Auctions with Dynamic Costly Information Acquisition

Transcription

1 Auctions with Dynamic Costly Information Acquisition Negin Golrezaei Hamid Nazerzadeh Marshall School of Business University of Southern California Los Angeles, CA We study the mechanism design problem for the seller of an indivisible good in a setting where buyers can purchase the additional information and refine their valuations for the good. This is motivated in part by information structures that appear in online advertising where advertisers can target certain demographics of users using cookie-matching services. For this setting, we propose a rich class of dynamic mechanisms, called Sequential Weighted Second-Price, which encompasses the optimal and the efficient mechanisms as special cases. We show that because information is costly, in the optimal and even the efficient mechanisms, not all the buyers would obtain the additional information. We also present a class of parameterized mechanisms that extends the commonly used second-price auction to dynamic settings with information acquisition. In our numerical experiments, we show that this mechanism can yield more than 95% of the optimal revenue. Key words : dynamic mechanisms, optimal auctions, online advertising, cookie matching, data markets, selling information 1. Introduction Traditionally, in the literature on auction theory, it is often assumed that bidders have full information about their valuations for the items that are sold at the auction, and the challenge for the auctioneer is to design a mechanism that elicits the preferences and valuations of the bidders. However, this assumption may not always hold, for instance, as we explain below, in the context of online advertising. The Internet has provided an unprecedented platform for advertisers to reach their consumers and customize their advertisements at an individual level. For instance, major retailers and department stores such as Macy s or Nordstrom target users who have previously visited their websites by showing them ads featuring products viewed by the users. 1 1 These ads usually include a picture of product(s), often with a promotional price or a discount linked to the retailer s website. 1

2 2 Operations Research 00(0), pp , c 0000 INFORMS This level of targeting has been made possible by using HTTP cookies that allow advertisers to track the users that visit their websites. A cookie contains the information that browsers communicate to the websites visited by the users (see RFC6265 (2011)). A growing trend in online advertising is the emergence of companies that provide information about Internet users, which we will refer as information providers. 2 The information is usually gathered about the users via third-party cookies, which are often installed by a website that a user may not have visited. Using this technology, it is possible to track a user across different websites in order to build a browsing history. The browsing history of the user provides valuable information for advertisers, including the user s interests and intentions. For example, consider a user who has searched on a website for flights to Hawaii. Later, when the user visits other websites, the advertisers can follow her and show her deals for vacation packages to Hawaii (cf. Helft and Vega (2010)). Some companies such as Epsilon and Acxiom take this one step further by merging the information gathered online with offline marketing databases (cf. Singer (2012)). The information providers, who have a detailed and up-to-date history for each user, sell their data to the advertisers, for instance, via cookie-matching services. The advertisers use this data to adjust their bids and target their potential customers. Our Contributions Modeling: Motivated by the applications above, we present a model to study costly information acquisition in auctions. Our model consists of an auctioneer (e.g., online publisher) that sells an item (e.g., advertising space) to a set of bidders (e.g., advertisers). The bidders have an initial private valuation for the item. The bidders can purchase additional information. However, access to this additional information is controlled by the auctioneer, and the mechanism may grant access only to a subset (or none) of the bidders. For instance, in online advertising, the publisher can release the identity of the user or the cookies to only a subset of the advertisers (see Section 2 for details). Sequential Weighted Second-Price Auctions: we introduce a class of parameterized sequential mechanisms for the aforementioned model. The mechanism has two parameters: a weight function β and a reserve price parameter ρ. Both parameters can be adjusted to 2 Examples include BlueKai, exelate, Acxiom, PulsePoint, LiveRamp, Neustar, DataLogix, and OpenTracker.

3 Operations Research 00(0), pp , c 0000 INFORMS 3 maximize the social welfare or the revenue. In fact, this class of mechanisms encompasses the optimal and the efficient mechanisms. The mechanism works as follows: Agents 3 bid in the initial round of bidding. Then, the auctioneer selects some of the agents to obtain the additional information. Each selected agent can acquire the additional information by incurring a cost. Then, the selected agents update their valuations and bid in the second round of bidding. Based on the initial and final bids and using weight function β, the mechanism assigns a weighted bid for each agent. Then, all agents participate in a weighted second-price auction in which the item is allocated to the agent with the highest weighted bid if his weighted bid is at least ρ. The selection rule of the mechanism controls agents access to information via prices. As we discuss later, even the efficient mechanism may price information in order to avoid over investment in the information cost. We prove that the mechanism above is incentive compatible; that is, agents bid truthfully, and, moreover, selected agents incur the information cost and update their valuations. In a static setting, it is rather easy to show that a truthful strategy is a dominant strategy; however, in a dynamic setting, establishing incentive compatibility is more involved because an agent optimizes his strategies over the two rounds of bidding and his decision on whether or not to incur the cost of the additional information. We identify three interesting and important special cases of our mechanism. Efficient Mechanism: a mechanism is efficient if it maximizes the sum of the social welfare of the auctioneer and advertisers minus the cost incurred to obtain the additional information. When there is no such cost, the efficient mechanism allows all agents to obtain the additional information. However, when the information is costly, the efficient mechanism grants access to the additional information only to a subset of the agents so that it maximizes the (expected) social welfare, taking into account the cost of the additional information. Sequential Second-Price Auctions: by setting the weight function β parameter equal to 0, we obtain a class of mechanisms where the item is allocated via a second-price auction with a reserve price parameter ρ provides a lower bound on the reserve price. We call this class of mechanism Sequential Second-Price (SSP) because they extend the second-price 3 We use advertiser, agent, and bidder interchangeably.

4 4 Operations Research 00(0), pp , c 0000 INFORMS auction to a setting with dynamic costly information acquisition. These mechanisms are appealing from a practical perspective because the second-price auction is the primary mechanism used in sponsored search auctions (Edelman et al. 2007, Varian 2007) and ad exchange markets (Muthukrishnan 2009, McAfee and Vassilvitskii 2012). Optimal Mechanism: we identify the optimal mechanism in our setting. It turns out the allocation stage of the optimal mechanism is a bit more complicated than a secondprice auction. The mechanism, using function β, assigns a weight to each agent based on his initial bid. These weights favor the agents with higher initial bids in the allocation stage of the mechanism. Insights from Numerical Studies: we numerically compare the above mechanisms in terms of number of selected agents, revenue, and social welfare. We observe that as the cost of obtaining information decreases, the mechanisms make access to information less restrictive, which increases the revenue and the social welfare obtained by the mechanisms. Interestingly, we observe that, on average, the optimal mechanism, compared with the efficient mechanism, allows fewer agents to obtain the additional information. It is well known that the optimal mechanism distorts the allocation and makes it inefficient in order to extract more revenue from agents with higher valuations. Our result implies that in our setting, the optimal mechanism distorts the revelation of information in addition to the allocation. In our simulation results, the SSP mechanism with the optimally chosen reserve price yields more than 95% of the optimal revenue. The application of this model is not limited to online advertising. In many other environments, bidders can obtain the additional information about their valuations at a cost. For instance, in the sale of complex financial or business assets, to make a better investment decision, the bidders invest heavily in the due diligence process to uncover the value of the assets Vallen and Bullinger (1999). In timber auctions, bidders need to examine the volume and composition of wood on tracts Roberts and Sweeting (2010) and Athey and Levin (1999). Similarly, in oil and gas auctions, bidders can conduct seismic studies to better assess the likelihood of finding oil and gas; see e.g., Hendricks et al. (2003). Related Work In this section, we briefly discuss the related literature to our work.

5 Operations Research 00(0), pp , c 0000 INFORMS 5 Dynamic Mechanism Design: our work belongs to the growing body of research on mechanism design; see Bergemann and Said (2011) for a survey. In particular, our work is closely related to Ëso and Szentes (2007) and generalizes their model to a setting where information acquisition is costly. In the absence of this cost, Ëso and Szentes (2007) show that the optimal mechanism grants all agents access to the additional information. In contrast, we show that when obtaining the additional information is costly, the auctioneer, even in the efficient mechanism, might not disclose the additional information to bidders. The selection rule of our mechanism determines the set of agents who could access (and obtain) the additional information. From a technical perspective, as we discuss later in Appendix A, this makes our proof a bit challenging. We point out to the extent of our knowledge, this is the first work that studies optimal dynamic mechanism design with costly information acquisition. Furthermore, even when the information is costless, we extend the results of Ëso and Szentes (2007) to a setting where the second signals of the agents are correlated. See Kakade et al. (2013), Pavan et al. (2014), Battaglini and Lamba (2012), Boleslavsky and Said (2013), and Lobel and Xiao (2013) for recent results on designing optimal dynamic mechanisms. Costly Information Acquisition: most of prior work on information acquisition consider settings where the bidders do not have any private information prior to entry to the auction. In such a setting, where the auctioneer controls the bidder s access to information, Crémer et al. (2003) show that the auctioneer can extract all the surplus by imposing an admission fee; see also Pancs (2011). Shi (2012) considers a setting where bidders do not have any private information prior to entry, yet they can obtain costly information. He shows that an optimal mechanism is the standard auctions (e.g., second-price) with a reserve price. The simple structure of the optimal mechanism in this setting follows from the fact that in contrast to our work, the only private signal of a bidder is his costly information; also McAfee and McMillan (1987, 1988). Information acquisition has also been studied rather extensively in the context of Principle-Agent models; see Cremer and Khalil (1992), Cremer et al. (1997), Szalay (2004). Ye (2007) and Quint and Hendricks (2012) study indicative bidding auctions which are commonly used in selling financial assets. The auction works as follows: bidders submit non-binding bids to indicate their interests in the assets. Then, the auctioneer selects some of the bidders that have higher valuations to proceed to the second round which involves

6 6 Operations Research 00(0), pp , c 0000 INFORMS costly due diligence process and final bidding. Ye (2007) shows that in the indicative bidding efficient entry of the bidders is not guaranteed and the most qualified bidders might not be selected by the auctioneer. Note that in contrast to our work, the number of selected bidders is predetermined. In addition, only selected bidders, who invest in obtaining the additional information, may participate in the allocation stage of the mechanism. For a discussion on computation issues and costly valuation computation in auctions see Larson and Sandholm (2001). Information Disclosure: Our work also contributes to the vast literature on information disclosure. Here, we briefly discuss this literature with the focus on the works motivated by applications in online advertising. Recently, several papers study the affect of sharing cookies and targeting in advertising. Abraham et al. (2011) show that in a common value setting, when some advertisers are better able to utilize information obtained from cookies, asymmetry of information can sometimes lead to low revenue in this market; see also Syrgkanis et al. (2013). The publisher might also suffer from sharing cookies with advertisers due to information leakage Mahdian et al. (2012). When the objective is to maximize the revenue, an important question is how to partition information and how much information to disclose (Milgrom and Weber 1982, Bergemann and Pesendorfer 2007, Ghosh et al. 2007, Rayo and Segal 2010, Bergemann and Bonatti 2011, Emek et al. 2012, Fu et al. 2012, Balseiro et al. 2013). As mentioned before, our proposed mechanisms control the access to information using prices. In the context of pricing information, Babaioff et al. (2012) study the selling information where both the buyer and the seller have private signals; see also Horner and Skrzypacz (2011). In a recent work, Bergemann and Bonatti (2013) study targeted advertising and pricing of cookies. In their model, there is a monopolist information provider from whom advertisers can buy data on individual customers. They show that the advertiser will get data for the customers with the highest and lowest match values. Furthermore, the optimal monopoly price is unimodular in cost of advertising. For a discussion on the privacy issues, see Johnson (2013), Goldfarb and Tucker (2011a,b).

7 Operations Research 00(0), pp , c 0000 INFORMS 7 Organization: In Section 2, we formally define our model. The weighted second-price mechanisms as well as the efficient mechanism are presented in Section 3. We study revenue maximization in Section 4. In Section 5, we evaluate our mechanisms via numerical simulations. Finally, in Appendix A, we show that our proposed mechanisms are incentive compatible and individually rational. 2. Setting We consider a setting with a seller of one (indivisible) item and n agents. Each agent i has an initial valuation v i,0 [v, v] for the item; v i,0 is private information of agent i and is drawn (independently) from distribution F, with p.d.f. f. Distribution F is known by the seller and all the agents. The seller may allow some of the agents to obtain additional information about their valuations. Suppose agent i is one of the agents to whom the seller has offered access to the additional information. In this case, an agent i may decide to incur cost c i and obtain signal δ i about his valuation. The second signals δ i, 1 i n, are private information. They might be correlated and their joint distributions, denoted by G : R n R + (independent of v i,0 ), are publicly known. Without loss of generality, we assume E[δ i ] = 0. 4 For an example, suppose an advertiser values males at $0 and females at $6. Assume that each user has the same chance of being male as of being female. Thus, when the user s gender is unknown, his expected value, that is, his initial valuation, is = 3. By revealing the gender, the valuation of the advertiser will change; with probability 1, his valuation is 2 increased by $3, and with probability 1, it is decreased by the same amount. That is, the 2 second signal is either 3 or 3 with equal probability. In the context of online advertising, our main motivating application, the seller (i.e., the online publisher) can control the access to additional information (i.e, δ i ) via cookies. If the publisher does not disclose the cookies, then the advertisers cannot obtain additional information. If the publisher releases the cookies to advertisers, the advertisers can subsequently take the cookie to the information provider and obtain (purchase) additional information. 5 4 Note that if E[δ i] = > 0, then we can add to v i,0 and then subtract from δ i. 5 Cookies are strings of characters that can only be interpreted by the party that has created them. In general, some of the cookies could have been created by the advertisers themselves. In this work, we focus on the third-party cookies created by the information providers.

8 8 Operations Research 00(0), pp , c 0000 INFORMS We denote the final valuation of agent i by v i,1. If agent i obtains signal δ i, his updated final valuation, v i,1, would be equal to v i,0 + δ i. For the agents who did not learn their second signals, either because the seller denied them the option or by their own choice, let v i,1 = v i,0. Throughout the paper, we denote the vector of the initial and final valuations of all agents by v 0 and v 1. Also, v i,0 and v i,1, respectively denote the vector of the initial and final valuations of all agents except for agent i. Using the revelation principle (cf. Myerson (1986)), we consider only direct mechanisms. A Direct Mechanism: each direct mechanism M is defined by a tuple (s, q, p), which respectively represents its selection, allocation, and payments rules, respectively. Here are the stages of the mechanism: 1. Initial Bidding: Agents bid in the first round. The initial bid of agent i is denoted by b i,0. 2. Selection: Based on the initial bids, the mechanism selects a set of agents that we call selected agents. To each selected agent i, the mechanism offers an option to access (and acquire) the additional information (signal δ i ) at price t i. More precisely, selection rule s : R n ({0, 1} R) n maps the initial bids to a pair (s i, t i ) for each agent. If agent i is selected, s i or s i (b 0 ) is equal to 1. Otherwise, it is equal to 0. Note that the selection rule depends on all initial bids. Each selected agent can pay amount t i to the mechanism to access his signal (the agent would still need to invest an additional cost c i to learn the signal) Obtaining Information: Each selected agent i chooses between the following alternatives: either taking the option and paying t i to the mechanism or rejecting its offer. If the agent accepts the offer from the mechanism, then he has to decide on whether to incur cost c i and learn δ i or not. We define d i as the decision variable for agent i for accepting the mechanism s offer; that is, d i is equal to 1 if the agent pays t i and is 0 otherwise. For any non selected agent 6 In general, one can think of a selection rule that discloses information sequentially (cf. McAfee and McMillan (1987)). During the selection stage, at each step, the mechanisms asks one of the agents to obtain the additional information. Then, based on the report of that agent, the mechanism makes a decision on obtaining more information or proceeding to the allocation stage. In this paper, we consider a two-stage information disclosure because it is more appealing for our motivating example from online advertising, where the mechanism should be executed in a fraction of a second, and sequential information disclosure might not be feasible.

9 Operations Research 00(0), pp , c 0000 INFORMS 9 j, d j is equal to 0. Similarly, we define e i to denote the decision variable for agent i for incurring cost c i and updating his valuations. e i is equal to 1 if d i = 1 and the agent learns δ i ; otherwise, e i is 0. Note that d i is observed by the mechanism but not e i. The mechanism cannot verify whether an agent has invested in obtaining information and updating his valuation or not. 4. Final Bidding: In the final round of bidding, the mechanism may ask a subset of agents (e.g., selected agents) to update their bids. 5. Allocation and Payments: Based on the initial and final bids of all agents (both selected and non selected) 7 and the decision of selected agents on taking the option, d, the seller decides to whom to allocate the item, allocation rule q : (R R {0, 1}) n R +, and how much to charge each agent, payment rule p : (R R {0, 1}) n R. Namely, given all the bids and decision variables, q i (b 0, b 1, d) is the allocation probability, and p i (b 0, b 1, d) is the payment of agent i Best Response Strategies and Incentive Compatibility We assume that all agents are risk neutral. The utility of agent i from participating in the mechanism is equal to q i v i,1 p i d i t i e i c i (more precisely, q i (b 0, b 1, d)v i,1 p i (b 0, b 1, d) d i t i (b 0 ) e i c i ). Each agent chooses a best-response strategy to the mechanism and strategies of the other agents in order to maximizes his expected utility, where the expectation is taken with respect to the second signals of the (selected) agents. More formally, the best response strategy of each agent i can be described with four mappings: b i,0 : R R, d i : R n {0, 1}, e i : R n {0, 1}, and b i,1 : R 2n 1 R. With slight abuse of notation, we denote the decision variables and best-response functions with the same notation. Function b i,0 maps v i,0, the initial valuation of the agent, to the bid in the first round b i,0. The decision on whether to accept the mechanism s offer (and pay t i ) or reject it is determined by d i, which is a function of the initial valuations of the agent himself (v i,0 ) and the vector of initial bids of other agents b i,0. Similarly, e i determines whether or not the agent invests in obtaining information and incurs cost c i. Finally, b i,1 is a function of the whole history and determines the final bid; it is a function of v i,0, b i,0, 7 This is in contrast with most of the previous work in the literature where only agents who obtained the additional information can participate in the second round, c.f., e.g., Ye (2007) and Quint and Hendricks (2012).

10 10 Operations Research 00(0), pp , c 0000 INFORMS and d i. 8 Given the strategy of other agents, agent i optimizes over tuple (b i,0, d i, e i, b i,1 ) to obtain his best (utility-maximizing) strategy. We denote the best-response strategy of the agent with (b i,0, d i, e i, b i,1). Definition 1 (Incentive Compatibility). Agent i is truthful if he bids truthfully in the first round, that is, b i,0 (v i,0 ) = v i,0. Also, if selected, then the agent would obtain the additional information, that is, d i = e i = 1, and finally, bids truthfully in the final round, that is, b i,1 = v i,1. A mechanism is (Bayesian interim) incentive compatible if for each agent i, the truthful strategy is a best response to the truthful strategies of other agents. We can now define the participation constraints for the mechanism. Definition 2 (Individual Rationality). An incentive compatible mechanism is individually rational if for each agent i, the expected utility under the truthful strategy is non-negative Objectives In this paper, we study the two most important objective functions in auction design: maximizing the seller s revenue and the social welfare. Definition 3 (Optimality). An incentive compatible and individually rational mechanism is optimal if it maximizes the revenue, equal to E [ n i=1 (t i + p i )], among all individually rational mechanisms. We present an optimal mechanism in Section 4.2. The social welfare of a mechanism is defined as the sum of the utility of the agents and the seller minus the information cost paid by all the agents. 9 Definition 4 (Efficiency). An incentive compatible and individually rational mechanism is efficient if it obtains the maximum social welfare equal to: { [ n ] E S q i v i,1 i=1 i S max S {1,,n} } { [ { c i = max E max max {v i,0 + δ i }, max {v i,0}, 0 S {1,,n} i S i/ S } ] c i }, i S 8 If the mechanism does not disclose the bids and actions of other agents to agent i, b i,0 and d i would correspond to the belief of the agent about other agents. Note that neither the mechanism nor other agents can observe decision e i of an agent; it is only known to agent i. 9 Note that the monetary transfers among the agents and the seller cancel out each other.

11 Operations Research 00(0), pp , c 0000 INFORMS 11 where in the l.h.s., E S denotes the expectation with respect to the realizations of the second signals when all agents in set S (and only those agents) obtain the additional information; i S c i is the information cost paid by the agents. This is equivalent to the r.h.s. where max i S {v i,0 + δ i } and max i/ S {v i,0 } are respectively the maximum (updated) bid of selected agents and the maximum bid of unselected agents. The item is not allocated if the maximum valuation is negative. To shed light on how the additional information changes the structure of the efficient and the optimal mechanisms, we consider a setting, called conflation, where no agent can obtain the additional information; for instance, this can happen because the mechanism does not grant the agents access or the cost of obtaining information is too large. In the conflation setting, the only available private information is v i,0. Therefore, the optimal and efficient mechanisms are simple. The highest social welfare is equal to max {max i {v i,0 }, 0} and can be obtained by allocating the item to the agent with the highest bid (e.g., a secondprice auction). Similarly, the optimal conflation mechanism corresponds to a second-price auction with a reserve price (Myerson 1981). 3. Sequential Weighted Second-Price Mechanism In this section, we present a parameterized class of mechanisms called Sequential Weighted Second-Price (SWSP), which is denoted by M(ρ, β). As we explain later, parameter ρ R and weight function β : R R can be used to adjust the social welfare and the revenue of the mechanism. Furthermore, we show that this class of mechanism includes the efficient and the optimal mechanisms as well as other interesting and practical mechanisms. We make the following assumption on function β. Assumption 1. Weight function β is non-decreasing and differentiable with bounded derivatives, that is, sup z {β (z)} <, z [v, v]. Note that as it will be clearer later, the non-decreasing function β alters the social welfare and the seller s revenue by distorting the allocation via favoring agents with higher valuations. We start with the description of the mechanism. At first glance, the mechanism may seem complicated, but, in the following, we will discuss each step of the mechanism in more detail and use examples to provide intuition behind each part.

12 12 Operations Research 00(0), pp , c 0000 INFORMS Sequential Weighted Second-Price Mechanism M(ρ, β): The selection, allocation, and payment rules are defined as follows: Selection: Select the following set of agents 10 [ { } S ρ,β (b 0 ) arg max S {1,,n} {E ] max max {b i,0 + β(b i,0 ) + δ i }, max{b i,0 + β(b i,0 )}, ρ } c i, i S where the expectation is with respect to the second signals of the selected agents. Each selected agent i who accepts the offer of the mechanism (to access the additional information) pays t i (b 0 ); see Eq. (4). Final Bidding: Only selected agents who accepted the offer of the mechanism get a chance to update their bids. For any such agent i, b i,1 denotes the updated (and final) bid of agent i. For all other agents, let b j,1 = b j,0. Allocation and Payments: Agents participate in a weighted second-price auction with a reserve price r ρ,β ρ, where r ρ,β will be defined later in Section 3.3. Consider an agent i argmax i {b i,1 + β(b i,0 )}. If b i,1 + β(b i,0) r ρ,β, then the item is allocated to agent i and he pays i/ S { } p i = max max {b i i i,1 + β(b i,0 )}, r ρ,β β(b i,0); otherwise, the item is not allocated. Moreover, for any agent i that does not receive the item, let p i = 0. Note that the last term is a function of the initial bid of the agent, b i,0, and not his final bid b i,1. We now discuss each part of the mechanism separately in the following subsections. i S (1) 3.1. Selection Rule The selection rule is arguably the most important part of the mechanism. The goal of the selection rule is to find a set of agents such that granting them access to information maximizes the objective of the mechanism, taking into account the cost of information. In order to explain the intuition behind the selection rule of the mechanism above, let us start with the special case where β( ) = 0 and ρ = In case of ties, we will choose one of the sets at random.

13 Operations Research 00(0), pp , c 0000 INFORMS The Efficient Selection Rule We define efficient selection rule, S Eff, as follows: [ { } S Eff (b 0 ) = S 0,0 (b 0 ) arg max S {1,,n} {E ] max max {b i,0 + δ i }, max {b i,0}, 0 } c i i S i/ S In Proposition 1, we show that M(ρ = 0, β( ) = 0) is in fact an efficient mechanism and maximizes the social welfare (see Definition 4). To observe this, consider the following scenario, assuming all the agents are truthful: the mechanism selects set S of agents and grants them access to their second signals; subsequently, each selected agent i updates his valuation at cost c i. The total cost of information is equal to i S c i. The efficient mechanism allocates the item to the agent with the highest final bid. The item is { { } allocated as follows. Consider agent i arg max max max i S {b i,0 + δ i }, max i/ S {b i,0 }, 0, where max i S {b i,0 + δ i } and max i/ S {b i,0 } are respectively the maximum updated bid of selected agents and the maximum bid of unselected agents. The item is allocated to i if his (final) bid is positive; otherwise, there would be no allocation. Therefore, in order to maximize the social welfare, the efficient selection rule grants access (to obtain the additional information) to a set of agents such that it maximizes the expected highest bid, after the selected agents updated their bids, taking into account the cost of information. As another example, as we show in Section 4.2, the following selection rule can maximize the revenue of the seller [ { } S Opt (b 0 ) arg max S {1,,n} {E ] max max {b i,0 + α(b i,0 ) + δ i }, max {b i,0 + α(b i,0 )}, 0 c i }, i S i/ S 1 F (y) where S Opt (b 0 ) = S 0,α (b 0 ) and α(y) =. f(y) 3.2. Payments in the First Round Mechanism M(ρ, β) offers selected agents access to their second signals at price t i. Precisely, for M(ρ, β), we have: t i (x) = E [q i v i,1 p i v 0 = x] xi v i S (2) i S (3) E [q i v i,0 = z, v i,0 = x i ] dz c i, (4) where the expectations are with respect to the second signals. Notation E [q i v i,0 = z, v i,0 = x i ] corresponds to the expected probability of allocation to agent i,

14 14 Operations Research 00(0), pp , c 0000 INFORMS where the initial valuation of agent i and other agents are, respectively, equal to z and x i, assuming all the agents, including agent i, bid truthfully and selected agents obtain the additional information. In all the expectations, unless stated otherwise, we assume that agents follow the truthful strategy. Note that the last term in the payment implies that the seller subsidizes the cost of information, c i, for the selected agents who take the option. In Lemmas 6 and 9 and their proofs, we discuss how this payment is calculated using the Envelope Theorem and show that it incentivizes the agent to be truthful. For non-selected agents, t i is equal to The Reserve Price Throughout the paper, when it is clear from the context, we denote S ρ,β (x) and r ρ,β, respectively, by S(x) and r. The reserve price r : (R {0, 1}) n R is a function of initial bids and decision variables d. On the equilibrium path, where all the selected agents obtain the additional information, the reserve price is simply ρ if all agents are selected. If not, define l arg max j / S(x) {x j,0 + β(x j,0 )} as an unselected agent with the highest weighted bid. Then, the reserve price r is equal to ρ if x l,0 + β(x l,0 ) < ρ. Otherwise, the reserve price ρ r x l,0 + β(x l,0 ) is the solution of the equation below 11 bl,0 +β(b l,0 ) max{r,ρ} [ { Pr z max bj,0 + δ j + β(b j,0 ) }] bl,0 ] vl,0 dz = E [q l = z, v l,0 = b l,0 dz (5) {j:d j =1} v where on the equilibrium path, b 0 = x and set {j : d j = 1} is equal to S(x). The below lemma, that is proved in the online appendix, shows that there exists an r that satisfies the above equation. Lemma 1. Suppose all selected agents take the mechanism s offer and ρ b l,0 +β(b l,0 ), l arg max {j:dj =0}{b j,0 + β(b j,0 )}. Then, there exists r [ρ, b l,0 + β(b l,0 )] that satisfies Eq. (5). Off the equilibrium path, let l arg max {j:dj =0}{b j,0 + β(b j,0 )} be an agent with the highest weighted bid that is selected but does not take the option. Then, the reserve price r = b l,0 + β(b l,0 ). The reserve price is chosen in this way to incentivize selected agents to update their valuations. 11 If there are multiple solutions to the above equation, we choose the largest one.

15 Operations Research 00(0), pp , c 0000 INFORMS 15 By construction of the reserve price, an agent i receives the item if his final bid, b i,1, is greater than or equal to max { max j i {b j,1 + β(b j,0 )}, ρ }. Selected agent i pays p i = max { max j i {b j,1 + β(b j,0 )}, ρ } β(b i,0 ) if he receives the item. That is, for the selected agents who take the option, the reserve price in the second round is simply ρ. If an unselected agent receives the item, his payment is equal to reserve r which might be larger than ρ Incentive Compatibility In this section, we present our main result, which shows that the proposed mechanism is incentive compatible and individually rational. Theorem 1 (Incentive Compatibility). Suppose ρ 0 and function β satisfies Assumption 1. Then, the Sequential Weighted Second-Price mechanism M(ρ, β) is incentive compatible and individually rational. The proof is given in Appendix A. Recall that by Definition 1, the theorem above implies that the selected agents take the option, pay the cost, and obtain the additional information. If Mechanism M(ρ, β) is incentive compatible, it maximizes the weighted surplus defined below as Ω(v 0 ) = argmax S {1,...,n} {Ω(v 0, S)}, where Ω(v 0, S) is the weighted surplus when agents with initial valuations v 0 are truthful and agents in set S are selected (and obtain the additional information). Precisely, [ { }] Ω(v 0, S) = E max max {v i,0 + β(v i,0 ) + δ i }, max {v i,0 + β(v i,0 )}, ρ c i. i S i/ S i S In Assumption 1, we assumed that the derivatives of the function β are bounded to ensure that the weighted surplus is absolutely continuous in the initial valuations of the agents. Observe that with ρ = 0 and β( ) = 0, Ω(v 0 ) is equal to the social welfare [ { }] max S {1,,n} {E max max i S {v i,0 + δ i }, max i/ S {v i,0 }, 0 } i S c i. Thus, the SWSP mechanism is efficient by letting ρ = 0 and β(.) = For the efficient mechanism, M(0, 0), the reserve price r 0,0 is in the range of [0, max{0, b l,0 }], where l arg max {i:di =0}{b i,0}. Thus, the mechanism allocates the item to an agent with the highest final bid as long as his bid is greater or equal to 0. In other words, in the second round, the selected agents participate in a second-price auction with reserve price 0. But, because r 0,0 max{0, b l,0 }, agent l might pay more than max{max i l {b i,1}, 0} if he receives the item.

16 16 Operations Research 00(0), pp , c 0000 INFORMS Proposition 1 (Efficiency). Mechanism M(ρ = 0, β( ) = 0) is efficient. Note that our efficient mechanism bears some resemblance to the efficient mechanism in the conflation setting, denoted by eff-c, which is simply a second-price auction with reserve price 0. However, unlike the eff-c mechanism, the reserve price in our efficient mechanism might not be always 0 (see our discussion in Section 3.3). The efficient mechanism M(0, 0) takes into account the cost of information in its selection rule. Thus, it may not allow all agents to update their valuations. In other words, it is not always socially optimal to release the information to all agents especially when they do not have high initial valuations for the item (see also Section 5.5) Who Will Be Selected? Intuitively, in the selection step, the seller may favor agents with higher initial bids because they are more likely to win the item in the second round. This intuition can be formalized under certain symmetric and independence assumptions. 13 Lemma 2. Assume that the second signals are independent from each other, the distribution of δ i, denoted by G i, is symmetric, and c i = c for 1 i n. Then, under Mechanism M(ρ, β), if an agent is selected, then all agents with higher initial bids would also be selected. A distribution G i is symmetric if G i ( y) = 1 G i (y). Note that we assumed E[δ i ] = 0. For instance, normal and uniform distributions satisfy the assumption. The proof is presented in Appendix C.1. Note that Lemma 2 provides a simple way to find the selected agents. One can sort the agents according to initial valuations (bids) in descending order and evaluate the value of the selection rule s objective function for each of the n subsets {1}, {1, 2},, and {1, 2,, n}, and then select the subset that maximizes the objective. To get more insight, in Section 5.5, via numerical analysis, we investigate the selected set of agents. We observe that when the valuations of the agents are close to each other, the seller allows both agents to obtain the additional information in order to differentiate the agent with the higher valuation. 13 See Guha et al. (2006) and Goel et al. (2010) for optimization problems similar to our selection rule.

17 Operations Research 00(0), pp , c 0000 INFORMS Maximizing the Revenue In Section 3.4, we described the efficient mechanism. Now, we present two subclasses of SWSP mechanisms where their objective is to maximize the revenue of the seller The Sequential Second-Price Mechanism The second-price auctions and their variations are prevalent in online advertising and are used by Google and all other major platforms. In this section, we present a class of mechanisms, called Sequential Second-Price (SSP), that extends the second-price auction to our setting with dynamic information acquisition. In these mechanisms, the allocation and payments are determined via a second-price auction, which makes the mechanisms practically appealing. The payment associated with the selection rule incentivizes the agents to obtain the additional information. An SSP mechanism corresponds to a special case of SWSP mechanisms with β( ) = 0. Therefore, the mechanism selects agents in set S ρ, where [ { } S ρ (b 0 ) = S ρ,0 (b 0 ) arg max S {1,,n} {E ] max max {b i,0 + δ i }, max {b i,0}, ρ c i }. i S i/ S In the second round, the item, if allocated, would be given to agent i argmax {i:bi,1 r ρ,0 }{b i,1 } with the highest (final) bid and is charged payment p i = max{max i i {b i,1 }, r ρ,0 }. The parameter of the SSP mechanisms, ρ, can be adjusted by the seller to trade off between social welfare and revenue (see Section 5). We point out that the SSP mechanism with ρ = 0 is indeed the efficient mechanism presented in Section The Optimal Mechanism The SSP mechanism is an attractive choice because it emulates the second-price auctions commonly used in practice. However, it may not be optimal. In this section, we present an optimal mechanism. Let α(v i,0 ) = (1 F (v i,0)) f(v i,0 ). We now define M Opt as an SWSP mechanism with ρ = 0 and β = α. In order to satisfy Assumption 1, we make the following assumption on α. Assumption 2 (Monotone Hazard Rate). Distribution F, with p.d.f. f, has a monotone hazard rate; that is, α( ) is increasing in v i,0. Furthermore, assume that α( ) is differentiable, and sup vi,0 [v, v]{α (v i,0 )} <. i S

18 18 Operations Research 00(0), pp , c 0000 INFORMS The above assumption is standard in the optimal mechanism design and ensures that the virtual valuation of the agents are increasing in their initial valuations (cf., Myerson (1981), Mas-Collel et al. (1995)). Mechanism M Opt selects the following set of agents: [ { } S Opt (b 0 ) = arg max S {1,,n} {E ] max max {b i,0 + δ i + α(b i,0 )}, max {b i,0 + α(b i,0 )}, 0 c i }, i S i/ S The mechanism allocates the item to agent i arg max i {b i,1 + α(b i,0 )} with the highest (non-negative) weighted bid. Unlike the SSP mechanism, in the optimal mechanism, the initial bids directly influence the allocation rule. Recall that in the SSP mechanism, initial bids determine the selected agents and only final bids specify the allocation for the selected agents. Mechanism M Opt is built upon the ideas of virtual value formulation of Myerson (1981) for static revenue-maximizing mechanism design. It allocates the item to the agent that has the highest (final) virtual valuation v i,1 +α(v i,0 ). Mechanism M Opt maximizes the expected highest virtual valuation minus the cost of the additional information. The following theorem establishes the optimality of M Opt (see Definition 3). i S Theorem 2. Suppose Assumption 2 holds. Mechanism M Opt compatible, individually rational, and optimal. described above is incentive The proof of incentive compatibility and individual rationality is followed from Theorem 1. As the next step in proving the above theorem, we find the revenue of the mechanism. Lemma 3 (Revenue of M Opt ). If all the agents follow the truthful strategy, then the expected revenue of M Opt is equal to [ { [ { } ] E max E max max {v i,0 + α(v i,0 ) + δ i }, max {v i,0 + α(v i,0 )}, 0 } ] c i, (6) S {1,2,...,n} i S i/ S where the inner expectation is with respect to the second signals and the outer expectation is with respect to the initial valuations. i S The proof is given in Appendix B. In the proof we show that the revenue of the seller is the expected virtual surplus where the expectation is with respect to initial valuations v 0. This justifies the seller s decision about granting the option in the selection step.

19 Operations Research 00(0), pp , c 0000 INFORMS 19 To complete the proof, we provide an upper bound on the revenue of any incentive compatible mechanisms and we show that this upper bound matches the revenue of mechanism M Opt. Lemma 4 (Upper Bound). The expected revenue of the seller is at most equal to Eq. (6). The upper bound is established using a simpler and closely related problem with fewer constraints called the relaxed problem, cf. Ëso and Szentes (2007), Kakade et al. (2013), and Pavan et al. (2014). In the relaxed problem, the mechanism can decide whether or not to obtain the additional information on the behalf of the agents. Namely, the mechanism can decide to incur cost c i, and then both the agent i and the mechanism learn δ i. It is easy to see that any mechanism that is incentive compatible in the original setting would also be incentive compatible in the relaxed setting. Therefore, the optimal relaxed mechanism obtains at least the same revenue as the optimal mechanism in the original setting (see Appendix B for details). 5. Numerical Comparison In this section, we present the results of our numerical studies where we compare the optimal (M Opt ), the efficient (M(0, 0)), and the SSP (M(0, ρ)) mechanisms in terms of their revenue, social welfare, and number of selected agents. As benchmarks, the results of the optimal and efficient mechanisms in the conflation environment, denoted by opt-c and eff-c, respectively, are also presented. Recall that in these mechanisms, no agents purchase the additional information. The eff-c mechanism corresponds to the second-price auctions with reserve price 0, and the opt-c mechanism corresponds to the second-price auctions with reserve price r 0, where r is the solution of r = 1 F (r). In this section, f(r) opt, eff, and ssp correspond to the optimal, the efficient, and the SSP mechanisms, respectively. Furthermore, unless stated otherwise, here ssp refers to the SSP mechanism with revenue-maximizing ρ, denoted by ρ. Setting of Numerical Experiments: We assume that the first and second signals are independent and are drawn from a normal distribution. 14 That is, v i,0 N(0.5, 0.5) and δ i N(0, σ 2 ), where σ 2 is variance of the second signals. We further assume that all signals are independent across agents. In all the numerical experiments, unless otherwise specified, 14 We get similar results for uniform distributions (see Appendix D).

20 20 Operations Research 00(0), pp , c 0000 INFORMS we assume that σ 2 = 0.5. For the sake of simplicity, we assume that the cost of information is the same for all agents and is equal to c = In this setting, the reserve price of opt-c mechanism is approximately In the next section, we find revenue-maximizing parameter ρ in the ssp mechanism. The impact of the cost of information is investigated in Section 5.2. Then, in Section 5.3, we study the effect of variance of the second signal, σ 2, on these mechanisms. In Sections 5.5 and 5.6, we present the number of selected agents and the initial payment, t i, for each realization of initial valuations. Finally, Section 5.7 examines how the information provider sets the price for the additional information Revenue-Maximizing Parameter of the SSP Mechanism The ssp mechanism is parameterized with a parameter ρ, which provides a lower bound on the reserve price in the second round. The parameter can be adjusted to maximize different criteria; for example, the ssp mechanism with ρ = 0 maximizes the social welfare. Here, the goal is to find revenue-maximizing parameter, ρ. We assume that the number of agents is 2 and the cost is Furthermore, as stated before, we assume that F = N(0.5, 0.5) and G i = N(0, 0.5), where G i is the distribution of second signal δ i. Figure 1a depicts how the revenue of the ssp mechanism changes compared with the revenue of the opt mechanism as a function of ρ. Observe that the revenue of the seller in the ssp mechanism is unimodular in ρ; that is, as ρ increases, the revenue first increases and then it decreases. The parameter ρ that maximizes the revenue is approximately 0.8, which is close to the reserve price in the opt-c mechanism (0.71). At revenue-maximizing ρ, the ssp mechanism obtains about 95% of the optimal revenue. Note that the revenue of the seller is very robust to parameter ρ, in a sense that 25% error in ρ only results in 6% decrease in revenue. Figure 1b shows the change in the social welfare as ρ increases. Not surprisingly, because raising ρ increases the reserve price of the second-price auction in ssp, the welfare of the mechanism is decreasing in ρ. Also, the choice of a large ρ can hurt both the revenue and the welfare of the mechanism. Interestingly, at revenue-maximizing ρ, the ssp and opt mechanisms yield almost the same social welfare.

21 Operations Research 00(0), pp , c 0000 INFORMS 21 Revenue SSP OPT Welfare SSP OPT (a) (b) Figure 1 The social welfare and revenue versus ρ with n = 2, c = 0.05, F = N(0.5, 0.5), and G i = N(0, 0.5) Effects of Cost The mechanisms react to an increase in the cost of information, c, by restricting the number of agents who can update their valuations. The change in the number of selected agents will influence revenue and social welfare as well. Here we study the impact of the cost of information on revenue, welfare, and average number of selected agents. Similar to the previous section, we assume that n = 2, F = N(0.5, 0.5), and G i = N(0, 0.5). We further assume that all agents incur the same cost to obtain the additional information. The impacts of the cost are shown in Figure 2. As a general observation, revenue, social welfare, and average number of selected agents decreases as the cost of the additional information increases. The opt mechanism offers the option to fewer agents, as expected. In terms of revenue and social welfare, the ssp mechanism has a similar performance compared with the opt mechanism. Revenue-maximizing ρ for the ssp mechanism, shown in Figure 2a, decreases as the additional information gets more costly because the mechanism compensates negative impacts of increase in the cost by reducing ρ. Furthermore, ρ converges to the reserve price of the opt-c mechanism as the cost increases. This follows from the fact that for the larger value of the cost, similar to opt-c, ssp does not allow any agents to obtain the additional information. Both opt and eff mechanisms beat their corresponding benchmarks in the conflation setting Effect of Uncertainty of the Second Signals The valuation of bidders might change significantly after purchasing the additional information if the second signal is very informative and has a high variance. Here we study how the variance of the second signal influences the mechanisms. The number of agents is

22 22 Operations Research 00(0), pp , c 0000 INFORMS Revenue Maximizing SSP OPT C Revenue SSP OPT EFF OPT C EFF C Welfare Cost (a) SSP OPT EFF OPT C EFF C Cost Figure 2 Average Number of Selected Agents Cost (c) (d) Revenue-maximizing ρ, revenue, social welfare, and average number of selected agents versus the cost with n = 2, F = N(0.5, 0.5), and G i = N(0, 0.5). (b) SSP OPT EFF Cost 2 and the cost of information is Furthermore, the distribution of the first and second signals are respectively N(0.5, 0.5) and N(0, σ 2 ). Figure 3 shows the impact of the variance of second signals on ρ, revenue, social welfare, and average number of selected agents. When the second signals bear more uncertainty, revenue, social welfare, average number of selected agents in the opt, eff, and ssp mechanisms increase. Revenue-maximizing ρ in the ssp mechanism is also an increasing function of σ. This follows from the fact that for a larger variance, the seller anticipates to see larger second signals. This would allow the seller to increase ρ to avoid over-investment in the additional information More Agents To this point, the number of agents participating in the auction is 2. In this section, we investigate how the number of agents can affect the outcome of opt, eff, and ssp (with revenue-maximizing ρ) mechanisms. Again, we assume that F = N(0.5, 0.5), G i = N(0, 0.5), and c i = 0.05 for 1 i n. Figure 4 shows revenue-maximizing ρ, 15 average number of selected agents, revenue and, social welfare versus number of agents, n. We observe that ρ is a decreasing function of 15 Because the ssp mechanism is very robust in ρ, we get almost the same results if we use ρ = 0.8 for all values of n.

23 Operations Research 00(0), pp , c 0000 INFORMS 23 Revenue Maximizing SSP OPT C Revenue SSP OPT EFF OPT C EFF C Welfare SSP OPT EFF OPT C EFF C (a) (c) Average Number of Selected Agents SSP OPT EFF (b) (d) Figure 3 Revenue-maximizing ρ, revenue, social welfare, and average number of selected agents versus σ with n = 2, c = 0.05, F = N(0.5, 0.5), and G i = N(0, σ 2 ). number of agents. A larger number of agents increases revenue and social welfare, as well as average number of selected agents in all considered mechanisms. However, even in the eff mechanism, average number of selected agents is sub linear (concave) in n Selected Agents One of the important aspects of our mechanisms is granting the agents access to the additional information. In the previous sections, we only investigate the impact of the cost and the variance of second signals on average number of selected agents. Here, we seek to understand how opt, eff, and ssp mechanisms offer the option for different realizations of initial valuations. Similar to the previous sections, we assume that n = 2, F = N(0.5, 0.5), and G i = N(0, 0.5). We further assume that c i = 0.05 for i = 1, 2. Figures 5a, 5b, and 5c show the number of selected agents in opt, eff, and ssp (with ρ = ) mechanisms, respectively, for all realizations of v 1,0 and v 2,0 in the range of [ 1.5, 2.5]. The x-axis is the initial valuation of the second agent, and the y-axis is the initial valuation of the first agent. The entire area is divided into several regions, in which the number of selected agents is the same. Precisely, in white, yellow, and green regions, 16 Recall that for c = 0.05 and σ 2 = 0.5, the revenue-maximizing parameter is approximately 0.8.

24 24 Operations Research 00(0), pp , c 0000 INFORMS Revenue Maximizing SSP OPT C Revenue SSP OPT EFF OPT C EFF C Welfare n SSP OPT EFF EFF C OPT C (a) n (c) Average Number of Selected Agents n 2 1 SSP OPT EFF (b) n (d) Figure 4 Revenue-maximizing ρ, revenue, social welfare, and average number of selected agents versus n with c = 0.05, F = N(0.5, 0.5), and G i = N(0, 0.5). the number of selected agents are zero, one, and two, respectively. 17 In all three figures, across the diagonal, (y = x), the seller might offer the option to both agents. The reason is that in this area, the initial valuation of agents are so close to each other that the seller is willing to select both agents to differentiate them. Optimal Mechanism versus Efficient Mechanism: As we have seen in the previous sections, on average, the opt mechanism releases less information to agents than the eff mechanism. This implies that the optimal mechanism distorts the revelation of information in addition to the allocation to extract rent from the agents Payments in the First Round Recall that the initial payment t i incentivizes agents to be truthful. In this section, we investigate how much opt, eff, and ssp (with ρ = 0.8) mechanisms charge each agent i upfront for different realizations of initial valuations. As usual, n = 2; c i = 0.05 for i = 1, 2; F = N(0.5, 0.5); and G i = N(0, 0.5). 17 By Lemma 2, when the cost of information c i, is the same for all agents, the seller allows agents with higher valuations to learn their second signals.

25 Operations Research 00(0), pp , c 0000 INFORMS 25 (a) Optimal mechanism (b) Efficient mechanism (c) SSP mechanism with ρ = 0.8 (d) Optimal mechanism (e) Efficient mechanism (f) SSP mechanism with ρ = 0.8 Figure 5 Number of selected agents (Figures 5a, 5b, and 5c) and the payment of agent 1 in the first round, t 1 (Figures 5d, 5e, and 5f), for different realizations of v 1,0 and v 2,0 with n = 2, c = 0.05, F = N(0.5, 0.5), and G i = N(0, 0.5). The initial payment for the first agent, t 1, in opt, eff, and ssp (with ρ = 0.8) mechanisms for all realizations of v 1,0 and v 2,0 in the range of [ 1.5, 2.5] is shown in Figures 5d, 5e, and 5f, respectively. The x-axis is v 2,0, and the y-axis is v 1,0. Here different shades of gray mean different initial payment as defined in color bars next to the figures. By construction, the initial payment of the first agent is zero if he is not granted the option. Furthermore, when he is selected, t 1 is an increasing function of v 1, Cost and the Information Provider The way that the information providers (IP) set the price for the additional information affects the revenue of the seller as well as the social welfare. IPs can charge agents in many different ways. In the simplest case, we can assume that there exists only one IP that charges all the agents in the same way, and the cost of information for all agents is equal to c. In this case, we numerically study the impact of the cost of information on the revenue

26 26 Operations Research 00(0), pp , c 0000 INFORMS Average Revenue of IP SSP, =0.8 OPT EFF Cost Figure 6 The average revenue of the IP versus the cost with n = 2, F = N(0.5, 0.5) and g = N(0, 0.5). of the IP. We assume that the IP knows the number of agents n, distributions of initial valuations F, and the additional information G. He is further aware of the mechanism used by the seller. The IP selects the cost to maximize his expected revenue R IP = arg max c { c E [ S (v 0 ) ] }, where S can be S Eff, S Opt, or S ρ, and the expectation is with respect to v 0. Note that S (v 0 ) depends on cost c and the distribution of the additional information G. Finding the solution of the aforementioned optimization problem in a closed form is difficult. However, the IP can find the optimal cost numerically as we illustrate in the following. Consider the same setting in previous sections. Figure 6 depicts the revenue of the IP versus cost for opt, eff, and ssp (with ρ = 0.8) mechanisms. In the opt mechanism, the IP yields the lowest revenue. The optimal cost that maximizes the revenue of the IP in opt and ssp mechanisms is approximately For the eff mechanism, the optimal cost is slightly smaller and is almost Conclusion The availability of individual-level information helps advertisers track customers and target them more efficiently. In fact, they can invest in obtaining this information to prepare their bids and tailor their spending. Motivated by this, we propose a class of parameterized dynamic mechanisms that selectively offers access to additional information to a subset of agents. Our proposed SWSP mechanisms can implement optimal and efficient mechanisms by choosing the parameters properly.

27 Operations Research 00(0), pp , c 0000 INFORMS 27 The class of SWSP mechanisms further includes a subclass of second-price auctions where information is revealed dynamically in two stages. In practice, this mechanism can be implemented via pre-negotiated contracts that, in advance, grant an advertiser access to additional information, and the advertiser bids only once in a second price auction. Appendix A: Proof of Theorem 1 In this section, we prove Theorem 1. We start with incentive compatibility and show that no agents would prefer to deviate from the truthful strategy, as long as all other agents are truthful. We prove this by going over the strategy of an agent in a backward manner. First, using Lemma 5, we show that agents bid truthfully in the second round. Then, we prove that a selected agent obtains the additional information (Lemma 6). Finally, in Lemma 9 we show that agents will be better off by being truthful in the first round. We present the proof of Lemma 9 in Section A.1. The proofs of other lemmas are relegated to the online appendix, Section C. The key challenging part is to show that agents bid truthfully in the first round. The reason is that the effects of initial bids are twofold. First, they determine the set of selected agents. Second, they influence the final allocation of the item. The following lemma shows that agents who can bid in the second round will be truthful even if they were untruthful in the first round. Note that unselected agents do not bid in the second round; that is, their initial bids are considered as their final bids. Lemma 5 (Truthfulness in the Second Round). Under Mechanism M(ρ, β), for any agent that is allowed to update his bid in the second round of bidding, truthfulness is a weakly dominant strategy, even if the agent has not been truthful in the first round. From a technical perspective, one of the aspects that differentiates our work from the previous work on dynamic mechanism design, in particular Ëso and Szentes (2007), is that the deviation strategies of the agents, in addition to misreporting his valuations, include the decision on obtaining information. In the following lemma, we show that a selected agent will purchase the additional information. Lemma 6 (Obtaining Additional Information). Consider selected agent i who bids truthfully in the first round. Assuming all other agents are truthful, agent i would accept the mechanism s offer to access the additional information (and pay t i ). Furthermore, agent i would incur cost c i to obtain signal δ i. We outline here the steps of the proof; see Section C.2 for the complete proof. First, we show that when agent i does not take the option, d i = 0, his utility is zero, which is less than or equal to

28 28 Operations Research 00(0), pp , c 0000 INFORMS his utility when he accepts the offer and obtains the additional information (see Corollary 1). The second step involves showing that selected agent i who accepts the offer prefers to incur cost c i and acquire the additional information; that is, e i = 1. This is obtained by showing that the incentive of the agent gets aligned with the selection rule given that he takes the option. The next lemma characterizes the utility of agent i when all agents, including agent i are truthful. Lemma 7. When the vector of initial valuations is given by x, and all the agents are truthful, the expected utility of truthful agent i, denoted by U i (x i, x i ), is equal to [ ] v0 U i (x i, x i ) = E (x i + δ i )q i d i t i e i p i c i = x = xi v ] vi,0 E [q i = z, v i,0 = x i dz, (7) where the expectation is with respect to the second signals. In addition, U i (x i, x i ) is non-decreasing in x i. We now consider the utility from a deviation strategy. Let U i (x i, ˆx i ) be the utility of agent i with initial valuation x i when he bids ˆx i in the first round and follows the optimal strategy (assuming other agents are truthful). More precisely, [ ] v0 U i (x i, ˆx i ) = E q i v i,1 d i t i e i c i p i = x, b i,0 = ˆx i, d i = d i, e i = e i, (8) where d i and e i refer to d i (v i,0 = x, b i,0 = ˆx i, v i,0 ) and e i (v i,0 = x, b i,0 = ˆx i, v i,0 ), respectively, and the expectation is with respect to the second signals. Note that untruthful agent i might not take the option and obtain the additional information given that he is selected. That is, d i and e i might not be equal to 1 if untruthful agent i is granted the option. The next lemma establishes an upper bound on the utility obtained from a deviation strategy. Lemma 8. Suppose the vector of initial valuations is given by x, and all agents except agent i are truthful. We have U i (x i, ˆx i ) max { U i (ˆx i, ˆx i ) + x i ˆx i { ω i = max max j S(ˆx i,x i ),j i and e i = e i (v i,0 = x i, b i,0 = ˆx i, v i,0 ). Pr [z + e i δ i + β(ˆx i ) ω i ] dz, 0 }, where {x j + β(x j ) + δ j }, max j / S(ˆx i,x i ),j i } {x j + β(x j )}, ρ Note that e i is equal to 1 if agent i gets selected, takes the option, and updates his valuation. Otherwise, it is equal to 0. The integral in the upper bound; that is, x i ˆx i is negative (only) when x i < ˆx i. (9) Pr [z + e i δ i + β(ˆx i ) ω i ] dz, In the following lemma, which is a key technical step in our proof, we show that agents prefer to bid truthfully in the first round. Note that the cost of the additional information and uncertainty in valuations introduce the additional contingent deviation alternatives for agents. Thus, it is harder to satisfy the incentive compatibility constraints.

29 Operations Research 00(0), pp , c 0000 INFORMS 29 Lemma 9 (Trustfulness in the First Round). Any agent i prefers to bid truthfully in the first round; that is, U i (x i, x i ) U i (x i, ˆx i ). As a corollary of Lemma 9 and Eq. (7), we obtain that Mechanism M(ρ, β) is individually rational; that is, if an agent i follows the truthful strategy, his utility is equal to U i (x i, x i ) 0. Corollary 1. Mechanism M(ρ, β) is individually rational. A.1. Proof of Lemma 9 In this section, using lemmas 7 and 8, we show that U i (x i, x i ) U i (x i, ˆx i ). Throughout this section, we assume that ˆx i x i. Similar arguments work when ˆx i > x i. To simplify the notation, we denote e i by e i. When U i (x i, ˆx i ) = 0, immediately we have U i (x i, x i ) U i (x i, ˆx i ) = 0. Now we show that even the upper bound of U i (x i, ˆx i ), that is, U i (ˆx i, ˆx i ) + x i ˆx i Pr [z + e i δ i + β(ˆx i ) ω i ] dz, is smaller than U i (x i, x i ). We show the result by contradiction. Suppose that, contrary to our claim, U i (ˆx i, ˆx i ) + xi ˆx i Pr [z + e i δ i + β(ˆx i ) ω i ] dz U i (x i, x i ). This implies that ˆQ(x i ) Q(x i ), where Q(y) = ] y vi,0 ˆx i E [q i = z, v i,0 = x i dz ˆQ(y) = ] y ˆx i Pr [z + e i δ i + β(ˆx i ) ω i dz. Note that ˆQ(x i ) is the upper bound of U i (x i, ˆx i ) U i (ˆx i, ˆx i ), given in Lemma 8, and Q(x i ) = U i (x i, x i ) U i (ˆx i, ˆx i ). We observe that functions Q(y) and ˆQ(y) are absolutely continuous in the range of [ˆx i, x i ]. 18 Because Q(ˆx i ) = ˆQ(ˆx i ) = 0 and ˆQ(x i ) Q(x i ), and both Q(y) and ˆQ(y) are non-decreasing and absolute continuous functions, there exists a nonzero measure interval [y 1, y 2 ] exists such that for any y in this interval ˆQ (y) Q (y). When the derivatives of Q(y) and ˆQ(y) exist, they are equal ] vi,0 to E [q i = y, v i,0 = x i and Pr [y + e i δ i + β(ˆx i ) ω i ], respectively. Thus, we have ] ] vi,0 E [q i = y, v i,0 = x i Pr [y + e i δ i + β(ˆx i ) ω i y [y 1, y 2 ]. Because weight function β is non-decreasing, the above equation implies that y2 ( 1 + β (z) ) ( ] ]) vi,0 E [q i = z, v i,0 = x i Pr [z + e i δ i + β(ˆx i ) ω i dz 0. (10) y 1 Next we show that there will be a suboptimal selection rule, denoted by Ŝ, that has a larger weighted surplus than that of the selection rule in Eq. (1). This contradicts the fact that the selection rule in Mechanism M(ρ, β) maximizes the weighted surplus. 18 The reason is that by Lemma 7, E [q i vi,0 = z, v i,0 = x i ] is a non-decreasing function of x i. Therefore, Q(y), which is an integral of a bounded and non-decreasing function, is absolutely continuous. By the same argument, ˆQ(y) = y ˆx i Pr [ z + d iδ i + β(ˆx i) ω i ] dz is absolutely continuous.

30 30 Operations Research 00(0), pp , c 0000 INFORMS For a given interval [y 1, y 2 ], the suboptimal selection rule Ŝ works as follows. When the initial valuation of agent i, which is z, is either less than y 1 or greater than y 2, the seller allows agents in set S(z, x i ) to obtain the additional information; that is, Ŝ(z, x i ) = S(z, x i ) for z < y 1 and z > y 2, where S is defined in Eq. (1). For y 1 z y 2, we have { S(ˆxi, x Ŝ(z, x i ) = i )\{i} if e i = 0; S(ˆx i, x i ) if e i = 1. In the online appendix, Section C.3, we prove the following lemma using the Envelope Theorem Milgrom and Segal (2002). Lemma 10. Consider the suboptimal selection rule Ŝ described above. Then, if ˆx i < x i, we have Ω(x, Ŝ(x)) Ω(x, S(x)) y2 ( = 1 + β (z) ) ( [ ] [ ]) vi,0 vi,0 E q i = z, v i,0 = x i, Ŝ(z, x i ) E q i = z, v i,0 = x i, S(z, x i ) dz, y 1 [ ] vi,0 where E q i = z, v i,0 = x i, S is the probability that truthful agent i with initial valuation z receives the item when other agents bid truthfully and agents in set S update their valuations. [ ] vi,0 Note that by definition, Ω(x, S(x)) = Ω(x) and E q i = z, v i,0 = x i, S(z, x i ) = ] vi,0 E [q i = z, v i,0 = x i. Then, according to the lemma above and the fact that the selection rule S maximizes the weighted surplus, that is., Ω(x, Ŝ(x)) Ω(x), we have y2 y 1 ( 1 + β (z) ) ( E [ ] ]) q i v i,0 = z, v i,0 = x i E [q i v i,0 = z, v i,0 = x i, Ŝ(z, x i ) dz 0. (11) Note that for any y 1 z y 2, we have [ ] vi,0 E q i = z, v i,0 = x i, Ŝ(z, x i ) = Pr [z + e i δ i + β(z) ω i ] ] [z + e i δ i + β(ˆx i ) ω i, (12) Pr where the equality follows from the construction of the suboptimal selection rule, and the inequality holds because β is non-decreasing. Applying Eq. (12) in Eq. (11), we obtain y2 y 1 ( 1 + β (z) ) ( ] ]) vi,0 E [q i = z, v i,0 = x i Pr [z + e i δ i + β(ˆx i ) ω i dz 0. The inequality above contradicts Eq. (10), and the contradiction completes the proof.

31 Operations Research 00(0), pp , c 0000 INFORMS 31 Appendix B: Proof of Theorem 2 In this section, we prove lemmas 3 and 4 from Section 4. Proof of Lemma 3 Let x be the vector of the initial valuations. Given that Mechanism M Opt is incentive compatible, the revenue of the seller is given by [ n ] [ n ] E t i + p i = E v i,1 q i c i s i U i (x i, x i ) v 0 = b 0 = x, (13) i=1 i=1 where v i,1 = x i + δ i if i S Opt (x) and x i otherwise, and the expectations are with respect to initial valuations and second signals. Note that the sum of the first and second terms is the social welfare of the agents and the seller. In the following, we compute the last term in the r.h.s. of the above equation, that is, E [U i (x i, x i )], where the expectation is with respect to initial valuations. By Lemma 7, we have E [U i (x i, x i )] = v [ ] xi E vi,0 x i q =v z=v i = z, v i,0 = x i dz df (x i ), where the expectation inside the integral is with respect to v i,0. Changing the order of integrals, we get v z=v v x i =z ] vi,0 df (x i )E [q i = z, v i,0 = x i dz = v z=v ] vi,0 (1 F (z))e [q i = z, v i,0 = x i dz. By multiplying and dividing the r.h.s. of the equation above by the probability density f(z), [ ] 1 F (x we obtain E [U i (x i, x i )] = E i ) f(x i q ) i v 0 = x = E [ α(x i )q i v 0 = x ]. Substituting E[U i (x i, x i )] in Eq. (13), the expected revenue of the seller is given by E [ n (v i=1 i,1 + α(x i ))q i c i s i v 0 = x ]. Finally, the result follows from applying the selection and allocation rules. Proof of Lemma 4 To find an upper bound, we consider a relaxed environment, in which the seller observes the additional information of selected agent i, and she can force agents to update their valuations. It is easy to see that the maximum achievable revenue in this environment is an upper bound on the revenue of the seller in the original environment. By the revelation principle, we focus on direct incentive compatible mechanisms that consist of transfer scheme t i : R n R, selection rule s i : R n {0, 1}, and allocation rule q i : R 2n R +, where s i is 1 when agent i is selected and is 0 otherwise. Note that the payment and selection rules are only functions of initial bids, and the allocation rule is a function of the initial bids and the second signals observed by the seller. To compute the upper bound on the revenue of any incentive compatible mechanism in the relaxed environment, we first need to characterize the utility of each agent i. Assume that agent i with initial valuation x i reports ˆx i, and other agents report truthfully. Then, his utility is given by ] bi,0 U i (x i, ˆx i ) = E [ q i (x i + δ i s i ) t i c i s i = ˆx i, v i,0 = x i, v i,0 = b i,0 = x i, where the expectation is with respect to the second signals. Incentive compatibility implies that ] bi,0 U i (x i, x i ) U i (ˆx i, ˆx i ) U i (x i, x i ) U i (ˆx i, x i ) = (x i ˆx i )E [ q i = x i, b i,0 = x i,

32 32 Operations Research 00(0), pp , c 0000 INFORMS ] bi,0 and similarly, U i (x i, x i ) U i (ˆx i, ˆx i ) (x i ˆx i )E [ q i = ˆx i, b i,0 = x i. Without loss of generality, we assume that x i > ˆx i. Then, using the above equations, ] E [ q i b i,0 = ˆx i, b i,0 = x i U i(x i, x i ) U i (ˆx i, ˆx i ) ] E [ q i b i,0 = x i, b i,0 = x i. x i ˆx i Finally by taking the limit as ˆx i x i, we get U i (x i, x i ) = U i (v, v)+ x i v E [ q i v i,0 = z, b i,0 = x i ] dz. We are now ready to compute the upper bound of the revenue. By using the same arguments as in Lemma 3, it can be shown that for any selection rule s i and allocation rule q i the revenue of the seller when agents are truthful is given by [ E i: s i (x)=1 ( xi + α(x i ) + δ i ) qi + i: s i (x)=0 ( xi + α(x i ) ) q i n c i s i i=1 n i=1 U i (v, v) ] b 0 = x, (14) where the expectation is with respect to the first and second signals. Because the mechanism should be individually rational, we set U i (v, v) = 0 for all i. Then, to maximize the revenue, the item should be allocated to the agent with the highest non-negative virtual valuation, that is, q i = 1 if i arg max j {( xj + α(x j ) + δ j s j (x) )} 19 and 0 otherwise. Therefore, the expected revenue of the seller can be written as [ { } E max max i: s i (x)=1 {x i + α(x i ) + δ i }, max i: s i (x)=0 {x i + α(x i )}, 0 ] n b0 c i s i = x. So, if agents in set S relaxed, defined below, obtain the additional information, the revenue gets maximized: [ { } S relaxed (x) = arg max S {1,,n} {E ] max max {x i + δ i + α(x i )}, max{x i + α(x i )}, 0 c i }. i S Finally, the result follows by plugging the selection rule S relaxed (x), and allocation rule q i in Eq. (14). i / S i=1 i S Acknowledgment We would like to thank S. Muthukrishnan for conversations that inspired this work. This work is supported in part by grants from Google Research and Integrated Media System Center of USC. 19 In case of ties, we choose one of them randomly.

33 Operations Research 00(0), pp , c 0000 INFORMS 33 References Ittai Abraham, Susan Athey, Moshe Babaioff, and Michael Grubb. Peaches, lemons, and cookies: Designing auction markets with dispersed information. Technical report, Tech. Rep. MSRTR , Microsoft Research. January, Susan Athey and Jonathan Levin. Information and competition in us forest service timber auctions. Technical report, National Bureau of Economic Research, Moshe Babaioff, Robert Kleinberg, and Renato Paes Leme. Optimal mechanisms for selling information. In Proceedings of the 13th ACM Conference on Electronic Commerce, pages ACM, Santiago R. Balseiro, Omar Besbes, and Gabriel Y. Weintraub. Auctions for online display advertising exchanges: approximations and design. In Michael Kearns, R. Preston McAfee, and Éva Tardos, editors, ACM Conference on Electronic Commerce, pages ACM, ISBN Marco Battaglini and Rohit Lamba. Optimal dynamic contracting. Economic Theory Center Working Paper, Dirk Bergemann and Alessandro Bonatti. Targeting in advertising markets: implications for offline versus online media. The RAND Journal of Economics, 42(3): , Dirk Bergemann and Alessandro Bonatti. Selling cookies. Working Paper, Dirk Bergemann and Martin Pesendorfer. Information structures in optimal auctions. Journal of Economic Theory, 137(1): , Dirk Bergemann and Maher Said. Dynamic auctions: A survey. Wiley Encyclopedia of Operations Research and Management Science, Raphael Boleslavsky and Maher Said. Progressive screening: Long-term contracting with a privately known stochastic process. Review of Economic Studies, 80(1):1 34, J. Cremer, F. Khalil, and J.-C. Rochet. Strategic information gathering before a contract is offered. Discussion Paper Series In Economics And Econometrics 9708, Economics Division, School of Social Sciences, University of Southampton, January URL Jacques Cremer and Fahad Khalil. Gathering information before signing a contract. The American Economic Review, pages , Jacques Crémer, Yossi Spiegel, and Charles Z Zheng. Optimal selling mechanisms with costly information acquisition. IDEI Working Paper, 205, Benjamin Edelman, Michael Ostrovsky, and Michael Schwartz. Internet advertising and the generalized second price auction: Selling billions of dollars of keywords. American Economic Review, 97(1): , Yuval Emek, Michal Feldman, Iftah Gamzu, Renato Paes Leme, and Moshe Tennenholtz. Signaling schemes for revenue maximization. In Boi Faltings, Kevin Leyton-Brown, and Panos Ipeirotis, editors, ACM Conference on Electronic Commerce, pages ACM, ISBN

34 34 Operations Research 00(0), pp , c 0000 INFORMS Peter Ëso and Balazs Szentes. Optimal information disclosure in auctions and the handicap auction. Review of Economic Studies, 74(3): , Hu Fu, Patrick Jordan, Mohammad Mahdian, Uri Nadav, Inbal Talgam-Cohen, and Sergei Vassilvitskii. Ad auctions with data. In Computer Communications Workshops (INFOCOM WKSHPS), 2012 IEEE Conference on, pages IEEE, Arpita Ghosh, Hamid Nazerzadeh, and Mukund Sundararajan. Computing optimal bundles for sponsored search. In Internet and Network Economics, Third International Workshop (WINE), pages , Ashish Goel, Sudipto Guha, and Kamesh Munagala. How to probe for an extreme value. ACM Transactions on Algorithms, 7(1):12, Avi Goldfarb and Catherine Tucker. Privacy regulation and online advertising. Management Science, 57(1): 57 71, 2011a. Avi Goldfarb and Catherine Tucker. Online display advertising: Targeting and intrusiveness. Marketing Science, 30(3): , 2011b. Sudipto Guha, Kamesh Munagala, and Saswati Sarkar. Optimizing transmission rate in wireless channels using adaptive probes. In Proceedings of the Joint International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS/Performance, pages , Miguel Helft and Tanzina Vega. Retargeting ads follow surfers to other sites. The New York Times, August URL Kenneth Hendricks, Joris Pinkse, and Robert H Porter. Empirical implications of equilibrium bidding in first-price, symmetric, common value auctions. The Review of Economic Studies, 70(1): , Johannes Horner and Andrzej Skrzypacz. Selling information. Cowles Foundation Discussion Paper, Garrett Johnson. The impact of privacy policy on the auction market for online display advertising. Working Paper, Sham M. Kakade, Ilan Lobel, and Hamid Nazerzadeh. Optimal dynamic mechanism design and the virtual pivot mechanism. Operations Research, 61(4): , Kate Larson and Tuomas Sandholm. Costly valuation computation in auctions. In Proceedings of the 8th Conference on Theoretical Aspects of Rationality and Knowledge, TARK 01, pages , Ilan Lobel and Wenqiang Xiao. Optimal long-term supply contracts with asymmetric demand information. Working Paper, Mohammad Mahdian, Arpita Ghosh, Preston McAfee, and Sergei Vassilvitskii. To match or not to match: Economics of cookie matching in online advertising. In Proceedings of the 13th ACM Conference on Electronic Commerce, pages ACM, Andreu Mas-Collel, Michael D Whinston, and Jerry Green. Microeconomic theory, 1995.

35 Operations Research 00(0), pp , c 0000 INFORMS 35 Preston McAfee and Sergei Vassilvitskii. An overview of practical exchange design. Current Science, 103(9): 12, November R Preston McAfee and John McMillan. Auctions with entry. Economics Letters, 23(4): , R Preston McAfee and John McMillan. Search mechanisms. Journal of Economic Theory, 44(1):99 123, Paul Milgrom and Ilya Segal. Envelope theorems for arbitrary choice sets. Econometrica, 70(2): , Paul R Milgrom and Robert J Weber. A theory of auctions and competitive bidding. Econometrica: Journal of the Econometric Society, pages , S Muthukrishnan. Ad exchanges: Research issues. In Internet and network economics, pages Springer, Roger Myerson. Multistage games with communications. Econometrica, 54(2): , Roger B Myerson. Optimal auction design. Mathematics of operations research, 6(1):58 73, Romans Pancs. Sequential negotiations with costly information acquisition. University of Rochester, Department of Economics, Alessandro Pavan, Ilya Segal, and Juuso Toikka. Econometrica, Dynamic mechanism design: A myersonian approach. Daniel Quint and Ken Hendricks. Selecting bidders via non-binding bids when entry is costly. Working Paper, Luis Rayo and Ilya Segal. Optimal information disclosure. Journal of Political Economy, 118(5): , RFC6265. Http state management mechanism. April James W Roberts and Andrew Sweeting. Entry and selection in auctions. Technical report, National Bureau of Economic Research, Xianwen Shi. Optimal auctions with information acquisition. Games and Economic Behavior, 74(2): , March Natasha Singer. Mapping, and sharing, the consumer genome. The New York Times, June URL acxiom-the-quiet-giant-of-consumer-database-marketing.html. Vasilis Syrgkanis, David Kempe, and Eva Tardos. Information asymmetries in common-value auctions with discrete signals. Working Paper, Dezs Szalay. Contracts with endogenous information. Cahiers de Recherches Economiques du Dpartement d Economtrie et d Economie politique (DEEP) 04.05, Universit de Lausanne, Facult des HEC, DEEP, April URL

36 36 Operations Research 00(0), pp , c 0000 INFORMS Marc A Vallen and Christopher D Bullinger. The due diligence process for acquiring and building power plants. The Electricity Journal, 12(8):28 37, Hal Varian. Position auctions. International Journal of Industrial Organization, 25(7): , Lixin Ye. Indicative bidding and a theory of two-stage auctions. Games and Economic Behavior, 58(1): , 2007.

37 Operations Research 00(0), pp , c 0000 INFORMS 37 Appendix C: Proofs of the Technical Lemmas C.1. Proofs from Section 3 Proof of Lemma 1 The basic idea is to establish an upper bound on the r.h.s. of Eq. (5) and a lower bound on the l.h.s. of Eq. (5). We will show the lower bound and the upper bound coincide at an r ρ. This will imply that Eq. (5) is satisfied at some r r. i) The upper bound of the r.h.s. of Eq. (5): we will show that the r.h.s. of Eq. (5), i.e., bl,0 E [ ] q v l bl,0 v l,0 = z, v l,0 = b l,0 dz, is less than or equal to Pr [ ] z + β(b v l,0 ) ω l dz, where ω l = max { { max j S(b0 ) bj,0 + δ j + β(b j,0 ) }, ρ }. Note that because agent l has the largest weighted bid among unselected agents, ω l = max { max j S(b0 ) { bj,0 + δ j + β(b j,0 ) }, max j / S(b0 ),j l { bj,0 + β(b j,0 ) }, ρ }. Thus, by Lemma 8, bl,0 v Pr [ z + β(b l,0 ) ω l ] dz Ul (b l,0, b l,0 ) U l (v, b l,0 ). Lemma 9 implies that even the upper bound of U l (v, b l,0 ), i.e., U l (b l,0, b l,0 ) b l,0 v ] ω l dz, is less than or equal to Ul (v, v). That is, bl,0 v Pr [ z + β(b l,0 ) ω l ] dz Ul (b l,0, b l,0 ) U l (v, v) = where the equality follows from Lemma 7. bl,0 v Pr [ z + β(b l,0 ) E [ q l v l,0 = z, v l,0 = b l,0 ] dz, ii) The lower bound on the l.h.s. of Eq. (5): note that by changing variable, the l.h.s. of Eq. (5) can be written as bl,0 max{r,ρ} β(b l,0 ) [ { Pr z + β(b l,0 ) max max {j:d j =1} bl,0 Pr [ ] z + β(b l,0 ) ω l dz, max{r,ρ} β(b l,0 ) { bj,0 + δ j + β(b j,0 ) }, ρ}] dz where the inequality follows from definition of ω l and the fact that set {j : d j = 1} S(b 0 ) and b l,0 max{r, ρ} β(b l,0 ). iii) The upper bound and the lower bound coincide: observe that the lower bound is a nonincreasing function of r and is zero at r = b l,0 + β(b l,0 ). Furthermore, the upper bound is not a function of r. In the following, we will show that at r = ρ, the lower bound is greater than the upper bound. This implies that there exists an r ρ at which the lower bound and the upper bound coincide. Because b l,0 + β(b l,0 ) ρ, we have bl,0 ρ β(b l,0 ) Pr [ z + β(b l,0 ) ω l ] dz bl,0 bl,0 = = v bl,0 v max{ρ β(b l,0 ),v} Pr [ z + β(b l,0 ) ω l ] dz Pr [ ] max{ρ β(bl,0 ),v} z + β(b l,0 ) ω l dz Pr [ ] z + β(b l,0 ) ω l dz Pr [ z + β(b l,0 ) ω l ] dz, v

38 38 Operations Research 00(0), pp , c 0000 INFORMS where the last equality holds because ω l ρ and for any z ρ β(b l,0 ), Pr [ z + β(b l,0 ) ω l ] = 0. iv) Existence of the reserve price: note that the l.h.s. of Eq. (5) is non-increasing function of r. Moreover, it is zero at r = b l,0 +β(b l,0 ), and is less than or equal to b l,0 Pr [ ] z +β(b r β(b l,0 ) l,0 ) ω l dz for any r ρ. Then, considering the fact that the r.h.s. of Eq. (5) is less than b l,0 Pr [ z + β(b v l,0 ) ] ω l dz, there exists an r [ r, bl,0 + β(b l,0 )] that satisfies Eq. (5). Proof of Lemma 2 Consider two unselected agents i, j such that b i,0 > b j,0. Assume that agents in set S are already selected. We will show that when the cost of information is the same for all agents and second signals are independent, Ω(b 0, S {i}) is greater than or equal to Ω(b 0, S {j}); that is, the seller prefers to add agent i to set S rather than agent j. By definition, Ω(b 0, S {i}) = E [ { }] max max {b k,1 + β(b k,0 )}, b i,0 + β(b i,0 ) + δ i, b j,0 + β(b j,0 ), ρ c ( S + 1), k i,j where the expectation is with respect to second signals and b k,1 = b k,0 + δ k if k S and is b k,0 otherwise. For any realizations of second signals, let Y i = max { max k i,j {b k,1 + β(b k,0 )}, b j,0 + β(b j,0 ), ρ } b i,0 β(b i,0 ). Then, Ω(b 0, S {i}) is given by (bi,0 ) E[ + β(b i,0 ) + δ i 1 {δi Y i } + ( Y i + b i,0 + β(b i,0 ) ) ] 1{δ i < Y i } c ( S + 1). After some manipulations, it can be rewritten as δ b i,0 + β(b i,0 ) + Y i G i (Y i ) + zdg i (z) c ( S + 1), Y i where G i is the distribution of δ i. Likewise, δ Ω(b 0, S {j}) = b j,0 + β(b j,0 ) + Y j G i (Y j ) + zdg i (z) c ( S + 1), Y j where Y j = max { max k i,j {b k,1 +β(b k,0 )}, b i,0 +β(b i,0 ), ρ } b j,0 β(b j,0 ). Using integration by part, one can easily show that Ω(b 0, S {i}) Ω(b 0, S {j}) = b i,0 + β(b i,0 ) b j,0 β(b j,0 ) Y j Y i To show the result we need to consider the following two cases. G i (z)dz. Y i Y j = b j,0 +β(b j,0 ) b i,0 β(b i,0 ): In this case, max { max k i,j {b k,1 +β(b k,0 )}, ρ } is greater than b j,0 + β(b j,0 ) and b i,0 + β(b i,0 ). That is, Y i = max { max k i,j {b k,1 + β(b k,0 )}, ρ } b i,0 β(b i,0 ) and Y j = max { max k i,j {b k,1 + β(b k,0 )}, ρ } b j,0 β(b j,0 ). Thus, Ω(b 0, S {i}) Ω(b 0, S {j}) b i,0 + β(b i,0 ) b j,0 β(b j,0 ) (b i,0 + β(b i,0 ) b j,0 β(b j,0 )) = 0 where the inequality follows from the fact that for any z, G i (z) 1.

39 Operations Research 00(0), pp , c 0000 INFORMS 39 Y i Y j b j,0 + β(b j,0 ) b i,0 β(b i,0 ): In this case, max { max k i,j {b k,1 + β(b k,0 )}, ρ } is less than b i,0 + β(b i,0 ). That is, Y j = b i,0 + β(b i,0 ) b j,0 β(b j,0 ) and Y i b j,0 + β(b j,0 ) b i,0 β(b i,0 ). Then, Ω(b 0, S {i}) Ω(b 0, S {j}) b i,0 + β(b i,0 ) b j,0 β(b j,0 ) bi,0 +β(b i,0 ) b j,0 β(b j,0 ) b j,0 +β(b j,0 ) b i,0 β(b i,0 ) G i (z)dz Note that the upper level of the integral equals negative the lower level of the integral. Thus, by the fact the G i ( z) = 1 G i (z), we have b i,0 +β(b i,0 ) b j,0 β(b j,0 ) b j,0 +β(b j,0 ) b i,0 β(b i,0 ) G i(z)dz = b i,0 + β(b i,0 ) b j,0 β(b j,0 ); that is, Ω(b 0, S {i}) Ω(b 0, S {j}) is at least zero, which is the desired result. C.2. Proofs from Appendix A Proof of Lemma 5 Consider a selected agent i with initial bid b i,0 who takes the option. If agent i wins the item, his payment in the second round, p i, would be equal to max{max j i {b j,1 + β(b j,0 )}, r} β(b i,0 ). Note that p i is independent of b i,1. Therefore, an agent cannot change his price for the item. However, the agent can change the probability of the allocation. It is easy to see that underbidding may only result in losing the item. On the other hand, over bidding may yield a negative utility. Note that overbidding can make a difference only when v i,1 + β(b i,0 ) max{max j i {b j,1 + β(b j,0 )}, r} b i,1 + β(b i,0 ). In this case, the utility of agent i would be non-positive: v i,1 p i = v i,1 + β(b i,0 ) max{max j i {b j,1 + β(b j,0 )}, r} v i,1 + β(b i,0 ) (v i,1 + β(b i,0 )) = 0. Therefore, a weekly dominate strategy of agent i is to be truthful. Proof of Lemma 6 Consider truthful agent i S(x) where x is the initial valuations of agents. We first show that agent i will take the option, d i = 1. Next, we will show given that he takes the option, he will learn his second signal, i.e., d i = e i = 1. First assume that agent i does not take the option; that is, d i = 0. If x i + β(x i ) is less than max{max j / S(x) {x j + β(x j )}, ρ}, he does not have any chance to receive the item. Thus, his utility is zero, which by Corollary 1 is less than or equal to his utility when he takes the option. When x i + β(x i ) max{max j / S(x) {x j + β(x j )}, ρ}, since the reserve price is r = x i + β(x i ), his utility is again zero. Therefore agent i prefers to take the option. We have established that selected agent i takes the option and pays t i, namely d i = 1. We now show that he would rather incur cost c i and update his valuation; that is, e i = 1. Let Y be the

40 40 Operations Research 00(0), pp , c 0000 INFORMS random variable corresponding to max { { max j S(x),j i xj +δ j +β(x j ) } {, max j / S(x) xj +β(x j ) } }, r. Then, when agent i does not update his valuation, his utility is given by E [ max { x i + β(x i ) Y, 0 }] t i = E [ max{y, x i + β(x i )} Y ] t i, (15) where the expectation is with respect to the second signals of selected agents except for agent i. By definition, the weighted surplus under set S(x)\{i}, i.e., Ω(x, S(x)\{i}), can be written as [ { }] E max max j S(x),j i {x j + β(x j ) + δ j }, max j / S(x) {x j + β(x j )}, x i + β(x i ), ρ c j S(x)\{i} j where the expectations are with respect to second signals. The parameter ρ in the above equation can be replaced by r because r is less than or equal to max j / S(x) {x j + β(x j )} if max j / S(x) {x j + β(x j )} ρ and it is equal to ρ otherwise. By the above equation and definition of variable Y, Ω(x, S(x)\{i}) can be written as E [max{y, x i + β(x i )}] c j S(x)\{i} j, which implies that the utility in Eq. (15) is given by Ω(x, S(x)\{i}) + c j S(x)\{i} j E[Y ] t i. Similarly, when agent i purchases the additional information, his utility is equal to Ω(x, S(x)) + c j S(x)\{i} j E[Y ] t i. Considering the fact that the SWSP mechanism maximizes the weighted surplus, i.e., Ω(x, S(x)) Ω(x, S(x)\{i}), we conclude that agent i prefers to learn his second signal. Proof of Lemma 7 We first show that for any x i [v, v] n 1 and any 1 i n, E [q i v i,0 = x i, v i,0 = x i ] is non-decreasing function of x i. Observe that the weighted surplus is maximum of affine functions of v i,0 + β(v i,0 ). Thus, it is convex function of v i,0 + β(v i,0 ). Furthermore, the weighted surplus is continuos function of v i,0 + β(v i,0 ) and its derivative with respect to v i,0 + β(v i,0 ) at v i,0 = x i, if exists, is equal to E [q i v i,0 = x i, v i,0 = x i ]. 20 This implies that E [q i v i,0 = x i, v i,0 = x i ] is non-decreasing function of v i,0 + β(v i,0 ). Finally, considering the fact β is non-decreasing, we can conclude that E [q i v i,0 = x i,0, v i,0 = x i ] is non-decreasing function of v i,0. Next we show that the utility of an agent i follows from Eq. (7) when all agents follow the truthful strategy. We consider the following cases: i) i S(x): By lemmas 5 and 6, selected agent i learns his second signal and reports it truthfully in the second round. Thus, his utility is given by E [ ] q i (x i + δ i ) p i t i c i v 0 = x 0. The claim follows from plugging t i from Eq. (4). 20 To see that note E [q i v i,0 = x i, v i,0 = x i] is equal to Pr[v i,1 + β(x i) max{max j S(x) {x j + δ j + β(x j,0)}, max j / S(x) {x j + β(x j,0)}, ρ}], where v i,1 = x i + δ i if i S(x) and it is x i otherwise.

41 Operations Research 00(0), pp , c 0000 INFORMS 41 ii) i / S(x) and x i + β(x i ) < max { max j / S(x),j i {x j + β(x j )}, ρ } : In this case, the utility of agent i and his allocation probability is 0. By the fact that E [ q i v i,0 = z, v i,0 = x i ] is an increasing function of z and E [ q i v i,0 = x i, v i,0 = x i ] = 0, we can write the utility of agent i as xi v E [ q i v i,0 = z, v i,0 = x i ] dz = 0. iii) i / S(x) and x i + β(x i ) max { max j / S(x),j i {x j + β(x j )}, ρ } : In this case, since in the second round agents participate in a weighted second price auction with reserve price r ρ, the utility [ ( { { of the agent is equal to U i (x i, x i ) = E q i x i + β(x i ) max max j S(x) xj + δ j + β(x j ) } })], r, where U i (x i, x i ) is defined in Eq. (8). Let Y = max j S(x) {x j + δ j + β(x j )} and let H be the distribution of Y. Then, U i (x i, x i ) can be written as (xi E[ + β(x i ) Y ) 1 { x i + β(x i ) Y r } + ( (x i + β(x i )) r ) { }] 1 x i + β(x i ) r Y = (x i + β(x i )) ( H(x i + β(x i )) H(r) ) x i +β(x i ) zdh(z) + (x r i + β(x i ) r)h(r) = ( x i + β(x i ) ) H(x i + β(x i )) rh(r) x i +β(x i ) zdh(z) = x i +β(x i ) H(z)dz, r r where in the first equation, the expectation is with respect to second signals. The last equality is followed from the integration by part. Therefore, using Eq. (5), we get U i (x i, x i ) = xi +β(x i ) Proof of Lemma 8 r [ { Pr z max xj + δ j + β(x j ) }] xi ] vi,0 dz = E [q i = z, v i,0 = x i dz. j S(x) v Throughout the proof, all the expectations are with respect to the second signals. Consider an untruthful agent i with initial valuation x i who bids ˆx i in the first round. We establish an upper bound on his utility. We consider the following two cases, d i = 1 and d i = 0. d i = 1: When agent i takes the option, d i = 1, he can either obtain the additional information or not. In both cases, by Lemma 5 his utility can be written as ] v0 U i (x i, x i ) = E [(x i + e i δ i ) p i t i e i c i = x, b i,0 = ˆx i, b i,0 = x i = E [max {x i + β(ˆx i ) + e i δ i ω i, 0} t i e i c i ], where ω i is defined in Eq. (9). 21 Note that for abbreviation, we omit the condition in the second equation and in the rest of the proof. Then, by adding and subtracting E [max {ˆx i + β(ˆx i ) + e i δ i ω i, 0}], the utility can be rewritten as [ U i (x i, x i ) = E max {ˆx i + β(ˆx i ) + e i δ i ω i, 0} t i e i c i (16) 21 Since { ρ r max{ρ, max j S(ˆxi,x i ){x j +β(x j)}}, agent i receives the item } if x i +β(ˆx i)+e iδ i is greater than ω i = max max j S(ˆxi,x i ),j i{x j + β(x j) + δ j}, max j / S(ˆxi,x i ),j i{x j + β(x j)}, ρ. That is, we do not need to consider the reserve price r in ω i.

42 42 Operations Research 00(0), pp , c 0000 INFORMS ( )] max {ˆx i + β(ˆx i ) + e i δ i ω i, 0} max {x i + β(ˆx i ) + e i δ i ω i, 0} [ ] xi = E max {ˆx i + β(ˆx i ) + e i δ i ω i, 0} t i e i c i Pr [z + β(ˆx i ) + e i δ i ω i ] dz When e i = 1 the first term in the last line is U i (ˆx i, ˆx i ). Otherwise, it is the utility of selected agent i with initial ˆx i who bids truthfully, gets selected, takes the option, but does not learn his second signal, which is by Lemma 6 is less than or equal to U i (ˆx i, ˆx i ). Thus, the utility is at most U i (ˆx i, ˆx i ) + x i ˆx i d i = 0: Pr [z + β(ˆx i ) + e i δ i ω i ] dz, which is the desired result. Note that d i = 0 whether agent i is not selected or he is selected, but he does not take the option. In both cases, if ˆx i + β(ˆx i ) < max { max j / S(ˆxi,x i ){x j + β(x j )}, ρ }, his utility is zero. If not, the utility of agent i given that he stays in the game can be written as [ ( E q i x i + β(ˆx i ) max { ω i, r })]. By adding and substracting E [q i ˆx i ], and by the fact that agent i receives the item if ˆx i + β(ˆx i ) is greater than ω i, we have E [ max { (ˆx i + β(ˆx i ) max { ω i, r }, 0 } + (x i ˆx i ) 1 {ˆx i + β(ˆx i ) ω i }]. If agent i is granted the option and rejects it, r = ˆx i + β(ˆx i ), which implies that the first term is simply zero. Otherwise, the first term is U i (ˆx i, ˆx i ). Since U i (ˆx i, ˆx i ) 0 and agent i can exit the game if his utility gets negative, he can at most yield max{u i (ˆx i, ˆx i ) + x i ˆx i Pr {ˆx i + β(ˆx i ) ω i } dz, 0}, which is less than max{u i (ˆx i, ˆx i ) + x i ˆx i Pr { z + β(ˆx i ) ω i } dz, 0}. C.3. Proofs from Appendix A.1 Proof of Lemma 10 Claim 1: For any set S {1, 2,..., n} We establish the following two claims. Ω(x i, x i, S) Ω(ˆx i, x i, S) = xi ˆx i ˆx i [ ] (1 + β vi,0 (z))e q i = z, v i,0 = x i, S dz. Claim 2: Suppose Assumption 1 holds. Then weighted surplus is an absolutely continuous and convex function of v i,0 + β(v i,0 ), and it is given by Ω(x) = Ω(ˆx i, x i ) + x i ˆx i (1 + β (z)) E [q i vi,0 = z, v i,0 = x i ] dz. The proof of claims follows from Theorems 1 and 2 in Milgrom and Segal (2002). So, we do [ ] vi,0 not repeat it here. By Claims 1 and 2 and the fact that E q i = z, v i,0 = x i, Ŝ(z, x i ) = ] vi,0 E [q i = z, v i,0 = x i for z < y 1 and z > y 2, we have y1 ( Ω(x, Ŝ(x)) = Ω(ˆx i, x i, Ŝ(ˆx i, x i )) β (z) ) ] vi,0 E [q i = z, v i,0 = x i dz ˆx i

43 Operations Research 00(0), pp , c 0000 INFORMS 43 y2 + + y 1 xi y 2 ( 1 + β (z) ) [ ] vi,0 E q i = z, v i,0 = x i, Ŝ(z, x i ) dz ( 1 + β (z) ) ] vi,0 E [q i = z, v i,0 = x i dz. Then, the result follows from Claim 2 and the facts that, by construction, Ω(ˆx i, x i ) = Ω(ˆx i, x i, Ŝ(ˆx i, x i )). Appendix D: Numerical Experiments: Appendix to Section Here, we compare the opt, eff and ssp (with revenue-maximizing ρ) when the first and second signals are uniformly distributed in [0, 1] and [ 0.5, 0.5], respectively. That is, v i,0 U(0, 1) and δ i U( 0.5, 0, 5). First and second signals are independent, and all signals are independent across i. Furthermore, the cost of additional information is 0.05 for all agents. In all figures except for Figure 7d, n = 2. Figure 7a depicts revenue of the ssp mechanism versus ρ with n = 2. Observe that the ssp mechanism yields its maximum revenue at ρ = 0.5. Furthermore, at revenue-maximizing ρ, it obtains more than 97% of the optimal revenue. The impacts of the cost on revenue and average number of selected agents are shown in Figures 7b and 7c. Since revenue-maximizing ρ is approximately 0.5 for all values of cost, we do not show it here. As expected, revenue and average number of selected agents are decreasing functions of the cost. Figure 7d shows that in all considered mechanisms, revenue increases as more agents participate in the auction. Figure 8 illustrates the number of selected agents, for the eff, opt, and ssp (with ρ = 0.5) mechanisms for different values of v 1,0 [0, 1] and v 2,0 [0, 1]. As we also saw in Section 5.6, both agents are selected when their valuations are close.

44 44 Operations Research 00(0), pp , c 0000 INFORMS Revenue Average Number of Selected Agents SSP OPT (a) n=2, =0.5, unifrom, = Cost (c) SSP OPT EFF Revenue Revenue n=2, =0.5, uniform, =0.5 SSP OPT EFF OPT C EFF C Cost SSP OPT EFF OPT C EFF C (b) n Figure 7 a- Revenue versus ρ, b- Revenue versus cost, c- Average number of selected agents versus cost, and d- Revenue versus n with F = U(0, 1) and G i = U( 0.5, 0.5). (d) (a) Optimal mechanism (b) Efficient mechanism (c) SSP mechanism with ρ = 0.5 Figure 8 Number of selected agents, for different realizations of v 1,0 and v 2,0 with n = 2, c = 0.05, F = U(0, 1), and G i = U( 0.5, 0.5).