Implementation by Gradual Revelation Gorkem Celiky July 29, 2014
Abstract We investigate the feasibility of implementing an allocation rule with a gradual revelation mechanism 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, an ex-post monotonicity condition is su¢ cient for budget-balanced implementation of any incentive compatible allocation rule with any gradual revelation scheme. When we extend the singlecrossing property over the set of randomized 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 incentive scheme that would generate an allocation 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. y ESSEC
Business School and THEMA Research Center,
[email protected]
1
fail to …nd romance. A direct 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 with gradual 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 – is incentive 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
1 For
many other examples of strategic thinking in Austen’s work, see the intriguing book by Chwe (2013).
2
sequential revelations as a stochastic 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 is dominant-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 transformed 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 the budget-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, balanced budget transfers depend on the entire path of revelations made by them. With these balanced 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 of revelation. Moreover, these transfers make the agents indi¤erent with respect to any revelation they
3
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 the expected externality method 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 a su¢ cient condition 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 continuation 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., …rstprice and second-price sealed-bid auctions). Others have a longer duration and the bidders are able to
4
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 mechanisms 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. These indirect 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 object to the highest valuation bidder – and the optimal allocation rule – which maximizes the expected revenue of the auctioneer – satisfy this condition. 3 See
Rajan and Reichelstein (2004) for a more general characterization of this problem as well as a review of the
subsequent literature.
5
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 A bifurcated trial, i.e., a trial in which the issues of liability and damages 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
4 An
example of a bifurcated trial is the patent infringement case between Polaroid and Kodak. Kodak’s liability was
established in 1985. The ruling on Polaroid’s damages was made in 1990.
6
decreasing in damages for the plainti¤.5 By an appropriate allocation of the burden of proof, the court can commit to how it will condition the level of recovery –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 a unitary 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
5 See
Froeb and Kobayashi (1996) and Yilankaya (2002) for similar costly evidence production models.
6 See
Spier (1994) and Neeman and Klement (2005) for examples of a similar mechanism design approach to litigation.
7 Landes’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. 8 See
Daughety and Reinganum (2000) for a model in which litigation expenditures are choice variables for the defendant
and the plainti¤ in a bifurcated trial.
7
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 their expected recovery rates are incentive compatible under the prior beliefs, then the court can …nd the actual 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 agent i 2 I has a …nite type set with a generic element
i.
Agent i’s type is distributed with respect to the prior distribution
independently of the other agents’types. In standard notation, the cross product space the set of type pro…les with a generic element priors on the agents’types. Similarly, and
0
i
represents
i
= f i gi2I . We de…ne
stands for
j2I fig
j
0
0 i i2I
=
2 i2I
i
i
,
is
as the collection of the
with generic element
0 j j2I fig .
=
0 i
R
i
i
= f j gj2I
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 function ui : 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 following single-crossing property: Assumption (SC) For any agent i and any two economic alternatives y and y^ 2 Y , 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 set Y , agent types can be relabeled to satisfy single-crossing. The single-crossing property implies an order on the set of economic alternatives Y for each agent. That is, for each agent i, there is a function hi : Y ! R such that for any two economic alternatives y
8
and y^ 2 Y , hi (y)
hi (^ y ) if and only if ui (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), function hi ( ) 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 for agent i, where y is the economic alternative,
i
(2)
is agent i’s type, and xi 2 R is 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 rule as a mapping from type pro…les into agents’
jIj
transfers, x :
! R , where xi yields the relevant dimension of the transfer rule for agent i. 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 satis…ed for all i 2 I and all E
ij
0
fui (y ( i ; i
^
2 i
pairs:
+ xi ( i ;
i )g
i; i
i) ; i)
2
E
ij
0
n ui y ^i ; i
0
i
if the following constraint is
;
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 whether we can implement an allocation rule –or a close variant of it –while delaying the full disclosure 9
of private information of the agents by making them reveal their types gradually. Our model is a discrete type model but perhaps this point is better explained by referring to a wellknown 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. The Dutch 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) = jIj times 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) = jIj times 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, t 2 f0; 1; :::; T g denotes
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 period t belief on agent i is denoted by distribution
t i
2
i
over the possible types of agent i, and
t
= f ti gi2I is a collection of
these beliefs. Technically, an agent’s information policy is a function that maps the current period t, 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 period t 9 As
1 is an unbiased estimate of what they will be believing in period t. Otherwise,
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 0 We
concentrate on …nite message sets to use sequential equilibrium as our solution concept.
1 1 Following
Ely, Frankel, and Kamenica, we assume that the information policy is memoryless, i.e., the signal depends
on the current period and the current belief rather than the full history.
10
the belief updates would have violated Bayes rule. Formally, let Mit 2
(
i)
be a distribution on
agent i’s type distributions and de…ne M t = fMit gi2I as a collection of these distributions. Following T
the terminology of Ely, Frankel, and Kamenica, a belief martingale is a sequence fM t gt=0 such that i) M 0 is degenerate at prior ii) E
t
j
0
; :::;
t 1
=
t 1
0
, and
for all t = 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, period T +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. A gradual revelation mechanism determines the economic alternative and the agents’transfers as functions of all the signals sent in T + 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
0
in 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 is high or with the cumulative distribution function 2 to the period 1 belief
1 i,
i
2 i
2 i
when the signal
when the signal is low.12 This corresponds
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
i
of agent i can 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 2 These
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 1 3 For
i
2 [0; 1].
instance, bidder i may reveal his type if and only if he is the only bidder who sent a "high" signal in period 1.
11
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 period T + 1 will 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 to T + 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 revelation. Suppose (y ( ) ; x ( )) is dominant-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 time T + 1 but 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 periods 1 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 compatibility 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 be ex-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 4 These
strategy pro…les constitute an ex-post equilibrium in which each agent’s strategy is optimal regardless of the
types of the other agents. As noted by Fadel and Segal (2009) in their Proposition 6, such strategy pro…les can be supported as equilibria under any prior beliefs. In fact, this observation extends to the interdependent values case where an agent’s payo¤ may depend on another agent’s type, as long as the allocation rule satis…es the ex-post version of the incentive compatibility constraints (see Van Zandt, 2007).
12
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 continuum of types and continuously di¤erentiable utility functions, these authors show that if allocation rule (y ( ) ; x ( )) is incentive compatible and decision rule y ( ) is ex-post monotone, then transfer rule x ( ) can be transformed into another transfer rule, xDS ( ), which constitutes a dominant-strategy incentive compatible allocation rule together with y ( ) 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 Assumption SC holds 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
0 i
xDS ( i; i
i)
=E
ij
0 i
xi ( i ;
i)
for all
i
2
i,
all i 2 I,
if and only if decision rule y ( ) 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 premium of type
^
i; ij
Note that function that E
ij
0 i
= ui (y ( i ;
i
^
i; ij
i
^
i; ij
i
i
i
i:
i) ; i)
+ xi ( i ;
gi ( i ;
i)
i
i)
ui y ^i ;
;
i
xi ^i ;
i
can take positive or negative values depending on
i,
i
.
(4)
but (3) implies
is non-negative.
The next step is to de…ne function gi ( i ; adjacent types
as the payo¤
i
for revealing his type truthfully instead of imitating an "adjacent" type ^i when
i
the other agents’types are given as i
^
i; ij
i
i ).
The rate of change of this function between any two
and ^i is given as:
gi ^i ;
i i
=
^i ; i j
i
E~ E~
ij
0
ij
0
i
i
^ ~
i
i; ij
i
i; ij
13
^ ~
i i
i
+ E~
^
i; ij ij
0 i
i i
E~
ij
^i ; i j~
0 i
i
^i ; i j~
i
i
(5)
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 for all
0 ij i
fgi ( i ;
i )g
=0
(6)
i.
When function gi is de…ned as above, its expectation with respect to its second argument
i
is also
zero: E for all over
i. i,
ij
0 i
fgi ( i ;
i )g
=0
(7)
To see this last point, observe that the expectation of the right-hand side of equation (5) 0
given
i,
is equal to zero. Hence, E
0
ij
i
fgi ( i ;
i )g
is a constant function of
i
with the
expected value of zero (equation (6)). We de…ne xDS ( ) with the equation xDS ( i; 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 dominantstrategy 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
rather than imitating an adjacent
i
type ^i under the allocation rule y ( ) ; xDS ( ) is non-negative for all i and all ui (y ( i ; =
i) ; i)
^
i; ij
i
i
E~
ij
0 i
+ gi ( i ;
i
^i ; i j
i
i; ij
=
+ xDS ( i; i
^ ~
i)
i
+
i
+ E~
ui y ^i ;
i)
gi ^i ; ^
i; ij
i
ij
0 i
;
i
^i ; xDS i
i
i
(8)
i i
i
i:
^i ; i j~
E~ i
ij
0 i
i
^ ~
i; ij
(9)
i
The terms in expectations are all non-negative because of the incentive compatibility of (y ( ) ; x ( )). Hence, the sign of i
^i ; i j
i
+
i
^
i; ij
=
i
h
i ui (y ( i ; i ) ; i ) ui y ^i ; i ; i h i ui y ( i ; i ) ; ^i ui y ^i ; i ; ^i
(10)
determines the sign of the updated payo¤ premium. This sum is a non-negative number due to the singlecrossing and the ex-post monotonicity conditions. (Either 1 5 If
the incentive constraints between types
i
^i and therefore ui y ( i ;
i
~
i) ; i
and ^i are binding in both directions (which would indeed be the case
when the allocation rule pools these types together), then both E~
ij
0
i
i
^ ~
i; ij
i
and E~
ij
0
i
i
^i ;
~
ij
i
will
be equal to zero. In this case, we adopt the convention that these terms cancel one another out such that the right hand (^ ; j ) ( ;^ j ) side of (5) is equal to i i i i 2 i i i i . We follow the same practice in de…ning similar functions in the proofs of our later results.
14
ui y ^i ;
i
; ~i is weakly increasing in ~i , or
i
^i and therefore ui y ( i ;
~
i) ; i
ui y ^i ;
i
; ~i
is weakly decreasing in ~i .) In the proof of the proposition, transfer rule xDS ( ) is constructed by the equation xDS ( i; i
i)
= xi ( i ;
i)
+ gi ( i ;
i) ,
where gi ( ) is a function that gives agent i the incentive to reveal his true type,
(11) i,
after he observes
i.
Another crucial property of function gi ( ) is that it has an expected value of zero when the expectation is taken either with respect to limit of our model, x
DS
i,
given any
i,
or with respect to
i,
given any
i.
In the continuous type
( ) 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 function ui (y; i ) is linear in
i
for all i, 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 …rstprice auction, which implies that a risk-averse seller would strictly prefer the …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. A similar criticism applies when this transformation is executed in an intra…rm resource allocation 15
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 condition to transform an incentive compatible allocation 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
16 A
gradual revelation mechanism induces a dynamic optimization problem for each type of each agent. For the
mechanism we construct, we will show that following an accurate revelation path is a solution to this dynamic optimization
16
T
Proposition 2 Suppose that the single-crossing condition in Assumption SC holds, fM t gt=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 T
i) types are gradually revealed according to martingale fM t gt=0 and decision rule y ( ) is implemented, ii) the interim expected payo¤ for type
of agent i is E i j 0 i fui (y ( i ; i )) + xi ( i ; i )g, and P iii) transfers are budget-balanced (they add up to i2I xi ( ) regardless of the path of revelation), i
if decision rule y ( ) is ex-post monotone.
The proof of the proposition will follow from the lemma below. When the agent types are independently 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 directly would involve replacing function gi ( ) that we used to construct xDS ( ) in (11) with the summation i of jIj di¤erent terms, each of which depends on the type report of each of the jIj agents: E~
ij
0 i
gi
i;
~
i
X j6=i
1 jIj
1
E~
jj
0 j
gj
j;
~
j
:
(12)
The …rst term above provides agent i with 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 function xi ( ),
which assures simultaneous revelation 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 7 See
d’Aspremont, Cremer, and Gerard-Varet (2004) and Kosenok and Severinov (2008) for an extension of this method
to correlated types. See Borgers and Norman (2009) for an extension to interdependent values. See Eso and Futo (1999) for how to smooth the transfers when the principal is risk averse, ambiguity averse (Bose, Ozdenoren, Pape, 2006), or is expecting agents to collude (Che and Kim, 2006, 2009, and Pavlov, 2008).
17
(12) will not be additively separable in the agents’signals. Agent i will 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 remaining jIj
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 last jIj
1 terms in this expression will be equal to zero in expectation for
agent i regardless of the signal he chooses to submit. Lemma 1 Suppose that the single-crossing property in Assumption SC holds, decision rule y ( ) is 1
ex-post monotone, allocation rule y ( ) ; x E
1
j
1
=
. There exists a belief-dependent transfer rule x ( ;
a) allocation rule (y ( ) ; x ( ; i,
c) x ( ;
all
, and that
) such that
)) is incentive compatible under belief
,
, and all i, E i j 1 E i j i xi i ; i ; i ; i = E i j P P ) is budget-balanced: ) = i xi 1 ( ) for all and all i xi ( ;
b) for all
1
( ) is incentive compatible under belief
1
i
i
xi
1
( i;
i ),
and
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; ij
i
= ui (y ( i ;
i
i) ; i)
Then, we de…ne function gi ( i ; gi ( i ;
where
i)
i
gi
^i ;
i
=
i)
E~
i)
ui y ^i ;
;
i
xi
i
^i ;
1
i
.
(13)
1
ij
i
ij
i
1
^ ~
i
i; ij
i
i; ij
^ ~
i i
^
i; ij
i
+ E~
1
ij
i
i i
E~
ij
1
i
^i ; i j~
^i ; i j~
i
i
(14)
and ^i are two adjacent types and
i.
i.
xi ( ; 1 8 Notice
ij i
1
fgi ( i ;
i )g
=0
(15)
As in the proof of Proposition 1, this de…nition implies that E
for all
E~
i
E for all
( i;
with equations
^i ; i j
i
1
+ xi
i:
Finally, we de…ne the period ) = xi
1
( ) + E~
ij
i
i
fgi ( i ;
i )g
=0
(16)
transfers as
gi
that we still must identify xi ( ;
1
ij
i;
~
1 i
jIj
1
E~i j
) when the type pro…le
i
X j6=i
E~
i
jj
i
j
gj
~ ~
j ; i;
is not in the support of belief
i j
:
(17)
. 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 P signal. In this case, our budget balance requirement still demands the sum of the transfers to be equal to i xi ( ), where will be determined by the …nal type reports.
18
i
Budget balancedness in part (c) of the lemma holds by construction. Part (b) follows from (15) and (16). Under Assumption SC and the ex-post monotonicity of y ( ), 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
E
= E~ under belief
i.
ij
ij
1
n
ij 1 i
i
i
i
^
n
i; ij i
i
+ gi ( i ;
^i ; i j ^ ~
i; ij
i
i
+
+ E~
^
1 i
^i ; o
gi
i; ij
i
ij
ij
1 i
i
^ ~
i; ij
i
and E~
( ) is incentive compatible under belief
SC and ex-post monotonicity of y ( ) imply that i
i)
i
^i ; i j~
i
i
o
E~ i
1
ij
i
i
^ ~
i; ij
i
(18)
To prove part (a) of the lemma, it is su¢ cient to show that this payo¤ premium is non-
negative. The terms E~ y( );x
i
i
ij
1
1 i
i
^i ; i j~
are both non-negative since
i
. As in the proof of Proposition 1, Assumption
^i ; i j
i
+
^
i; ij
i
i
is non-negative for all
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 resulting belief-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 which T = 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 rule y ( ). 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 agent i in period 1 as the beliefs in the support of Mi1 . The gradual revelation mechanism maps the sent signals and the reported types into transfers by using the belief-dependent transfer rule x1 ;
1
described in Lemma 1 (setting
0
x ( ) in the lemma equal to x ( ) 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 by M 1 constitute an equilibrium behavior. Another implication of this indi¤erence condition is that 19
the equilibrium yields the interim payo¤ E
0
ij
i
fui (y ( i ;
i ))
+ xi ( i ;
i )g
for agent i with 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 period t as the beliefs in the support of the distribution Mit and then allowing the transfers to depend on signals sent in all periods and types reported in period T +1. These transfers are determined iteratively by using the evolution of the equilibrium beliefs and function xT 1 (set function xT
1
( ) equal to xT
1
;
T
1
, function xT
2
;
( ) equal to xT
T 2
described in Lemma ;
T
2
, etc.). This
extended gradual revelation mechanism has an equilibrium in which the agents’randomizations on the T
signals respect martingale fM t gt=0 and where they report their true types in period T + 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 their balanced 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 a full revelation property, 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 9 Recall
that we only consider martingales that are fully revealing by the end, such that each agent’s type is made public
in period T + 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 0 As
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 period T + 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 period T + 1. 2 1 In
fact, Fadel and Segal (2009) refer to Athey and Segal’s work (their footnote 18) to suggest that the budget-
unbalanced mechanism in their Proposition 6 can be transformed into a budget-balanced mechanism.
20
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 exante 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 the deterministic economic alternatives. As demonstrated by Strausz (2006), this assumption does not imply a regularity regarding how di¤erent types of an agent would evaluate the randomizations 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 q^ 2
Y , the expected utility di¤ erence Eyjq fui (y; i )g
weakly decreasing in
Eyj^q fui (y; i )g is either weakly increasing or
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 set Y . With an arbitrary number of agent types and economic alternatives, this condition holds if the utility functions are multiplicatively separable in type and economic alternative, i.e., if ui (y; i ) can be written as vi (y) +
i li (y).
Multiplicative separability is satis…ed for many applications involving auctions with independent private
21
values or production functions with constant unit costs. In the same manner in which Assumption SC implies an order on set Y , the extended single-crossing property in Assumption ESC implies an order on the set of randomized economic alternatives
Y for
each agent i. 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 Eyjq f
i
i
: Y ! R such that for any two randomized economic alternatives q and q^ 2
(y)g
Eyj^q f
i
(y)g if and only if Eyjq fui (y; i )g
Y,
Eyj^q fui (y; i )g is weakly increasing in
i:
In Section 3, we noted that the ex-post monotonicity of the decision rule y ( ) is a necessary condition for dominant-strategy incentive compatibility under Assumption SC. Below, we identify a similar monotonicity property with respect to y ( ), 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 T
to the belief martingale fM t gt=0 . Take an arbitrary period t and a belief that
t i
assigns positive probabilities to types
t
in the support of M t such
and ^i of agent i. Beginning with this period, a
i
is to follow the equilibrium strategy of type ^i in the continuation of the n o mechanism. For this deviation not to be pro…table, E i j t i ui (y ( i ; i ) ; i ) ui y ^i ; i ; i possible deviation for type
i
must be at least as large as the di¤erence between the expected equilibrium transfers to type ^i and where the expectation is taken under period t belief t i . Similarly, the same expected transfer n o di¤erence must be at least as large as E i j t i ui y ( i ; i ) ; ^i ui y ^i ; i ; ^i for type ^i not type
i,
to …nd it optimal to begin imitating type inequalities imply that E
ij
t i
f
i
[y ( i ;
in period t. Assuming that i is larger than ^i , these two n h io E i j t i i y ^i ; i under Assumption ESC. i )]g
i
This discussion yields the following condition on the decision rule as a necessary condition for gradual T
revelation: Decision rule y ( ) is monotone with respect to martingale fM t gt=0 if, for all periods t
T , all
t
in the support of M t , and all i 2 I, E
the domain of
i
is restricted to the support of
ij
t i
i
[y ( i ;
i )]
is weakly increasing in
i
when
t i.
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 1). 22
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, and
B
A
are given as below: A
A
^A
A
B
A
^A
B
0
1/2
0
B
0
0
0
^B
0
0
1
^B
1/2
0
0
B
0
0
0
B
0
1
0
A
(19)
1
Suppose that we want to implement this decision rule gradually with a belief martingale fM t gt=0 , where 0 i
T = 1. Because the types are assumed to be equally likely,
is uniform. Moreover, we construct the
belief martingale such that bidder B’s type is fully revealed in period 1, i.e., MB1 consists only of the three degenerate beliefs. The decision rule is monotone for both bidders in period 0 because E increasing in
i.
0 jj j
i
[y ( i ;
j )]
is weakly
This requirement is also trivially satis…ed for bidder B in period 1 because the supports
of the beliefs on his type are singletons. We now discuss the restrictions on distribution MA1 of period 1 beliefs on bidder A that would ensure monotonicity for this bidder in period 1. First, note that this decision rule cannot be implemented in dominant strategies because is not monotone in
A
A
[y (
A ; B )]
and therefore ex-post monotonicity fails. This observation implies that MA1 can0 A:
not assign full weight on the prior belief
Bidder A cannot wait until after hearing bidder B’s type
to make his …rst revelation. However, monotonicity of A would be restored if the domain of A is o n ^ or to in period 1. This observation implies that the decision restricted either to A; A A; A T
rule is monotone with respect to fM t gt=0 if the support of MA1 consists only of beliefs assigning positive probabilities either only to types
A
and ^A or only to types
A
and
A.
In other words, the monotonicity
condition demands that bidder A sends a signal fully separating his types ^A and type of bidder B, but it allows for type
A
A
before he hears the
to 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. T
Proposition 3 Suppose that the single-crossing property in Assumption ESC holds, fM t gt=0 is 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 T
i) types are gradually revealed according to martingale fM t gt=0 and decision rule y ( ) is implemented, 23
ii) the interim expected payo¤ for type
of agent i is E i j 0 i fui (y ( i ; i ) ; i ) + xi ( i ; i )g, and P iii) transfers are budget-balanced (they add up to i2I xi ( ) regardless of the path of revelation),22 i
T
if and only if decision rule y ( ) is monotone with respect to martingale fM t gt=0 .
The "only if" part of the proposition was previously discussed above. The "if" part will follow from the lemma below: Lemma 2 Suppose that the single-crossing property in Assumption ESC holds, that 0 < 1
rule y
1
( );x
) for all
a) allocation rule (y ( ; b) for all
with equality when
1
ij
1 i
1
i
i
ui y
1
ui y
i;
( i;
i) ; i
i;
i
;
i
+ xi
i
)=
P
i
xi
is in the support of
,
;
+ xi
i
i;
1
( i;
1
( ) for all ( ;
i)
i;
i
;
i
,
(20)
i
),
for all i, and
T fM t gt=0 .
, we de…ne function
i
:
i
!
i
as follows. Suppose that under belief
of agent i must choose a type to imitate in the support of belief
incentive compatibility of y i
ij
( ) when
Proof. Given belief
hand, if
E
) is monotone with respect to
i
)
i
)=y
e) y ( ;
, type
)) is incentive compatible under belief
is in the support of i , P ) is budget-balanced: i xi ( ;
d) y ( ;
. There exist a decision rule y ( ;
in the support of M such that
);x ( ;
ij
E
c) x ( ;
1
and all i,
i
E
i
T
( ) is monotone with respect to martingale fM t gt=0 , and allocation
( ) is incentive compatible under belief
and a transfer rule x ( ;
1
1
T , decision rule y
is a time period such
1
( );x
1
i
. If
i
is in this support,
( ) implies that he will choose his own type. On the other
does not assign a positive probability to
i,
he chooses the type which would minimize his
payo¤ loss. Formally, i
where supp ( we de…ne 2 2 The
i i
( i ) = arg
max E ^i 2supp( ) i
) is the support of belief
ij
i
1 i
n ui y
1
^i ;
i
;
i
+ xi
1
^i ;
i
o
,
. If there are multiple types maximizing the above expression,
( i ) as the closest one to type
i,
according to Euclidean distance. In the event that there
equilibrium is consistent with the intended allocation rule in the sense that the agents expect the same interim P i2I xi ( ). O¤ the
payo¤s, the equilibrium economic alternative is y ( ), and the sum of the equilibrium transfers is
equilibrium path (for instance, when an agent …rst sends a signal and later reports an inconsistent type with this signal), the mechanism will determine an economic alternative, such as y ^ balanced budget constraint requires the sum of the transfers to equal
in the range of the original decision rule. The P is determined by i2I xi ( ) in this case (where
the …nal type reports of the agents). The proposition remains valid if we impose an alternative budget constraint and ask P ^ . this sum to be equal to i2I xi
24
are two such types (one below above
i
and the other above), we adopt the convention that
( i ) is the one
i
i. 1
The single-crossing condition ESC and the incentive compatibility of y imply that
( i ) is weakly increasing in
i
lower type than
i:
It cannot be that a higher type than
( i ). We de…ne decision rule y ( ;
i
) by setting y ( ;
1
( );x i
1
( ) under
i
chooses to imitate a 1
)=y
i
( i)
i2I
,
which proves parts (d) and (e) of the lemma. As the …rst step in constructing transfer rule x ( ; adjacent types,
and ^i , of agent i under the condition that both types must imitate types within the
i
support of belief
i
:
^
i; ij
i
i
=
i
h
ui y 1 h ui y
It follows from the de…nition of
i
gi i ( i ;
i
E~ E~
where
i
^ ~
i
i
1
i
^i ;
i
i
i
i
.
(21)
is non-negative. Similar to our earlier with equations
i
i
^
i; ij
i
+ E~
ij
i
1
i
i
E~
i
1
ij
^i ; i j~
i
i
i
^i ; i j~
i
(22)
i
and all
)
n gi i ( i ;
ij i
o ) =0 i
(23)
This de…nition implies that
i
1
= xi +
n gi i ( i ;
1
ij
i
o ) =0 i
(24)
. To see this last point, take the expectation of both sides of equations (22) and (23).
Finally, we de…ne the period xi ( ;
i
( i) ;
and ^i are two adjacent types, and
i.
i
i
+ xi
i
i
i
i; ij
i
1
E for all
^ ~
i; ij
i
E for all
i
^
i
i
;
i; ij
i
1
ij
i
i
ij
^i ;
i
1
+ xi
i
i 1
ij
1
;
i
for all agents i and all beliefs
gi i ^i ;
i)
^i ; i j
i
i
i)
( i) ;
i
that E
proofs, we de…ne function gi i ( i ;
=
), we de…ne the payo¤ premium between two
i
1 jIj
1
transfers with the following equation:
( i) ;
Xh xj
i
1
+ E~
( j;
ij
i
j)
gi i
xj
i;
1
1
~
j
i
jIj
( j) ;
j
j6=i
1
E~i j
i
i
X
E~
j6=i
i
jj
i
j
gj j
~ ~
j ; i;
(25)
Budget balancedness in part (c) of the lemma holds by construction. To see the proof for part (b), notice that equations (23), (24), and E
ij
1
E
ij
i
j
xi
( j) = i;
j
i;
for
j
2 supp
;
i
=E
i
25
j
ij
imply that 1 i
n xi
i j
1
i
( i) ;
i
o
,
(26)
and y for all
i
i;
in the support of
i.
E
n ui y
equal to 1
ij
i
i;
i
;
1
=y
i
i
( i) ;
(27)
i
It follows from the last two equations that the left hand side of (20) is
1
1
The incentive compatibility of y
( i) ;
i
1
( );x
;
i
1
+ xi
i
( ) under belief
i
i
( i) ;
i
o
.
(28)
implies that this last expression is
weakly smaller than the right hand side of (20) and exactly equal to it for
2 supp (
i
i
), which thus
proves part (b) of the lemma. Assumption ESC and the monotonicity of function
imply that the incentive compatibility con-
i
straints 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 imitating an adjacent type ^i (for allocation rule (y ( ; E
ij
E
= E~ under belief
i
i.
n
ij
i
1
ij
i
^
i; ij
i
i
n
^i ; i j
i
i
^ ~
i; ij
i
i
+ gi i ( i ;
i
i
+
i
+ E~
i)
^
i; ij
i
i
i
i
))) is
gi i ^i ; o
i
i
^i ; i j~
i
1
ij
);x ( ;
rather than
i
o
E~
ij
i
1 i
^ ~
i; ij
i
i
i
(29)
To prove part (a) of the lemma, it is su¢ cient to show that this payo¤ premium is non-
negative. We have previously seen that the terms E~
1
ij
i
^ ~
i; ij
i
i
i
and E~
ij
1 i
i
i
^i ; i j~
i
are both non-negative. Moreover, E = E
ij
ij
i
i
n
8 <
i
i
:
h
^i ; i j
ui y h ui y
i
+
i
( i) ; i
i
i
( i) ;
i
o
i; ij
^
i
;
i
ui y
i
; ^i
^i ;
i
ui y
i
;
i
^i ;
is also non-negative due to Assumption ESC and the monotonicity of function
i
i
; ^i i
.
i
i
9 = ;
(30)
Like Lemma 1, the current lemma also addresses transforming an incentive compatible allocation rule under belief
1
into a family of allocation rules that are all incentive compatible under the
corresponding updated beliefs
. According to part (b) of the lemma, the resulting allocation rules
yield the same expected payo¤ as the initial allocation rule only for the types that are in the support of the updated belief
i
. The types that are not in the support of this belief may have a strictly lower
expected payo¤. Hence, under allocation rule (y ( ;
);x ( ;
)), whenever an agent is given the
opportunity to send a signal, he will be indi¤erent with respect to all the signals which lead to beliefs assigning a positive weight to his realized type. Moreover, he will (weakly) prefer these signals to any other signal that would generate an inconsistent belief with his type.
26
In the proof for Lemma 2, we address the possibility of inconsistency of the beliefs and the type reports as follows: For each type
i
that is not assigned a positive probability by belief
i
, we designate
a "best type to imitate" within the support of this belief. In the event that agent i sends a signal leading to belief
i
and then reports his type as
i,
the constructed allocation rule (y ( ;
would treat this agent as if he were the designated type in the support of
i
);x ( ;
))
.23 An implication of this
construction is that function gi ( ), which also plays a crucial role in the proof of this lemma, depends on belief
unlike in the proofs of our earlier results. In Appendix B, we provide a numerical example
i
for the construction of these modi…ed allocation rules. In order to illustrate how Lemma 2 implies the "if" part of Proposition 3, we begin with a revelation scheme with T = 1, as we did in our discussion of Proposition 2. In this case, the gradual revelation mechanism would ask the agents to send some signals in period 1 and to report their types in period 2. We relabel the signals available to agent i as the beliefs
1 i
in the support of Mi1 . We set y 0 ( ) and x0 ( )
in Lemma 2 equal to y ( ) and x ( ) in Proposition 3. The mechanism would determine the economic alternative and the transfers by using the signals sent and the types reported as arguments for functions y 1 ( ; ) and x1 ( ; ) mentioned in Lemma 2. The gradual revelation mechanism introduced above has an equilibrium in which each agent i reports his type truthfully in period 2 and follows a type-dependent randomization over his period 1 signals such that Mi1 is the distribution over the equilibrium beliefs on his types. Part (a) of the lemma implies the sequential rationality of truthful reporting in period 2, regardless of the signals sent in the earlier period. The optimality of the signaling behavior in period 1 follows from part (b). When agents follow their equilibrium path of play, note that the implemented economic alternative is y ( ) (part d), and the agents receive the type-dependent interim payo¤ E
ij
0 i
fui (y ( i ;
i) ; i)
+ xi ( i ;
i )g
(part b).
Finally, budget balance follows directly from part (c). The gradual revelation mechanism above can be extended to deal with longer horizons in which T > 1. The extended version of the mechanism conditions the decision rule and the transfers to the signals sent in all periods t = 1; :::; T and the reported types in period T + 1 through the functions yT
T
;
and xT
;
T
identi…ed by Lemma 2. These functions are iteratively determined by using the
evolution of the equilibrium beliefs and setting functions xT and y T
1
;
T
1
, functions xT
2
( ) and y T
2
1
( ) and y T
( ) equal to xT
2
;
T
2
1
( ) equal to xT
1
and y T
2
2
;
T
;
T
1
, and so
on. This extended gradual revelation mechanism has an equilibrium in which the agents’randomizations T
on the signals respect martingale fM t gt=0 and where their type reports are truthful in period T + 1. 23 A
crucial step here is ruling out potential deviations in which an agent deviates from his equilibrium behavior and
conditions his type report on the revelations of other agents. The static incentive compatibility constraints do not exclude the pro…tability of these dynamic deviations. Under the allocation rule we construct, these deviations are dominated by imitating the designated type regardless of the other agents’revelations.
27
As discussed in the Introduction, Mookherjee and Tsumagari (2013) consider a setting in which gradual revelation is an implication of communication costs. The economic decision in question is the production level for each of the two productive agents. These agents have linear utility functions (in their respective production levels and types) and their types (production costs) are drawn from a continuum. The agents use deterministic communication strategies such that each round of communication can be represented as a partition of their type spaces. Mookherjee and Tsumagari are mainly interested in …nding the optimal output function that maximizes a principal’s objective subject to the incentive and communication constraints. They make important contributions to the debate on organizations by comparing the performance of centralized versus decentralized production decisions and simultaneous versus sequential communication protocols from the perspective of such a principal under di¤erent communication cost structures. Because the principal is risk-neutral in monetary transfers, balancing the budget is not an issue for their analysis. As a preliminary result, Mookherjee and Tsumagari identify a monotonicity condition that is necessary and su¢ cient for the implementability of an output function. To understand how our Proposition 3 relates to Mookherjee and Tsumagari’s characterization, consider a belief martingale that is generated by deterministic information policies, such that each agent’s type-dependent strategy speci…es a single signal in each period rather than a non-degenerate distribution over signals. For this belief martingale, the monotonicity condition we introduce in this section boils down to the monotonicity condition in Mookherjee and Tsumagari’s paper. In this sense, Proposition 3 complements Mookherjee and Tsumagari’s analysis by extending their characterization to environments in which the agents have discrete types, communication strategies are stochastic, economic decisions are not necessarily one dimensional, and monetary transfers are budget-balanced.24
2 4 The
main di¤erence between the proofs of the Mookherjee-Tsumagari result and our Proposition 3 is as follows.
In order to implement decision rule y ( ), we construct a gradual revelation mechanism which uses only the economic alternatives in the range of y ( ). Even if an agent deviates from his equilibrium behavior and sends inconsistent signals in di¤erent stages of this mechanism, he will still be facing an economic alternative within the range of the original decision rule. By contrast, Mookherjee and Tsumagari make use of the single dimensional nature of their economic alternatives and generate an auxiliary decision rule for o¤-the-equilibrium-path events. The resulting mechanism makes the agents in their model indi¤erent among all the signals that are available to them, even though each type of each agent chooses a single signal (per period) with probability one in equilibrium. In our case, the gradual revelation mechanism ensures that each type of each agent is indi¤erent among only the signals he would send with positive probability on the equilibrium path, but this indi¤erence does not necessarily extend to the signals which are not supposed to be sent by this particular type.
28
6
Appendix A
In this Appendix, we illustrate the construction of the budget-balanced transfers described by Proposition 2 and Lemma 1 with the help of a numerical example based on an independent private values auction. The two bidders are referred to as A and B. As in the continuous type example mentioned in the text, the private values of the bidders are uniformly distributed on the interval [0; 1]. We are interested in the …rst-price auction allocation rule, in which the auctioned object is e¢ ciently allocated to the highest value bidder and this bidder pays half of his value as the price of the object.25 The resulting transfer rule for bidder A is described as xi ( i ;
j)
=
8 <
i =2
:
0
if
i
>
j
:
(31)
otherwise
We want to implement this allocation rule or a budget-balanced close variant of it with a gradual im1
plementation mechanism, which would generate the symmetric belief martingale fM t gt=0 . Distribution Mi0 is degenerate at the uniform prior low i ,
2
in its support. Beliefs 2 i,
i
high i
and
0 i,
and distribution Mi1 has two equally likely beliefs,
low i
high i
and
2 i
and
are represented by cumulative distribution functions
respectively, with a full support on [0; 1]. These beliefs can be sustained by an information
policy in which bidder i with value probability 1
i
sends a "high" signal with probability
i
and a "low" signal with
i.
We now concentrate on the derivation of the gradual revelation transfers for bidder A. The analogous 1 transfers for bidder B can be identically constructed. The proof of Lemma 1 refers to function gA , which
can transform an implementable transfer rule into a dominant-strategy implementable rule. When the type space is discrete, this function can be constructed by referring to the payo¤ premium
1 A
which is
de…ned in the same proof. The continuous type assumption in this example allows us to follow a more direct approach. The only transfers that would achieve incentive compatibility of the e¢ cient allocation with dominant strategies in a continuous type setting are the Vickrey Clarke Groves transfers below: 8 < if A > B B + k ( B) xDS , (32) A ( A; B ) = : k ( B) otherwise
where k ( ) gives a constant term that does not depend on the type of bidder A. (Note that when k ( )
is set to zero, these transfers can be implemented by the second-price auction.) Accordingly, function 1 gA (
A; B )
will have the form
1 gA (
2 5 To
A; B )
= xDS A (
A; B )
xA (
A; B )
=
8 < :
A =2
B
k(
+k( B)
B)
if
A
>
B
.
(33)
otherwise
simplify the exposition, we assume that the object is not allocated to either bidder in the zero-probability event
that both bidders have the same value.
29
1 Equation (15) in the proof of the lemma yields the constant k ( B ) and therefore pins down function gA : 8 3 2 1 < A if A > B 2 4 B 4 1 . (34) gA ( A; B ) = 3 2 1 : otherwise B
4 B
4
The next step involves taking the expectation of this function under the equilibrium beliefs that bidder A may have regarding the type of bidder B: E~B j E~B j
high B
n 1 gA
o
~
A; B
n 1 gA low B
1
h i 2~B d~B =
~
1 gA
A; B
1 gA
~ A; B
0
o
~ A; B
Z
=
Z
=
1
h 2
0
1 6
3 A
+
i 1 2~B d~B = 6
1 , 24 1 , 24
3 A
(35)
where the terms in the square brackets are the density functions derived from the cumulative distribution functions. Now we are ready to give the belief-dependent transfer rule x1A
A; B ;
1 B
1 A;
by using equation
(17) in the proof of Lemma 1: x1A
A;
x1A
A;
x1A
B;
x1A
1 6
3 B
3 A
1 1 + 24 6
3 B
3 A
1 24
3 B
1 6
3 A
+
low B
1 = xA ( A ; B ) + 6
low B
= xA (
1 6
low A ;
high B
= xA (
high ; A
low A ;
A; B ;
1 24
+
= xA (
A; B ;
3 B
3 A
high B
B;
1 1 + 24 6
1 6
high ; A
A;
A;
B)
B)
A; B )
+
1 6
8 <
1 3 6 A 1 3 6 A
A
1 = 24 : 8 < 1 + = 24 : 8 < 1 = 24 : 8 < 1 + = 24 :
2
A
A
2
+
1 3 6 A 1 3 6 A
A
2
2
+
1 3 6 A + 61 3A
+
1 3 6 B 1 3 6 B 1 3 6 B 1 3 6 B
+ +
1 3 6 A + 16 3A
+
if
A
>
or
low A ,
E
1 j 0 B B
E
Bj
1 B
x1A
A; B ;
1 A;
1 B
=E
Bj
0 B
xA (
A; B )
=
2 A
2
1 A
+
1 12
if
+
1 12
otherwise
1 3 6 B 1 3 6 B
1 12
if
1 12
otherwise
1 3 6 B 1 3 6 B
if
A
>
high A
equals to
.
(36)
either the high signal or the low signal in period 1 and reveal their type in period 2. The object is allocated to the bidder with the highest reported type. The transfers for the bidders are determined by A
and
B
are the reported types in period 2 and beliefs
1 A
and
1 B
are
determined by the signals in period 1. This gradual revelation mechanism has an equilibrium in which each bidder i sends the high signal with probability
i
in period 1 (because he is indi¤erent between the
two signals) and reveals his type truthfully in period 2.
30
A
A
B
otherwise
We can use function x1A above to construct a gradual revelation mechanism, in which the bidders send
function x1A where arguments
,
otherwise
These transfers are budget-balanced and they make the e¢ cient allocation of the object incentive compatible under the updated beliefs in period 1. Moreover, regardless of whether
B
>
B
>
B
.
,
,
7
Appendix B
In this Appendix, we illustrate the construction of the gradual revelation mechanism described by Proposition 3 and Lemma 2 with the help of a numerical example that is based on an independent private values auction. The two bidders are referred to as bidders A and B. As in the example worked out in the text before the statement of the proposition, we assume that bidder i’s private value for the auctioned object can take one of three equally likely values: and
i
i,
^i , or
i.
We assume further that
i
= , ^i = 2 ,
= 3 for both bidders. We want to implement the symmetric decision rule described in table
(19) that is given in the text, with
i
representing the probability that agent i receives the auctioned
object. The transfer for each bidder depends on his own value but not on the value of the other bidder, as described by the transfer rule below:
xi ( i ;
for all
j.
j) =
8 > > > < > > > :
0
if
i
=
i
1 4
if
i
= ^i
2 3
if
i
=
(37)
i
Note that these decision and transfer rules constitute an incentive compatible allocation
rule. We want to implement this allocation rule or a budget-balanced close variant of it with a gradual 1
revelation mechanism generating the belief martingale fM t gt=0 . Distribution Mi0 is degenerate at the uniform prior
0 i,
and distribution ~ 1i has two equally likely beliefs in its support:
assigns probability 2/3 to type
i
and probability 1/3 to type
type ^i and probability 1/3 to type that types
i
i.
i.
1 i
and ^ 1i . Belief
1 i
Belief ^ 1i assigns probability 2/3 to
These beliefs can be supported with an information policy such
and ^i send separate signals in period 1, and type
i
randomizes between these signals
with equal probabilities. The …rst point we establish is that the decision rule in table (19) is monotone with respect to the belief martingale above. The following table gives the expected probability of receiving the object for bidder A under the equilibrium beliefs he may hold during the implementation: A
A
^A
0 B
0
1/6
1/3
^ 1B
0
1/6
2/3
1 B
0
1/6
0
A
Note that the acquisition probabilities are increasing in bidder A’s type under beliefs not under belief
1 B.
(38)
0 B
and ^ 1B but
Yet, monotonicity with respect to the martingale in question is satis…ed because
this martingale fully separates types ^A and
A
(the two types responsible for the non-monotonicity) in
period 1. In the following, we concentrate on the construction of the gradual revelation mechanism transfers 31
for bidder A. The analogous transfers for bidder B can be constructed identically. Under the targeted allocation rule and the prior belief type
A
0 B,
type
prefers to imitate type ^A rather than type
A
would mimic type ^A if he is restricted to choose a type in the support of ^ 1A
of the proof of Lemma 2, this indicates that 1 A; A
pairs,
1 A
(
A)
is equal to
1 A
= ^A . Similarly,
A
^ 1A .
^A =
A.
Therefore,
In the language
26 A.
For all other
A.
Following the proof of Lemma 2, we calculate the values of the payo¤ premium functions ^ 1A A
1 A
A
and
as de…ned in (21):
1 A
A
^ 1A A
0 A; Aj B
A;
0 Aj B
These payo¤ premium functions n 1 o A ^ ~ E~B j 0 ; j A A B A B n 1 o A ^A j~B E~B j 0 ; A A B n 1 o ^A ^ ~ E~B j 0 A; Aj B A B o n 1 ^A ^ ~ E~B j 0 A; Aj B A B
=
=
8 > > > > > > > > > > < > > > > > > > > > > :
8 > > > > > > > > > > < > > > > > > > > > > :
^
or ^A ;
if
0 A; A
if
A;
0 A
= ^A ;
A
and
B
= ^B
2 3
if
0 A; A
= ^A ;
A
and
B
1 3
if
0 A; A
=
^
6= ^B
A; A
and
B
= ^B
2 3
if
0 A; A
=
A; A
^
and
B
6= ^B
0
if
0 A; A
=
A; A
^
or ^A ;
if
A;
0 A
= ^A ;
A
and
B
=
B
0 A
= ^A ;
A
and
B
B
^
6=
and
B
=
B
^
and
B
6=
B
0 4 3
3 4
=
A; A
1 4
if
A;
1 4
if
0 A; A
=
A; A
if
0 A
=
A; A
1 4
A;
yield the following expected values: n 1 o A ^A ; A j~B = E~B j 0 = E~B j A B
=
1 3
= E~B j
0 B
= E~B j
0 B
n n
^ 1A A
^A ;
^ 1A A
~
Aj B
^ ~
A; Aj B
o o
0 B
n
A
(39)
A
1 A
A
^A ;
(40)
~ Aj B
o
=0
=0 =
1 12
(41) 1
When we substitute the expressions above in (22) and use equation (23), we identify functions gAA and ^1
gAA as 1 A
gA (
^ 1A
gA (
2 6 To
as
A
A; B )
A; B )
=
=
8 > > > <
> > > : 8 > > > < > > > :
8 9
if
A
=
A
and
B
= ^B
2 9
if
A
6=
A
and
B
6= ^B ,
4 9
otherwise
4 12
if
A
=
A
and
B
=
B
1 12
if
A
6=
A
and
B
6=
B
2 12
(42)
.
(43)
otherwise
be accurate, type ^A is indi¤erent between imitating type
A
or type
A.
because this function designates the highest of the "best types to imitate."
32
The proof of the lemma sets
1 A
^A
1
^1
The next step is …nding the expected values of gAA and gAA under the equilibrium beliefs on bidder B’s type:
E
n 1 gAA ( B
Bj
and E
Bj B
o
A; B )
n 1 ^ gAA (
=
o
A; B )
8 > > > > > > < > > > > > > :
4 9
if
A
=
A
and
B
=
1 B
2 9
if
A
A
and
B
=
1 B
4 3
6=
if
A
=
A
and
B
= ^ 1B
if
A
6=
A
and
B
= ^ 1B
and
B
= ^ 1B or
2 3
= 0 for all
Now we are ready to identify function x1A
A; B ;
A
1 A;
1 B
(44)
1 B.
(45)
which yields the transfer to bidder A as
a function of the signals and the type reports by using equation (25). The values of this function are reported in the tables below: x1A ;
x1A ;
1 B
1 A;
1 A;
A
^A
x1A ; ^ 1A ;
A
1 B
A
^A
A
19 12
19 12
B
0
0
0
B
4 3
^B
1 4
1 4
1 4
^B
13 12
5 6
5 6
B
2 3
2 3
2 3
B
2 3
5 12
5 12
A
^A
A
B
4 3
4 3
4 3
B
^B
4 3
4 3
4 3
^B
B
11 12
7 4
7 4
B
^ 1B
x1A ; ^ 1A ; ^ 1B
^A
A
0
1 4
1 4
0
1 4
1 4
5 12
2 3
2 3
A
As an example of the calculation of this function, consider the o¤-the-equilibrium-path situation in which both bidders send a signal leading to belief
1 i
and then reveal their types as ^i . In this case, the transfer
to bidder A is determined as x1A ^A ; ^B ; ^
A; B
= xA |
{z 2 3
1 A;
+E } |
1 B 1 Bj B
n 1 gAA ^A ; {z 2 9
B
o }
E |
1 Aj A
n 1 gBA {z
^
A; B
2 9
o
+ xB ^A ; ^B } | {z }
xB ^A ; | {z
1 4
2 3
B
= }
This yields the number in the middle of the upper left table above. All the other entries in these tables are calculated in the same manner. As discussed above, transfer function x1B can be constructed symmetrically to function x1A . The proof of Lemma 2 yields the belief-dependent decision rule as y 1 Allocation rule
x1A ; x1B ; y 1
A; B ;
1 A;
1 B
=y
1 A
(
A) ;
1 B
(
B)
.
satis…es conditions (a) to (e) listed in Lemma 2. We can use this allocation
rule to construct a gradual revelation mechanism in which the bidders send either the high signal (leading to belief
1 i
in equilibrium) or the low signal (leading to ^ 1i ) in period 1 and report their types in period 2. 33
1 . 4
The allocation decision and transfers are determined by the signals and type reports through x1A ; x1B ; y 1 . This gradual revelation mechanism has an equilibrium in which bidders report their true types in period 2, type
i
sends the high signal in period 1, type ^i sends the low signal in period 1, and type
i
randomizes between these two signals with equal probabilities.
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