Sapienza University of Rome

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1 Sapienza University of Rome Ph.D. program in Engineering in Computer Science XXVII Cycle/ Approximation Algorithms in Mechanism Design: an application to sponsored search auctions Riccardo Colini-Baldeschi

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3 Sapienza University of Rome Ph.D. program in Engineering in Computer Science XXVII Cycle/ Riccardo Colini-Baldeschi Approximation Algorithms in Mechanism Design: an application to sponsored search auctions Thesis Committee Prof. Stefano Leonardi (Advisor) Prof. Roberto Baldoni (Co-advisor) Reviewers Prof. Elias Koutsoupias Prof. Guido Schäfer

4 Author s address: Riccardo Colini-Baldeschi Dipartimento di Ingegneria Informatica, Automatica e Gestionale (DIAG) Sapienza Università di Roma Via Ariosto 25, Roma, Italy colini@dis.uniroma1.it www: dottoratoii/ students/riccardo-colini-baldeschi

5 Contents I Overview and Preliminaries 1 1 Introduction Approximation algorithms and Mechanism design A case of study: sponsored search auctions Auction with budget constraints Social efficiency in Auctions with budgets Revenue maximization in Auctions with budgets Double Auctions Main inquiry of this thesis Structure of the thesis Technical Preliminaries Basic notation Basics of Mechanism Design Solution concepts Revelation principle Mechanism in Bayesian model Quasi-linear model Desirable properties Social welfare Auctioneer revenue Sponsored-search environment VCG auction Example of VCG auction VCG with budgets Ausubel clinching auction Example of Ausubel ascending auction II Social-efficiency in auctions with budgets 29 3 Preliminary: Budgets and social-efficiency Keywords and slots in sponsored search auctions Results overview From social-welfare to Pareto-optimality Pareto optimality and other desirable properties i

6 3.3 Set the boundary Impossibility result for diminishing marginal valuations Identical keywords with many copies and different qualities Outline The Setting Pareto-optimality characterization Deterministic clinching auction for the divisible case Randomized clinching auction for the indivisible case Pareto-optimality in expectation Non-identical keywords with identical copies Outline The Setting Pareto-optimality characterization Deterministic clinching auction for matching-markets with budgets.. 64 III Revenue Maximizing Envy-free auctions with budgets 73 6 Preliminary: Envy-freeness and fixed-price Envy-freeness vs Incentive-compatibility Discriminatory vs fair prices Basic notation Fixed-price auctions Multi-good Auctions for Unit-demand Bidders Multi-unit Auctions with Budgets If many items are allocated by the optimum If few items are allocated by the optimum Multi-good Auctions with Budgets and Matching Preferences Hardness result An FPTAS for a Constant Number of Types of Goods A c-approximation for a Constant Number of Types of Goods Non-fixed price, non discriminating auctions Outline Limits of fixed-price auctions Multi-good Auctions with Budgets and Matching Preferences in flea market VL allocating procedure NVL allocating procedure Envy-freeness Approximation Multi-good Auctions with Budgets and Matching Preferences in luxury market

7 IV Double Auctions Preliminaries Environment description Bilateral trade Bilateral-trade Previous results Improved bounds Auctions in two-sided markets Extending Sequential Posted Price Mechanisms to Two-Sided Markets Two-sided Sequential Posted Price Mechanisms with Matroid Constraints Two-sided Fixed Price Mechanisms V Conclusions Future directions and Open problems 157 VI Appendices 159 A Complementary notions in Mechanism Design 161 A.1 Incentive-compatibility characterization in single-parameter domain. 161 A.2 Revenue equivalence theorem B Fiat et al. auction 163 C b-matching 165 Bibliography 169

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9 Part I Overview and Preliminaries 1

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11 Chapter 1 Introduction 1.1 Approximation algorithms and Mechanism design In the last two decades many computer scientists focus on mechanism design problems. Mechanism design gives a theory on how design rules of interactions - when players act strategically and selfishly - that obtain good outcomes. A good outcome is with respect to some objective function defined by the mechanism designer. Natural objective functions are the maximization of the participants utilities - social efficiency - or the maximization of the sum of the payments given to the mechanism - the revenue (or profit). This field was mainly investigated by the economists and a huge literature exists in what is sometimes called classical mechanism design. What brings the attention of the computer science community into mechanism design problems is the Internet growth and the increasing number of Internet applications. The design of applications that was previously interpreted as a pure algorithmic problem, now has to deal with selfish agents that want to maximize their own objective functions. Thus, an algorithm to behave properly has to consider these strategic behaviours and a way to steer them. With an algorithmic approach to mechanism design we start to be concern about computational issues. The new challenge is to obtain mechanisms that retain some game-theoretic properties and are tractable. This field is sometimes referred as computational mechanism design. If for any reasons we cannot optimize over the objective function, we can go for a near-optimal solution. Informally speaking, we say that a mechanism (or an algorithm) is a β-approximation with respect to a optimal solution OP T if the solution provided by our mechanism (or algorithm) is at least OP T β. 1.2 A case of study: sponsored search auctions The research on computational mechanism design was mainly driven by sponsored search auctions. Sponsored search auctions deal with the problem of allocate ad slots available in a web page to the advertisers. Usually, a web page has a certain number 3

12 4 CHAPTER 1. INTRODUCTION of slots - say k - and each slot has a click-through rate 1 value (CTR). The goal for each advertiser is to put his ad on the slot that is more effective for his campaign - the slot with the highest CTR. Advertisement has a fundamental role in the web since the first GUI browser was released in At that time ads were sold on a per-impression basis 2 and the advertisers were charged by a flat fee usually for a big stock of ads impressions. So, this result in a market that is accessible only to the big companies that are able to afford big payments for their advertisement campaigns. Later in 1997, Overture 3 introduced a new way to deal with online advertisement that made the market more flexible and gave to everyone the opportunity to participate. In the Overture s auction every advertiser (or bidders) submits a bid that represents his willingness to pay for a click on his ads and the keywords of interest for his campaign. Then, when an user submits a query to the search engine, the mechanism selects the top bids for the searched keywords and allocates to every winning advertiser a slot in the page of interest (higher is the bid better is the obtained slot). The auction from Overture is known as Generalized first-price (GFP) auction because to each winning bidder is charged an amount equal to the submitted bid. Example 1. Assume there are two slots A and B with a CTR values α A = 100 and α B = 50 respectively. Three bidders compete for this slots and have a per-click values v 1 = $5, v 2 = $2, and v 3 = $1. Then, the GFP auction allocates the first slot to the bidder 1 that pays 100 $5 = $500 and the second slot to bidder 2 that pays 50 $2 = $100. Since a bidder s payment depends directly from the bidder s submitted bid and everyone wants to pay as less as possible, the bidders will try to strategize on their reported values. In the previous example if bidder 1 lowers his bid to $2.1, he still wins the best slot but he pays much less. Thus, the GFP auction is not stable and is undesirable for both users and search-engine. To address this problem a different mechanism was introduced by Google, the Generalized second-price (GSP) auction. The difference between GSP and GFP is not in the selection of the winning bidder - also in GSP the same top bids are selected - but in the payments. To avoid a direct correlation between the bid and the payment of a bidder, each bidder pays an amount equal to the bid 4 of the advertiser ranked below him. Example 2. Using the same bidders and slots values as in the previous example, the GSP auction still allocates the slots in the same way. But now, each bidder pays the bid of the bidder ranked immediately next to him. That is bidder 1 pays 100 $2 = $200 and bidder 2 pays 50 $1 = $50. Notice that even in the GSP auction the bidders may have incentive to lie. A bidder may prefer to lower his value and obtain a worse slot at a low price than 1 We refer to slots CTRs to indicate how the position of the slot affects the click received by a given advertisement. CTR values are also used to indicate the quality of an ad, i.e. how many users are interested in that ad. 2 Advertiser pays every time the ads are shown on the page of interest. 3 Overture was a sponsored search internet advertising provider acquired by Yahoo! in Now the service is named Yahoo! Search Marketing. 4 Actually, the amount that a bidder pays is sometimes related also to the quality of the ads.

13 1.2. A CASE OF STUDY: SPONSORED SEARCH AUCTIONS 5 Figure 1.1: Screen shot from TV ads targeting interface from [Nisan et al., 2009]. declare an higher value and obtain a good slot paying much more. Nonetheless, the GSP auction guarantees good stable allocations - bidding cycles are avoided - and nice revenue properties (see [Edelman et al., 2005, Varian, 2007]). Thus, GSP becomes a golden standard in sponsored search auctions and it is the key monetization tool for many big firms (Google, Yahoo!, Microsoft, Facebook). According to the Google s financial reports 5, the total advertising revenues for Google reaches $59.6 billion dollars in What is missing in the GSP auction is the notion of budgets. Theoretical analysis of GSP consider bidders without financial constraints that compete for a single keyword. Recently, a different example of online advertisement that involves budgets has been offered in [Nisan et al., 2009]. They describe the Google s auction for TV advertising. A set of time slots for advertisement are available on different networks. Each bidder has to choose a subset of networks of interest and some preferences about dayparts, programs and target audience (see Figure 1.1). Then the bidder has to submit, among other parameters, a valuation (indicated as value per thousand of views) and a maximum budget for the daily campaign (see Figure 1.2). In the article the authors point out how the theoretical literature on auction theory has no insight in many practical problems involving budget constraints design. Main difficulties arise in providing an exact characterization of the goals and to understand what is a good trade-off between them. An informal list of desirable goals in sponsored search auctions are: Efficiency: an auction should try to maximize the utility of the participants (including the auctioneer), Revenue: the auctioneer should aim to maximize his revenue, Fairness: the bidders should not be discriminated (somehow), 5

14 6 CHAPTER 1. INTRODUCTION Figure 1.2: Screen shot from TV ads bidding interface from [Nisan et al., 2009]. Incentive compatibility: the bidders should not report their bids in a strategical way. If a bidder can increase his utility lying on the submitted bid extremely inefficient equilibria may be reached. So, it is clear that research in sponsored search auctions has extremely important practical applications for many Internet companies. On the theoretical side, sponsored search auctions present numerous challenging problems. For example, computational aspects represent a big issue. A huge number of auctions need to be executed daily and millions participants are present in each auction. Even if auction theory and mechanism design are widely studied in classical economics, computational complexity problems were approached only in the last decades. Another important gap that arises between the available theoretical results and the necessary practical requirements is about budget constraints for bidders. In economics auctions with budgeted bidders are studied in different perspectives. A first branch focuses on the analysis of standard auctions - first-price, second-price, and allpay auctions - with budgets [Che and Gale, 1998, Benoit and Krishna, 2001]. A second branch studies optimal (revenue) single-item auction [Laffont and Robert, 1996, Pai and Vohra, 2014]. The third branch addresses the problem of maximize the (expected) social efficiency in Bayesian setting and with a Bayesian incentive-compatible constraint [Maskin, 2000]. Auctions in which budgeted, non-unit-demand bidders compete for many identical goods in a non-bayesian setting (multi-unit auctions) have been addressed from computer science community. The goal is to design mechanisms that approximate the optimal revenue or the social efficiency under the incentive compatible constraint [Borgs et al., 2005, Abrams, 2006, Dobzinski et al., 2012, Goel et al., 2012, Colini- Baldeschi et al., 2012, Dobzinski and Leme, 2014]. The design of social efficiency auctions with incentive compatible constraints in (generalized) multi-unit auctions is one of the research directions addressed by this thesis. On the revenue optimization side we consider a different setting. The incentive-compatibility constraint is

15 1.3. AUCTION WITH BUDGET CONSTRAINTS 7 dropped and a pure algorithmic approach is adopted 6. Then we focus on computational complexity problems and provide positive (approximation algorithms) and negative (hardness proofs) results in different settings. Finally, a challenging environment is offered by the double auctions. In double auction there are two set of players: the buyers - that want some items - and the sellers - that own some items for sale. This setting has a variety of applications, just to stay in the realm of online advertisement let us consider Google AdSense and DoubleClick AdExchange. There we have a set of advertisers (buyers) that want their ads displayed on web pages and a set of webmasters (sellers) that give web pages slots for sale. Using this tools the buyers can choose to appear not only on the Google s result pages but also on many web pages - relevant for their campaigns - spread around Internet. To give some numbers: over the $59.6 billions obtained by online advertisement, $14.5 are obtained through double auctions. 1.3 Auction with budget constraints So far we have considered sponsored search auctions as a motivating example for many problems that we address in this thesis. Now, it is worthy to consider other settings in which the bidders may be financial constrained and how the literature deals with that. Auctions are a widely used tool to sell a large variety of goods. Works of art, extraction rights, online advertisements, timber harvesting, and Personal Communication Services (PCS) are few examples of how auctions work in the real world. Moreover, in many part of the world privatize socially held assets is done using auctions. Some examples cited in [Benoit and Krishna, 2001] are sale of industrial enterprises in the Soviet Union and Eastern Europe, public transportation system in Britain and Scandinavia and radio spectra frequencies in the United States. For their practical and theoretical interests, auctions have been deeply studied in economic theory but the focus was mainly set to an environment with bidders without financial constraints. Financial constraints for the auctions participants seems a fundamental property. As Pai and Vohra wrote in [Pai and Vohra, 2014] : Not every potential buyer of a David painting who values it at million dollars has access to a million dollar to make the bid. Indeed in many situations arose that bidders had not enough liquidity availability to afford some of the auctioned goods. An easy example is provided in government auctions. Here sale prices are often high compared to the bidders liquid assets, thus the bidders need to have access to credit (external financing). An example of how financial constraints impact on bidding strategies in real auction is provided by David Salant ([Salant, 1997] p.567). During one of the FCC auctions for the spectrum licenses sale, he was in the GTE bidding team and reporting his experience he says: We were very concerned about how budget constraints could affect bidding. Most of the theoretical literature ignores budget constraints. In the MTA [Major Trading Area] auction, budget constraints appeared to limit bids. 6 In the subsequent chapters this choice will be extensively motivated.

16 8 CHAPTER 1. INTRODUCTION But many reasons could justify a financial constraint: a bidder may be conditioned from moral hazard problems, subtracting resource from other productive activities could result in a rising opportunity cost, or again a financial constraint could be imposed externally to the participants (think about the salary caps in the professional sports leagues in the United States). Looking at [Salant, 1997] we also notice that budget constraint often arise as an endogenous constraint. Indeed seems that it is a kind of risk-control parameter for the bidders. Specifically, we want to emphasize how in the FCC auction the bidding strategy of the GTE used budget with this philosophy ([Salant, 1997] p.553): budget parameters determined by GTE s management were in part based on these valuations. Thus, in order to study auction with financial constrained bidders, we distinguished two parameters for each bidder: the valuation of an item that represents the willingness to pay and the budget that represents the ability to pay. Notice that separating the ability to pay and the willingness to pay have highlighted new questions about the notion of social efficiency. For example in the context of single unit auctions without budgets, the social efficiency is not an issue since the item is always assigned to the bidder that values it most. Now, the item is not necessarily assigned to the bidder with the highest valuation but could be assigned to the bidder with the greater financial asset. At first glance, a budget can be viewed as an upper bound on a bidder s bid, but one important insight that arises from [Che and Gale, 1998, Che and Gale, 2000] is that a bidder with a valuation v and a budget b is not the same as a bidder with valuation min{v, b}. We cite an example from [Che and Gale, 1998, Che and Gale, 2000] to have a better understanding of the previous claim. Assume there are two buyers that compete for an item. Both bidders have a valuation drawn uniformly from [0, 1] and a budget equal to 2. Notice that the budgets are not a constraint for the buyers. Thus, from standard economic literature results we know that both first-price auction and second-price auction obtain an expected revenue of 1/3. Now, assume that the two buyers have a valuation equal to 2 and a budget drawn uniformly at random from [0, 1]. Even if the minimum of the two parameters is the same, now the equilibrium strategy for the buyers changes. Since for the buyers is an equilibrium to bid their budgets in both first-price and second-price auctions, the expected revenue are 2/3 and 1/3 respectively. The same result can be obtained for the single-buyer case. Moreover, it is interesting to notice that without binding constraints offering a menu of lotteries does not pay [Riley and Zeckhauser, 1983]. Instead, with binding constraints the optimal mechanism typically involves a menu of lotteries [Che and Gale, 2000]. The critical role of budgets is again demonstrated when we note that the revenue equivalence theorem 7 does not hold any more [Pitchik and Schotter, 1986, Che, 1996, 7 For brief discussion on the revenue equivalence theorem refer to Appendix A.2.

17 1.4. SOCIAL EFFICIENCY IN AUCTIONS WITH BUDGETS 9 Che and Gale, 1998, Milgrom, 2004]. Notice also that budget constraints create a new way to strategize for the bidders. In particular it is noted that many auctions behave in unexpected way when bidders can declare a budget ([Brusco and Lopomo, 2008, Benoit and Krishna, 2001]). For example, imposing a reservation price on the items result in excluding some bidders from an auction. When bidders are not constrained by budgets, measures that exclude some participants reduce the social surplus, because some gain from trade could not be realized. Instead, with budgeted bidders a reserve price could help to rule out bidders with a small budget that can induce dangerous strategic behaviours. Thus, a reserve price excluding low-budgeted bidders increase the competition. Moreover, even if budgets are public knowledge (thus bidders cannot strategize on that) could be convenient for a bidder to bid aggressively on an item in order to deplete the budget of a competitor. This kind of behaviour is confirmed in ([Salant, 1997] p. 561) that describe how it was convenient to bid strategically on licenses of secondary importance in order to make rivals spend more on some markets, leaving them with less to spend in other markets. Notice that this comes into play only when multiple items are auctioned. Now that we have analyzed how the budget constraint is interpreted and considered, we can analyze how the classical goal of social efficiency and revenue change in a financial constrained setting. 1.4 Social efficiency in Auctions with budgets Social efficiency is a standard requirement in mechanism design. In a world without budgets (unconstrained buyers), it is easy to see that an outcome is social efficient if the goods are allocated to the buyers that value them most, i.e. maximize the social-welfare ([Krishna, 2009, Klemperer, 1999, Vickrey, 1961]). On the other hand, a world with budgeted bidders changes this paradigm of social efficiency. This is mainly due to a different concept of utility for the buyers. Until a buyer is unconstrained his utility is measured as the difference between the valuation for the obtained set of goods and the payment (the quasi-linear model). But now we have one more parameter to consider. As we saw in the previous section, we cannot use a modified valuation (the minimum between the valuation and the budget) to preserve a single parameter setting. Thus we need to switch to a different utility model. A standard model when buyers have budgets is the non-quasilinear model ([Dobzinski et al., 2012]). Here the utility of a buyer is the difference between the valuation for the obtained set of goods and the payment until the payment is not greater than the budget and minus infinity otherwise. With the non-quasilinear model the notion of social efficiency that seemed natural and well defined becomes a little ambiguous. We will discuss the underlying subtleties of the non-quasilinear model in Chapter 3. However, we need a new way to understand if an outcome is socially efficient or not, thus we will focus on Pareto-optimal outcomes ([Dobzinski et al., 2012, Fiat et al., 2011, Colini-Baldeschi et al., 2012, Goel et al., 2012]).

18 10 CHAPTER 1. INTRODUCTION An outcome is said to be Pareto-optimal if there is no alternative allocation that makes at least one player strictly better off without making any other player worse off. Thus, in Part II we will discuss mechanisms that are Pareto-optimal and incentivecompatible in a wide variety of settings. 1.5 Revenue maximization in Auctions with budgets A different common goal for auctions is the maximization of the revenue of the auctioneer. Revenue maximization is a standard property and the existing literature widely focus on that ([Krishna, 2009, Klemperer, 1999]). In particular, [Myerson, 1981] represents a cornerstone in optimal (revenue) mechanism design. Notice that classical economical results assume that the participants valuations are drawn from common known distributions. Instead, we are interested in a worst-case approach to design and analyse optimal (and near-optimal) algorithm for any possible participants valuations. But bad news come when bidders are budget constrained. In this setting the auction proposed in [Dobzinski et al., 2012] is proved to be the only possible incentivecompatible and Pareto-optimal auction. Thus any auction that satisfies these two requirements produce the same allocation and the same prices. But the prices computed by [Dobzinski et al., 2012] are not socially acceptable (and thus not implementable) for many reasons as pointed out in [Feldman et al., 2012]. The main drawback is the discrimination that the prices make among the participants. Specifically, different participants can obtain the same items at very different prices and this is forbidden by many law institutions. Therefore, in order to avoid price discrimination behaviour we need to drop the incentive-compatibility and move to envy-free, non-incentive-compatible auctions setting. Envy-free allocations were defined in [Foley, 1967] and [Varian, 1974]. A key property of such allocations is that no one envies anyone else. Informally, no agent wants to switch his allocation with another agent. In recent literature the more restricted version of envy-free pricing has been studied for unit-demand bidders [Guruswami et al., 2005, Demaine et al., 2008, Briest and Krysta, 2006, Cheung and Swamy, 2008, Balcan et al., 2008, Chalermsook et al., 2012]. Here, identical items must be given per item prices, and the allocation to an agent must be a set of (identical or not) items that maximize the agent utility, given these item prices. In these item pricing settings it was possible to show logarithmic approximations for general prices [Guruswami et al., 2005] or fixed-prices [Balcan et al., 2008], which are accompanied by log 1 ɛ (n+m) approximation hardness [Chalermsook et al., 2012]. In this setting envy-freeness can be seen as a generalization of Walrasian equilibrium. The main difference is that for envy-free allocations we do not insist that the market clears. In the context of auctions this means that there may be unsold items. Another difference that is not present in all works is that one is allowed to bundle the items [Fiat and Wingarten, 2009, Feldman et al., 2012]. In this case, agents are offered bundles of items each at a specific price and choose the best bundle for

19 1.6. DOUBLE AUCTIONS 11 them. In many of the problems, one distinguishes between the limited supply [Guruswami et al., 2005, Cheung and Swamy, 2008] setting (generally more difficult) and the unlimited supply [Guruswami et al., 2005, Balcan et al., 2008, Demaine et al., 2008]. In a quasi-linear world, where agents do not have a budget constraint, an allocation is envy-free (i.e., prices can be assigned that make it envy-free), if and only if it is locally efficient; i.e., social welfare cannot be increased by permuting the bundles amongst the agents. On the other hand, the envy-free auction in the case of budgets was considered only in [Feldman et al., 2012], where the multi-unit case was studied. Note that for the multi-unit auctions without budget constraints or demand constraints (the standard quasi-linear model), it is trivial to give an envy-free allocation: give all m items to some agent with maximal valuation, say v i, set the price of these m items to be mv i. It is easy to see that this is also the outcome that maximizes both social welfare and revenue. The situation becomes significantly more difficult if agents have budget constraints since it might be possible to allocate only a small subset of the goods and therefore to obtain only a very limited revenue. Unfortunately, the envy-free, revenue-maximization problem with budgets cannot be optimally solved ([Feldman et al., 2012]). In the third part of this thesis we focus on design near-optimal results for revenue maximizing, envy-free auctions with budgeted bidders. The problem is studied in various settings, various participants constraints and different pricing schema. The exposition is completed with hardness results for many of these settings. 1.6 Double Auctions As opposed to one-sided markets, which were extensively studied by numerous economists (and since more than a decade, by computer scientists as well), two-sided markets did not have the same spread. Most of recent work in algorithmic mechanism design has indeed concentrated on one-sided markets having the buyers as the only agents playing in the market. In two-sided markets, both buyers and sellers play the role of strategic agents. Two-sided markets naturally arise in selling display-ads in web advertisement, the New York Stock Exchange (NYSE), the US FCC spectrum license reallocation, and many other markets with multiple buyers and sellers. For example, ad exchange platforms for selling display-ads face asymmetric information regarding both the valuations of buyers - the value per advertiser s impression shown - as well as about the reservation prices of sellers - the profit that the publisher could obtain by sending the pageviews to competing ad exchanges. An ideal goal in market design is to find individually rational, incentive compatible mechanisms that maximize the social welfare of all agents in the market. In two-sided markets, a further important requirement is strong budget-balance (SBB), which states the payments of the buyers must entirely and exclusively be transferred to the sellers, i.e., the buyers and the sellers are allowed to trade without leaving to the mechanism any share of the payments, and without the mechanism adding money into the market. For example, in ad exchange auctions, the intermediation profit of the broker must be limited to a fixed percentage of the revenue of the

20 12 CHAPTER 1. INTRODUCTION publishers. This is an important feature of the market, as otherwise it will be perceived as unfair if the mechanism keeps an additional arbitrary share of the payments charged to the advertisers. Though SBB is a most desired requirement in many applications, it is hard to achieve in practice since it imposes a constraint on the payments that is difficult to satisfy. A weaker version of SBB often considered in literature is weak budget balance (WBB), which only requires that the mechanism does not inject money into the market. In this thesis we study a standard and simple model of a two-sided market. In its basic form, there is a single type of items for sale. The buyers want to acquire a single unit of this item, and the sellers have a single unit to sell. So, to each buyer and seller there is a single number associated, called her valuation, which describes how much a buyer or seller values having an item in possession. The valuations of the buyers and sellers are drawn from independent, possibly distinct distributions. Mechanisms that work in such a setting are referred to as double auctions. 8 Unfortunately, Myerson and Satterthwaite [Myerson and Satterthwaite, 1983] proved that it is impossible for an invidually-rational, Bayesian incentive-compatibility (BIC), 9 and WBB mechanism to maximize social welfare in such a market. This result implies that an individually-rational, BIC, social welfare maximizing double auction necessarily subsidizes the market. Since then, much of the literature on double auctions [McAfee, 1992, Satterthwaite and Williams, 1989, Satterthwaite and Williams, 2002] has focused on trading off social welfare, incentive compatibility and budget balance. 1.7 Main inquiry of this thesis As you have already seen in the previous sections, this thesis focuses on theoretical problems that can be applied in the sponsored search auctions environments. We can distinguish three main research directions: social efficiency in auctions with budgets, revenue maximization in envy-free auction with budgets, and double auctions. The study of social efficiency in auctions with budgets was inspired by [Dobzinski et al., 2012, Fiat et al., 2011]. [Dobzinski et al., 2012] studied multi-unit auctions with budgeted bidders. They prove that if budgets are private knowledge no incentive compatible, Pareto-optimal auction can be obtained. They also present an auction that is individually-rational, Pareto-optimal, and incentive compatible when budgets are public knowledge. Moreover they prove that if we insist on Pareto optimality and incentive compatibility this is the only possible auctions - for both allocations and payments. 8 Some economics literature uses a stricter definition of a double auction where the mechanism may only charge a single common price at which each trading seller/buyer-pair trades. In the present thesis however, we refer to this stricter notion as a fixed price double auction, and instead use the term double auction for any mechanism that works in the two-sided market setting. 9 Bayesian incentive compatibility is a less restrictive form of incentive compatibility. Informally, it only requires that reporting truthfully one s valuation to the mechanism gives an agent the best expected utility conditioned on the assumption that all other agents also report their valuations truthfully (taking into account the valuation distributions of the other agents).

21 1.7. MAIN INQUIRY OF THIS THESIS 13 This work is extended by [Fiat et al., 2011] to the setting of different items. Now, the biddershave to declare an preference set over the items for sale in addition to a valuation and a budget. In this setting they prove that if preference sets are private knowledge no incentive compatible, Pareto-optimal auction exists. Instead, they present an individually-rational, incentive-compatible and Pareto-optimal auction when budgets and preference sets are public knowledge. Part of the work of this thesis is dedicated to extend previous setting to multiple keyword sponsored search auctions with budgets. There, each keyword has multiple ad slots with a click-through rate. The bidders have additive valuations, which are linear in the click-through rates, and budgets, which are restricting their overall payments. Additionally, the number of slots per keyword assigned to a bidder is bounded. We show the following results: We give the first mechanism for multiple keywords with slots, where clickthrough rates differ among slots. Our mechanism is incentive compatible in expectation, individually rational in expectation, and Pareto-optimal. We study the matching market setting, where each bidder is only interested in a subset of the keywords and each keyword has many identical slots. We give an incentive compatible, individually rational, Pareto-optimal, and deterministic mechanism for identical click-through rates. We give an impossibility result for incentive compatible, individually rational, Pareto-optimal, and deterministic mechanisms for bidders with diminishing marginal valuations. This setting is studied in On Multiple Keyword Sponsored Search Auctions with Budgets, a joint work with Monika Henzinger, Stefano Leonardi, and Martin Starnberger. A preliminary version of this work was presented at 39th International Colloquium, ICALP 2012 ([Colini-Baldeschi et al., 2012]). A full version of the paper was recently accepted to The ACM Transactions on Economics and Computation journal ([Colini-Baldeschi et al., 2016b]). The second research line regards revenue maximization in envy-free auctions with budgets. Traditional incentive-compatible auctions [Vickrey, 1961, Dobzinski et al., 2012] for selling multiple goods to unconstrained and budgeted bidders can discriminate between bidders by selling identical goods at different prices. For this reason, Feldman et al. [Feldman et al., 2012] dropped incentive compatibility and turned the attention to revenue maximizing envy-free item-pricing allocations for budgeted bidders. Envy-free allocations were suggested by classical papers [Foley, 1967, Varian, 1974]. The key property of such allocations is that no one envies the allocation and the price charged to anyone else. In recent studies on envy-freeness ([Guruswami et al., 2005] and much subsequent work) a much more restrictive definition of envy-freeness has been adopted. In this paper we consider this classical notion of envy-freeness and study fixed-price mechanisms which use nondiscriminatory uniform prices for all goods. Feldman et al. [Feldman et al., 2012] gave an item-pricing mechanism that obtains 1/2 of the revenue obtained from any envy-free fixed-price mechanism for identical goods. We improve over this result by presenting an FPTAS for the problem that returns an (1 ɛ)-approximation of the revenue obtained by any envy-free fixed-price mechanism for any ɛ > 0 and runs in polynomial time in the

22 14 CHAPTER 1. INTRODUCTION number of bidders n and 1/ɛ even for exponential supply of goods m. Subsequently, we consider the case of budgeted bidders with matching-type preferences on the set of goods, i.e., the valuation of each bidder for each item is either v i or 0. In this more general case, we prove that it is impossible to approximate the optimum revenue within O(min(n, m) 1/2 ɛ ) for any ɛ > 0 unless P = NP. On the positive side, we are able to extend the FPTAS for identical goods to budgeted bidders in the case of constant number of different types of goods. Our FPTAS gives also a constant approximation with respect to the general envy-free auction. These results are presented in joint work with Stefano Leonardi, Piotr Sankowski, and Qiang Zhang entitled Revenue Maximizing Envy-free Fixed-price Auctions with Budgets appeared in The 10th Conference on Web and Internet Economics, WINE 2014 ([Colini-Baldeschi et al., 2014]). We then generalize this setting to the matching martkets case, i.e. each bidder is interested only in a subset of the items. The problem turns out to be much harder for matching markets. In matching markets, agents have a single valuation for any item in the agent s preference set. In the previous work we prove that the optimum revenue in envy-free matching markets cannot be approximated within O(min{n, m} 1 2 ɛ ), for any ɛ > 0, unless P = NP, for n agents and m items. In this work we show how to circumvent this hardness result by restricting to instances with all agents with budget larger than valuation. We present in this case a 4-approximated polynomial time auction that uses multiple prices for the items on sale. Our auction has the important property that each bidder is charged at the minimum of the price of the items in her preference set. It is also shown that using multiple prices it is strictly needed in order to achieve a constant approximation. For the case of agents with budget smaller than valuation, we present a fixed-price envy-free auction for matching markets that achieves an O(log n) approximation of the optimum revenue. This joint work with Stefano Leonardi and Qiang Zhang is not yet published ([Colini-Baldeschi et al., 2015]). Finally we focus on the double auctions in which unit-demand buyers and unitsupply sellers act strategically. An ideal goal in double auction design is to maximize the social welfare of buyers and sellers with individually rational (IR), incentive compatible (IC) and strongly budget-balanced (SBB) mechanisms. The first two properties are standard. SBB requires that the payments charged to the buyers are entirely handed to the sellers. This property is crucial in all the contexts that do not allow the auctioneer retaining a share of buyers payments or subsidizing the market. Unfortunately this goal is known to be unachievable even for the special case of bilateral trade, where there is only one buyer and one seller. Therefore, in subsequent papers, meaningful trade-offs between these requirements have been investigated [McAfee, 2007, McAfee, 1992, Satterthwaite and Williams, 1989, Satterthwaite and Williams, 2002, Dütting et al., 2014]. Our main contribution is the first IR, IC and SBB mechanism that provides an O(1)-approximation to the optimal social welfare. This result holds for any number of buyers and sellers with arbitrary, independent distributions. Moreover, our result continues to hold when there is an additional matroid constraint on the sets of buyers who may get allocated an item. To prove our main result, we devise an extension of sequential posted price mechanisms [Chawla et al., 2010, Kleinberg and Weinberg,

23 1.8. STRUCTURE OF THE THESIS ] to two-sided markets. In addition to this, we improve the best-known approximation bounds for the bilateral trade problem [Blumrosen and Dobzinski, 2014]. These results are presented in a joint work with Bart de Keijzer, Stefano Leonardi, and Stefano Turchetta accepted to ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 entitled Approximately Efficient Double Auctions with Strong Budget Balance ([Colini-Baldeschi et al., 2016a]). 1.8 Structure of the thesis The first two chapters provide an introduction of the model. In this first chapter we give an high-level introduction about these problems and the Chapter 2 will introduce many concepts and definitions in a formal way. Moreover, basic intuitions about two famous auctions (VCG auction and Ausubel s ascending auction) will be provided. The second part of this thesis deals with the problem of maximize the social efficiency when participants have budget constraints. In Chapter 3 we introduce the problem of social efficiency in auctions with budgeted participants. The differences between a setting without budgets and the main limitations of this setting are discussed. In Chapter 4 we study a setting with many divisible goods and budgeted participants. In this setting we give a characterization for a special measure of social efficiency (Pareto-optimality) and then design a mechanism that satisfies it. Moreover, we prove that this mechanism can be used with randomization techniques to produce social efficient allocations in a setting with many indivisible goods. In Chapter 5 we study a setting in which each bidder is interested only in a (possibly different) subset of the auctioned goods. Also in this setting a social efficient mechanism is designed. In the third part of the thesis we focus on a different perspective: we look at the revenue maximization while basics social efficiency properties are guaranteed to the (budgeted) participants. In Chapter 6 the problem of revenue maximization is introduced. In Chapter 7 we present various mechanisms (in different settings) that are guaranteed to obtain at least a constant fraction of the revenue obtained by the optimal mechanism. Moreover, all the presented mechanisms implement a fix-pricing schema. Fix-price mechanisms are important because do not discriminate among the participants with different prices. We also present hardness results in some settings. In Chapter 8 we move beyond fix-price mechanisms. In certain settings we cannot use fix-price schema in order to approximate the optimal revenue. Thus we present mechanism that implements non-discriminatory non-fix-price schema and approximate the optimal revenue in these settings. In the fourth part the two-sided mechanisms are studied. In Chapter 9 we formally introduce this setting and describe a specific instance of one-seller-one-buyer problem named bilateral trade problem. In Chapter 10 we present improved lower and upper bounds for the bilateral-trade problem. In Chapter 11 we present mechanisms for two-sided markets in different settings that have a constant approximation to the optimal social-welfare. Finally, in Chapter 12 the results are summarized and future research directions are presented.

24 16 CHAPTER 1. INTRODUCTION

25 Chapter 2 Technical Preliminaries 2.1 Basic notation Throughout this thesis we will use the following notation. We use R to denote the set of real numbers. R 0 = {x R x 0} represents the set of non-negative real numbers and R >0 = {x R x > 0} represents the set of strictly positive real numbers. Similarly, the set of natural numbers is denoted by N, thus N 0 and N >0 are the set of non-negative natural numbers and the set of strictly positive natural numbers respectively. For a number a N >0, we use [a] to denote the set {1,..., a}. Let x = {x 1,..., x t } be a vector of size t N >0, we use x i with i [t] to denote the vector x without the i-th element. Furthermore, let L be a set, then we use L i where i L as a shorthand for L\{i}. Now, assume that x is a totally-ordered vector or L is a totally-ordered set, then x (i) denotes the i-th highest element of x and l (i) denotes the i-th highest element of L respectively. Moreover, when we need to handle a distribution function with support F R, we use f as the probability density function. Let F (x) := x f(x) dx be the corresponding cumulative distribution functions (cdf). We also define the inverse cumulative distribution function as F 1 (y) = inf{x F F (x) y}. Consequently, the median of the cumulative distribution function F (x) is defined as m := F 1 ( 1 2 ). Let T : F R be a random variable, then we use E F [T ] to denote the expected value of T over F. Moreover, if D = D 1... D n, where D i is a probability distribution, then we use D i where i [n] to denote the probability distribution j [n]:i i D j. 2.2 Basics of Mechanism Design Now, we discuss the basic notions of mechanism design. Notice that the following discussion is specific to the context of auction design that is the topic of this thesis. Mechanism design is a game of private information. There are n players (or bidders or agents) represented by the set I, i.e. I = n. The players compete for a set of goods (or items or keywords). X is the set of possible allocations and an allocation is represented by X. Each player i has a private information (type) θ i 17

26 18 CHAPTER 2. TECHNICAL PRELIMINARIES about the realizations of the set in X. The valuation v i (θ i, X) is the value that bidder i assigned to the allocation X given the private type θ i. A player (called principal or auctioneer) decides the payoff structure of the game. Each player has his own strategy s i, i.e. given his type he decides what he prefers to report to the auctioneer. Therefore, we use s to define the strategy vector of the players, i.e. s = {s 1,..., s n }. The goal is to design a protocol such that it is specified what players can report to the auctioneer and what is done in each case. Formally, Definition 1. A mechanism (A, ρ) for n player is defined by: a players type space Θ = Θ 1... Θ n (private information), a players reported-type space T = T 1... T n, a set of allocations X (each element X X is an n dimensional vector), a valuation function for each player i, v i : Θ i X R, that is how much the player i values the allocation X, an allocation rule A : T X, that given the reported-type of the players returns an allocation, a payment rule ρ : T R n, that given the reported-type of the players returns the payments, an utility function for each player i, u i : Θ i T R, and an utility function of the auctioneer is the sum of the payments by the players, i.e. ρ ρ(t) ρ, where t T. Notice that the outcome of a mechanism is a pair (X t, p t ). In order to shorten the notation, when the vector of reported types is clear from the context, we simply use (X, p). The assignment X X is the output of the allocation rule, i.e. A(t) = X. The structure of X will be given in each specific setting. The payment vector p = {p 1,..., p n } R n is the output of the payment rule, i.e. ρ(t) = p. In each setting p has n components and the i-th component is p i the payment of bidder i Solution concepts Given a mechanism (A, ρ) we are interested in the solution concepts that it can implement. A solution concept Sol is a formal rule that given the set of true types predicts the behaviour of the player in the game, i.e. Sol : Θ T, where T is a set of distributions over T. Now, let us examine the most common solution concepts used in this thesis. Dominant Strategy Solution. If in a mechanism each player has a best strategy regardless of what other players are reporting, then we say that the mechanism has a dominant strategy solution concept.

27 2.2. BASICS OF MECHANISM DESIGN 19 Definition 2. A vector of players reported-type t T is a dominant strategy solution if for each player i, for each alternative reported-type of player i t i T i and for each alternative reported-type vector t T, we have: u i (θ i, (t i, t i) u i (θ i, (t i, t i)) Notice that a dominant strategy solution may not be optimal for any of the players. Having a single dominant strategy is an extremely stringent requirement for a mechanism. But in mechanism design it is often desirable that a dominant strategies are implemented in a way that an efficient solution (socially efficient or efficient for the auctioneer) is achieved. Pure Strategy Nash Equilibrium. Since only few games have a dominant strategy solutions, we look at a weaker solution concept: the pure Nash equilibrium. The Nash equilibrium captures the idea of stable solution. Informally, given a state of the mechanism (all the reported actions of the players) no one can increase his utility deviating unilaterally from his reported action. Definition 3. A vector of players reported-types t is a Nash-equilibrium if for each player i and each alternative i s reported-type t i, we have u i (θ i, (t i, t i )) u i (θ i, (t i, t i )) Notice that a dominant strategy solution is a Nash equilibrium. This also implies that a Nash equilibrium may not be optimal for the players. Another important aspect that we have to notice is that multiple Nash equilibria can exist. Due to the existence of multiple Nash equilibria some doubts may arise in terms of goodness as solution concept. If multiple equilibria exist which one can we expect that the players play? Nonetheless, the solution concept of pure Nash equilibrium is the most natural concept of equilibria. Besides its simplicity, pure Nash equilibria not always exist and are not easy to find Revelation principle In this thesis often we will look at the class of direct-revelation mechanisms. A directrevelation mechanism is a mechanism where the action space of the players collapse on their type space, i.e. T i = Θ i. Definition 4. A direct-revelation is a single-round, sealed bid mechanism (A, ρ) for n player is defined by: a players type space Θ = Θ 1... Θ n (private information), a set of alternatives X, each element of X is an n dimensional vector, a valuation function for each player i, v i : Θ i X R, that is how much the player i values the allocation X,

28 20 CHAPTER 2. TECHNICAL PRELIMINARIES an allocation rule A : Θ X, that given the reported-type of the players returns an allocation, a payment rule ρ : Θ R n, that given the reported-type of the players returns the payments, an utility function for each player i, u i : Θ i Θ R, and the utility function of the auctioneer, i.e. ρ ρ(t) ρ, that is the sum of the payments. It seems that a direct-revelation mechanism is a more stringent mechanism than a general mechanism, but this is not true. The revelation principle states that if we are looking for a mechanism with a specific equilibrium we can restrict our attention to a direct-revelation mechanism in which reporting his own type is an equilibrium. Theorem 1 (Revelation Principle). Any mechanism (A, ρ) with an implemented equilibrium can be converted in a mechanism (A, ρ ) in which the players report their true type and has the same equilibrium outcome. This is an extremely useful tool in mechanism design. It allows any researcher that want to implement a specific outcome or property to restrict his attention to the mechanisms in which players report their true type. If such a mechanism does not exist, then no (generic) mechanism can implement the desired outcome or property. Due to the revelation principle, therefore we will focus on direct-revelation mechanism. Therefore, we will use θ i to denote the true type of a player i and θ i to denote the type player i reports (possibly untruthful). Analogously, θ = {θ 1,..., θ n } is the vector of true-types of all players and θ = { θ 1,..., θ n } is the vector of reported-type of all players Mechanism in Bayesian model In the mechanisms described above we assumed that each player has not any information about the private types of the other players. This assumption allows us only to operate in worst case. A different model is the Bayesian mechanism design. It is standard in the economic literature to deal with a Bayesian approach. In this model there are commonly known prior distributions an the types of the players are drawn from these distributions. Now, each player has some beliefs on the other agents types, thus he tries to optimize in a Bayesian sense exploiting his informations. Formally a mechanism in a Bayesian model is described as follows: Definition 5. A direct-revelation Bayesian mechanism (A, ρ) for n player is defined by: a players type space Θ = Θ 1... Θ n (private information), prior distributions on the type of the players D = D 1... D n (commonly known), a set of allocations X, each element of X is an n dimensional vector,

29 2.2. BASICS OF MECHANISM DESIGN 21 a valuation function for each player i, v i : Θ i X R, that is how much the player i values the allocation X, an allocation rule A : Θ A, that given the reported-type of the players returns an allocation, a payment rule ρ : Θ R n, that given the reported-type of the players returns the payments, an utility function for each player i, u i : Θ i Θ R, and the utility function of the auctioneer, i.e. ρ ρ(t) ρ, that is the sum of the payments. If in the non-bayesian model a possible solution concept for a mechanism is the Nash equilibrium, in the Bayesian model there is the notion of Bayesian-Nash equilibrium. Definition 6. A vector of players reported-type θ is in Bayesian-Nash equilibrium if for each player i, each type θ i, and each θ i, we have E D i [u i (θ i, (θ i, θ i )] E D i [u i (θ i, (θ i, θ i ))] Notice that an ex-post-nash equilibrium is a Bayesian-Nash equilibrium for any distribution D Quasi-linear model Until now, we have not defined formally the utility functions of players. The utility gives a measure of the happiness of a player when the outcome of the mechanism is realized. Notice that the former (informal) definition applies to the Bayesian model in terms of ex-post utility. Otherwise, in Bayesian model the notion of expected utility is used for the social efficiency analysis of a mechanism. Now, the formal definition of the utility function in the quasi-linear model is: u i (θ i, θ) = v i (θ i, A( θ)) p i where p i is the i-th component of ρ( θ) Desirable properties We can now depict the most desirable properties that a mechanism should implement. The most important property for a mechanism is to avoid that the players can strategize on their action inducing an equilibrium that is not efficient for the auctioneer or for the players themselves. A mechanism implementing an allocation rule and a payment rule that do not admit manipulations is called incentive-compatible or truthful or strategy-proof (we use these terms interchangeably). A formal definition of incentive-compatibility is the following:

30 22 CHAPTER 2. TECHNICAL PRELIMINARIES Definition 7. A mechanism (A, ρ) is called incentive-compatible if for every player i, every θ 1 Θ 1,..., θ n Θ n, every θ i Θ i, and every X = A(θ), p = ρ(θ), X = A( θ), and p = ρ( θ) then v i (θ i, X) p i v i (θ i, X ) p i. Essentially, a mechanism is incentive compatible if every player is better off reporting his true type. A complete characterization of incentive-compatibility in single parameter domain is given in Appendix A.1 for completeness. Other important properties for a mechanism are individual-rationality and nopositive-transfer. Definition 8. A mechanism satisfies individual-rationality (IR) if every player i has a non negative utility after the outcome is computed, i.e. u i (θ i, θ) 0 i I. And, Definition 9. A mechanism satisfied no-positive-transfer (NPT) if the payment of every player is non-negative, i.e. p i 0 i I. The former properties guarantees that all the bidders have no regrets in being a participant of the mechanism. This is why individual-rationality is also often referred as voluntary participation. The latter guarantees that the mechanism does not give money to any player Social welfare A standard goal in mechanism design is to maximize the social efficiency of all the players. The most natural way to measure the social efficiency of a mechanism s outcome is to sum the utilities of the participants, i.e. the social-welfare. We use SW(A( θ), p( θ)) to denote the social-welfare of an outcome (A( θ), p( θ)) (or equivalently SW(X, p) to denote the social-welfare of an outcome (X, p)). If the utility model is quasi-linear, the social welfare of an outcome is equal to the sum of the players valuations for that outcome. SW(A( θ), p( θ)) = u i (θ i, θ) + i [n] i [n] p i = ) (v i (θ i, A( θ)) p i + i [n] = v i (θ i, A( θ)) i [n] Thus we can maximize the social-welfare computing an outcome that maximize the sum of the players valuations Auctioneer revenue Another standard goal in mechanism design is to maximize the revenue of the auctioneer. Maximize the revenue is equivalent to maximize the payments and this is i [n] p i

31 2.3. SPONSORED-SEARCH ENVIRONMENT 23 easily achievable if all players pay an amount equal to their reported-types regardless of the allocation. But notice that a mechanism with a payment rule like that is not appealing for the participants. Thus a mechanism designer should maximize the auctioneer s revenue and guarantees some kind of social efficiency to the players. Notice also that with a payment rule as the one discussed above, the players have incentive to underbid in order to achieve a good allocation and pay less. This kind of behaviour can induce dangerous bidding strategies that result in extremely low revenue. An example of the mentioned behaviour is depicted in [Edelman et al., 2005] through two diagrams provided by Overture. Figure 2.1 refers to the bidding strategies of two bidders competing for the top slot on a keyword in the first-price auction implemented by Overture. The first diagram shows how the top bids change in a 14 hours selling process. Since in a first-price auction the winning bidder pays the submitted bid, each bidder has incentive to bid just an epsilon more than the other bidders. Notice how these bidding strategies lead to a sawtooth pattern. The same behaviour is observable in a larger period, i.e. the second diagram shows the top bids in one week. As the authors state in [Edelman et al., 2005], this sawtooth patter is common in many keywords when first-price auctions are implemented. We use R(p θ) = i [n] p i to denote the revenue extracted by the payment rule with players reported-type vector θ. 2.3 Sponsored-search environment Now, we provide a description of the different environments that are discussed in this thesis. Single divisible item: there is one item and all the bidders are interested in it. The item is divisible thus each bidder i can obtain a fraction x i [0, 1] of the item. The feasibility constraint imposes that i I x i 1. Multi-unit: there are m identical and indivisible items. The bidders compete to obtain items as much as possible. If each bidder i allocates x i [m] items, the feasibility constraint imposes that i I x i m. Matching markets: there are J = m non-identical and indivisible items. Each bidder i is interested in a subset S i J of the items and competes to win items in his preference set. If x i S i is the items allocated to bidder i, the feasibility constraints impose that i I x i J and x i x j = for each i, j I. We will discuss specializations of these setting the different chapters. 2.4 VCG auction Now, we present the most famous result in auction design: the Vicrey-Clarke-Groves (VCG) auction [Vickrey, 1961, Clarke, 1971, Groves, 1973]. This auction is remarkably important for many reasons: it is incentive-compatible, individually-rational, guarantees no-positive-transfer and optimize the social-welfare in many environment.

32 24 CHAPTER 2. TECHNICAL PRELIMINARIES (a) 14 hours (b) 1 week Figure 2.1: Example of bidding behaviour by two advertisers over a single keyword in Overture first price auction.

33 2.4. VCG AUCTION 25 Let I be the bidders (players), let J be the set of goods auctioned and let X be the set of allocation feasible to the auctioneer (each element X of X is a vector as described in the previous sections). Now, each bidder i I has different preferences over different subset of J. The valuation function v i : X R describe this preferences and is bidder s private information. The goal is to compute the allocation X that maximizes the social-welfare: i I v i(x). Formally, Definition 10. A mechanism (A, ρ) is a VCG mechanism if: A( θ) maximized the social-welfare, i.e. A( θ) arg max X X i I v i(x), and for all θ Θ: ρ i ( θ) = h i ( θ i ) j I:i j v j(a( θ)), where h 1,..., h n are functions such that h i : Θ i R for all i I. It is immediate to note that the allocation rule maximizes the social-welfare when the bidders report their true-type, i.e. θi = θ i for all i I. Observe also that the payment of each bidder has a component j I:i j v j(a( θ)) that has a negative impact on the payment. It means that higher is that component lower will be the payment of the bidder. Notice that this component is the sum of all the valuations for the produced outcomes (except the valuation of the bidder itself), thus maximize this term is equivalent to maximize the social-welfare that is obtained reporting the true-type. Theorem 2. Every VCG mechanism is incentive compatible. We still did not discuss the term h i that is in the payments. Since h i is dependent only on the type reported by the other bidders it is just a constant for each bidder. Thus, even it is not important for the incentive-compatibility (the bidder cannot strategize on that), it is important to achieve other desirable properties: individuallyrationality and no positive transfer. These properties are achievable when h i is defined accordingly to the Clarke pivot rule. Definition 11. A VCG mechanism implements the Clarke pivot rule if the function h i is defined as h i (θ i ) = max b X The final payment for each bidder i is ρ i ( θ) = max b X j I:j i j I:j i v j (b) v j (b) j I:i j v j (A( θ)) that is the difference between the social-welfare achievable by all the bidders except i when i does not participate in the auction and the social-welfare achieved by all the bidders except i when i participate. In other words, i pays the loss of value he causes to the other bidders. Or, each bidder internalize the externalities that he causes. Thus, we can finally state:

34 26 CHAPTER 2. TECHNICAL PRELIMINARIES Lemma 1. A VCG mechanism that implements the Clarke pivot rule has the no positive transfer property. Moreover, if for every θ i Θ i and every X X, v i (X) 0 then the VCG mechanism is also individually-rational Example of VCG auction Assume that there are two bidders, b 1 and b 2, two items, j 1 and j 2, and each bidder is allowed to obtain one item. We let v i,t be bidder b i s valuation for item j t. Assume v 1,1 = 10, v 1,2 = 5, v 2,1 = 5, and v 2,2 = 3. We see that both b 1 and b 2 would prefer to receive item j 1 ; however, the socially optimal assignment gives item j 1 to bidder b 1 (so his achieved value is 10) and item j 2 to bidder b 2 (so his achieved value is 3). Hence, the total achieved value is 13, which is optimal. If person b 2 were not in the auction, person b 1 would still be assigned to j 1, and hence person b 1 can gain nothing. The current outcome is 10 hence b 2 is charged = 0. If person b 1 were not in the auction, j 1 would be assigned to b 2, and would have valuation 5. The current outcome is 3 hence b 1 is charged 5 3 = VCG with budgets We have seen how the V CG auction optimize the social welfare of the participants. Since its remarkable properties it became a cornerstone in the mechanism design, but an important limitation arises when there are budget constraints. Assume a simple setting with two identical items and two bidders. Each bidder i has an additive valuation function v i, i.e. the valuation for k items is k v i, and a budget b i. The utility of bidder i for k items is min(k v i, b i ). Now, assume that bidders 1 and 2 have valuations and budgets v 1 = 4, v 2 = 1, b 1 = 4, and b 2 = 4 respectively. The modified V CG allocates one item to bidder 1 at price 1 and one item to bidder 2 at price 0. Thus, the utility of bidder 1 is u 1 = 4 1 = 3 and the utility of bidder 2 is u 2 = 1 0 = 1. But, if bidder 1 declares a valuation v 1 = 2 the mechanism allocates both items to him at a price equal to 2. Now, the utility of bidder 1 is u 1 = = 6 > 3 = u 1. Thus bidder 1 has incentive to lie. 2.5 Ausubel clinching auction Another important mechanism that inspires many of the auctions presented in this thesis is the ascending auction by Ausubel for selling multiple identical items to bidders. We give in the following an informal description and an example of the execution of the auction. The input to the auction by bidder i is a vector v i ( ) of marginal valuations. The value v i (k) is the valuation of bidder i for the k-th item when k 1 items have already been allocated to him. At the beginning of the auction, the price per item π is initially set to zero and then it is monotonically increased by the auctioneer until all items are sold. For each price π the bidders implicitly submit a demand: the demand at price π of bidder i having already allocated k items is equal to the minimum of (1) the number of unsold items and (2) the number of marginals of order bigger than k and value strictly bigger than π. An item is allocated to bidder

35 2.5. AUSUBEL CLINCHING AUCTION 27 Bidder A Bidder B Bidder C Marginal valuation (unit 1) Marginal valuation (unit 2) Marginal valuation (unit 3) Table 2.1: Marginal valuations. i at price π if the total demand of all other bidders is strictly less than the number of unsold items, i.e., at least one item is under demanded. If no item can be allocated then the price gets increased. More formally: I is the set of bidders. m is the number of unsold items. The marginal valuations are given by v i : [1,..., m] N 0. The values v i (k) are non-increasing in the number k of items already sold to bidder i. If bidder i is already allocated k items at price π or lower, the demand of bidder i at price π is D i (π) = min{m, {j > k v i (j) > π} }. At the end of the auction bidder i is allocated x i items for a total payment p i b i. The utility of bidder i is u i = x i j=1 v i(j) p i Example of Ausubel ascending auction We give in the following an example of the execution of the auction with three bidders and three items. The marginal valuations of the bidders are in Table 2.1. No item is sold at price lower than 2. The demand at price 2 is: Price Bidder A Bidder B Bidder C No item is sold since the total demand of any two bidders is greater than or equal 3. The demand of the bidders at price 4 is: Price Bidder A Bidder B Bidder C One item is sold to bidder A since the total demand of the other bidders is 2. The demand of the bidders at price 4 is now equal to Price Bidder A Bidder B Bidder C No more items are sold at this price. At price 5 the demand is:

36 28 CHAPTER 2. TECHNICAL PRELIMINARIES Price Bidder A Bidder B Bidder C No item is sold at price 5. At price 8 the demand is: Price Bidder A Bidder B Bidder C Bidders A and B get allocated to one item each at price 8.

37 Part II Social-efficiency in auctions with budgets 29

38

39 Chapter 3 Preliminary: Budgets and social-efficiency 3.1 Keywords and slots in sponsored search auctions Informally speaking, a sponsored search auction has a set of bidders that compete to obtain a set of keywords on which display their ads. Each bidder submits a valuation that describes his willingness to pay, a budget that describes his ability to pay, and a preference set that describes the set of keywords in which he is interested. Each keyword has some number of slots (copies) available to the bidders. Each slot has a (distinct) associated quality. Notice that this setting is a generalization of the standard position auctions setting [Varian, 2007, Edelman et al., 2005, Ashlagi et al., 2010] Results overview In these part of the thesis we focus on studying efficient auction for sponsored search auctions. For a discussion on what we mean by efficiency in sponsored search auction we refer to Section 1.4 and to the following sections. Specifically we present the following results: In this chapter we prove that if bidders have private diminishing valuation functions then no incentive-compatible, invidually-rational and Pareto-optimal auction exists, even when budgets are public knowledge. In Chapter 4 a characterization of Pareto optimality is provided in the single keyword with many divisible slots setting. Then we present an auction that is individually-rational, incentive-compatible and Pareto-optimal in this setting when budgets are public knowledge and the bidders have additive valuation functions. Moreover, we show how to probabilistically round the fractional allocation produced in the divisible case to an integer allocation for the indivisible case with multiple keywords (i.e., the adwords setting) and get an auction that 31

40 32 CHAPTER 3. PRELIMINARY: BUDGETS AND SOCIAL-EFFICIENCY is incentive-compatible in expectation, individually-rational in expectation, and Pareto-optimal. In Chapter 5 we study a setting with multiple, indivisible keywords with identical slots (i.e. having the same quality) and budgeted bidders that have preferences over the set of keywords (i.e. are not interested in all the keywords). In this setting we characterize the Pareto-optimality and provide an auction that is individually-rational, incentive-compatible and Pareto-optimal. 3.2 From social-welfare to Pareto-optimality As we already discussed in the former chapters of this thesis, budget is a crucial point in actual implementations of auctions. The main problem with budget is related to the utility function and thus to the notion of efficiency. Despite the quasilinear setting, now the utility function becomes non-linear and discontinuous. The utility is a function of the bidder s payment and since it is lower than the budget the utility decreases linearly. When the bidder s payment exceeds the budget it goes to negative infinity. { v i (θ i, X) p i if b i p i u i (X, p) = if b i < p i Notice that the assumption that the utility function is quasilinear underlies most of the mechanism design s literature. Initially there have been many efforts to maintain a quasilinear utility function despite budget constraints. A model with each bidder i has a modified valuation equal to min{v i, b i } maintains the quasilinear setting. However, the modified valuations do not capture the real nature of the budget constraints. For example the VCG mechanism loses its revenue and incentive compatibility properties. Moreover, the produced allocation are not even Pareto-efficient since an allocation in which each bidder reaches his budget is considered efficient (maximize the social-welfare) despite the real valuations. Thus, we need to move from a quasilinear model to a non quasilinear model and to a different notion of (social) efficiency. In auctions with budgets there are no individually-rational and incentive compatible auction that maximize the social welfare [Dobzinski et al., 2012]. Indeed a folklore result shows that even if the budgets are common knowledge, no truthful auction approximates the social welfare better than n, where n is the number of bidders [Dobzinski and Leme, 2014]. Thus, we concentrate on a weaker measure of social efficiency: Pareto-optimality Pareto optimality and other desirable properties Given a feasible allocation (X, p) and the utility u i of bidder i an allocation (X, p ) is Pareto-superior to the feasible allocation (X, p) if 1. the utility of no bidder in (X, p ) is less than his utility in (X, p), 1 Since we are in a non quasilinear setting, different allocation could be preferred by different bidders.

41 3.3. SET THE BOUNDARY the utility of the auctioneer in (X, p ) is no less than his utility in (X, p), and 3. at least one bidder or the auctioneer is better off in (X, p ) compared with (X, p). We study auctions that select allocations obeying the following conditions: (Bidder rationality) u i 0 for all bidders i I, (Auctioneer rationality) the utility of the auctioneer fulfills i I p i 0, and (No-positive-transfer) p i 0 for all bidders i I. An auction that on all inputs outputs an allocation that is both bidder rational and auctioneer rational is called individually rational (IR). A feasible allocation is Pareto-optimal (PO) if there is no other feasible allocation (X, p ) that is Paretosuperior to (X, p). An auction is incentive compatible (IC) if it is a dominant strategy for all bidders to reveal their true valuation. An auction is said to be Pareto-optimal (PO) if the allocation it produces is PO. A randomized auction is IC in expectation, IR in expectation, respectively PO in expectation if the above conditions hold in expectation, and IC ex post, IR ex post, respectively PO ex post if the conditions hold for every realized outcome. 3.3 Set the boundary In this section we want to depict the boundaries in mechanism design when bidders have budgets and there are many (indivisible) items to sell. We will discuss impossibility results for different settings and different auctions properties. We start looking at the impossibility result provided in [Dobzinski et al., 2012]. They discuss a setting with m indivisible items and n bidders with additive valuation functions and budgets. Thus, they prove that there is no deterministic IC auction that satisfies individual rationality, no positive transfer, and Pareto-optimality for any number of bidders when budgets are private knowledge. The impossibility result is because the combination of the truthfulness and Paretooptimality when budgets are private knowledge and each bidder desires more than one item. The condition of non-unit-demand bidders is crucial since if we move in a unit-demand setting a positive result exists. In [Ashlagi et al., 2010] the authors show that for position auctions (many different copies of a single item) an auction that is truthful, Pareto optimal, and envy-free exists also with private budgets. Moreover, they prove that their auction is the only possible auction. Thus, if we insist on incentive-compatibility, Pareto-optimality, and not unitdemand bidders we need to assume that budgets are public knowledge. But also for public budgets there are problems. Also the valuation functions of the bidders may originate some problems for the combination of truthfulness and Paretooptimality. In [Lavi and May, 2012] is proved that no truthful, individually-rational, no-positive-transfer, and Pareto-optimal auction exists for private free-disposal 2 valuation functions. 2 A valuation function v : N >0 R is said free-disposal (or monotone) if v(i) v(j) if i j.

42 34 CHAPTER 3. PRELIMINARY: BUDGETS AND SOCIAL-EFFICIENCY Setting Valuation Auction Multiunit Value function (f i ) No Budgets Multiunit Additive Value v i Finite Budgets (b i ) Matching markets (S i ) Additive Value (v i ) Finite Budgets (b i ) Matching markets (S i ) Additive Value (v i ) Finite Budgets (b i ) Slots (α j ) f i (# items) Vickrey [Vickrey, 1961] Multiunit Private Data f i Public Data Properties Maximizes Social Welfare v i #items DLN [Dobzinski et al., 2012] v i b i Pareto Optimal v i #(items S i ) FLSS [Fiat et al., 2011] v i b i, S i Pareto Optimal v i #(items S i ) This thesis v i b i, S i, α j Pareto Optimal Table 3.1: Incentive compatible multi-unit and matching-makertes Auctions Free-disposal valuation functions are a very general class of valuation functions, but the same impossibility result is obtained in the more restrictive setting of diminishing marginal valuation function (see next section for a formal description). Our result in [Colini-Baldeschi et al., 2012] for indivisible items is stronger as it applies to bidders with non-negative and diminishing marginal valuations. The result will be presented in Section 3.4. In [Goel et al., 2012] the same impossibility result for divisible items and bidders with monotone and concave utility functions was given. Note that neither their result nor ours implies the other. These two results limit the possibility of truthful, IR, NPT and Pareto-optimal auction to a small class of valuation functions: therefore we focus on the additive valuation functions 3. Finally, an impossibility result when each bidder is interested in only a subset of the auctioned items is presented in [Fiat et al., 2011]. They show that there are no truthful, individually-rational, no-positive-transfer, and Pareto-optimal auctions when budgets are public knowledge but the preference sets are private knowledge. Finally in [Borgs et al., 2005] the authors show that if there are many items, the bidders have a value for each unit of goods and a budget, both private informations then no trivial auction can be achieved Impossibility result for diminishing marginal valuations We consider in this section the setting of m 2 identical indivisible items and 2 bidders with private diminishing marginal valuations and public budgets. We show that there is no IC, IR, and PO deterministic mechanism for this case. As in the previous sections we use I := {1,..., n} to denote the set of bidders. Furthermore, for each bidder i I and k {1,..., m} let v i (k) 0 be bidder i-th marginal valuation for the k-th item assigned to him. A bidder i has diminishing marginal 3 A valuation function is said to be additive if given a value v R >0 the valuation of k items is v k. 4 A trivial auction is defined as an auction that satisfies natural properties like consumer sovereignty, independence of irrelevant, and strong non-bundling.

43 3.4. IMPOSSIBILITY RESULT FOR DIMINISHING MARGINAL VALUATIONS 35 valuations if v i (k) v i (k + 1) for all k {1,..., m 1}. theorem. We show the following Theorem 3. There is no IC, IR, and PO deterministic mechanism for m 2 identical indivisible items and 2 bidders with private diminishing marginal valuations and public budget constraints. Proof. Assume by contradiction that such a mechanism exists and we are given m items and 2 bidders. The proof uses the valuation profiles given by the following diminishing marginal valuations: v 1 (1) = v 1 (2) = = v 1 (m) = 6 v 2 (1) = v 2 (2) = = v 2 (m) = 5 v 2(1) = v 2(2) = = v 2(m) = 4 v 2 (1) = v 2 (2) = = v 2 (m 1) = 5 and v 2 (m) = 0 The budget of bidder 1 is fixed to b 1 = 6 and the budget of bidder 2 is fixed to b 2 = 5m. We first show that b 1 and b 2 are generic according to the following definition in [Dobzinski et al., 2012]. Definition 12 ([Dobzinski et al., 2012]). Let S = (S 1, S 2 ) be a partition of {1,..., m}. Given b 1, b 2 0, define b k,s i recursively, for each 1 k m: for k = m, b m,s 1 = b 1, b m,s 2 = b 2. For each 1 k m 1, if k S 1 then: b k,s 1 = b k+1,s 1, b k,s 2 = b k+1,s 2 bk+1,s 1 k+1. If k S 2 then: b k,s 1 = b k+1,s 1 bk+1,s 2 k+1, b k,s 2 = b k+1,s 2. We say that b 1 and b 2 are S-generic if for each 1 k m we have that b k,s 1 b k,s 2. We say that b 1 and b 2 are generic if they are S-generic for all S. Claim 1. The budgets b 1 and b 2 are generic. Proof. Fix a k {1,..., m} and a partition S = (S 1, S 2 ) of {1,..., m}. Observe that b k,s 2 b m,s 2 b m,s 1 and bk,s 1 b m,s 1. Thus, b k,s 2 5m 6( (m 2) 1 3 ) = m 1 l=1 1 l+1 3m + 1 > 6 b k,s 1 which proves the claim. Hence, it holds by Claim 1 and Theorem 4.1 in [Dobzinski et al., 2012] that given budgets b 1 and b 2 the outcomes of all deterministic mechanisms that are IC, IR, and PO have to coincide with that of the clinching auction when the valuations differ between the bidders and are additive (i.e., the marginal valuations are constant). 5 Note that v 1, v 2, and v 2 are additive. Thus, given the bids v 1 and v 2 (v 1 and v 2) the outcome has to be identical with the outcome of the clinching auction. Let p 2 be the payment by bidder 2 in the clinching auction given the bids v 1 and v 2. Furthermore, let m 2 be the number of items assigned to bidder 2 in that case. Claim 2. Given the bids v 1 and v 2 then bidder 2 gets m 2 = m 1 items and pays p 2. 5 Note that no positive transfer in [Dobzinski et al., 2012] is called auctioneer rationality in this work and is a necessary condition for individual rationality.

44 36 CHAPTER 3. PRELIMINARY: BUDGETS AND SOCIAL-EFFICIENCY Proof. Consider the clinching auction in [Dobzinski et al., 2012]. In both cases (v 1, v 2 ) and (v 1, v 2) bidder 2 clinches items at prices 6 m, 6 m 1,..., 6 2 and bidder 1 clinches a single item when bidder 2 leaves the auction. Claim 3. Assume we are given the bids v 1 and v 2 and let m 2 be the number of items that bidder 2 obtains. It holds that m 2 m 2. Proof. Consider the case where bidder 2 truthfully bids v 2. We show that bidder 1 gets at least 2 items: (a) Assume that bidder 1 gets no item and all his marginal valuations are positive. By PO all items have to be assigned to the bidders, and by IR the payment of bidder 1 is zero. Thus, bidder 1 could increase his utility if he bought an item from bidder 2 with his remaining budget. Contradiction to PO! (b) Assume that bidder 1 gets one item and his marginal valuations are v 1. By (a) bidder 1 gets at least one item when he states arbitrarily small positive marginal valuations. Thus, by IR and IC his payment has to be zero. Hence, bidder 1 could increase his utility if he bought an item from bidder 2 with his remaining budget. Contradiction to PO! Claim 4. Assume we are given the bids v 1 and v 2 and let p 2 be the payment of bidder 2. It holds that p 2 p 2 (m 1 m 2) 5. Proof. Consider the case where bidder 2 truthfully bids v 2, and assume by contradiction that p 2 > p 2 (m 1 m 2) 5. By Claim 2 it holds that m 2 = m 1. Thus, bidder 2 can increase his utility by bidding v 2 ; that is, m 2 5 p 2 < m 2 5 p 2. Contradiction to IC! We can now show a contradiction to the assumption that an IC, IR and PO mechanism for diminishing marginal valuations exists. Assume we are given the truthful bids v 1 and v 2. By IC and Claim 2 it has to holds that (m 1) 4 p 2 m 2 4 p 2. This and Claim 4 implies (m 1) 4 p 2 m 2 4 p 2 + (m 1) 5 m 2 5 which is equivalent to m 2 m 1. Thus, Claim 3 gives a contradiction.

45 Chapter 4 Identical keywords with many copies and different qualities 4.1 Outline In this chapter we present positive results for two different settings: the divisible case with only one keyword with different slots and the indivisible case with many identical keywords with different slots. In the former each bidder is allowed to buy (or clinch) a fraction of the slots. In the latter the slots are indivisible, thus each bidder can allocate only complete slots but there are many keywords available. Notice that the latter is obtained through a randomization of the algorithm presented for the former case The Setting Using the same notation as in Chapter 2 we specialize the setting for sponsored search auctions as follows. We have n bidders and m slots. We call the set of bidders I := {1,..., n} and the set of slots J := {1,..., m}. Each bidder i I has a private valuation v i 0, a public budget b i 0, and a public slot constraint κ i N >0 (i.e. maximum number of slots desired). Each slot j J has a public quality α j Q 0. The slots are ordered such that α j α j if j > j, where ties are broken in some arbitrary but fixed order. We assume that the number of slots m fulfills m = i I κ i as we could add dummybidders with valuation v i = 0, if m > i I κ i, or we could add dummy-items with quality α j = 0, if m < i I κ i. Then settings are formally described as follow. Divisible case: In the divisible case we assume that there is only one keyword with infinitely divisible slots. We will discuss the divisible case in Section 4.2 and Section 4.3. Thus the goal is to assign each bidder i a fraction x i,j 0 of each slot j and charge him a payment p i. An assignment matrix X = (x i,j ) (i,j) I J and a payment vector p are called an allocation (X, p). We call c i = j J α jx i,j the weighted capacity allocated to bidder i. An allocation is feasible if it fulfills the following conditions: 37

46 38 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES 1. the sum of the fractions assigned to a bidder does not exceed his slot constraint ( j J x i,j κ i i I); 2. each of the slots is fully assigned to the bidders ( i I x i,j = 1 j J); and 3. the payment of a bidder does not exceed his budget limit (b i p i i I). Indivisible case: The indivisible case is addressed in Section 4.4 and assume that we additionally have a set R of keywords, where R is public, and each keyword has the set of slots J. The goal is to assign each slot j J of keyword r R to one bidder i I while obeying various constraints. An assignment X = (x i,j,r ) (i,j,r) I J R where x i,j,r = 1 if slot j is assigned to bidder i in keyword r, and x i,j,r = 0 otherwise, and a payment vector p form an allocation (X, p). We call c i = α j j J R ( r R x i,j,r) the weighted capacity allocated to bidder i. An allocation is feasible if it fulfills the following conditions: 1. the number of slots of a keyword that are assigned to a bidder does not exceed his slot constraint ( j J x i,j,r κ i i I, r R); 2. each slot is assigned to exactly one bidder ( i I x i,j,r = 1 j J, r R); and 3. the payment of a bidder does not exceed his budget limit (b i p i i I). 4.2 Pareto-optimality characterization In this section we present a novel characterization of PO allocations that allows to address the case of multiple divisible slots with different CTRs. Like previous characterizations of PO for other settings [Bhattacharya et al., 2010, Dobzinski et al., 2012, Fiat et al., 2011] our characterization ensures that no bidder can resell items (i.e., weighted capacity) to another bidder to increase his utility. However, in our setting, we have to consider that transferring weighted capacity between two bidders might result in the fractional exchange of slots between many bidders. We use the characterization to prove the PO of the auction given in Section 4.3. Given a feasible allocation (X, p), a swap between two bidders i and i is a fractional exchange of slots, i.e., if there are slots j and j and a constant τ > 0 with x i,j τ and x i,j τ then a swap between i and i can give a new feasible allocation (X, p) with x i,j = x i,j τ, x i,j = x i,j τ, x i,j = x i,j + τ, and x i,j = x i,j + τ. If α j < α j then the swap increases i s weighted capacity and reduces i s weighted capacity. To characterize PO allocations we first define for each bidder i the set N i of bidders such that for every bidder a in N i there exists a swap between i and a that increases i s weighted capacity. Given a feasible allocation (X, p) we use h (i) := max{j J x i,j > 0} for the slot with the highest quality that is assigned to bidder i and l (i) := min{j J x i,j > 0} for the slot with the lowest quality that is assigned to bidder i. To consider the case of slots with equal α-value we define h(i) := min{j J α j = α h (i)} and l(i) := min{j J α j = α l (i)}. Thus, N i is the set of all the bidders a I with h(a) > l(i).

47 4.2. PARETO-OPTIMALITY CHARACTERIZATION 39 slot number h(1) l(1) h(2) l(2) h(3) l(3) h(4) l(4) h(5) l(5) N 1 = {1, 2} N 3 = {1, 2, 3, 4} N 5 = {1, 2, 3, 4, 5} N 2 1 = {1, 2, 4} N 3 1 = {1, 2, 3, 4} Ñ 1 = {2, 3, 4} Ñ 3 = {1, 2, 4} Ñ 5 = {1, 2, 3, 4} N 2 = {1, 2, 4} N 4 = {1, 2, 3, 4} N 2 2 = {1, 2, 3, 4} Ñ 2 = {1, 3, 4} Ñ 4 = {1, 2, 3} Figure 4.1: Example of desired trading partners. To model sequences of swaps we define furthermore Ni 1 := N i and Ni k := a N k 1 i k=1 N i k for all n n. We for k > 1. Since we have only n bidders, n k=1 N k i = n define Ñi := n k=1 N i k \ {i} as the set of desired (recursive) trading partners of i. See Figure 4.1 for an example with five bidders. The bidders a in Ñi are all the bidders such that through a sequence of trades that starts with i and ends with a, bidder i could increase his weighted capacity, bidder a could decrease his weighted capacity, and the capacity of the remaining bidders involved in the sequence would be unchanged. The following lemma shows the non-reflexive transitivity of Ñ i. Lemma 2. Given an arbitrary assignment, if b Ña, c Ñb, and a c then c Ña. Proof. Let us assume that b Ña, c Ñb, and a c. It follows that there exists an integer k b with b N k b a and an integer k c with c N kc b. We first show by induction that Nb l N k b+l a for all l 1. For l = 1 we have Nb 1 = N b i N k b N i = N k b+1 a. a For l > 1 we assume inductively that N l 1 N k b+l 1 a. Then Nb l = i N N l 1 i i N k b +l 1 a N k b+k c a N i = N k b+l a. Thus, N kc b N k b+k c a, and moreover, since a c it follows that c Ña. b. Since c N kc b it follows that c b N a Given a feasible allocation (X, p) we use B := {i I b i > p i } to denote the set of bidders who have a positive remaining budget (i.e., each bidder i with budget b i larger than payment p i ). Let ṽ i = min a Ñ i (v a ), if Ñ i, and ṽ i = else. Note that ṽ i depends on Ñi and, thus, on the current assignment. As we show below the ṽ i -value and the remaining budget b i p i for each bidder i suffices to decide whether a given assignment is PO or not. We say that a feasible allocation (X, p) contains a trading swap sequence (δ, a) (for short tss), where δ > 0 is the swapped amount of weighted capacity and a is a sequence a 0, a 1,..., a k of bidders in I, if the following conditions hold: (S1) the sequence has no cycles, i.e., a l a l if l l, (S2) bidder a 0 has a higher valuation than bidder a k, i.e., v a0 > v ak,

48 40 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES (S3) we can swap weighted capacity δ from a l+1 to a l for all l {0, 1,..., k 1} in a certain way, in particular, (α h (a l+1 ) α l (a l )) min{x al,l (a l ), x al+1,h (a l+1 )} δ for all l {0, 1,..., k 1}, and (S4) bidder a 0 has a remaining budget that could compensate bidder a k s loss of weighted capacity δ, i.e., b a0 p a0 δv ak. Furthermore, we say that the allocation (X, p ) results from the allocation (X, p) through the tss (δ, a) where the length of the sequence a is k + 1 if (A1) p i = p i for all I \ {a 0, a k }, p a 0 = p a0 + δv ak, and p a k = p ak δv ak, (A2) x a l,l (a l ) = x a l,l (a l ) τ l x a l,h (a l+1 ) = x a l,h (a l+1 ) + τ l x a l+1,l (a l ) = x a l+1,l (a l ) + τ l x a l+1,h (a l+1 ) = x a l+1,h (a l+1 ) τ l where τ l = δ/(α h (a l+1 ) α l (a l )) for all l {0, 1,..., k 1}, and all other entries of X are identical to the entries of X. Slot 3 Slot 2 Slot 1 (δ, a) Bidder 1 (α 1, α 2, α 3) = (2, 4, 6) Bidder 2 (v 1, v 2, v 3) = (1, 2, 3) Bidder 3 (b 1, b 2, b 3) = (2, 1, 0) (p 1, p 2, p 3) = (0, 0, 0) (δ, a) = (1, (Bidder 2, Bidder 3, Bidder 1)) Figure 4.2: Example of a trading swap sequence that transfers weighted capacity from bidder 1 to bidder 2. Before the tss slot 1 is assigned to bidder 1 and 3, slot 2 is assigned to bidder 1 and 2, and slot 3 is assigned to bidder 2 and 3. The tss swaps the half of slot 3 assigned to bidder 3 with the half of slot 2 assigned to bidder 2, and the half of slot 2 assigned to bidder 1 with the half of slot 1 assigned to bidder 3. In the remainder of this section we prove the following characterization of Paretooptimal allocations. Theorem 4. A feasible allocation is Pareto-optimal, if and only if it contains no trading swap sequence. We show first that if an allocation (X, p) is PO, then it contains no trading swap sequence. Proposition 1. An allocation (X, p ) that results from the allocation (X, p) through a trading swap sequence (δ, a) is Pareto-superior to (X, p).

49 4.2. PARETO-OPTIMALITY CHARACTERIZATION 41 Proof. Let k + 1 be the length of the sequence a. By (A1) the sum of the payments, which is equal to the utility of the auctioneer, does not change. By (A1) and (A2), for all bidders except a 0 and a k neither the payment, nor the weighted capacity changes, and thus, their utility does not change. By (A2) the weighted capacity assigned to a k decreases by δ and by (A1) the payment of a k decreases by δv ak. Thus, the utility of a k does not change. Finally, the utility of a 0 increases because p a 0 p a0 = v ak δ by (A1) and v ak δ < v a0 δ by (S2) implies p a 0 p a0 < v a0 δ, and by (A2) a 0 s weighted capacity increases by δ. Next we show that if a feasible allocation does not contain a trading swap sequence then it is PO. We show this in two steps, namely, Proposition 2 proves that the nonexistence of a trading swap sequence depends on a certain condition for the ṽ i s of the bidders, and Proposition 3 shows that if this condition is fulfilled, then the allocation is PO. Proposition 2. A feasible allocation contains no trading swap sequences, if and only if (a) ṽ i v i for each bidder i B and (b) ṽ i > 0 for each bidder i I with v i > 0. Proof. (A) We show first that given a feasible allocation (X, p) that contains a tss (δ, a) where a has length k+1, it either holds that a 0 B with ṽ a0 < v a0 or that a 0 I with ṽ a0 = 0 and v a0 > 0. This is a consequence of the definition of a tss as follows. Since δ > 0 it follows by (S3) that α h (a l+1 ) > α l (a l ) for all l {0, 1,..., k 1}, and thus, a l+1 N al for all l {0,..., k 1}. It follows that a k Ña 0 by Lemma 2. Hence, by (S2) holds that v a0 > v ak ṽ a0. If a 0 B this directly proves (A). If a 0 I \ B then b a0 = p a0 and thus by (S4) holds v ak = 0. It follows v a0 > ṽ a0 = 0, which proves (A) for a 0 I \ B. (B) We will show now the other direction, i.e., that given a feasible allocation (X, p) such that there exists a bidder a 0 B with ṽ a0 < v a0 or a bidder a 0 I with ṽ a0 = 0 and v a0 > 0, there exists a tss in (X, p). Thus, there is a bidder a 0 I with ṽ a0 < v a0. We select the smallest k {1, 2,..., n} for which there is a bidder a k Na k 0 who has v ak = ṽ a0. We define for all l {1, 2,..., k 1} the bidder a l such that a l Na l 0 and a l+1 N al. By using this construction there cannot be a cycle in the sequence a which proves (S1) for any tss with sequence a. Furthermore, it holds that v ak = ṽ a0 which proves together with ṽ a0 < v a0 (S2) for any tss with sequence a. We next define the value δ S as the largest possible value for δ in (S3). We define τ l = min{x al,l (a l ), x al+1,h (a l+1 )} for all l {0, 1,..., k 1} and δ S = min l {0,1,...,k 1} τ l (α h (a l+1 ) α l (a l )). Observe that δ S > 0 because a l+1 N al for all l {0, 1,..., k 1} implies α h (a l+1 ) > α l (a l ). We next define δ B = (b a0 p a0 )/v ak if a 0 B and v ak > 0 and we define δ B = δ S otherwise. Thus, [0, δ B ] are feasible values for δ in (S4). Hence, it follows that (min{δ S, δ B }, a) satisfies all conditions for a tss.

50 42 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES To show Proposition 3 we first need to extend the sets Ñi to deal with agents that occupy slots with identical quality. For this purpose we introduce the sets T i. The containment relation on the sets T i gives a total order on these sets (Lemma 3). Additionally these sets are tight in the sense that no other feasible assignment can assign more weighted capacity to them (Lemma 4). This fact is crucial when showing Proposition 3. Definition 13. If h(i) = l(i) let L i = {u I (v u v i ) (h(u) = l(u) = h(i))}, and let L i = otherwise. Let T i = Ñi {i} L i for all i I. The next lemma shows that for a given assignment X the relation defines a total order on the sets T i with i I. Lemma 3. Given agents i, u I, then T i T u or T u T i. Proof. We can restrict our analysis to the case i u as otherwise T i = T u. Let us first assume that h(u) = l(u) = h(i) = l(i). It follows that N i = N u, and thus, Ñ i {i, u} = Ñu {i, u}. Let us assume additionally that v u v i. Then u L i and L u L i, and thus, T u = Ñu {u} L u Ñu {i, u} L u Ñi {i} L i = T i. For the same arguments v i v u implies T i T u. Next assume that h(u) = l(u) = h(i) = l(i) does not apply. If both h(u) l(i) and h(i) l(u) then h(u) l(u) h(i) l(i) h(u) implies that all are = which gives us a contradiction. Thus we can assume wlog that h(u) > l(i). It follows that u N i Ñi {i} and i u implies u Ñi. By Lemma 2 it follows that all c Ñu with c i satisfy c Ñi, and thus, Ñ u Ñi {i}. Furthermore, all x L u satisfy h(x) = h(u) > l(i), and thus, x N i Ñi {i}. Hence, we obtain T u = Ñu {u} L u Ñi {i} T i. Moreover, we can show in the next lemma for any i I that given the set T i determined by a feasible assignment X, no other assignment allocates more weighted capacity to the set of agents T i. Lemma 4. Given a feasible assignment X and the set T i for an agent i I determined by X, then for any other feasible assignment X it holds that α j x u,j α j x u,j u T i u T i j J Proof. We first fix an i I. Let κ = u T i κ u and let a = min u Ti l(u). Since l(i) = l(u) for all u L i it follows that (Fact a) a = min u Ñ l(u). Recall i {i} that since X is a feasible assignment it holds u I x u,j = 1 for all j J and j J x u,j κ u for all u I by definition. Further, recall that we assume in this section that u I κ u = m. Thus, and j J x u,j κ u = κ u T i u T i j J

51 4.2. PARETO-OPTIMALITY CHARACTERIZATION 43 x u,j = κ u = m = 1 = u I j J j J u I x u,j + u T i j J u I\T i j J u T i x u,j. x u,j x u,j + u T i j J u I\T i κ u. It follows that (Fact b) κ = j J Notice that (Fact c) for all j J with α j > α a it holds that u T i x u,j = 1: Assume by contradiction that u T i x u,j < 1. Then there exists a w I \ T i with x w,j > 0 because feasibility of X implies u I x u,j = 1. Thus, h(w) > a = l(u) for some u Ñi {i} by Fact a, which implies w Ñi {i} T i by Lemma 2. This leads to a contradiction to the assumption w I \ T i. Next we argue that (Fact d) for all j J with α j < α a it holds that u T i x u,j = 0. This is the case because otherwise there would be an agent u T i with x u,j > 0 which implies l(u) < a. This is a contradiction to the definition of a. Note that m m m m a + 1 = 1 x u,j = x u,j = κ j=a j=a u T i j=1 u T i where the first inequality follows from feasibility of X, the second equality follows from Fact d, and the third equality follows from Fact b. Thus, m κ+1 a. Together with the ordering of the slots by α, this implies (Fact e) α j α a for all j m κ+1. We now define j = min{j J α j > α a }. By the following arguments we obtain the next sequence of equalities: The first equality follows from Fact d; the second equality follows from Fact b and Fact c; and the fourth equality follows from Fact e and α j α a for all j j 1 implied by the definition of j. x u,j = x u,j u T i j J u T i j J:α j=α a x u,j j J:α j>α a u T i = κ (m j + 1) = j 1 j=m κ+1 1 = (1/α a ) j 1 j=m κ+1 In the next sequence of equalities the second equality follows from Fact c and Fact d, and the third equality from the above sequence. x u,j α j = m m α j x u,j = α j + α a ( x u,j ) = α j j J j J u T i j=j u T i j=m κ+1 u T i j J:α j=α a Thus, the agents in T i have an aggregated weighted capacity equal to the weighted capacity of the assignment where the most valuable slots from m down to m κ + 1 and no fraction of a slot below are occupied by T i. This is the optimal assignment for T i, i.e., for any other feasible assignment X it holds that α j x u,j α j x u,j u T i u T i j J j J α j

52 44 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES Now we use the previous lemmata in the next proposition that gives a sufficient condition for the Pareto-optimality of a feasible allocation (X, p). Proposition 3. 1 Given a feasible allocation (X, p), if (a) ṽ i v i for each i B and (b) ṽ i > 0 for each i I with v i > 0 then the respective feasible allocation is Pareto-optimal. Proof. Let us assume by contradiction that we have a feasible allocation (X, p ) that is Pareto-superior to (X, p) and (a) and (b) hold. The utility of the auctioneer does not decrease. Thus, the sum of the payments of the bidders fulfills i I p i i I p i. If i I p i > i I p i then an allocation (X, p ) where i I p i = i I p i exists, which is Pareto-superior compared to (X, p) as well: simply give the additional payments back to some of the bidders. We can therefore restrict our analysis to the cases with (Fact a) i I p i = i I p i. In the following parts of the proof we study a set I that contains all agents with positive valuations, and we define a sequence of subsets of I ordered by that starts with and ends with I. First we show that all agents in I who have not spent all their budget in (X, p) and appear the first time in a subset S of the sequence have the lowest valuation among all agents in S. Then we show that all agents in I who spent all their budget in (X, p) cannot get more weighted capacity in X than in X. Furthermore, we use Lemma 4 to show that no subset of agents in the sequence can get more weighted capacity in X than in X. After this, we can show by induction over the sequence that the social welfare of X cannot be higher than the social welfare of X. This leads immediately to a contradiction to the assumed Pareto-superiority of (X, p ) over (X, p). For all i I let the set T i from Definition 13 be determined by X. We define I = i I:v T i>0 i and show first some facts about I and I \ I. By Lemma 3, induces a total order on the sets T i and thus there is a largest T i in this order. For this set T i it holds that T i = I. Thus, (Fact b) there exists an i I with v i > 0 for which I = T i. Let i I be an agent with v i > 0. By (b) ṽ i > 0 holds, implying that no u with v u = 0 is in Ñi. Furthermore, no u with v u = 0 is in L i by the definition of L i. Thus, no u with v u = 0 is in T i. It follows that (Fact c) I = {i I v i > 0}. The agents i I \ I have v i = 0 and by Pareto-superiority of (X, p ) over (X, p) it follows that x i v i p i x iv i p i, and thus, (Fact d) p i p i for all i I \ I. Now we introduce an ordered sequence of subsets of I that we use later in an induction. By Lemma 3 the relation forms a total order on the sets T i with i B := B I. Reorder the bidders, such that T 1,..., T B are the sets T i with i B ordered by and that T 1 is the smallest set. We define T 0 = and T B +1 = I. Furthermore, we let v Ti = min u Ti v u for i = 1,..., B + 1, v T0 = v T1, and v T B +2 = 0. We will use T i for T i \ T i 1. It is easy to see that (Fact e) for an agent u T i B it holds that v u = v Ti : Since u T i B it follows that u T 0,..., T i 1. Thus, T u = T i for some i i because u B and u T u, and 1 The idea of the proof for Proposition 3 is based on the proof of in Lemma 3.8 of [Goel et al., 2012]. An independent proof of the proposition can be found in earlier versions of this work.

53 4.2. PARETO-OPTIMALITY CHARACTERIZATION 45 moreover, T i T i = T u = Ñu {u} L u since i > i implies that T i T i. The definition of L u implies that all w L u satisfy v w v u and (a) implies that all w Ñu satisfy v w v u. Thus all w T i satisfy v w v u which implies v u = min x Ti v x = v Ti. We partition I into three sets, namely I B, C +, and C. Recall that c i = j J α jx i,j and c i = j J α jx i,j for all i I. Formally we define C = I \B, C + := {i C c i > c i}, and C := C \ C + and show (Fact f): C + =. Which implies that the agents with positive valuations who spent their full budget under X cannot get more weighted capacity under X. The sequence of inequalities below follows by the following arguments: The first inequality holds since p u = b u p u for all u C C + ; the second inequality follows from Pareto-superiority of (X, p ) compared to (X, p); and the third inequality follows because T i \ C + = ( T i B ) ( T i C ), all u T i B have v u = v Ti by Fact e, and all u C have c u c u and all u T i have v Ti v u. u T i (p u p u) u T i\c + (p u p u) u T i\c + v Ti (c u c u) = u T i\c + v u (c u c u) u T i v Ti (c u c u) + u T i C + v Ti (c u c u ). (4.1) The next sequence of inequalities holds for the following reason: The first inequality follows from Fact a and Fact d; the second inequality follows by summing (4.1) for i = 1,..., B + 1; and the third inequality holds since B +1 i=1 u T i v Ti (c u c u) = B +1 i=1 (v Ti v Ti+1 ) u T i (c u c u) 0 as u T i (c u c u) 0 for all T i by Lemma 4, which applies by Fact b also for T B B +1 i=1 u T i (p u p u) B +1 i=1 u T i v Ti (c u c u) + B +1 i=1 B +1 i=1 u T i C + v Ti (c u c u ) u T i C + v Ti (c u c u ). (4.2) It follows that B +1 i=1 u T v i C + T i (c u c u ) 0. Since v u > 0 for all u I and T i I it holds that v Ti > 0. Thus, since c u > c u for all u C + it follows that C + has to be empty and C = C. We can prove now by induction that (Fact g) u T i v u (c u c u) v Ti u T i (c u c u) for all T i. For i = 0 we have that T 0 = and thus the claim holds. For i > 0 we have that

54 46 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES u T i\b v u (c u c u) v Ti u T i\b (c u c u) because for all u C = C T i \ B (by Fact f) holds c u c u 0 and for all u T i holds v u v Ti. Furthermore, we have that u T i B v u (c u c u) v Ti because of Fact e and by induction it holds that u T i 1 v u (c u c u) v Ti 1 u T i B (c u c u) u T i 1 (c u c u) It follows that u T i v u (c u c u) v Ti u T i (c u c u) for all T i. Now we finish the proof by generating a contradiction. By Lemma 4, which applies by Fact b also for T B +1, it holds that u T i (c u c u) 0 for all i = 1,..., B + 1, and thus, v Ti u T i (c u c u) 0. Consequently, it holds u I v uc u u I v uc u because u T i v u (c u c u) v Ti u T i (c u c u) 0 by Fact g, T B +1 = I, and v u = 0 for all u I \ I by Fact c. Thus, the social welfare under (X, p) is at least as large as under (X, p ). This implies that u I (v uc u p u ) u I (v uc u p u) by Fact a. Paretosuperiority of (X, p ) compared to (X, p) implies that for one agent w I it holds that v w c w p w < v w c w p w. Hence, u I\{w} (v uc u p u ) > u I\{w} (v uc u p u), which implies that for a u I \ {w} it holds that v u c u p u > v u c u p u. This, contradicts our assumption that (X, p ) is Pareto-superior. By Proposition 2 and Proposition 3 we know that a feasible allocation that contains no trading swap sequence is Pareto-optimal. Moreover, by Proposition 1 an allocation resulting from a trading swap sequence is Pareto-superior, and thus, a feasible allocation that contains a trading swap is not Pareto-optimal. This concludes the proof of Theorem Deterministic clinching auction for the divisible case We describe next our deterministic clinching auction for divisible slots and show that it is IC, IR, and PO. The auction repeatedly increases a price per weighted capacity and gives different weights to different slots depending on their CTRs. To perform the check whether all remaining unsold weighted capacity can still be sold we solve suitable linear programs. We will show that if the allocation of the auction did not fulfill the characterization of Pareto-optimality given in Section 4.2, i.e., if it contained a trading swap sequence, then one of the linear programs solved by the auction would not have computed an optimal solution. Since this is not possible, it will follow that

55 4.3. DETERMINISTIC CLINCHING AUCTION FOR THE DIVISIBLE CASE 47 the allocation is PO. A formal description of the auction is given in the procedure Auction. The input values of procedure Auction are the bids v, budget limits b, and slot constraints κ that the bidders communicate to the auctioneer at the beginning of the auction, and information about the qualities of the slots α. We assume that bidders bid their valuation because Proposition 4 shows that bidding the valuation is a dominant strategy; thus, we use v i for bidder i s bid and valuation. Note that the auction is a so called one-shot auction ; the bidders are asked once for their bids at the beginning of the auction and then they cannot input any further data. In particular, the demand of bidder i for weighted capacity is computed by the mechanism based on i s remaining budget b i p i, the current price π, and the bid v i : The demand D i (b i p i, π) is (b i p i )/π if v i π > 0, it is infinite if π = 0, and it is zero otherwise. Algorithm 1 Clinching auction for divisible slots 1: procedure Auction(I, J, α, κ, v, b) 2: π 0; c i 0, p i 0, d i i I 3: while i I ci < j J αj do unsold weighted capacity exists 4: for all i in I with d i > D i (b i p i, π + 1) do for all bidder whose demand has to be updated 5: (X, γ) Compute solution of LP 1 for (c, d, i ) solve linear program for i 6: (c i, p i ) (c i + γ i, p i + γ i π) update capacity and payment of bidder i 7: d i D i (b i p i, π + 1) update demand variable of bidder i 8: end for 9: π π + 1 increase price 10: end while 11: return (X, p) 12: end procedure In the linear program defined next, bidder i and the coefficients c and d change during the auction, while the coefficients α and κ are fixed. Linear Program 1. minimize γ i s.t.: (a) i I xi,j = 1 j J assign all slots (b) j J xi,j = κi i I slot constraint (c) j J xi,j αj γi = ci i I assign value to γi (d) γ i d i i I demand constraint (e) x i,j 0 i I, j J (f) γ i 0 i I We assume throughout this section that v i N 0 and b i Q 0 for all i I. 2 Furthermore, we assume that the input I is ordered such that all i I with bid v i = 0 are in the order before all i I with bid v i > 0, and all i I with v i > 0 are ordered independently of their bids. This order is used by the for-loop in Line 4 and is needed 2 All the arguments go through if we simply assume that v i Q 0 for all i I and there exists a publicly known value z R + such that for all bidders i and i either v i = v i or v i v i z.

56 48 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES to show PO in Theorem 5; it is necessary to avoid the existence of trading swaps that do not require monetary compensation. Finally, we assume that m = i I κ i. If m > i I κ i we could add dummy-bidders with valuation v i = 0 and budget b i = 0, that is, they have to pay no money and they are not competing with the other bidders. If m < i I κ i we could add dummy-items with quality α j = 0, that is, they have no value for any bidder. Thus, the slot constraints imply j J x i,j = κ i for all i I. The state of the auction is defined by the current price π, the weighted capacity c i that bidder i I has clinched so far, and the payment p i that has been charged so far to bidder i. The crucial point of the auction is that it sells only weighted capacity γ i to bidder i at a certain price π, if it can sell j J α j i I c i γ i to the other bidders but not more. The auction computes γ i by solving an LP. We use an LP as there are two types of constraints to consider: the slot constraint in Line (b) of the LP, which constrains unweighted capacity, and the demand constraint in Line (d) of the LP, which is implied by the budget limit, and which constrains weighted capacity. In the homogeneous item setting in [Dobzinski et al., 2012, Bhattacharya et al., 2010] there are no slot constraints and the demand constraints are unweighted (i.e., α j = 1 for all j J). Thus, no LP is needed to decide what amount to sell to whom. At the beginning of the auction the price π is zero and the demand variable for each bidder i is set to d i =. For each iteration of the while-loop the auction first solves an LP for one of the bidders i who has d i > D i (b i p i, π + 1). It sells bidder i the respective γ i for price π. Next it updates i s demand variable d i. If bidder i reported a valuation v i less than π +1 the auction sets d i = 0 and i cannot get further weighted capacity. Otherwise i might get further weighted capacity in the next iteration of the while-loop but has to pay a price of at least π + 1 for it. The auction continues the previous step until d i of each bidder i corresponds to his demand for price π + 1. Then it sets π to π + 1. To illustrate the mechanism we give the following example. Example 3. There are two slots with qualities α 1 = 1 and α 2 = 2. Bidder 1 has valuation v 1 = 1, budget b 1 = 1, and slot constraint κ 1 = 1. Bidder 2 has valuation v 2 = 2, budget b 2 = 0.5, and slot constraint κ 2 = 1. The auction starts for both bidders with a price of zero and thus their demand is infinite. First we solve an LP for bidder 1. He is assigned a weighted capacity of one for price zero, since the most weighted capacity that we can assign to bidder 2 is the quality of slot 2. Then by updating his demand variable we implicitly set the price of bidder 1 to one. Next, we solve an LP for bidder 2. After this we sell a weighted capacity of one to bidder 2, since the most weighted capacity that we can assign to bidder 1 is the quality of slot 2 and he can also afford just an additional weighted capacity of one. Then we set the price of bidder 2 implicitly to one and continue with the next iteration. We solve an LP for bidder 1; bidder 2 can only afford an additional weighted capacity of one half. Hence, we have to sell the other half that is left to bidder 1. Next we sell the other half to bidder 2. Each bidder gets a weighted capacity of one and a half and pays a half. The only possible assignment is that each bidder gets half of the first slot and half of the second slot. It is crucial for the progress and the correctness of the mechanism that there is a feasible solution for each LP we try to solve.

57 4.3. DETERMINISTIC CLINCHING AUCTION FOR THE DIVISIBLE CASE 49 Lemma 5. There exists a feasible solution for all the linear programs that Algorithm 1 has to solve. Proof. We show the claim by induction on the linear programs that Algorithm 1 solves. Let LP t be the t-th such LP. There is a feasible solution for LP 1 as the demand d i of every bidder is unlimited. Hence, we can set X such that j J x i,j = κ i i I and can make γ i as large as necessary for every bidder i. Next let us inductively assume that there was a feasible solution for LP t. As there exists a feasible solution for LP t, we obtain an optimal solution (X, γ) by solving LP t. After the call, c i is increased by γ i, and thus, (X, γ f ) with γi f = γ i for i i and γi f = 0 for i = i is a feasible solution of LP t+1, which uses the new c-values. Since γi f = 0, we know that (X, γ f ) is a feasible solution for LP t+1 even if the demand variable d i was decreased. Thus the inductive claim holds. The previous lemma implies that the final assignment X gives a feasible solution for the final LP. Thus, X fulfills conditions (1) and (2) for a feasible allocation. Condition (3) is also fulfilled as by the definition of the demand of a bidder, the auction guarantees that b i p i for all i I. Thus, the allocation (X, p) computed by the auction is a feasible allocation. As no bidder is assigned weighted capacity if the price is above his valuation and the mechanism never pays the bidders, the auction is IR. As it is an increasing price auction, it is also IC. We show this formally in the next proposition. Proposition 4. The auction in Algorithm 1 is individually rational and incentive compatible. Proof. Since no bidder will ever pay a higher price than his reported valuation and the demand is set so that b i p i, individual rationality follows. We next show incentive compatibility and use v for the bids and v for the real valuations. First observe that a bidder i with v i = 0 cannot increase his utility by bidding v i > 0. Bidder i s utility is zero when bidding v i = 0 and cannot become positive by bidding v i > 0. Let us now consider a bidder i with v i > 0. We first show that our ordering assumption causes no problems. Observe that if i bids v i = 0 then i is selected in an earlier or the same iteration of the for-loop during the first iteration of the while-loop when the price for the bidder is zero. That is, the set of bidders processed before i when v i = 0 is a subset of the set of bidders processed before i when v i = v i. Thus, all the bidders with positive bid still have infinite demand and the optimal solution of the LP for i cannot increase. More formally, assume first that i bids his valuation v i = v i, let the solution of the first LP for bidder i be (X, γ), and let the parameters of the LP be d and c. Next consider the case where i bids v i = 0, let the first LP for bidder i be LP, and let the parameters of LP be d and c. It holds that c c and d d. Moreover, for all u with c u > c u holds d u =. It follows that (1) γ := γ +c c d by γ d and (2) 0 γ γ. Thus, (X, γ ) is a feasible solution for LP with objective value γ i = γ i. It follows that the optimal value is at most γ i. Since bidder i cannot obtain weighted capacity in the next iterations of the while loop if he bids v i = 0 and a bidder never pays a higher price than his reported valuation it follows that bidding v i = 0 does not increase i s utility.

58 50 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES Again consider a bidder i with v i > 0 and recall that by the construction of the auction, each bidder i with v i > 0 never pays a higher price than his reported valuation. If bidder i s reported valuation is v i and 0 < v i < v i, his demand variable d i is zero for all prices larger than v i. Thus, his utility cannot increase by reporting v i as the weighted capacity he gets for each price π v i cannot increase and he will lose all weighted capacity that he clinched at a price larger than v i. Moreover, if his reported valuation is v i > v i, he gets the same weighted capacity for each price π v i. He might receive additional weighted capacity at a price larger v i, but this cannot increase his utility. Thus, the auction is IC. We show finally that the allocation (X, p) our auction computes does not contain any trading swap sequence, and thus, by Theorem 4 it is PO. The proof shows that every trading swap sequence in (X, p) would lead to a superior solution to one of the linear programs solved by the mechanism. Since the mechanism found an optimal solution this leads to a contradiction. Theorem 5. The allocation returned by the auction in Algorithm 1 is Pareto-optimal. Proof. We will show that the allocation does not contain any tss. Let (X f, p f ) be the final allocation computed by the auction and assume by contradiction that there exists a tss (δ, a) where the length of a is k + 1. We define u = a 0 and w = a k. Consider the allocation (X, p ) that results from (X f, p f ) through the tss (δ, a) where δ = δ/2 and that is Pareto-superior to (X f, p f ) by Proposition 1. Define c f i := j J α jx f i,j and c i := j J α jx i,j for all bidders i. Note that c w = c f w δ, c u = c f u + δ, and c i = cf i i I \ {u, w} by (A2). We will show that (X, p ) can be used to construct a smaller feasible solution to one of the linear programs solved by the algorithm. Since the linear program has found the minimal solution this leads to a contradiction with the assumption that there exists a tss in (X f, p f ). The value c w of bidder w increases only when an LP was trying to minimize γ w and returns a non-zero value for γ w. Since c f w > c w, there exists a unique LP solved for bidder w by the auction that has parameters c and d, for which c w c w, and where the solution (X, γ ) satisfies c w + γw > c w. We name the linear program LP and show the contradiction for it. Let π be the price and p be the payment vector at the time when we solve LP. We first show that the outcome of the auction (X f, p f ) corresponds to a feasible solution (X f, γ f ) for LP where γi f = cf i c i for all i I and that (Xf, γ f ) actually fulfills a stronger version of Constraint (d). Claim 5. The solution (X f, γ f ) is feasible for LP. It holds that (1) if π > 0 then γ f i d i bi pf i π for all i I with d i > D i(b i p i, π + 1) and (2) for all i I with d i D i (b i p i, π + 1) and v i π + 1 it holds that γi f d i bi pf i π +1. Proof. Let i be a bidder in I. First we show that (X f, γ f ) fulfills the constraints of LP. Since the allocation (X f, p f ) is derived from the final linear program executed by the algorithm, X f fulfills the Constraints (a), (b), and (e). Constraint (c) holds by definition of γ f and Constraint (f) holds because γ f = c f c 0. It remains to show

59 4.3. DETERMINISTIC CLINCHING AUCTION FOR THE DIVISIBLE CASE 51 that (d) is fulfilled as well. Recall that if π = 0 the case d i > D i(b i p i, π + 1) takes place before the demand of bidder i has been updated in the first iteration of the while-loop. Thus, if π = 0 and d i > D i(b i p i, π + 1) then d i = and (d) holds. Moreover, d i D i(b i p i, π + 1) and v i < π + 1 imply d i = D i(b i p i, π + 1) = 0. It follows that γi f has to be zero by the condition of the for-loop, and thus, (d) holds. Otherwise Constraint (d) will follow from (1) and (2). For (1) and (2) notice that (b i p f i ) is the remaining budget of bidder i at the end of the auction, that is, the money not spent by i, and that bidder i clinched γi f = cf i c i weighted capacity after LP was solved. To (1): Consider first the case where π > 0 and d i > D i(b i p i, π + 1). Note that bidder i has a remaining budget of d i π when LP is solved. Thus, bidder i pays d i π (b i p f i ) for all the weighted capacity γf i that was not clinched before LP was solved. Moreover, the price that he pays per weighted capacity in this and the following iterations is at least π. It follows that γ f i π d i π (b i p f i) To (2): Consider next a bidder i I with d i D i(b i p i, π +1) and v i π +1. By Line 7 it holds that d i is equal to D i (b i p i, π ) or D i (b i p i, π + 1). Since D i (b i p i, π + 1) D i (b i p i, π ) it holds that d i = D i (b i p i, π + 1). Note that every such bidder has a remaining budget of d i (π + 1) when LP is solved. Thus, bidder i pays d i (π + 1) (b i p f i ) for all the weighted capacity γf i that was not clinched before LP was solved. Moreover, the price that he pays per weighted capacity in this and the following iterations is at least π + 1. It follows that γ f i (π + 1) d i (π + 1) (b i p f i) Next we define γ i = c i c i for all i I and show that (X, γ ) is a feasible solution of LP and that γ w < γw, thus leading to a contradiction. By (A2) and Claim 5 it holds that X satisfies Constraints (a) and (b) for LP. By the definition of γ Constraint (c) also holds. Constraint (e) is satisfied for X by Claim 5 and because (S3) holds for (δ, a), and thus, also for (δ, a). Constraint (f) is satisfied because c i c i for all i I: LP was selected such that c w c w, and we have c i cf i c i for all i I \ {w}. We next show that also Constraint (d) is satisfied. First note that for all i I \ {u, w} we know that γ i = γf i, and thus, Constraint (d) holds for such i by Claim 5. For i = w, it holds that c i < cf i, and thus, γ w < γw f d w by Claim 5. Hence Constraint (d) also holds for i = w. For i = u, we know that γ u = γu f + δ as c u = c f u + δ and we have to show that d u γ u. We first consider the case π = 0. Since v u > v w and v w 0 we know that v u π + 1. Assume first that d u > D u (b u p u, π + 1). Thus, the demand of bidder u was not updated when LP was called and is still infinite. Hence, d u γ u. Next assume that d u D u (b u p u, π + 1). Thus, the demand of bidder u was already updated when LP was called for w; by the ordering of the input this implies that v w > 0, i.e., v w π + 1. Hence, b u p f u p u p f u = v w δ (π + 1)δ. By Claim 5 and v u π + 1 it follows that

60 52 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES d u γ f u + b u p f u π + 1 γf u + δ = γ u Next, we consider the case π > 0. Since we solve LP for w we know that d w > 0 when LP is solved, and thus, v w π. Hence, b u p f u (p f u + v w δ) p f u = v w δ π δ Assume first that d u > D u (b u p u, π + 1). By Claim 5 it follows that d u γ f u + b u p f u π γ f u + δ = γ u + δ δ > γ u Next assume that d u D u (b u p u, π + 1). By Claim 5 and v u π + 1 it follows that π d u γu f + b u p f u π + 1 γf u + δ π + 1 γf u + δ 1 2 = γf u + δ = γ u It remains to show that γ w < γ w. Recall that by the definition of LP it holds that c w + γ w > c w, while, by definition of γ w, c w = c w + γ w. Thus γ w < γ w, which leads to the desired contradiction. 4.4 Randomized clinching auction for the indivisible case We will now use the allocation computed by the deterministic auction for divisible slots to give a randomized auction for multiple keywords with indivisible slots that ensures that bidder i receives at most κ i slots for each keyword. The randomized auction has to assign to every slot j J exactly one bidder i I for each keyword r R. We call a distribution over allocations for the indivisible case Pareto-superior to another such distribution, if the expected utility of a bidder or the auctioneer is higher, while the expected utilities of the others are at least as large. If a distribution has no Pareto-superior distribution, we call it Pareto-optimal. The basic idea is as follows: given the PO allocation for the divisible case, we construct a distribution over allocations of the indivisible case such that the expected utility of every bidder and of the auctioneer is the same as the utility of the bidder and the auctioneer in the divisible case. The mechanism for the indivisible case would, thus, first call the mechanism for the divisible case (with the same input) and then convert the resulting allocation (X d, p d ) into a representation of a distribution over PO allocations for the indivisible case. It then samples from this representation to receive the allocation that it outputs. As during all these steps the expected utility of the bidders and the auctioneer remains unchanged and the mechanism for the divisible case is IR and IC, the mechanism for the indivisible case is IR in expectation and IC in expectation. To show that the final allocation is PO in expectation and also PO ex post we use the following lemma.

61 4.4. RANDOMIZED CLINCHING AUCTION FOR THE INDIVISIBLE CASE 53 Lemma 6. For every probability distribution over feasible allocations in the indivisible case there exists a feasible allocation in the divisible case such that the utility of the bidders and the auctioneer in the divisible case equals their corresponding expected utilities in the indivisible case. Proof. We show first that for every feasible allocation (X, p) in the indivisible case there exists feasible allocation (X d, p) in the divisible case where all the bidders and the auctioneer have the same utility. The utility of the auctioneer stays unchanged because we leave the payments unchanged. We set x d i,j = 1 R r R x i,j,r for all i I and j J. The utility of bidder i is the same for (X, p) and (X d, p), since the utility of bidder i is ( α j x i,j,r )v i p i = α j x d R i,jv i p i j J r R j J for (X, p). The slot constraint for (X, p) implies κ i max x i,j,r ( 1 x i,j,r ) = x d i,j r R R j J j J r R j J for all i I, and therefore it implies the slot constraint in (X d, p). Since all the slots are fully assigned to the bidders in (X, p), and consequently for (X d, p), it follows that (X d, p) is feasible. Given a probability distribution over feasible allocations for the indivisible case, transform each feasible allocation that has a non-zero probability into a feasible allocation for the divisible case. Then create a new allocation for the divisible case by adding up all of these feasible allocations for the divisible case weighted by the probability distribution. Since the weights are created by a probability distribution, they add up to one, and thus, the resulting combined allocation fulfills Conditions (1) and (2) of a feasible allocation. As the payment is identical to the payment for the indivisible case, Condition (3) is also fulfilled. Lemma 6 implies that any probability distribution over feasible allocations in the indivisible case that is Pareto-superior to the distribution generated by our auction would lead to a feasible allocation for the divisible case that is Pareto-superior to (X d, p d ). This is not possible as (X d, p d ) is PO. Thus, the mechanism for the indivisible case described above is PO in expectation. Additionally, since our auction selects only allocations having a positive probability, each realized allocation is ex-post Pareto-optimal: if in the indivisible case there existed a Pareto-superior allocation to one of the allocations that gets chosen with a positive probability in our auction, then a Pareto-superior distribution would exist, and thus, a Pareto-superior allocation would exist in the divisible case. By the same argument as above this would lead to a contradiction. See Section for a discussion of the differences between PO ex post and PO in expectation. We still need to explain how to use the PO allocation (X d, p d ) for the divisible case to give a probability distribution for the indivisible case such that the expected

62 54 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES utility of every bidder for the probability distribution is equal to their utility in the divisible case. We will use the following steps: (a) We will reduce the computation of the probability distribution to a scheduling problem with preemption on uniform processors with the objective to minimize the finishing time. (b) We use Birkhoff s theorem [Schrijver, 2003] to show that an optimal schedule exists and has finishing time one. (c) Then we argue that an algorithm by Gonzalez and Sahni [Gonzalez and Sahni, 1978] can be used to compute a schedule with finishing time one. This schedule represents a probability distribution on feasible allocations in the indivisible case and we show how to use it to sample from the probability distribution. Computing the probability distribution and sampling from it can be done in time linear in the number of slots m. We first define the input and the output clearly. For the computation of the probability distribution the input is the set of slots J, the set of bidders I, the slot constraints κ i for all i I, the qualities α j for all j J, and a feasible divisible allocation (X d, p d ) that also defines the weighted capacities c d i for all i I. The output is a function that gives us for each number t (0, 1] an assignment X(t) of slots to bidders, where each bidder i I gets κ i slots. The assignment X(t) is a binary matrix where (X(t)) i,j = 1 if and only if slot j is assigned to bidder i. For a random number T that is uniformly distributed on (0, 1], the expected weighted capacity E[ j J (X(T )) i,jα j ] for each bidder i I has to be equal to c d i. Given the assignment function X(t) it suffices to draw R numbers t 1,..., t R uniformly from (0, 1], use the assignment X(t r ) for keyword r, and set p = p d. The expected utility of all bidders i I is equal to their utility in (X d, p d ) because E[( j J α j R (X(T )) i,j )v i p i ] = r R ( 1 R r R E[ j J (X(T )) i,j α j ])v i p i = ( 1 R c d i )v i p i = c d i v i p i (a) In the scheduling problem we consider, we have m jobs with length l 1,..., l m and m processors with speed s 1,..., s m as input. Jobs can be processed on multiple processors, but not at the same time, and preemption is allowed. In a feasible schedule every job has to be finished. That is, if t λ,j is the time length that job λ is processed on processor j in the schedule then m j=1 t λ,js j = l λ for all λ {1,..., m}. The finishing time of a schedule is the time it takes until every job is finished. The goal in the scheduling problem is to find a schedule with minimal finishing time. We first show how to convert the input for the computation of the probability distribution to the input of the scheduling problem. Recall that we suppose m = n i=1 κ i. We set λ 0 := 0, replace each bidder i I with κ i jobs λ i 1 + 1,..., λ i 1 + κ i =: λ i having length cd i κ i, set Λ(i) := {λ i 1 + 1,..., λ i }, and set Λ := i I Λ(i). Furthermore, each slot j J is a processor with speed α j. r R

63 4.4. RANDOMIZED CLINCHING AUCTION FOR THE INDIVISIBLE CASE 55 Next we show that a schedule with finishing time one gives us the desired assignment function. Suppose that we have an assignment of jobs to processors on the interval (0, 1] where no job is processed on multiple processors at the same time. As each processor represents one of the slots in J, we get an assignment function X(t) if we replace for each time t (0, 1] and for each bidder i I the jobs in Λ(i) with bidder i. As Λ(i) = κ i each bidder i gets κ i slots assigned and each slot is assigned to one bidder. For T U(0, 1] we have E[(X(T )) i,j ] = λ Λ(i) t λ,j for all i I and j J. Thus, E[ (X(T )) i,j α j ] = E[(X(T )) i,j ]α j = j J j J t λ,j α j = t λ,j α j = j J λ Λ(i) λ Λ(i) j J λ Λ(i) c d i κ i = c d i (b) We now argue that the minimal finishing time t is one for scheduling problems when the inputs are generated by the above reduction. First we restate Birkhoff s theorem, which we use for the argument. Theorem 6 (Birkhoff s theorem in [Schrijver, 2003]). Each doubly stochastic matrix is a convex combination of permutation matrices. Recall that Λ = J because Λ = i I κ i = m = J. We build an m m- dimensional square matrix T as follows. We assign for each bidder i I each job λ Λ(i) to processor j J for time t λ,j = xd i,j κ i. The matrix T has the entries t λ,j where λ Λ and j J. We show next that T is doubly stochastic, that is, the entries of the matrix are non-negative and for each column and for each row the sum of the entries is one. The sums are non-negative because x d i,j 0 and κ i 0 for all i I and j J. As the assignment X d is feasible and j J xd i,j = κ i for all i I because m = i I κ i, it follows that for each column j J of T it holds t λ,j = λ Λ i I x d i,j κ i λ Λ(i) = i I x d i,j λ Λ(i) 1 = x d i,j = 1 κ i i I and for each row λ Λ(i) with i I of T it holds t λ,j = j J j J x d i,j κ i = 1 x d i,j = 1 κ i j J Since T is doubly stochastic, we can decompose T by Birkhoff s theorem into a convex combination of permutation matrices. In a permutation matrix there is one entry in each column and each row that is one and all other entries are zero. Let k be the number of permutation matrices in the convex combination and ζ l be the coefficient of the l th permutation matrix P l for all l {1,..., k}. We construct our schedule in the following way: for the time interval ( l 1 s=1 ζ s, l s=1 ζ s] we assign job λ to processor j if (P l ) λ,j = 1. Every job λ Λ(i) for all i I is finished because

64 56 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES k ( l ) λ,j ζ l )α j = j J l=1(p t λ,j α j = j J j J x d i,j κ i α j = 1 κ i j J x d i,jα j = cd i κ i By the definition of a permutation matrix, it follows that each job is computed on exactly one processor at the same time, and each processor computes exactly one job at the same time. The finishing time is one because the time intervals of the schedule are the coefficients of a convex combination, and thus, k l=1 ζ l = 1. As the schedule has no idle time, the finishing time cannot be less than one. Thus, every optimal schedule has finishing time exactly one. (c) We can use the scheduling algorithm by Gonzalez and Sahnithat minimizes the finishing time to compute an optimal schedule with finishing time one. The schedules computed by the algorithm have at most 2(m 1) preemptions, and the computation has a time complexity that is linear in the number of jobs Λ = m. The algorithm outputs a schedule for each job, which is represented by a list of the processors on which the job gets processed; the lists contain the start time and the end time of the assignments of the jobs to the processors. Thus, we can represent the assignment function X(t) by merging the lists of all the jobs in Λ(i) to a list for each bidder i I. We can evaluate the i th row of X(t) for a certain t by traversing the list of bidder i and setting (X(t)) i,j = 1 if and only if processor j is in the list and t (a, b] where a is the start time and b is the end time in the list entry. To sample from the probability distribution for the indivisible case we pick R random numbers t r, 1 r R, uniformly at random from (0, 1] and set for each bidder i and each slot j the value x i,j,r = X(t r ) i,j. The following theorem summarizes the results in this section. Theorem 7. A PO and IR allocation for the divisible case can be converted in polynomial time without a change of the (expected) utilities into a randomized allocation for the indivisible case that is PO in expectation, PO ex post, and IR in expectation. This results in a mechanism that is PO in expectation, PO ex post, IR in expectation, and IC in expectation Pareto-optimality in expectation We show that neither PO in expectation implies PO ex post, nor PO ex post implies PO in expectation. Let us assume that we have two bidders, a single indivisible item (one slot with α 1 = 1), and a uniformly distributed random variable Y U(0, 1) that represents the random decisions taken by an auction. Consider first the case that bidder 1 has valuation v 1 = 1 and budget b 1 = 1, bidder 2 has valuation v 2 = 2 and budget b 2 = 1, and we have a value ỹ (0, 1). If we sell the item to bidder 2 for the payment p 2 = 1 (and p 1 = 0) for every realization y of Y with y ỹ the allocation is PO in expectation. However, only if we sell the item to bidder 2 also for y = ỹ every possibly realized allocation is PO. Hence, PO in expectation does not imply that each realized allocation is PO. Next consider the case that bidder 1 has valuation v 1 = 1 and budget b 1 = 1, and bidder 2 has valuation v 2 = 2 and budget b 2 = 0.5. If we sell the item to bidder 1 for the payment p 1 = 1 (and p 2 = 0) for every realization y (0, 1) each realized allocation is PO because v 1 > b 2. However, we

65 4.4. RANDOMIZED CLINCHING AUCTION FOR THE INDIVISIBLE CASE 57 could select the bidder who gets the item with probability one half, and both bidders have to pay p 1 = p 2 = 0.5 independent of the assignment. Hence, the allocation is not PO in expectation, and therefore, PO in expectation is not implied if every realized allocation is PO.

66 58 CHAPTER 4. IDENTICAL KEYWORDS WITH MANY COPIES AND DIFFERENT QUALITIES

67 Chapter 5 Non-identical keywords with identical copies 5.1 Outline This chapter is committed to the study of auctions when different keywords are on sale. We consider auctions in matching markets setting with different keywords and multiple slots of the same quality available on each keyword; we refer to this also as the keyword-problem. This is a generalization of the setting proposed in [Fiat et al., 2011] to multiple slots case. First, we present a characterization of Pareto-optimality for the given setting. Later, we propose an incentive-compatible auction that is individually-rational and Pareto-optimal. To prove the properties of the auction we formally defined two problems: the item-problem that is the problem presented in [Fiat et al., 2011] (different keywords without slots) and the keyword-problem that is the problem presented in this thesis (different keywords with identical slots). Then we prove that the our auction is IR, IC and PO through a reduction of the keyword-problem to the itemproblem The Setting We denote the keyword-problem by K = (I, R, m, v, b, S). The set I = {1,..., n} is the set of bidders. R is the set of keywords. The vector m = (m r ) r R denotes the number of slots (copies) for each keyword r R. We denote by S = (S 1,..., S n ) the vector of preference sets, where S i R is the set of keywords of interest of bidder i I. All keywords in S i are valued v i by agent i I and v = (v 1,..., v n ) is the vector of valuations. All keywords not in S i are valued zero by bidder i. Finally, b = (b 1,..., b n ) is the vector of budgets. Preference sets S i and budgets b i are public knowledge. As stated above, all slots are identical, i.e., they have same quality. Every bidder is therefore indifferent between obtaining one or the other slot of the same keyword. However, at most one slot per keyword is allocated to a single bidder (i.e., κ i = 1). We also require that at least m r bidders are interested in keyword r R, as some 59

68 60 CHAPTER 5. NON-IDENTICAL KEYWORDS WITH IDENTICAL COPIES slot would otherwise stay unsold. Thus, our setting can be described as the problem of allocating a multiset of keywords, with m r copies for keyword r R, under the constraint that each bidder i I can receive at most one copy of a keyword r S i. We denote by G the bidder/keyword bipartite graph G = (I R, E), E = {(i, r) I R r S i }. In [Fiat et al., 2011] the authors solved this problem for the case of m r = 1. The approach in [Fiat et al., 2011] was based on 1. making decisions on which items to sell to which agent by solving B-matching instances on the node-weighted bidder/keyword bipartite graph G, where the bipartite graph has a weight on each bidder equal to her demand at the current per-item price, and on 2. characterizing Pareto-optimality by the non-existence of trading paths (simple alternating paths that satisfy additional conditions) in the bipartite graphs. Along this section we will refer to an instance of the problem described in [Fiat et al., 2011], where all the keywords have exactly one slot, as the item-problem, in contrast to our keyword-problem where many slots for each keyword exist. We extend their approach by 1. considering selling slots on the basis of the computation of B-matchings in a bipartite graph that is also weighted on the keywords by the number of unsold slots, and by 2. showing that an instance of a keyword-problem can be reproduced as an ad-hoc constructed instance of an item-problem. A feasible allocation (X, p) of the keyword-problem is denoted by a tuple X = (X 1, X 2,..., X n ), where X i S i represents the keywords allocated to bidder i, and p = (p 1, p 2,..., p n ) is the vector of payments. It holds that {i I r X i } m r r R, p i b i for all i I, and X i v i p i. By assumption, for each item there are not more copies than the number of bidders interested in it, i.e., {i I r S i } m r r R. We also represent X as a B-matching in the bipartite graph G. The utility of bidder i is defined by u i := v i X i p i, and the utility of the auctioneer is n i=1 p i. Pareto-optimality has been related in previous works [Dobzinski et al., 2012, Fiat et al., 2011] to the non-existence of trading options between bidders. We use a definition of a trading path in graph G that also admits non-simple paths. Definition 14. A path σ = (i 1, r 1, i 2, r 2,..., i t 1, r t 1, i t ), where i k I for k = 1,..., t and r k R for k = 1,..., t 1, is an alternating path for allocation X if (i k, r k ) X, r k S ik+1, and r k X ik+1 for all 1 k < t. Definition 15. A path σ = (i 1, r 1, i 2, r 2,..., i t 1, r t 1, i t ) is a trading path with respect to allocation (X, p) if the following holds: 1. σ is an alternating path in X, 2. the valuation of bidder i t is strictly greater than the valuation of bidder i 1 (i.e., v it > v i1 ),

69 5.1. OUTLINE 61 Figure 5.1: Example of an alternating path. 3. the remaining (unused) budget b i t of bidder i t at the conclusion of the auction is at least the valuation of bidder i 1 (i.e., b i t v i1 ). Note that a keyword or an item could appear multiple times in the trading path. In this case the trading path contains cycles. Now we formally define the concept of the item-problem I. An item-problem I is an instance of a keyword-problem K where each keyword has exactly one copy, i.e., for all r R holds m r = 1. An item-problem represents the setting defined and studied in [Fiat et al., 2011]. We say that an item-problem is an equivalent item-problem I K of a keywordproblem K and an allocation (X, p) if I K respects the definition below. Definition 16 (Item-problem). An equivalent item-problem is denoted by I K = (Ī, R, v, b, S) and it is derived by an instance of K and an outcome (X, p) of K. In particular: The set of bidders of the equivalent item-problem I K is the same as in K, i.e., Ī = I. The valuation vector of the equivalent item-problem I K is the same as in K, i.e., v = v. The budget vector of the equivalent item-problem I K is the same as in K, i.e., b = b. The set of items R of the equivalent item-problem I K is derived by the set R and the vector m of the keyword-problem K. For each r R exists a set C r containing m r items such that C k and C w are disjoint if k w. We define R = r R C r. The preference set vector S of the equivalent item-problem I K is derived from K and (X, p). For each bidder i I we define the preference set S i as follows: For each keyword r S i in the keyword-problem that is not allocated to i, i.e., r X i, we add all m r items in C r to S i. For each keyword r S i in the keyword-problem that is allocated to i, i.e., r X i, we add only one item in C r to S i. We select this item such that for any two bidders i and i with r X i and r X i it holds that S i C r and S i C r are disjoint.

70 62 CHAPTER 5. NON-IDENTICAL KEYWORDS WITH IDENTICAL COPIES (a) Keyword-problem (b) Item-problem Figure 5.2: Example of an item-problem constructed by a keyword-problem and its outcome. Therefore, we will use I, v and b to describe the set of bidders, the vector of valuations and the vector of budgets respectively, in both the keyword-problem and the item-problem. Figure 5.2 shows an instance of a keyword-problem, the outcome of the keywordproblem, and the equivalent item-problem. We denote by G IK the bidder/keyword bipartite graph G IK = (I R, Ē) where Ē = {(i, j) I R j S i }. Therefore we refer to G IK as Ḡ. The reader that is interested in a brief discussion of the auction provided in [Fiat et al., 2011] can refer to Appendix B. 5.2 Pareto-optimality characterization Now we provide a characterization of Pareto-optimality in Theorem 8. In order to prove the theorem, we initially prove that if the allocation is Pareto-optimal then all the slots of the keywords are sold and there are no trading-paths. In order to prove the other direction, we have to show that for every trading-path of an equivalent itemproblem I K there is a corresponding trading-path in the original keyword-problem K. Theorem 8. Any allocation (X, p) is PO if and only if (1) all slots of the keywords are sold in (X, p), and (2) there are no trading paths in (X, p). Proof. Only if direction. In order to prove the only-if-direction we show that if the allocation is Pareto-optimal then all slots of the keywords are sold and there are no trading paths. Assume by contradiction that (X, p) is PO and there are some unallocated slots or there are some trading paths. If the allocation (X, p) does not assign all items then it is clearly not PO. We can get a better allocation by assigning unsold items to any bidder i with such items in S i \ X i. This increases the utility of bidder i.