Robustly E¢ cient Auctions and Informational Size under Ambiguity Kyungmin Kimy

Antonio Pentaz

HKUST

Penn

September, 2009

Abstract This paper considers an interdependent value auction environment with a set of priors. We show that an augmented Vickrey auction with conditional payments is robustly e¢ cient. Furthermore, we extend the notion of informational size, de…ned by McLean and Postlewaite (2002, 2004), to this environment and show that (approximate) implementational e¢ ciency will be achieved if agents are su¢ ciently informationall small in our sense. The more stringent de…nition of informational size in this paper, relative to that of McLean and Postlewaite, indirectly quanti…es the degree of additional complication faced by the mechanism designer due to ambiguity.

1

Introduction

This paper considers a familiar interdependent value auction environment: a unit of indivisible good is to be sold. Bidders’ valuations depend on the underlying quality of the good 2 (common payo¤ state), but bidders do not observe the quality. Instead, they receive private signals, si 2 Si . In addition, they may have di¤erent tastes about the good, ci 2 Ci (idiosyncratic preference parameters). For example, wildcatters are competing for the drilling right on an oil …eld. They are commonly interested in the quantity of oil, , but may have We are indebted to Andrew Postlewaite and Steven Matthews for their encouragement and helpful comments. We also have bene…ted from discussions with Kim-sau Chung, David Dillenberger, and David Schmeidler. y Hong Kong University of Science and Technology, . z University of Pennsylvania,

1

di¤erent costs of extraction, ci . Each wildcatter performs a test and receive a private signal si about . The standard approach proceeds by assuming a commonly known distribution, P , over the set of payo¤ states, signals, and preference parameters, S C. Upon observing their “type”, which includes both information and preference parameters, bidders update their prior, recovering their beliefs about the underlying space of uncertainty as well as their opponents’preferences and beliefs. Bidders’(ex ante, interim, and ex post) expected valuations are easily recovered from such updated beliefs and the underlying preferences. The modelling assumption of a single, commonly known prior, imposes strong restrictions on the agents’in…nite hierarchies of beliefs: it implicitly assumes that the “designer” knows what the agents’hierarchies would be, conditional on each possible realization of their “type.” This paper departs from the standard approach by relaxing the single-prior assumption. We assume that there exists a commonly known set of priors, , over S C, but no further common knowledge is available. Our motivations are twofold. First, in reality, it is rarely the case that there is precise information about the physical environment, as typically and implicitly assumed in the standard approach. The singleprior assumption, though not innocuous, is made only to simplify analysis. However, some problems are hard to be cast within the standard framework. For example, suppose a new technology to test the quantity of oil is available. It is scienti…cally proven that it generates better estimates than the old technology, but it is not known how much the new improves upon the old. In this paper, we will compare our results to those that are obtained from the single-prior assumption and examine the extent to which the single-prior assumption simpli…es the mechanism design problem. Second, we aim to disentangle some conceptual issues that are hidden in the single-prior framework. The assumption that there exists a commonly known probability distribution over S can be interpreted as there being some “scienti…c knowledge” about the signal generating process. The fact that a single such probability distribution exists is an assumption about the form that such scienti…c knowledge takes. Given this, the further assumption that the designer knows as much as the agents about it does not seem particularly problematic to us, as both are based on “objective” knowledge. The standard approach, however, goes beyond that. The probability distribution that is typically assumed common knowledge not only determines the signal generating process, but also pins down agents’ beliefs and preferences. When it comes to beliefs and preferences, we …nd it hard to argue that there exists anything like an “objective” knowledge, which with no harm can be assumed to be commonly known by the agents and known to the designer. However, the standard (single-

2

prior) approach disables us to distinguish between these two di¤erent aspects by imposing strong consistency requirements on beliefs and preferences. Our multiple-prior assumption necessitates explicit restrictions on the mechanism designer’s knowledge regarding agents’ beliefs and preferences, and thus allows us to treat objective knowledge and subjective beliefs di¤erently. In terms of analysis and results, our benchmark is McLean and Postlewaite (2004) (MP, hereafter), who considered the single-prior case. They showed that an augmented Vickrey auction with conditional payments induces allocative e¢ ciency. In addition, more importantly, they provided a notion of informational size and showed that the necessary side payments for allocative e¢ ciency will be small if agents are informationally small. In other words, they identi…ed conditions on the information structure under which (approximate) full e¢ ciency is also achieved. We study a multiple-prior version of their model and obtain results analogues to theirs. We …nd that a generalized version of their mechanism is robust and generalize the de…nition of informational size into the multiple-prior environment. We show that full e¢ ciency is achieved if agents are informationally small in our sense. The main advantages of our approach are twofold. First, we establish robustness in a very strong sense. In our problem, the mechanism designer does not possess any information about agents’preferences and beliefs beyond that given by the physical environment. Agents are allowed to have arbitrary attitudes toward ambiguity and arbitrary beliefs, all of which are private information to the agents. In this sense, our analysis is closely related to the recent work on robust mechanism design.1 The di¤erence is the degree of independence of the information structure. Most of the literature on robust mechanism design takes a completely belief-free approach: treating all beliefs as purely subjective, it focuses on outcomes that are fully independent of the beliefs and information structure. Our mechanism instead is independent of beliefs, but only within an objectively given information structure (a set of priors). Our view is that though a well-de…ned and commonly known unique prior rarely exists in reality, it is also true that often some information, regarded as “objective”, is indeed commonly known. We feel that the explicit distinction between “objective” and “subjective”features of the environment that we propose improves on the existing literature in the direction of more realism. Second, we indirectly quantify the degree of additional complication faced by the mechanism designer due to ambiguity (and consequent smaller extent of common knowledge). In order to ensure allocative e¢ ciency, the mechanism designer must provide a su¢ cient in1

See, in particular, Bergemann and Morris (2005, 2008a, 2008b, 2009).

3

centive for agents to reveal their private information. Under ambiguity, this is harder for the mechanism designer, because she does not possess information about agents’preferences and beliefs. In the auction context, we not only demonstrate that this is indeed the case, but also show its degree through the formal de…nitions of informational size and variability of beliefs. Our de…nitions are equivalent to the corresponding de…nitions of MP in the degenerate case where the set of priors is a singleton. With non-trivial multiple priors, our de…nition of informational size is stronger than that of MP, while that of variability of beliefs is weaker than that of MP. This is intuitive because informational size measures how much incentive an agent must be provided, while the variability of beliefs measures how e¤ectively the mechanism designer can provide an incentive. This argument is further strengthened through our consideration of an intermediate case, motivated by the decision theory literature, where agents’valuation functions are known to have some particular forms. We show that the following de…nition of informational size is in-between, that is, stronger than that of the single-prior case and weaker than that of our general setup. Besides MP and the robust mechanism design literature, this paper contributes to two strands of the literature. Several papers studied e¢ ciency in interdependent value auctions: Dasgupta and Maskin (2000), and Perry and Reny (2002) constructed ex post incentive compatible e¢ cient auctions, but the mechanisms crucially depend on common knowledge assumptions about bidders’valuation functions. That is, the mechanism designer does not have to possess information about bidders’beliefs and preferences, but bidders must know their opponents’valuations, as functions of the realized signals. There is a fairly large literature on the implication of ambiguity in strategic environments. In the auction contexts, Bose, Ozdenoren, and Pape (2006) studied an independent value auction with a set of priors where bidders have maximin preferences of Gilboa and Schmeidler (1989). Lopomo, Rigotti, and Shannon (2008) considered the case where bidders have incomplete preferences of Bewley (1986). All of the previous works have assumed that agents’valuations functions are commonly known to have particular forms (typically, those of Gilboa and Schmeidler, 1989). This paper identi…es and highlights another possible channel, lack of common knowledge about agents’beliefs and preferences, through which ambiguity a¤ects strategic situations. The remainder of the paper is organized as follows. Section 2 presents the formal setup and the maintained assumptions. Section 3 introduces the mechanism we are interested in. We illustrate the main idea through a simple example in Section 4 and present the main result in Section 5. We conclude by discussing several relevant issues in 6.

4

2 2.1

Environment and Assumptions Baseline Setup

A set of agents N = f1; :::; ng compete for the allocation of a single unit of an indivisible object. Agents have possibly heterogeneous preferences over the good, but they all care about the realization of some unknown state 2 = f 1 ; :::; m g R+ , referred to as the payo¤ state. Agents’ underlying preferences are represented by a utility function u : Ci ! R, where Ci is the set of possible personal characteristics of agent i. Each idiosyncratic preference parameter ci 2 Ci determines agent i’s valuation of the good conditional on payo¤ state realization. Let C i2N Ci denote the set of personal characteristic pro…les. The set Ci is common across all agents, and therefore in a …ctitious ex-ante stage agents are symmetric. The set = f 1 ; :::; m g R+ is ordered so that k < k+1 for each k, and Ci has an upper bound c. The function u is strictly increasing in and weakly decreasing in ci . We assume that u ( 1 ; c) > 0 so that the object is always valuable. Prior to the auction, each agent observes the realization of the private component ci 2 Ci and receives a private signal si 2 Si = fsi;1 ; si;2 ; :::; si;ni g about the payo¤ state 2 . Insofar as the private signals are correlated with the payo¤ state, their observation conveys information on . This generates interdependence among agents’valuations: agent i’s expected valuation depends on the opponents’signals s i . We assume that conditional on s i , each ci does not provide any additional information about and s i . For each …nite set X, we denote by (X) be the set of probability distributions over X. In addition, for a product set X = A B, we denote by pA (pB ) the marginal of p over A (B).

2.2

Ambiguity and Valuation Functions

Let vi : S Ci ! R be agent i’s valuation function where vi (s; ci ) is agent i’s willingnessto-pay for the object when the realized signal pro…le is s and his personal characteristic is ci . When there is a single common prior over p 2 ( S), the relationship between agents’ underlying preferences and valuation functions is straightforward. Agent i’s valuation is equal to the expected value of the good based on the posterior beliefs that are updated from the prior p with a realized signal pro…le. In other words, agent i’s valuation function

5

vi : S

Ci ! R is given by vi (s; ci ) =

X 2

p ( js) u ( ; ci ) .

(1)

Furthermore, given si , agent i’s belief over s i is updated from p. This belief can be used to induce the agent to take some action and to calculate the interim expected valuation of agent i, which is given by Ep [vi (s; ci ) jsi ; ci ] =

s

X

i 2S i

p (s i jsi ) vi (s; ci ) =

s

X

i 2S i

p (s i jsi )

X 2

!

p ( js) u ( ; ci ) .

Now suppose there is imprecise information regarding the distribution over S. More precisely, there is a commonly known set of priors over S (that is, ( S)) and no further common knowledge is available. In this case, the relationship between agents’ underlying preferences and valuation functions are no longer straightforward. The problem can be decomposed into two steps. First, suppose a signal pro…le s is known. Unlike in the single-prior case, agents’valuation functions cannot be immediately recovered from the underlying preferences. This opens the door to the consideration of non-expected utility (NEU) theories. Since Ellsberg (1961), the importance of ambiguity (or Knightian uncertainty) in economic problems is well recognized. Several works have investigated the decision rules of agents facing ambiguity,2 and some of them have been successfully applied to problems involving competitive as well as strategic environments. However, there are several competing theories, and each theory is subject to some criticisms. In addition, more importantly, there is an issue concerning other agents’(and the mechanism designer’s) knowledge about agents’preferences in a strategic environment. We think that agents’preferences (in particular, their attitudes toward ambiguity) are private insofar as they are subjective. For these reasons, in this paper we do not impose any particular theory on agents’attitudes toward ambiguity. Instead, we assume that each agent privately observes his own attitude toward ambiguity, which is represented by a valuation function vi : S Ci ! R. Formally, let Vi be the set of feasible valuation functions and V be the set of feasible valuation function pro…les (V i2N Vi ). Prior to the auction, agent i observes his own valuation function vi 2 Vi (but not the opponents’, v i 2 V i ). In other words, we treat agents’ attitudes toward ambiguity in complete analogy with their personal characteristics ci . 2

See Bewley (1986), Gilboa and Schmeidler (1989) and Schmeidler (1989) among others.

6

Second, consider the problem agent i faces after he observes si . Unlike in the single-prior case, there does not exist a single consistent belief over S i . This raises a question regarding the mechanism designer’s knowledge about agents’ beliefs. In the single-prior case, due to each agent’s private information, the mechanism designer does not know what (interim) belief an agent possesses. However, the mechanism designer does know what belief the agent would have if he received a particular signal. Such mechanics simpli…es analysis but is not particularly compelling in the presence of imprecise information. We think that agents’ beliefs are not less subjective and, therefore, private than their preferences. In this paper, we assume that the mechanism designer does not know agents’interim beliefs conditional on their private signals. What he knows is that every agent knows that the true prior lies in the commonly known set of priors, ( S).

2.3

Environments with Ambiguity

An auction environment with ambiguity is de…ned by the set E = N; ; ; u; (Si; Ci ; Vi )i2N , which is assumed to be common knowledge. The set ( S) is the “objectively given”set of priors and we assume that it is compact and convex. Each agent i’s payo¤ type, ti , consists of three components: the signal about the common payo¤ state, si , the preference parameter, ci , and his valuation function, vi . The assumption that E is common knowledge imposes no restrictions on agents’higher order beliefs about their characteristic and attitudes (i.e. their preferences): priors in the set are probability distributions over payo¤ states and signals only. Given the set of priors, , each player’s signal is associated with a set of probability distributions over the opponents’signals, each of which in turn is associated to a set of probability distributions over his opponents’signals. Thus, the assumption of common knowledge of does entail restrictions on the agents’hierarchies of ambiguous beliefs about the payo¤ state,3 and therefore it assumes the designer’s knowledge of such restrictions. But it imposes no restrictions on the designer’s knowledge of the agents’ hierarchies of beliefs about the preference parameters, C and V .

2.4

Assumptions

The main result of the paper (Theorem 1) is based on the following two assumptions on the sets Vi , i 2 N . 3

See Ahn (2007).

7

Assumption 1 (Monotonicity) For all i and j, all ci and all vi 2 Vi , if sj ; s0j 2 Sj and sj < s0j , then vi (s j ; sj ; ci ) vi (s j ; s0j ; ci ). That is, for each bidder and for each personal characteristic ci , the valuation on the object (weakly) improves as agents receive higher signals. Assumption 2 (Boundedness) For all i, all vi 2 Vi , all ci 2 Ci and for all s 2 S, v p (s; ci )

vi (s; ci )

v o (s; ci )

where v o (s; ci ) and v p (s; ci ) are given by v p (s; ci ) = min Ep [u ( ; ci ) js], p2

and v o (s; ci ) = max Ep [u ( ; ci ) js]. p2

The function v p : S Ci ! R can be envisioned as the “fully pessimistic” valuations and corresponds to Gilboa and Schmeidler’s (1986) maximin criterion applied with the set . Similarly, the function v o : S Ci ! R speci…es the “fully optimistic” valuations with the set . This is a rational requirement in the sense that agents in our model do not derive any (dis-)utility from ambiguity itself. They are concerned with ambiguity only insofar as it complicates their probabilistic assessments. From now on, we let Vi be the set of valuation functions of agent i that satisfy Assumptions 1 and 2.

3 3.1

Mechanism Mechanisms

An auction mechanism is a collection fqi ; xi gi2N where qi is the probability that agent i is awarded the object and xi is a transfer function given a vector of bidders’ reports. The domains of the two functions depend on the auction mechanism. Denote by Ri the space of n bidder i’s reports and let R i=1 Ri . De…nition 1 An auction mechanism fqi ; xi gi2N where qi : R ! [0; 1] and xi : R ! ( 1; 1) is

8

ex post individually rational (XIR) (with respect to qi (r)vi (s; ci )

) if

0 for all i, for all s 2 S, for all ci 2 Ci and for all r 2 R,

xi (r)

ex post e¢ cient (XE) (with respect to vi (s; ci )

max vj (s; cj ) j

) if 0 whenever qi (r) > 0, and

incentive compatible (IC) (with respect to ) if every bidder has no incentive to falsely report independently of his (higher-order) beliefs about p 2 and v i 2 V i , that is, given his payo¤ type ti = (si ; ci ; vi ), for any beliefs of his, Ep [qi (ri ; r i )vi (s; ci )

xi (r)jti ]

Ep [qi (ri0 ; r i )vi (s; ci )

xi (ri0 ; r i )jti ] for all ri0 2 Ri .

The incentive compatibility condition is fairly strong.4 But it is necessary in the current problem due to the mechanism designer’s lack of knowledge about bidders’valuation functions and, more importantly, their subjective beliefs.

3.2

Discussion on E¢ ciency Concept

In the standard setup with a single common prior P , ex-post e¢ ciency unambiguously requires the good to be allocated to the agent with the lowest ci . This is because, under the single-prior assumption, agents’valuations of the common component are symmetric in the setup of section 2.1. With imprecise information, di¤erent agents may have di¤erent valuations of the good even with the same signal pro…le and same personal characteristic. Our notion of e¢ ciency requires the object to be assigned to the bidder with the highest willingness-to-pay for the object, for each realization of preferences and signals. Agent i’s willingness-to-pay is equal to his (subjective) valuation of the object, vi (s; ci ). Notice that willingness-to-pay is an “ex-post”concept, in the sense that it is de…ned with respect to the pooled information available to the agents. But it is not “ex-post” in the stronger sense of conditioning on the realization of . If e¢ ciency after the realization of the state is an issue, the object should always be given to the bidder with the smallest private component ci . In our setup, the realized payo¤ state is never observed. Thus, according to our notion, it may be the case that an e¢ cient auction does not assign the object to the agent with lowest ci . If agent i is particularly “optimistic”, his willingness to pay can be higher than another 4

This incentive compatibility is similar, but not equivalent, to the "optimal incentive compatibility" in Lopomo, Rigotti, and Shannon (2008).

9

agent j with cj < ci .

3.3

Augmented Vickrey Auction with Conditional Payments

We are interested in the following mechanism, which is an augmented Vickrey auction with conditional payments: each bidder reports his own payo¤ type ti that consists of his own signal si , his personal preference parameter ci , and valuation function vi 2 Vi . That is, ri = (b si ; b ci ; vbi ); 8i 2 N and r = (ri ; i 2 N ). For each r 2 S C V , let I(r) = fi 2 N j vbi (b s) = max vbj (b s)g, j

and de…ne

wi (r) = max vbj (b s; b cj ). j6=i

The winning probability is determined so that qi (r) =

(

1 jI(r)j

if i 2 I(r),

0 if i 2 = I(r).

The transfer rule consists of two parts, the Vickrey-auction price, x , and conditional payment scheme, z . More precisely, xi (r) = qi (r)wi (r) and z = fzi gi2N where zi : S ! [0; 1) is a function from the set of signal pro…les to the set of nonnegative real numbers. We focus on this mechanism for two reasons. First, this mechanism is essentially identical to that of MP.5 Therefore, we can isolate from other possible compounding factors and, therefore, highlight the marginal e¤ect on the mechanism due to ambiguity. Second, this mechanism is easy to implement. As argued in McLean and Postlewaite, this mechanism is equivalent to the two-stage mechanism in which agents report their signals and get stochastic rewards in the …rst stage, and the standard Vickrey auction is run in the second stage.

4

Example

Three wildcatters are competing for the right to drill for oil on a tract of land. The amount of oil is either H = 1 or L = 0, each equally likely. Each wildcatter i receives a noisy signal si 2 fH; Lg on the amount of oil . The probability that each wildcatter receives the correct signal 5

The only di¤erence is that agents report only si and ci in their mechanism, while agents are required to report vi in our mechanism. The latter part is necessary because agents’attitudes toward ambiguity also matter for allocational e¢ ciency.

10

about the amount of oil is (> 1=2), that is, Pr fsi = Hj = H g = Pr fsi = Lj = L g = for all i. In addition, each bidder observes his own cost of extraction ci . If the object is assigned to wildcatter i, his payo¤ from the oil …eld is ci . Otherwise, his payo¤ is 0. If is commonly known, all bidders agree on the expected value of the object given a vector of signals. However, there is imprecise information over . In particular, the only common knowledge among bidders and the mechanism designer is 2 0 ; for some 1=2 < < . With imprecise information, allocative e¢ ciency is not straightforward even when ci is constant across agents. Bidders may have di¤erent attitudes toward ambiguity and, therefore, may di¤er in their willingness-to-pay for the object. More importantly, the mechanism designer does not have su¢ cient knowledge about bidders’valuations and beliefs under ambiguity. One solution is to divide the problem into two stages: (1) information extraction stage and (2) Vickrey auction. Each bidder’s cost of extraction and attitude toward ambiguity are purely private, while each bidder’s signal has a common value. Therefore, if the mechanism designer can collect and disseminate each bidder’s private information on , the problem reduces to a purely independent values problem, where the standard Vickrey auction ensures allocative e¢ ciency. Now the question is how to induce bidders to truthfully report their signals.

4.1

Common Prior

As a benchmark, suppose is commonly known to all bidders and the mechanism designer. MP made the following two observations. 4.1.1

Information Extraction through Conditional Payments

Consider the following simple mechanism: the mechanism designer pays some side payment z ( 0) to wildcatters who report a signal that is the majority of reported signals. In the three-wildcatter case, this means that wildcatters receive a positive reward if and only if there is at least one other wildcatter who reports the same signal. This scheme utilizes correlations among wildcatters’ signals, similarly to Cremer and McLean (1988). When a wildcatter receives a signal s, it is more likely that the other wildcatters get the same signal. Provided that the others report truthfully, wildcatters have an incentive to truthfully report their signals. Of course, wildcatters may misreport in order to increase their expected payo¤s in the Vickrey auction. When z is su¢ ciently large, however, the former incentive dominates the latter and then the mechanism designer can extract wildcatters’private information about

11

. 4.1.2

Informational Size and Variability of Beliefs

In the above mechanism, the required amount of side payment z can be quite large. MP were also concerned with full e¢ ciency and examined under what condition z can be small. Importantly, they related the condition for full e¢ ciency to the notion of informational size. Consider …rst the cases where = 1 or = 1=2. In the former, the private information of any single wildcatter is redundant given the combined information of other wildcatters.6 In the latter, wildcatters’private information does not provide any information about . Therefore, in both cases, wildcatters cannot alter the expected value of the object and thus have no incentive to misreport in the information extraction stage. Consequently, the necessary amount of side payment z is equal to 0. Agents are informationally negligible. For 2 (1=2; 1), any single wildcatter can alter the expected valuations of all bidders. When is su¢ ciently close to 1=2 or 1, however, it is unlikely that the posterior is changed a lot by any single wildcatter’s misrepresentation. Therefore, the incentive for wildcatters to misrepresent their signals is small. On the other hand, it is more likely that other wildcatters receive the same signal. Therefore, wildcatters have a positive incentive to truthfully report their signals so as to get side payment z > 0. Overall, the two forces determine the size of side payment z. One is how much each wildcatter’s signal can a¤ect the posterior and hence the expected valuations of other wildcatters (Informational Size). The other is how informative each wildcatter’s signal is about the others’signals (Variability of Beliefs). The previous discussion suggests that the necessary side payment z is small when the latter e¤ect dominates the former one, that is, when each agent’s informational size is small relative to his variability of beliefs.

4.2

A Set of Priors

We generalize MP’s insights into the case where there is a set of priors over . It is commonly known that the true signal-generating process is in the ; for some 1=2 < < . By Assumption 2, this implies that the only common knowledge about wildcatters’ valuation functions are v p (s) vi (s) v o (s) where (H; H; H)

6

v

p

v

o

(H; H; L)

(H; L; L)

3 3 +(1

)3

1

)3

1

3 3 +(1

(L; L; L) (1

)3

3 +(1

)3 (1 )3 3 +(1 )3

This is the case of "nonexclusive information" introduced by Postlewaite and Schemeidler (1986).

12

Consider the same side payment mechanism in the information extraction stage as in the single-prior case. Suppose a wildcatter receives a signal. Since > 1=2, no matter what prior 2 0 is true, it is more likely that other wildcatters receive the same signal.7 Therefore, applying the same reasoning as in the single-prior case, when z is su¢ ciently large, wildcatters are willing to truthfully report their signals. In order to determine under what conditions z can be small, suppose a wildcatter receives a signal H. His deviation gain in the Vickrey auction by reporting L instead of H is determined by the extent to which he alters the expected valuations of other wildcatters. In our problem, however, bidders’valuations and their beliefs over others’valuations are not known. Hence the mechanism designer cannot pin down how much each wildcatter (believes he) can alter others’ expected valuations. But it is commonly known that bidders can be neither too pessimistic nor too optimistic. This implies that the maximal (subjective) deviation gain of a wildcatter with signal H is given by max E [v o (si ; s i ) 2[ ; ]

v p (s0i ; s i )jsi = H]

3

=

max 2[ ; ]

2 3

+ (1

)3

+ 2 (1

) (2

1) + (1

(1 )3 3 + (1 )3

)2 1

On the other hand, the minimal (subjective) deviation loss in the information extraction stage is given by min ((n 1) 2[ ; ] 1) 2z. = (2

(n

1)(1

)) z

As tends to 1, the former converges to zero, while the latter tends to z. Therefore, the necessary side payment z can be small. As in the single-prior setup, the amount of necessary side payment is determined by two forces, informational size and variability of beliefs. However, the mechanism designer does not possess information about wildcatters’ expected valuations, their beliefs over the others’ valuations, and so on. This makes it harder for the mechanism designer to elicit private information from wildcatters and thus to achieve (approximate) full e¢ ciency. The insight that informational size must be small relative to variability of beliefs for full e¢ ciency, however, is still transparent. In Section 5, we provide the formal de…nitions of informational size and variability of beliefs under imprecise information. Comparison of those de…nitions to 7

The probability that another wildcatter receives the same signal is bounded below by

13

2

+ 1

2

.

.

those in MP will clearly demonstrate the extent to which a lack of knowledge about bidders’ attitudes toward ambiguity and their beliefs restricts the mechanism designer’s ability to achieve allocative e¢ ciency.

5

Main Result

This section presents the formal notions of informational size and variability of beliefs under ambiguity and the main result of the paper.

5.1

Informational Size

Given a probability distribution p 2 , a signal vector s = (s i ; si ) 2 S induces a posterior distribution p( js i ; si ) on . If agent i unilaterally deviates and reports s0i instead of si , this posterior distribution will change. MP formalized each agent’s ability to a¤ect a posterior distribution as a notion of informational size. With a set of priors, each agent can alter the set of posterior distributions. Therefore, the de…nition of informational size must re‡ect agents’ ability to a¤ect the set of posterior distributions. One important issue here is how to measure the distance between sets of posterior distributions. One may compare each pair of posteriors that are generated from the same prior and apply some criterion (for example, maximum) for integration. This is a natural candidate but does not perform well in our problem. It is because we allow agents to have arbitrary valuation functions and arbitrary beliefs over the others’valuations. It is possible that (agent i believes) that some agent, say agent j 6= i, has the most pessimistic valuation under information (s i ; si ), while he has the most optimistic valuation under information (s i ; s0i ). The appropriate metric must re‡ect this possibility and thus must be more sensitive to each piece of information.8 Suppose bidder i receives a signal si but announces s0i 6= si . Measure the change of the set of posteriors by max kp1 ( je s i ; sei ) p2 ( je s i ; s0i )k . p1 ;p2 2

Then the set

A(si ; s0i ) = 8

s

i

2S

i

: max kp1 ( je s i ; sei ) p1 ;p2 2

p2 ( je s i ; s0i )k > "je si = si

This discussion suggests the possibility that we can weaken the de…nition of informational size if the mechanism designer has better knowledge about bidders’valuation functions. In the last section, we investigate this issue more.

14

consists of those s

i

for which agent i has at least “ 0 i (si ; si )

e¤ect”on the set of posteriors. Let

2 [0; 1]j max PrfA(si ; s0i )g

= min

p2

.

The probability distribution in the last expression plays a role as the stochastic process generating s i given si . To show that i (si ; s0i ) is well-de…ned, let F ( ) = min Prfs p2

i

2S

i

: max k p1 ( je s i ; si ) p1 ;p2 2

p2 ( je s i ; s0i ) k

je si = si g

Hence f 2 [0; 1]j1 F ( ) g is nonempty (since 1 F (1) 1), bounded and closed (since F is right continuous with left hand limits using the fact that is convex and compact). Finally, de…ne the informational size of agent i as

i

= max 0

si ;si 2Si

i

(si ; s0i ) .

If there is a single prior, the de…nition shrinks to i

= max min f 2 [0; 1]j fs 0 si ;si 2Si

i

2S

i

: kp( je s i ; sei )

p( je s i ; s0i )k > "je si = si g

g,

which coincides with the de…nition of informational size under a single prior given by MP.

5.2

Variability of Beliefs

Under a common prior p, the mechanism designer’s ability to provide each agent with an incentive to reveal his private information through conditional payment scheme depends on the di¤erence between the conditional distributions on S i given di¤erent signals. To see this, consider a conditional payment scheme zi : S ! R such that zi (s) = pS i (s i jsi )= pS i ( jsi ) for some constant > 0. Given agent i’s signal si , his belief over the others’signals S i is given by pS i ( jsi ). Therefore, agent i’s expected reward when he receives si but announces s0i is given by X pS i (s i js0i ) pS i (s i jsi ). 0 p ( js ) S i i s 2S i

i

As long as pS i (s i js0i ) and pS i (s i jsi ) are di¤erent, this expression is uniquely maximized at s0i = si . By adjusting , the mechanism designer can induce bidders to truthfully report

15

their signals. Based on this idea, MP de…nes agent’s variability of belief as i

= min 0min

si 2Si si 2Si nsi

2

pS i ( js0i )

pS i ( jsi )

.

With a set of priors, a natural analogue would be the di¤erence between the sets of S conditional distributions over the others’ signals induced by di¤erent signals. Let i i be S the set of probability distributions over S i . De…ne i i (si ) by S i

i

(si ) =

That is, Let

S i

0 i (si ; si )

i

(

pS i ( ) 2

S i

i

: 9p 2

s.t. p(s i jsi ) =

X

p(s i j ; si )p( jsi ); 8s

(si ) is the set of conditional distributions over S

= max

(

p;r2

min

S i (si );q2 i

S i 0 (si ) i

(

q k q k2

r k r k2 S

0 i (si ; si )

2 2

i

i

2S

obtained from

p k p k2

r k r k2

i

)

given si .

2 2

)

)

;0 .

S

can be thought as the distance between i i (s0i ) and i i (si ), normalized by the S S internal size of i (si ). If i i (s0i ) is not too di¤erent from i i (si ) (the …rst term) and S the size of i i (si ) (the second term) is big enough, then i (si ; s0i ) may be zero. De…ne the variability of agent i’s belief by i

= min 0min

si 2Si si 2Si nsi

0 i (si ; si ).

The de…nition is weaker than that of MP. Intuitively, with a set of priors, the mechanism designer does not know bidders’beliefs about the stochastic process that generates signals. Consequently, in order to ensure truthful reporting, the mechanism designer must provide a su¢ cient incentive for bidders so that they are willing to truthfully reveal their private information, no matter what beliefs they possess. This restricts the mechanism designer’s ability to provide an incentive through conditional payment schemes and, ultimately, translates into weaker de…nition of variability of beliefs.

5.3

Main Result

Theorem 1 Under Assumptions 1 and 2, (i) if satis…es i > 0 for each i, then there exists an incentive compatible augmented Vickrey auction fqi ; xi zi gi2N for the auction problem , and 16

(ii) for every

> 0, there exists a

> 0 such that, whenever

max i

min

i

i

satis…es

i,

there exists an incentive compatible augmented Vickrey auction fqi ; xi problem satisfying 0 zi (t) for every i and t.

zi gi2N for the auction

Part (i) of Theorem 1 could be easily predicted. As long as it is possible to provide agents with an incentive ( i > 0), we can induce bidders to truthfully report by making the conditional payment scheme su¢ ciently important. Part (ii) of the theorem deals with the size of the conditional payments. The necessary payments can be small if the informational size is relatively smaller than the variability of beliefs. We brie‡y provide the idea of the proof. First, notice that provided that each bidder truthfully report his signal si , the problem is a Vickrey auction in a purely private value environment. Therefore, bidders have no reasons to lie about their valuation functions and personal preference parameters. Given this, there are two main components in the proof. One is to show that the expected deviation gain in signal reporting is bounded above by his informational size i times some constant. The other part is to show that bidder i’s expected deviation loss from a reward competition fzi gi2N is bounded below by his variability of beliefs i times some constant. The key in the …rst part is the upper bound of each agent’s deviation gain by misrepresentation due to Assumption 1. Suppose agent i’s true signal is si but he reported s0i < si . His deviation gain is maximal when he lowers other agents’valuations as much as possible. Since agents are allowed to have arbitrary beliefs about others’ valuations, the maximum is achieved when agent i believes that all other agents will switch from the most optimistic valuation to the most pessimistic one by his misrepresentation. More precisely, independent of agent i’s belief, his deviation gain is bounded by v o (s i ; si ) v p (s i ; s0i ). Our de…nition of informational size is speci…cally designed to accommodate this bound. In the second part, the main concern is whether agents are provided a su¢ cient incentive to truthfully report their signals. More precisely, it must be that agents are willing to report their true signals, no matter what signal generating process they believe is true. Our de…nition of variability of beliefs identi…es the minimal incentive agents possibly perceive from the conditional payment scheme.

17

6 6.1

Discussion More Restrictions on Attitudes toward Ambiguity

Our discussion shows that the de…nition of informational size is closely related to the amount of common knowledge about agents’valuation functions. This suggests the possibility that if there are more restrictions (or common knowledge) on valuation functions, the de…nition can be weakened. This is indeed true and we discuss two speci…c cases that are motivated from the decision theory literature. In both cases, the exact valuation functions are still private information to each agent. Also, both are only about representation of agents’ valuation functions, and do not restrict agents’beliefs. Case 1 It is commonly known that each agent’s valuation function can be represented by a unique probability distribution, that is, for each i, there exists pi 2 such that for all s 2 S, vi (s) = Epi [ js] =

X 2

pi ( js).

Case 2 It is commonly known that each agent’s valuation function is represented by vi (s) =

i

min Ep [ js] + (1 p2

i )Epi [

js]

where pi is agent i’s center of the set and i is the value of the constant relative imprecision aversion premium (could be di¤erent between agents) as de…ned in Gajdos, Tallon, and Vergnaud (2004). The two cases lead to the same de…nition of informational size. Precisely, the de…nition of the set A(si ; s0i ) is now given by A(si ; s0i ) = fs

i

2S

i

: max k p( je s i ; si ) p2

p( je s i ; s0i ) k> je si = si g.

The intuition is as follows. In both cases, valuation functions are more closely tied to a speci…c probability distribution. This generates a kind of Lipschitz condition in the change of agents’valuation functions. A jump between the most optimistic valuation and the most pessimistic valuation no longer occurs. Then in measuring the di¤erence between the sets of posteriors, we can compare posterior by posterior. This ultimately translates into a de…nition that is weaker than the one in Section 5. But it is still stronger than that of MP, because agents are allowed to have arbitrary valuation functions and beliefs within each class.

18

6.2

Subjective view: Individual Sets of Priors

In our model, agents share a common set of priors. In the context of our model, such assumption is actually weaker than it might seem. The more general case would be the one in which each player has his own set of priors. Imposing common knowledge of such sets of priors would entail some restrictions on the in…nite hierarchies of ambiguous beliefs. In that case, our approach would be that of considering the union of such sets, and imposing common knowledge that the set of priors is from such union. Together with our minimal assumptions on the nature of the agents’ attitudes toward ambiguity, this can be done without loss of generality, allowing to relax such extra assumptions on higher-order ambiguous beliefs. For example, we can let the common set of priors equal to the union of the individual sets, and impose the “optimistic”and “pessimistic”bounds on the valuations for each player according to their own sets of priors.

7

APPENDIX

This appendix provides the proof of Theorem 1. Let i (ri ; ri0 ; r i ) be agent i0 s deviation gain from the Vickrey auction fqi ; xi gi2N when his true report is ri , but he reported ri0 , given other bidders’true reports r i . Lemma 1 Let fqi ; xi gi2N be a Vickrey auction mechanism. Then for every i 2 N and for each r 2 R and ri0 2 Ri , 0 i (ri ; ri ; r i )

= (qi (r)vi (s; ci ) xi (r)) (qi (r i ; ri0 )vi (s; ci ) xi (r i ; ri0 ))

jwi (r i ; ri0 )

wi (r)j .

Proof. Remember that wi (r) = max vbj (b s; b cj ). j6=i

Suppose vbi (r i ; ri0 ) < wi (r i ; ri0 ). Then qi (r i ; ri0 ) = xi (r i ; ri0 ) = 0, so (qi (r)vi (s; ci )

xi (r))

(qi (r i ; ri0 )vi (s; ci )

= (qi (r)vi (s; ci )

xi (r))

0

jwi (r i ; ri0 )

Now suppose vbi (r i ; ri0 ) wi (r i ; ri0 ). Then 0 < qi (r i ; ri0 ) If vi (s; ci ) > wi (r), then qi (r)vi (s; ci )

xi (r) = vi (s; ci )

wi (r) 19

xi (r i ; ri0 ))

wi (r)j .

1 and xi (r i ; ri0 ) = qi (r i ; ri0 )wi (r i ; ri0 ).

qi (r i ; ri0 )(vi (s; ci )

wi (r)).

If vi (s; ci )

wi (r), then qi (r)vi (s; ci )

qi (r i ; ri0 )(vi (s; ci )

xi (r) = 0

wi (r)).

Therefore, (qi (r)vi (s; ci )

xi (r))

qi (r i ; ri0 )(vi (s; ci ) qi (r i ; ri0 )(wi (r) jwi (r i ; ri0 )

(qi (r i ; ri0 )vi (s; ci )

xi (r i ; ri0 ))

qi (r i ; ri0 )(vi (s; ci )

wi (r))

wi (r i ; ri0 ))

wi (r i ; ri0 ))

wi (r)j .

Lemma 2 Suppose all other agents except agent i report truthfully. Then wi (r i ; ri )

vjo (s i ; si ; cj )

wi (r i ; ri0 )

vjp (s i ; s0i ; cj ),

where j is s.t. vj (s i ; si ; cj ) = wi (r). Proof. Let j and j 0 be vj (s i ; si ; cj ) = wi (r) and vj 0 (s i ; s0i ; cj 0 ) = wi (r i ; ri0 )9 . Then by de…nition, vj 0 (s i ; s0i ; cj 0 ) vj (s i ; s0i ; cj ). Therefore, wi (r i ; ri )

wi (r i ; ri0 ) = vj (s i ; si ; cj ) vj (s i ; si ; cj ) vjo (s; cj )

vj 0 (s i ; s0i ; cj 0 ) vj (s i ; s0i ; cj )

vjp (s i ; s0i ; cj )

Lemma 3 For all r 2 R and ri0 2 Ri , let j is s.t. vj (s i ; si ; cj ) = wi (r) and let j 0 is s.t. vj 0 (s i ; si ; cj 0 ) = wi (r) Then: if si > s0i , 0 i (ri ; ri ; r i ) 9

vjo (s i ; si ; cj )

vjp (s i ; s0i ; cj ),

Since we are supposing all the other agents report truthfully, sj = sbj ; cj = b cj ; uj ( ) = u bj ( ).

20

and if si < s0i , vjp0 (s i ; si ; cj 0 ).

vjo0 (s i ; s0i ; cj 0 )

0 i (ri ; ri ; r i )

Proof. Let j and j 0 be vj (s; cj ) = wi (r) and vj 0 (s i ; s0i ; cj 0 ) = wi (r i ; ri0 ). Suppose si > s0i and …x r i . Then, wi (r) = vj (s; cj )

vj 0 (s i ; s0i ; cj 0 ) = wi (r i ; ri0 ).

vj 0 (s; cj 0 )

Therefore; 0 i (ri ; ri ; r i )

jwi (r i ; ri0 )

vjo (s i ; si ; cj )

wi (r i ; ri0 )

wi (r)j = wi (r)

vjp (s i ; s0i ; cj ).

The …rst inequality comes from Lemma 1, while the last inequality uses Lemma 2. Now suppose si < s0i . Then wi (r) = vj (s; cj )

vj (s i ; s0i ; cj )

vj 0 (s i ; s0i ; cj 0 ) = wi (r i ; ri0 ).

Therefore, 0 i (ri ; ri ; r i )

jwi (r i ; ri0 )

Lemma 4 Suppose constant ,

wi (r)j = wi (r i ; ri0 )

> 0, p; r 2

i

S i

p k p k2 Proof. By de…nition of

i

S i

(si ) and q 2

q k q k2

vjo0 (s i ; s0i ; cj 0 )

wi (r)

(s0i ). Then for some positive

i.

r

i, 2

i

i

q r p k q k2 k r k2 2 k p k2 2q r 2p r = + k q k2 k r k2 k p k2 k r k2 1 p q = r. k r k2 k p k2 k q k2

r k r k2

2 2

Therefore, p k p k2

q k q k2

r

i

k r k2

21

min

i t2

vjp0 (s i ; si ; cj 0 )

S i (si ) i

k t k2

Proof of Theorem 1 The proof consists of two parts. (i) Bidder i’s deviation gain from the Vickrey auction is bounded above by i times some constant Fix si ; s0i 2 Si . Let A i (si ; s0i ) = fs i 2 S i : maxp1 ;p2 2 k p1 ( je s i ; sei ) p2 ( je s i ; s0i ) k> je si = si g. Suppose si > s0i and …x a probability distribution p 2 . From the previous lemmas, we know that for j : vj (s; cj ) = wi (r) Ep [vjo (s i ; si ; cj ) vjp (s i ; s0i ; cj )] X = pS i (s i jsi )[vjo (s i ; si ; cj ) vjp (s i ; s0i ; cj )]

Ep [ i (ri ; ri0 ; r i )]

A

= A

i

X

0 i (si ;si )

+ S

vjp (s i ; s0i ; cj )]

pS i (s i jsi )[vjo (s i ; si ; cj )

X

0 i nA i (si ;si )

p(s i jsi )[vjo (s i ; si ; cj )

vjp (s i ; s0i ; cj )].

P 0 First notice that A i (si ;s0 ) p(s i jsi ) = p(S 0 i jsi ) i (si ; si ) i . Therefore, the …rst i term of the above inequality is less than 2M i (si ; s0i ) where M = maxs2S vjo (s; cj ). For the second term, let’s look at vjo (s i ; si ; cj ) v p (s i ; si ; cj ) when s i 2 S i nS 0 i . First let po and pp be the probability distributions which make vjo (s i ; si ; cj ) = max Ep [ js i ; si ] = Epo [ js i ; si ] = p2

vjp (s i ; s0i ; cj ) = min Ep [ js i ; si ] = Epp [ js i ; s0i ] = p2

X

X

uj ( ; cj ) po ( js i ; si ),

uj ( ; cj ) pp ( js i ; s0i ).

Then vjo (s i ; si ; cj )

vjp (s i ; s0i ; cj )

X X i

uj ( ; cj ) jpo ( js i ; si ) uj ( ; c j )

X

0 i (si ; si )

uj ( ; c j ) .

The inequality comes from the fact that when s 22

pp ( js i ; s0i )j

i

2 S i nA i (si ; s0i ), k po ( je s i ; si )

0 pp ( je s i ; s0i ) k i (si ; si ). Similar argument applies to the case si < s0i , letting j be such that vj 0 (s i ; s0i ; cj 0 ) = wi (r i ; ri0 ). Thus, overall, we know

Ep [ i (si ; s0i )] where

= 2M +

P

i,

8p 2

; 8si ; s0i 2 Si

.

(ii) Bidder i’s deviation loss from the "reward" competition is bounded below by i times some constant S For each si 2 Si , let pS i ( jsi ) be a conditional probability in i i (si ) such that pS i ( jsi ) 2 arg

min p2

De…ne i (s i ; si )

max

S i (si ) i

S i (si ) i

q2

pS i ( jsi ) pS i ( jsi )

!

p q .

, 2

and let zi (s) Suppose i > 0 and …x p 2 from si to s0i is p

i (s).

. Applying Lemma 4, agent i’s expected loss by deviating

(si ; s0i ) = Ep [zi (s i ; si )] Ep [zi (s i ; s0i )] X 0 = [ i (s i ; si ) i (s i ; si )] p(s i jsi ) s

=

i

X s

i

p (s i jsi ) kp (s i jsi )k2

p (s i js0i ) p(s i jsi ) kp (s i js0i )k2

i

where

= mint2

i (si )

k t k2 .

Part (i) is immediate from (i) and (ii). For part (ii), letting because by construction jzi (s)j j i (s)j

=

= completes the proof,

and for any agent and any deviation, the expected deviation gain from a Vickrey auction 23

is less than the expected deviation loss from a reward competition independent of probability distributions.

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[13] Harsanyi, J. (1967/1968), "Games with Incomplete Information Played by Bayesian Agents," Management Science 14, 159-182, 320-334, 486-502 [14] Hayashi, T. (2005), "Information, Subjective Belief and Preference," mimeo, University of Texas at Austin. [15] Knight, F. H. (1921), Uncertainty and Pro…t. Boston: Houghton Mi- in. [16] Lopomo, G., L, Rigotti and C. Shannon (2008), "Uncertainty in Mechanism Design," mimeo [17] McLean, R. and A. Postlewaite (2002), "Informational Size and Incentive Compatibility", Econometrica 70, 2421-2453 [18] McLean, R. and A. Postlewaite (2004), "Informational Size and E¢ cient Auctions", Review of Economic Studies 71, 809-827 [19] Perry, M. and P. J. Reny (2002), "An E¢ cient Auction", Econometrica 70, 1199-1212 [20] Postlewaite, A. and D. Schemeidler (1986), "Implementation in Di¤erential Information Economies", Journal of Economic Theory 39, 14-33. [21] Rigotti, L., and C. Shannon (2005), "Uncertainty and Risk in Financial Markets", Econometrica 73, 203-243 [22] Schmeidler, D. (1989), "Subjective probability and expected utility without additivity," Econometrica 57, 571-587

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