumes that it was Edward who was married. Near the end
of the novel, Edward shows up unexpectedly and announces to Elinor that he is no longer burdened by
his earlier engagement nor by his family obligations.
To generate suspense throughout the novel, Austen does not reveal the traits of her characters at once.
Instead, the revelations are supported by a sequence of decisions made by the characters. These decisions
make sense from the characters’perspective: Each decision re‡ects the preferences of the decisionmaker
given what he/she knows about the other characters at the time that the decision is made. The decisions
not only serve to reveal the qualities of the characters to readers (and to the other characters in the
book) but also lead to the outcomes that the characters will face at the end of the novel.1
In this paper, we investigate the feasibility of implementing an allocation rule withgradual revelation
schemes, such as the one that serves as the foundation of the story line in Sense and Sensibility. We study
this question in the context of an economy with independently distributed types, private values, and
transferable utilities satisfying a single-crossing property. A well-known condition which is indispensable
– both for implementation by simultaneous revelation (as prescribed by the revelation principle) and
for implementation by the gradual revelation that we attempt to elucidate in this paper – isincentive compatibility: An implementable allocation rule should provide each agent with the incentive to report his type truthfully under his prior beliefs regarding the types of the other agents. The results of this
paper identify other conditions that guarantee implementation of an allocation rule by using a gradual
revelation mechanism. These results are presented in three propositions with closely related proofs.
In a recent paper, Ely, Frankel, and Kamenica (2013) discuss the idea that gradual revelation of
information has entertainment value for a Bayesian audience. These authors o¤er many examples from
real-life auctions, political primaries, mystery novels, sports events, gambling, and reality TV that
support the contention that receivers of information have preferences regarding the evolution of their
beliefs over time. The model we develop in Section 2 bene…ts a great deal from their formalization of
sequential revelations as astochastic path of beliefs. Ely, Frankel, and Kamenica identify the revelation sequences to which a designer would commit to maximize the “suspense”or “surprise”that the audience
would experience. The current paper complements the work of these authors by detailing how such a
revelation sequence can be supported by the behavior of self-interested agents instead of by the actions
of a central designer with a commitment power.
Sustaining suspense and surprise is not the only economic motivation for pursuing gradual revelation.
Earlier literature provides a number of rationales, such as partial veri…ability of agent type (Bull and
Watson, 2007; Deneckere and Severinov, 2008), communication costs (Van Zandt, 2007; Fadel and Segal,
2009; Mookherjee and Tsumagari, 2013), agents’outside options (Celik and Peters, 2011, 2013), and a
lack of commitment to monetary transfers (Horner and Skrzypacz, 2012). At the end of this section, we
present additional applications that rely on the use of gradual revelation mechanisms.
Results
There is one class of allocation rules that can be trivially implemented through gradual revelation
schemes. When an allocation rule isdominant-strategy incentive compatible, truthful revelation remains the optimal strategy of an agent regardless of what he learns about the types of the other agents.
Mookherjee and Reichelstein (1992) establish that an incentive compatible allocation rule can be
trans-formed into a dominant-strategy incentive compatible allocation rule when the original allocation rule
is ex-post monotone. The allocation rule resulting from this transformation is identical to the original allocation rule up to a transfer function that yields the same interim expected payo¤ for the agents
as in the original allocation rule. Mookherjee and Reichelstein prove this result in a continuous type
environment by invoking the revenue equivalence theorem. In Section 3, we extend this transformation to our discrete type model (Proposition 1).
In Section 4, we turn our attention to the analysis of thebudget-balanced mechanisms, which remain the main focus for the rest of the paper. It is well known that dominant-strategy implementation is
generally incompatible with balancing the budget. Therefore the transformation discussed above requires
outside intervention to provide agents with the appropriate transfers. However, in Proposition 2, we show
that the ex-post monotonicity condition, which is su¢ cient for dominant-strategy implementation of an
incentive compatible allocation rule, is also su¢ cient for its budget-balanced implementation under any
possible sequence that the private information may be revealed.
Unlike dominant-strategy transfers, which depend only on the …nal type reports of the agents,
bal-anced budget transfers depend on the entire path of revelations made by them. With these balbal-anced
budget transfers, the agents …nd it optimal to report their types truthfully not only at the beginning of
the game under their prior beliefs but also at any information set they may reach during the process
may make prior to their …nal reporting of the exact types. Thanks to this indi¤erence condition, it is
an equilibrium behavior for each type of each agent to randomize between di¤erent revelations with the
appropriate frequencies that would generate the targeted sequence of information disclosures.
We construct these balanced budget transfers by following theexpected externalitymethod introduced by Arrow (1979) and d’Aspremont and Gerard-Varet (1979). The crucial aspect of supporting a balanced
budget is ensuring that each agent’s transfer re‡ects the expected level of the externality he imposes
on the other agents at each step of the revelation procedure. However, achieving this in our dynamic
setting is a more involved exercise than what is required in a simultaneous revelation game because the
expectations in our case are based on endogenous probability distributions that are determined by earlier
revelations made by the same set of agents.
Ex-post monotonicity is asu¢ cientcondition to reconcile gradual revelation with a balanced budget, but it is not a necessary condition. In Section 5, we introduce a weaker monotonicity condition that is de…ned with respect to the intended sequence of revelations. When the single-crossing property is
extended over the stochastic allocations, this weaker monotonicity condition is necessary and su¢ cient
for budget-balanced implementation by gradual revelation (Proposition 3). Mookherjee and Tsumagari
(2013) refer to a similar monotonicity condition in characterizing implementable output functions in the context of communication costs. We discuss the relation between these monotonicity conditions in
Section 5.
One complication in proving the su¢ ciency part of Proposition 3 is identifying the optimal
contin-uation behavior of an agent who makes an o¤-the-equilibrium-path revelation that is inconsistent with
his type. We resolve this complication by designating a "type to imitate" for each deviation that a
particular type can follow. We show that a gradual revelation mechanism can provide the incentives for
a deviating type to imitate only these designated types after he makes the decision to deviate from the
equilibrium behavior. This mechanism ensures that each type of each agent is indi¤erent with respect to
the revelations he makes with a positive probability on the equilibrium path, but this indi¤erence does
not necessarily extend to the o¤-the-equilibrium-path revelations for this type.
Economic Applications
We now discuss the real-life relevance of gradual revelation mechanisms and the implications of our
results in the contexts of auctions, intra…rm resource allocation problems, and civil litigation.
Auctions
There are many di¤erent ways an auctioneer can sell goods and services to potential buyers. Some
auction forms ask each bidder to submit a single bid at the same time as the other bidders (e.g.,
move at di¤erent points in time (e.g., Dutch and English auctions). One of the primary teachings of the
mechanism design theory is that, whatever allocation rule an auction creates, there is a direct revelation
mechanism that supports the very same allocation rule. Bidders reveal their private information to this
mechanism simultaneously.
Recent developments in information technology have enabled such simultaneous revelation
mech-anisms to be designed at a low cost. Nevertheless, we still observe many auctions in which bidders’
private information is gradually revealed into the public domain. Theseindirect revelation auctions are used everyday in sales of antiques, artwork, ‡owers, …sh, etc. The existence of such auctions may serve
an additional purpose, such as creating a cultural and entertainment value or they may simply be the
product of a historical accident.
Regardless of the rationale for these gradual revelation auctions, an important question is whether
they succeed in attaining desirable allocations, which would have been available under direct revelation
mechanisms. This paper answers this question in the a¢ rmative for the independent private values
speci…cation. As long as the auctioned object is assigned to the bidders in an ex-post monotonic manner,2
the auctioneer can choose any arbitrary sequence that the bidders reveal their private information. In
this paper, we construct the gradual revelation mechanism which makes this sequence of revelations
incentive compatible for the bidders (Proposition 1) and which brings the same revenue to the auctioneer
(pointwise across all value pro…les) as does a direct revelation mechanism (Proposition 2).
Intra…rm resource allocation
As another example of gradual revelation mechanisms, consider the intra…rm resource allocation
problem as formalized by Harris, Kriebel, and Raviv (1982) (hereafter, HKR).3 The …rm consists of
N product divisions and a single resource division. Each product division is tasked with producing a pre-determined level of output by using either its own e¤ort or an intermediate good that is sourced
from the resource division. The productivity of each product division in utilizing the intermediate good
is its private information, as is the e¢ ciency of the resource division in transforming a primary resource
into the intermediate good. The management of the …rm must decide how to allocate the intermediate
good across di¤erent divisions and how much monetary transfer to make to each division to compensate
for the cost of the e¤ort.
HKR show that monetary transfers to the divisions can be minimized in expectation with a direct
revelation mechanism that involves transfer pricing. According to the HKR mechanism, the product
2 Ex-post monotonicity is a mild restriction for auctions with independent private values. The e¢ cient allocation
rule – which assigns the ob ject to the highest valuation bidder – and theoptimal allocation rule – which maximizes the expected revenue of the auctioneer – satisfy this condition.
3See Rajan and Reichelstein (2004) for a more general characterization of this problem as well as a review of the
divisions and the resource division simultaneously report all their private information to the …rm’s
management. Each product division is provided with the intermediate good and asked to pay a transfer
price if and only if its productivity is higher than a certain threshold. The optimal productivity threshold
is determined by the resource division’s e¢ ciency, but the transfer price for each product division depends
only on that division’s own productivity.
For many …rms, a mechanism that requires management to simultaneously communicate with all
its divisions is not a realistic solution to the resource allocation problem. A more practical design
may involve a sequential form of communication. According to such an alternative design, the resource
division …rst reports its e¢ ciency and this information is used to determine the productivity threshold
for the product divisions. Next, each product division is asked (in any arbitrary order) whether its
productivity is lower than this threshold. Only in the event that the product division’s productivity is
higher than the threshold, it is asked to reveal its productivity level completely (so that the appropriate
transfer price can be calculated). Such a gradual revelation mechanism will involve a less-demanding
communication procedure than the HKR mechanism because a product division must report its exact
productivity only when it is relevant (when it is higher than the threshold).
Unfortunately, HKR’s transfer pricing scheme is not compatible with this type of gradual revelation
because a high productivity division will now have an incentive to understate its productivity to reduce
the transfer price it pays. Nonetheless, the results we present in this paper identify an alternative set of
monetary transfers that provide the right incentives for the divisions. These alternative transfers can be
designed to be equal to (from each division’s perspective) the HKR transfers in expectation (Proposition
1). Moreover, the total transfer – from the …rm to its divisions – can be set equal to the total HKR
transfer in all states of nature (Proposition 2) such that the overall budget requirements of the …rm are
respected by the new transfers.
Bifurcated trials
Abifurcated trial, i.e., a trial in which the issues ofliability anddamages are litigated sequentially,4
can be thought as another example of a gradual revelation mechanism. Consider an environment of civil
dispute in which a defendant is potentially responsible for a plainti¤’s injuries. As in Daughety and Reinganum (1994), the defendant is privately informed on the extent of her liability and the plainti¤
is privately informed on the magnitude of the damages he has su¤ered. These two litigants can invest in evidence production technologies that allow them to collect costly evidence. Similar to the model
developed by Rubinfeld and Sappington (1987), this cost is increasing in liability for the defendant and
4An example of a bifurcated trial is the patent infringement case between Polaroid and Kodak. Kodak’s liability was
decreasing in damages for the plainti¤.5 By an appropriate allocation of theburden of proof, the court
can commit to how it will condition the level ofrecovery–the transfer from the defendant to the plainti¤ –on the evidence provided by the litigants. Assume that the defendant minimizes the recovery plus her
evidence collection costs and the plainti¤ maximizes the recovery minus his evidence collection costs in
expectation.
In aunitary trial, in which the two parties make their evidence collection decisions simultaneously, the court can be considered as a designer deciding on
i) the defendant’s evidence level as a function of her liability;
ii) the plainti¤’s evidence level as a function of his damages; and
iii) the recovery rate as a function of both of these parameters.
The court can implement a collection of these three functions, if they collectively satisfy the incentive
compatibility constraints of the two litigants under their prior beliefs regarding one another.6
By contrast, in a bifurcated trial, the court …rst considers the evidence on liability (which is provided
by the defendant) and makes a judgement on it. Only after the liability issue is settled does the trial
proceed to a second stage, in which the court begins hearing evidence regarding damages (provided by
the plainti¤) and makes a …nal decision on recovery. Under bifurcation, the court can link the plainti¤’s
evidence level to the extent of the defendant’s liability, which is revealed by the evidence unearthed at the
…rst stage of the trial. According to Landes (1993), this process generally leads to lower total litigation
costs than under unitary trials.7 For instance, if the defendant’s evidence indicates that there is a low
level of liability, the plainti¤ may voluntarily choose not to collect any evidence on the magnitude of
damages.
A bifurcated trial should also respect the incentive compatibility of the evidence levels and the
recovery rates, under the litigants’prior beliefs about their rivals’types. However, unlike in a unitary
trial, this condition is not su¢ cient for implementability: Even if incentive compatibility is satis…ed at
the interim stage, once the defendant is found to be liable, the plainti¤ may have a more pronounced
incentive to overspend on evidence collection and to exaggerate his damages. Landes suggests that this
latter e¤ect may cancel out the cost-cutting features of bifurcation discussed above.8
The current paper identi…es a monotonicity condition that would allow us to compare unitary and
5See Froeb and Kobayashi (1996) and Yilankaya (2002) for similar costly evidence production models.
6See Spier (1994) and Neeman and Klement (2005) for examples of a similar mechanism design approach to litigation. 7Landes’s model relies on a framework of mutual optimism in which the defendant and the plainti¤ share the same
information but have di¤erent estimates of the court decision. See Chen, Chien, and Chu (1997) for an alternative model with asymmetric information.
8See Daughety and Reinganum (2000) for a model in which litigation expenditures are choice variables for the defendant
bifurcated trials more clearly. Suppose that the court would like the plainti¤’s choice of evidence to be
(weakly) increasing in his damages, regardless of the extent of the defendant’s liability. Our Proposition
3 implies that, sequentiality of the bifurcated trials does not introduce any additional incentive problems
for the legal system, once this monotonicity condition is satis…ed. If the type-dependent evidence levels
of the litigants and theirexpected recovery rates are incentive compatible under the prior beliefs, then the court can …nd theactual recovery levels that will give the right incentives to the litigants –both for the liability and the damages stages of the trial.
2
The Model
In an environment with incomplete information, I is the set of agents and jIj is the number of agents
in this set. We refer to the private information of an agent as his type, and agents’ types are drawn
independently from …nite sets of real numbers. Formally, each agenti2I has a …nite type set i R
with a generic element i. Agenti’s type is distributed with respect to the prior distribution 0i 2 i, independently of the other agents’types. In standard notation, the cross product space = i2I i is
the set of type pro…les with a generic element =f igi2I. We de…ne 0= 0i i2Ias the collection of the priors on the agents’types. Similarly, istands for j2I fig jwith generic element i=f jgj2I fig,
and 0
i represents 0j j2I fig.
Y is the …nite set of economic alternatives. An agent’s preferences for economic alternatives depend
on his own type but not on the types of the other agents. Accordingly, the preferences of agent i are
represented by the utility functionui:Y i!R. The continuous type model, in which agents’types
are drawn from a continuous distribution and their utility functions are continuously di¤erentiable, can
be understood as a limit of our discrete type model. We will refer to this limit case to better illustrate
some of our points.
A mechanism is a means of separating agent types by exploiting di¤erences in the intensity of their
preferences on economic alternatives. We put some structure on such di¤erences in taste by assuming
the followingsingle-crossing property:
Assumption(SC)For any agentiand any two economic alternatives yandy^2Y, the utility di¤ erence ui(y; i) ui(^y; i)is either weakly increasing or weakly decreasing in i.
Note that this assumption is trivially satis…ed when agents have at most two types. Similarly, when
there are only two economic alternatives in setY, agent types can be relabeled to satisfy single-crossing.
The single-crossing property implies an order on the set of economic alternativesY for each agent.
andy^2Y,
hi(y) hi(^y) if and only ifui(y; i) ui(^y; i) is weakly increasing in i: (1)
If function hi( ) satis…es condition (1), any positive monotone transformation of it also satis…es this
condition. Non-uniqueness of function hi( ) will not be a problem for our analysis because we will
be exclusively concerned with its monotonicity properties, which are robust under such a monotone
transformation. For many settings in which preferences satisfy a one dimensional condensation condition
(Mookherjee and Reichelstein, 1992), functionhi( )has a natural interpretation such as the probability
of receiving an object, the level of consumption, or the amount of production.
In addition to the economic alternative and his type, an agent’s payo¤ is also a¤ected by the monetary
transfer he receives. We assume that the payo¤ functions are quasilinear in transfers and can be written
as
ui(y; i) +xi (2)
for agenti, whereyis the economic alternative, i is agenti’s type, andxi2Ris his monetary transfer.
Following the earlier literature, we de…ne a decision rule as a mapping from type pro…les into
economic alternatives,y : ! Y, and a transfer ruleas a mapping from type pro…les into agents’
transfers,x: !RjIj, wherexi yields the relevant dimension of the transfer rule for agenti. We refer
to (y( ); x( ))as an allocation rule.
In an auction, the decision rule and the transfer rule determine which bidder will receive the auctioned
object and how much each bidder would pay as functions of the realized valuations of all the bidders. For
the intra…rm resource allocation problem, the decision rule guides the assignment of the intermediate
good to di¤erent divisions of the …rm and the transfer rule sets monetary compensations as functions
of the divisions’productivities. In our litigation example, the decision rule controls the evidence levels
and the transfer rule regulates the recovery rates as functions of the litigants’types.
Allocation rule (y( ); x( ))is incentive compatible under belief 0 if the following constraint is
satis…ed for all i2I and all i;^i 2 2i pairs:
E ij 0ifui(y( i; i); i) +xi( i; i)g E ij 0i
n
ui y ^i; i ; i +xi ^i; i o
: (3)
If (y( ); x( )) satis…es the incentive compatibility constraints above, we know from the proof of the
revelation principle that there is a direct revelation mechanism in which the agents reveal their types
all at once, knowing that the economic alternative and their transfers will be chosen according to this
allocation rule. Of course, the revelation principle does not rule out the possibility of implementing the
same allocation rule in alternative ways.
What we study in this paper is these alternative ways of implementation. In particular, we investigate
of private information of the agents by making them reveal their typesgradually.
Our model is a discrete type model but perhaps this point is better explained by referring to a
well-known example where types are distributed on a continuum. Consider a …rst-price sealed-bid auction
such that the type (the private value for the auctioned object) of each bidder is uniformly distributed
on the interval [0;1]. This auction has an equilibrium in which each bidder bids (jIj 1)=jIj times
his value, such that the types of the bidders are revealed simultaneously with the observation of their
bids. TheDutch auction (the descending price auction in which the asking price of the object lowers in time) is another mechanism that implements the same allocation rule with a gradual revelation scheme.
The Dutch auction has an equilibrium in which each bidder waits until the asking price declines to the
(jIj 1)=jIjtimes his value and agrees to buy the object at that price if no other bidder chooses to buy
it earlier at a higher price. As time passes and the asking price declines, the bidders in the Dutch auction
(and the outside observers) continue to update their beliefs about their rivals until a bidder buys the
object and reveals his type completely. This makes the Dutch auction more exciting than the …rst-price
auction, even though both auction formats generate the same economic allocation in which the bidder
with the highest value receives the object and pays(jIj 1)=jIjtimes his value.9
The idea that an audience may demand suspense and surprise in the revelation of information has
recently been studied by Ely, Frankel, and Kamenica (2013), and we refer to their model to integrate
the notion of gradual revelation in the mechanism design framework. Ely, Frankel, and Kamenica are
concerned with information disclosures by a single informed party, but their approach can be extended
to accommodate revelations made by multiple agents. In this extended model, t2 f0;1; :::; Tgdenotes
the period. In each period, each agent chooses a signal from a …nite set.10 All agents observe the signals
sent and update their beliefs regarding the sending agents. The resulting periodt belief on agent iis
denoted by distribution t
i 2 i over the possible types of agent i, and t=f tigi2I is a collection of these beliefs. Technically, an agent’sinformation policyis a function that maps the current periodt,
the agent’s type i, and the current belief t into a distribution over the signals.11
The agents’ information policies together generate a stochastic path of beliefs on their types. This
path of beliefs satis…es the following martingale property: What the other agents believe about agent
i’s type in periodt 1 is an unbiased estimate of what they will be believing in periodt. Otherwise,
9As alluded to in this example, gradual revelation may be intended to keep an audience in suspense. In this case, it
is conceivable that the agents participating in the mechanism may have preferences over suspense as well. For instance, the bidders may prefer an immediate resolution with a …rst-price auction rather than su¤ering anxiety in a Dutch auction. Our analysis applies to this latter case as long as the participants’ preferences over suspense (over the evolution of their beliefs) are separable from their preferences over the economic alternatives.
1 0We concentrate on …nite message sets to use sequential equilibrium as our solution concept.
1 1Following Ely, Frankel, and Kamenica, we assume that the information policy ismemoryless, i.e., the signal depends
the belief updates would have violated Bayes rule. Formally, let Mt
i 2 ( i) be a distribution on
agent i’s type distributions and de…neMt =fMitgi2I as a collection of these distributions. Following
the terminology of Ely, Frankel, and Kamenica, a belief martingaleis a sequence fMtgT
t=0 such that
i)M0 is degenerate at prior 0, and
ii)E tj 0; :::; t 1 = t 1 for allt= 1; :::; T.
The Bayes rule implies that beliefs evolve according to a belief martingale under any information
policy. Ely, Frankel, and Kamenica argue that the converse of this statement is also true: any belief
martingale can be induced by some information policy. This last point follows from an extension of
Kamenica and Gentzkow’s (2011) Proposition 1, which is established in a static model, to multi-period
settings.
We also add a …nal period to Ely, Frankel, and Kamenica’s timing, periodT+1, where each agent can
send a last signal. In our formulation, this …nal period is when the agents will be given the incentive to
fully reveal their private information if they have not done so in earlier periods. Agradual revelation
mechanismdetermines the economic alternative and the agents’transfers as functions of all the signals
sent inT+ 1 periods.
To illustrate how a gradual revelation mechanism would work, we reconsider the independent private
values auction setting. The bidders in the example begin with the prior belief that their rivals’ types
are uniformly distributed on the interval[0;1]. This corresponds to the period 0 belief 0in our model. In period 1, the gradual revelation mechanism may ask each bidder to send either a "high" signal or a
"low" signal. After observing the period 1 signal of a bidder, his rivals update their belief regarding the
bidder’s value to a triangular distribution with the cumulative distribution function 2i when the signal is high or with the cumulative distribution function2 i 2i when the signal is low.12 This corresponds to the period 1 belief 1
i, which happens to have two possible realizations in this example. In period 2,
depending on the period 1 revelations, some bidders may be asked to send a second signal revealing their
type completely.13 Finally, in period 3, all the remaining bidders reveal their types and the mechanism
determines the economic alternative and the transfers by processing the data generated by the bidders
in all three time periods.
Any allocation rule implemented by a gradual revelation mechanism would still respect the incentive
compatibility conditions: Beginning with the …rst period of the mechanism, type iof agentican choose
to imitate type^i by following the equilibrium strategy of this latter type. For this strategy not to be
a pro…table deviation, condition (3) must hold in expectation. However, a gradual revelation scheme
(such as the one described in the preceding paragraph) introduces many additional conditions to be
1 2These cumulative distribution functions are derived from Bayes rule with the assumption that each bidder sends a
high signal with a probability equal to his value i2[0;1].
satis…ed in the construction of an equilibrium. First, consider an agent who waits until the last period
to reveal his type. Such an agent will have information about the other agents that is superior to what
he knew in period 0. Therefore his truthtelling constraints in periodT+ 1will be more stringent than
the interim version of the incentive compatibility constraints in (3). Moreover, a gradual revelation
mechanism should also provide the incentive to send accurate signals in periods prior toT + 1. For
instance, if an agent is supposed to randomize between di¤erent signals (as is the case in period 1 in
the example described above), his expected continuation payo¤ from these signals must be equal to one
another and weakly larger than the continuation payo¤ from any other signal that he is not supposed to
send in equilibrium.
3
Dominant-Strategy Incentive Compatibility
There is one class of allocation rules that are easily shown to be implementable with gradual
revela-tion. Suppose(y( ); x( ))isdominant-strategy incentive compatible, i.e., it satis…es the incentive
constraints in (3) for all i 2 i rather than satisfying them in expectation only. In this case, we
can construct a gradual revelation mechanism in which the chosen allocation depends only on the type
reports at timeT+ 1but not on the sent signals or the updated beliefs from earlier periods. Regardless
of what an agent learns regarding the types of the other agents in these earlier periods, he would prefer
to report his type truthfully at the end of the procedure. Moreover, because the payo¤s do not depend
on the revelations made in periods1 to T, all types of all agents would be indi¤erent with respect to
all the information policies available to them. Accordingly, sending their signals in a type-dependent
manner to generate any given martingale would constitute an equilibrium behavior for the agents.14
Under the single-crossing property, it is well known that dominant-strategy incentive
compatibil-ity demands a monotonic relation between agent i’s type and function hi. If decision rule y( ) is
dominant-strategy incentive compatible with some transfers, then it must beex-post monotone, i.e.,
hi[y( i; i)] must be weakly increasing in i for all i 2 i and all i 2 I. For instance, for
auc-tions with private values, ex-post monotonicity corresponds to the requirement that the probability of
assigning the object to any given bidder increases in the private value of this bidder, regardless of the
values of all the other bidders. For intra…rm resource allocation, ex-post monotonicity demands that the
amount of the intermediate good received by each product division increases in this product division’s
1 4These strategy pro…les constitute an ex-post equilibrium in which each agent’s strategy is optimal regardless of the
productivity, regardless of the resource division’s e¢ ciency. Similarly, the resource division’s production
of this intermediate good must be increasing in its e¢ ciency. For the litigation environment, ex-post
monotonicity means that the level of evidence collected by each litigant is increasing in the strength of
his/her information.
Mookherjee and Reichelstein (1992) argue that there is a sense in which ex-post monotonicity is
also su¢ cient for dominant-strategy incentive compatibility. In the context of a model with a
contin-uum of types and continuously di¤erentiable utility functions, these authors show that if allocation rule
(y( ); x( ))is incentive compatible and decision ruley( )is ex-post monotone, then transfer rule x( )
can be transformed into another transfer rule, xDS( ), which constitutes a dominant-strategy
incen-tive compatible allocation rule together withy( )and which yields the same interim transfers as x( ).
Mookherjee and Reichelstein construct xDS( ) by invoking the revenue equivalence theorem in their
continuous type environment. We show that this result extends to our model with discrete types.
Proposition 1 Suppose that the single-crossing condition in AssumptionSCholds and that(y( ); x( ))
is an incentive compatible allocation rule under belief 0. There exists a transfer rule xDS( ) such that i) allocation rule y( ); xDS( ) is dominant-strategy incentive compatible, and
ii) E ij 0ix
DS
i ( i; i) =E ij 0ixi( i; i)for all i 2 i, all i2I,
if and only if decision ruley( )is ex-post monotone.
Proof. The "only if" part of the proposition is a standard result from screening theory, which can be
proven by adding the two dominant-strategy incentive compatibility constraints between any two types
under allocation rule y( ); xDS( ) .
As the …rst step in proving the "if" part of the proposition, de…ne i i;^ij i as the payo¤
premium of type i for revealing his type truthfully instead of imitating an "adjacent" type ^i when
the other agents’types are given as i:
i i;^ij i =ui(y( i; i); i) +xi( i; i) ui y ^i; i ; i xi ^i; i . (4)
Note that function i i;^ij i can take positive or negative values depending on i, but (3) implies
thatE ij 0
i i i;
^ij i is non-negative.
The next step is to de…ne functiongi( i; i). The rate of change of this function between any two
adjacent types i and^i is given as:
gi( i; i) gi ^i; i =
i ^i; ij i E~ ij 0
i i i;
^ij~ i i i;^ij i E~
ij 0i i
^i; ij~ i
E~ ij 0
i i i;
^ij~ i +E~ ij 0i i
^i; ij~ i
for all i. This equation determines gi(; i)up to a constant15 for any i. The following equation
yields this constant term and thus pins down the function:
E ij 0ifgi( i; i)g= 0 (6)
for all i.
When functiongi is de…ned as above, its expectation with respect to its second argument i is also
zero:
E ij 0ifgi( i; i)g= 0 (7)
for all i. To see this last point, observe that the expectation of the right-hand side of equation (5)
over i, given 0i, is equal to zero. Hence, E ij 0ifgi( i; i)g is a constant function of i with the expected value of zero (equation (6)).
We de…nexDS( )with the equationxDSi ( i; i) =xi( i; i) +gi( i; i). Part (ii) of the
propo-sition follows from (7). Under condition SC and the ex-post monotonicity of y( ), the
dominant-strategy incentive compatibility constraints between the adjacent types will be su¢ cient for all the other
dominant-strategy incentive compatibility constraints. Therefore, to prove part (i), it is su¢ cient to
show that the updated payo¤ premium for revealing the type as i rather than imitating an adjacent
type^i under the allocation rule y( ); xDS( ) is non-negative for alliand all i:
ui(y( i; i); i) +xDSi ( i; i) ui y ^i; i ; i xDSi ^i; i
= i i;^ij i +gi( i; i) gi ^i; i (8)
= i
^i; ij i + i i;^ij i
E~ ij 0
i i i;
^ij~ i +E~ ij 0i i
^i; ij~ i E~ ij 0i i i; ^
ij~ i (9)
The terms in expectations are all non-negative because of the incentive compatibility of (y( ); x( )).
Hence, the sign of
i ^i; ij i + i i;^ij i = h
ui(y( i; i); i) ui y ^i; i ; i i
h
ui y( i; i);^i ui y ^i; i ;^i i
(10)
determines the sign of the updated payo¤ premium. This sum is a non-negative number due to the
single-crossing and the ex-post monotonicity conditions. (Either i ^i and therefore ui y( i; i);~i
1 5If the incentive constraints between types
i and^iare binding in both directions (which would indeed be the case
when the allocation rule pools these types together), then bothE~ ij 0
i i i;
^ij~ i and E~
ij 0i i
^i; ij~ i will
ui y ^i; i ;~i is weakly increasing in~i, or i ^iand thereforeui y( i; i);~i ui y ^i; i ;~i
is weakly decreasing in ~i.)
In the proof of the proposition, transfer rulexDS( )is constructed by the equation
xDSi ( i; i) =xi( i; i) +gi( i; i), (11)
wheregi( )is a function that gives agentithe incentive to reveal his true type, i, after he observes i.
Another crucial property of functiongi( )is that it has an expected value of zero when the expectation is
taken either with respect to i, given any i, or with respect to i, given any i. In the continuous type
limit of our model,xDS( )de…ned in (11) corresponds to the unique (up to a constant depending on i)
transfers that the revenue equivalence theorem yields for the equivalent implementation of (y( ); x( )) in dominant strategies (see Mookherjee and Reichelstein, 1992, Proposition 1). In the special case of our
model in which the utility functionui(y; i)is linear in ifor alli,xDS( )is the same as the transfer rule
derived by Gershkov et al. (2013) in their Theorem 2. Gershkov et al. use this transfer rule to construct
a dominant-strategy incentive compatible allocation rule which delivers the same interim expected payo¤
as an incentive compatible (but possibly ex-post non-monotonic) allocation rule.
If we apply the transformation described in the proof of Proposition 1 to the independent private
values auction example of the previous section, we end up converting the allocation rule of the …rst-price
auction into the allocation rule of the second-price auction, in which the highest value bidder receives
the object and pays a monetary amount equal to the second highest value. This allocation rule has the
advantage of being implementable through a gradual revelation mechanism. Whatever the bidders learn
about their rivals in the earlier stages of this mechanism, they still …nd that it is optimal to report their
types truthfully at the …nal stage. Furthermore, because the transfers will depend only on these …nal
period type reports, bidders would be indi¤erent with respect to any of the signals they can submit
in the earlier periods. This last point makes it possible to construct an equilibrium in which bidders’
randomizations over the signals would generate any desired belief martingale.
Both the …rst-price and the second-price auction allocation rules yield the same decision regarding
the identity of the winning bidder, the same expected transfers from the bidders at the interim stage,
and the same expected revenue for the seller at the ex-ante stage. However, the realized level of the
revenue has di¤erent distributions under these two auctions. As …rst noted by Vickrey (1961, Appendix
1), the second-price auction revenue has a larger support and a higher variance than that of the
…rst-price auction, which implies that a risk-averse seller would strictly prefer the …rst-…rst-price auction. In other
words, although risk-neutral agents are indi¤erent between the original and the modi…ed allocation rules,
these two rules may not necessarily describe equivalent outcomes for a principal with more elaborate
risk preferences.
setting. In fact, as one of the examples in their paper, Mookherjee and Reichelstein (1992) establish that
the HKR allocation rule can be transformed into a dominant-strategy incentive compatible allocation
rule. The expected monetary transfers from management to the divisions are identical under the original
and the transformed allocation rules. Nevertheless, the total transfers di¤er. Therefore, the latter
allocation rule may violate the budget restrictions of the …rm even when the former rule does not.
Perhaps a more important problem with the applicability of this result is manifest in design settings in
which there is no principal capable of covering the di¤erences between the original monetary transfers and
their dominant-strategy incentive compatible variants. For instance, in the context of our civil litigation
example, Proposition 1 implies that any allocation rule that results from a unitary trial can also be
transformed into a dominant-strategy incentive compatible rule. However, under this transformation,
what the plainti¤ receives as a recovery from the legal system is not necessarily equal to what the
defendant is asked to pay. Therefore Proposition 1 would be relevant to this setting only if the legal
system is ready to assume the role of a budget breaker for these two parties.
These observations indicate that balancing the budget is a desirable property for gradual revelation
mechanisms – just as it is for direct revelation mechanisms. The problem is that balancing the budget
and attaining dominant-strategy incentive compatibility are two objectives that cannot generally be
achieved together. In the following section, we will abandon the latter objective and concentrate on
sustaining gradual revelation without insisting on dominant-strategy implementation. We will see that
it is possible to balance the budget and at the same time to provide incentives for gradual revelation.
4
Gradual Revelation with Budget Balance:
A Su¢ cient Condition
In the previous section, we argued that the ex-post monotonicity of the decision rule is a su¢ cient
condi-tion to transform an incentive compatible allocacondi-tion rule into a dominant-strategy incentive compatible
rule, without considering the budget balance. In this section, we will see that ex-post monotonicity
is also su¢ cient for the construction of a gradual revelation mechanism that adheres to the balanced
budget requirement.
Unlike the transfers identi…ed in the previous section, transfers ensuring a balanced budget will not
result in dominant-strategy incentive compatibility. Nevertheless, these transfers will provide the agents
with the incentive to make accurate revelations in all periods of the constructed mechanism, including
the last period in which they will fully reveal their types.16
1 6A gradual revelation mechanism induces a dynamic optimization problem for each type of each agent. For the
Proposition 2 Suppose that the single-crossing condition in Assumption SC holds, fMtgTt=0 is an arbitrary belief martingale, and that (y( ); x( )) is an incentive compatible allocation rule under belief
0. There exist a gradual revelation mechanism and a sequential equilibrium of this mechanism such that
i) types are gradually revealed according to martingalefMtgT
t=0and decision ruley( )is implemented,
ii) the interim expected payo¤ for type i of agent iisE ij 0ifui(y( i; i)) +xi( i; i)g, and
iii) transfers are budget-balanced (they add up toPi2Ixi( )regardless of the path of revelation),
if decision ruley( )is ex-post monotone.
The proof of the proposition will follow from the lemma below. When the agent types are
inde-pendently distributed, the typical method to attain budget-balanced implementation – while keeping
the Bayesian incentives intact – is outlined by the expected externality approach of Arrow (1979) and d’Aspremont and Gerard-Varet (1979).17 In our context, applying the expected externality method
di-rectly would involve replacing functiongi( )that we used to constructxDSi ( )in (11) with the summation
ofjIj di¤erent terms, each of which depends on the type report of each of thejIjagents:
E~ ij 0
igi i;
~ i X
j6=i
1
jIj 1E~ jj 0jgj j;
~ j : (12)
The …rst term above provides agentiwith the incentive to report truthfully, and the remaining terms
are included just to balance the budget. However, because the expected level of gi( ) is zero in both
i and~ i in our setting, this construction amounts to reverting to the original transfer functionxi( ),
which assures simultaneousrevelation of the types but is not necessarily suitable for gradual revelation. We break this tension between the provision of the right incentives and the balanced budget by
allowing the transfers to depend on the signals sent by the agents in the …rst T periods as well as the
type reports submitted in period T+ 1. Construction of these transfers will bene…t from the expected
externality approach discussed above. We will continue to refer to an analogous expression to (12)
to determine the evolution of the transfers in each time period. However, the relevant probability
distribution in the calculation of the expectations in this expression will not be the prior belief, 0.
Instead, the gradual revelation mechanism transfers will depend on the expected values of functions
gi ;~ i , conditional on the updated belief of the period in question. Because the beliefs evolve on
the path of play according to the signals sent by the agents in our setting, the analogous expression to
problem. Solving this problem involves identifying an optimal action in each round, contingent on the history of the earlier signals. The complication is that, as Fadel and Segal (2009) observe, these histories must include histories that are not supposed to be reached on the equilibrium path, given the type of the agent.
1 7See d’Aspremont, Cremer, and Gerard-Varet (2004) and Kosenok and Severinov (2008) for an extension of this method
(12) will not be additively separable in the agents’signals. Agentiwill have an e¤ect not only in the
value of its …rst term through his …nal type report but also in the values of the remainingjIj 1 terms
through his information policy. Nevertheless, we will be able to provide a proof for the lemma below by
invoking the fact that the lastjIj 1 terms in this expression will be equal to zero in expectation for
agentiregardless of the signal he chooses to submit.
Lemma 1 Suppose that the single-crossing property in Assumption SC holds, decision rule y( ) is ex-post monotone, allocation rule y( ); x 1( ) is incentive compatible under belief 1, and that
E j 1 = 1. There exists a belief-dependent transfer rule x (; )such that
a) allocation rule(y( ); x (; ))is incentive compatible under belief ,
b) for all i, all i, and alli,E ij 1 E ij ixi i; i; i; i =E ij i1x
1
i ( i; i), and
c)x (; )is budget-balanced: Pixi ( ; ) =Pixi 1( ) for all and all .18
Proof. The proof of the lemma follows similar steps as in the proof of the "if" part of Proposition
1. We …rst de…ne the payo¤ premium of type i for revealing his type truthfully instead of imitating an
adjacent type^i when the other agents’types are given as i:
i i;^ij i =ui(y( i; i); i) +xi 1( i; i) ui y ^i; i ; i xi 1 ^i; i . (13)
Then, we de…ne functiongi ( i; i)with equations
gi ( i; i) gi ^i; i =
i ^i; ij i E~ ij 1
i i i;
^
ij~ i i i;^ij i E~ ij 1
i i
^
i; ij~ i
E~ ij 1
i i i;
^
ij~ i +E~ ij 1
i i ^
i; ij~ i
(14)
where i and^i are two adjacent types and
E ij i 1fgi ( i; i)g= 0 (15)
for all i. As in the proof of Proposition 1, this de…nition implies that
E ij i1fgi ( i; i)g= 0 (16)
for all i. Finally, we de…ne the period transfers as
xi ( ; ) =xi 1( ) +E~ ij
igi i;
~ i 1
jIj 1E~ij i
X
j6=i E~ i jj
i jgj j;
~i;~ i j : (17)
1 8Notice that we still must identifyx
i( ; )when the type pro…le is not in the support of belief . In the equilibrium
we will construct to prove Proposition 2, this situation would correspond to an o¤-the-equilibrium-path event that an agent …rst sends a signal, and later reports a type which is not in the support of the equilibrium belief generated by the earlier signal. In this case, our budget balance requirement still demands the sum of the transfers to be equal toPixi( ), where
Budget balancedness in part (c) of the lemma holds by construction. Part (b) follows from (15) and (16).
Under AssumptionSC and the ex-post monotonicity ofy( ), the incentive compatibility constraints in
(3) between the adjacent types are su¢ cient for all the other incentive compatibility constraints. The
expected value of the updated payo¤ premium to revealing the type as i rather than imitating an
adjacent type ^i is
E ij i
n
i i;^ij i +gi ( i; i) gi ^i; i o
= E ij i
n
i ^i; ij i + i i;^ij i o
E~ ij 1
i i i;
^ij~ i +E~
ij i1 i
^i; ij~ i E~ ij i1 i i;
^ij~ i (18)
under belief i. To prove part (a) of the lemma, it is su¢ cient to show that this payo¤ premium is
non-negative. The terms E~ ij 1
i i i;
^
ij~ i and E~ ij 1
i i
^
i; ij~ i are both non-negative since
y( ); x 1( ) is incentive compatible under belief 1. As in the proof of Proposition 1, Assumption
SC and ex-post monotonicity ofy( )imply that i ^i; ij i + i i;^ij i is non-negative for all
i as well.
This lemma establishes the following: Start with a transfer rule which makes a decision rule incentive
compatible under a certain belief. If this belief is updated in a Bayesian fashion, it is possible to
modify the transfer rule to make the original decision rule incentive compatible under the updated
belief. Moreover, the resultingbelief-dependent allocation rules yield the same interim expected payo¤ as the initial allocation rule, which ensures that the agents will be indi¤erent among all the belief
updates they may generate about their own types. In Appendix A, we provide a numerical example to
this construction.
To see how Proposition 2 follows from Lemma 1, …rst consider the case in whichT = 1. In this case,
agents are given an opportunity to send some signals simultaneously in period 1 and then they report
their types in period 2. The gradual revelation mechanism maps the reported types into the economic
alternative prescribed by the decision ruley( ). The transfers depend on both the signals sent in period
1 and the types reported in period 2. We relabel the signals available to each agentiin period 1 as the
beliefs in the support ofM1
i. The gradual revelation mechanism maps the sent signals and the reported
types into transfers by using the belief-dependent transfer rulex1 ; 1 described in Lemma 1 (setting
x0( )in the lemma equal tox( )in the proposition).
This gradual revelation mechanism has an equilibrium in which each agent reports his type truthfully
in period 2 and randomizes between the signals in period 1 such that the resulting updated belief
after sending a signal is identical to the label of the signal. Optimality of truthful reporting under the
updated beliefs follows from part (a) of the lemma. Moreover, part (b) implies that agents are indi¤erent
among all the signals available to them in period 1, which proves that the randomizations prescribed
the equilibrium yields the interim payo¤E ij 0ifui(y( i; i)) +xi( i; i)gfor agentiwith type i.
Finally, budget balance follows from part (c) of the lemma.
The gradual revelation mechanism above can be extended to longer horizons(T >1), by relabeling
signals at each periodtas the beliefs in the support of the distributionMt
i and then allowing the transfers
to depend on signals sent in all periods and types reported in periodT+1. These transfers are determined
iteratively by using the evolution of the equilibrium beliefs and functionxT ; T described in Lemma
1 (set functionxT 1( )equal to xT 1 ; T 1 , functionxT 2( ) equal toxT 2 ; T 2 , etc.). This
extended gradual revelation mechanism has an equilibrium in which the agents’randomizations on the
signals respect martingalefMtgT
t=0and where they report their true types in periodT + 1.19
Note that Proposition 2 does not place any restriction on the belief martingales. As long as the
decision rule is ex-post monotone, we can construct the transfers that would induce the martingale
we choose.20 Unlike the transfer rule in Proposition 1, however, the transfers we construct here are
contingent on the belief martingale that we intend to support.
How we use Arrow (1979) and d’Aspremont and Gerard-Varet (1979)’s expected externality approach
to balance the budget here is similar to the way that Athey and Segal (2013) calculate the transfers for
theirbalanced team mechanism in an in…nite horizon design setting.21 In each period of their dynamic
setting, agents acquire additional private information and decide on a new economic alternative. The
distribution of private information is a¤ected by both past information and past decisions. Athey and
Segal are interested in implementing the e¢ cient decision rule, which requires designing a mechanism with afull revelationproperty, such that the agents would reveal all the private information they hold at each period. In the process of identifying this mechanism, Athey and Segal show that, for any mechanism
1 9Recall that we only consider martingales that arefully revealing by the end, such that each agent’s type is made public
in periodT+ 1, at the latest. Under this modeling choice, the last stage of gradual revelation resembles a direct revelation mechanism. However, if the aim is to implement an allocation rule that pools certain types of an agent conditional on a certain event, then this allocation rule can be implemented without fully revealing all agents’ types. For instance, we can construct a gradual revelation mechanism to implement the …rst-price auction allocation without revealing the exact valuations of the losing bidders (the Dutch auction), the HKR allocation without asking for low productivity divisions’ exact productivity levels, or the bilateral trial allocation without learning the extent of the plainti¤’s damages in the case that the defendant’s liability is too low for any reasonable compensation.
2 0As the agents play this gradual revelation mechanism, their beliefs regarding their rivals evolve according to the chosen
martingale. Note that we do not need to specify any o¤-the-equilibrium-path beliefs here. By design, any of the signals available to an agent in any given period can be sent with a positive probability by a type of this agent that has not been ruled out earlier. Only in periodT+ 1, can an agent make a type report that is inconsistent with the earlier belief on his type. However, how the beliefs are updated at this stage is irrelevant because no agent would have a further move in the mechanism after periodT+ 1.
2 1In fact, Fadel and Segal (2009) refer to Athey and Segal’s work (their footnote 18) to suggest that the
satisfying the full revelation property, there is another full revelation mechanism that implements the
same decision rule with a balanced budget. The analysis in this section suggests that Athey and Segal’s
result extends to mechanisms in which agents are induced to follow more general information policies
than fully revealing their information at every opportunity.
The results derived in this section refer to an ex-post monotonicity condition that is satis…ed by
many allocation rules of particular interest. For instance, Mookherjee and Reichelstein (1992) show that
the incentive compatible allocation rule that maximizes the objective function of a principal (the
ex-ante expected value of the gross bene…t from the chosen economic alternative minus transfers to agents)
is ex-post monotone. In the context of linear utility functions (such as the bidders’ value functions
in auctions), Manelli and Vincent (2010) and Gershkov et al. (2013) argue that, for any incentive
compatible allocation rule, there is an ex-post monotone and incentive compatible rule that generates
the same interim expected payo¤s (but not necessarily the same economic alternatives and the same
interim transfers).
We close this section by noting that the ex-post monotonicity condition is su¢ cient for constructing
a gradual revelation mechanism but is not necessary. Characterizing a necessary and su¢ cient condition
is the subject of the following section.
5
Gradual Revelation with Budget Balance:
A Necessary and Su¢ cient Condition
The single-crossing property in Assumption SC is concerned with the agents’ preferences only with
respect to thedeterministiceconomic alternatives. As demonstrated by Strausz (2006), this assumption does not imply a regularity regarding how di¤erent types of an agent would evaluate therandomizations
on the economic alternatives. We begin this section by extending the single-crossing property over these
randomized alternatives.
Assumption(ESC) For any agent i and any two (possibly randomized) economic alternatives q and
^
q2 Y, the expected utility di¤ erence Eyjqfui(y; i)g Eyjq^fui(y; i)g is either weakly increasing or
weakly decreasing in i.
Note that this extended version of the single-crossing property is also trivially satis…ed when agents
have at most two types or when there are only two economic alternatives in setY. With an arbitrary
number of agent types and economic alternatives, this condition holds if the utility functions are
multi-plicatively separable in type and economic alternative, i.e., ifui(y; i)can be written asvi(y) + ili(y).
values or production functions with constant unit costs.
In the same manner in which AssumptionSCimplies an order on setY, the extended single-crossing
property in AssumptionESC implies an order on the set of randomized economic alternatives Y for
each agenti. Moreover, because the expected utility is linear in utility levels from deterministic economic
alternatives, this order satis…es the independence condition. Accordingly, for each agent i, there exists
a function i :Y !Rsuch that for any two randomized economic alternativesqandq^2 Y,
Eyjqf i(y)g Eyjq^f i(y)g if and only if Eyjqfui(y; i)g Eyjq^fui(y; i)g is weakly increasing in i:
In Section 3, we noted that the ex-post monotonicity of the decision ruley( )is a necessary
condi-tion for dominant-strategy incentive compatibility under Assumpcondi-tion SC. Below, we identify a similar
monotonicity property with respect toy( ), which would be a necessary condition for sustaining gradual
revelation under Assumption ESC.
Consider a gradual revelation mechanism in which agents reveal their private information with respect
to the belief martingalefMtgT
t=0. Take an arbitrary periodtand a belief tin the support ofMtsuch
that t
i assigns positive probabilities to types i and ^i of agent i. Beginning with this period, a
possible deviation for type i is to follow the equilibrium strategy of type^i in the continuation of the
mechanism. For this deviation not to be pro…table, E ij ti
n
ui(y( i; i); i) ui y ^i; i ; i o
must be at least as large as the di¤erence between the expected equilibrium transfers to type ^i and
type i, where the expectation is taken under periodt belief ti. Similarly, the same expected transfer
di¤erence must be at least as large asE ij ti
n
ui y( i; i);^i ui y ^i; i ;^i o
for type^inot
to …nd it optimal to begin imitating type i in period t. Assuming that i is larger than ^i, these two
inequalities imply that E ij tif i[y( i; i)]g E ij ti
n i
h
y ^i; i io
under Assumption ESC.
This discussion yields the following condition on the decision rule as a necessary condition for gradual
revelation: Decision rule y( ) ismonotone with respect to martingale fMtgT
t=0 if, for all periods
t T, all t in the support ofMt, and alli2I,E
ij ti i[y( i; i)]is weakly increasing in i when
the domain of i is restricted to the support of ti.
For an illustration of this monotonicity requirement, reconsider the bifurcated trial example discussed
in the Introduction. If we think of this trial as a gradual revelation mechanism in which the defendant
reveals her liability in the …rst period and the plainti¤ makes his damages known in the second one,
the monotonicity requirement from above is reduced to two conditions. First, the evidence level that
the defendant presents in the …rst period must be weakly decreasing with respect to the extent of her
liability. Note that this condition is also an implication of incentive compatibility under the prior beliefs.
Second, the plainti¤’s evidence level in the second period must be weakly increasing in the magnitude of
his damages, regardless of the defendant’s liability (which is revealed by the evidence observed in period
For an additional illustration, consider a private values auction with two bidders, bidders A and
B. Each bidder’s private value for the auctioned object can take one of three equally likely values:
<^ < . A natural choice for function i here is to allow it to be equal to bidder i’s probability of
receiving the object. Consider the symmetric decision rule which assigns the object to a type bidder
with probability 1 if and only if the rival’s type is^, and to a type^bidder with probability 1/2 if and
only if the rival’s type is . In all the remaining cases, no bidder receives the object. Accordingly, A
and B are given as below:
A A ^A A
B 0 1/2 0
^
B 0 0 1
B 0 0 0
B A ^A A
B 0 0 0
^
B 1/2 0 0
B 0 1 0
(19)
Suppose that we want to implement this decision rule gradually with a belief martingalefMtg1t=0, where
T = 1. Because the types are assumed to be equally likely, 0
i is uniform. Moreover, we construct the
belief martingale such that bidderB’s type is fully revealed in period 1, i.e., M1
B consists only of the
three degenerate beliefs.
The decision rule is monotone for both bidders in period 0 because E jj 0j i[y( i; j)] is weakly
increasing in i. This requirement is also trivially satis…ed for bidderBin period 1 because the supports
of the beliefs on his type are singletons. We now discuss the restrictions on distributionM1
A of period 1
beliefs on bidderAthat would ensure monotonicity for this bidder in period 1.
First, note that this decision rule cannot be implemented in dominant strategies because A[y( A; B)]
is not monotone in A and therefore ex-post monotonicity fails. This observation implies thatMA1 can-not assign full weight on the prior belief 0
A: BidderA cannot wait until after hearing bidderB’s type
to make his …rst revelation. However, monotonicity of A would be restored if the domain of A is
restricted either to n
A;^A o
or to A; A in period 1. This observation implies that the decision
rule is monotone with respect tofMtgT
t=0if the support ofMA1 consists only of beliefs assigning positive probabilities either only to types Aand^Aor only to types Aand A. In other words, the monotonicity
condition demands that bidderAsends a signal fully separating his types^Aand A before he hears the
type of bidderB, but it allows for type Ato stay mixed with either one of the other two types.
With the following proposition, we establish that this monotonicity condition is also su¢ cient for
gradual revelation.
Proposition 3 Suppose that the single-crossing property in AssumptionESCholds,fMtgTt=0is a belief martingale, and that(y( ); x( ))is an incentive compatible allocation rule under belief 0. There exist
a gradual revelation mechanism and a sequential equilibrium of this mechanism such that i) types are gradually revealed according to martingalefMtgT
