Double-Sided Markets with Strategic Multi-dimensional Players1footnote 11footnote 1This work is supported by the Horizon 2020 funded project TYPES (Project number: 653449. Call Identifier H2020-DS-2014-1). We are submitting this paper for confidential review to be considered for publication in 2017.

Double-Sided Markets with Strategic Multi-dimensional Players111This work is supported by the Horizon 2020 funded project TYPES (Project number: 653449. Call Identifier H2020-DS-2014-1). We are submitting this paper for confidential review to be considered for publication in 2017.

Moran Feldman Department of Mathematics and Computer Science, the Open University of Israel. E-mail: moranfe@openu.ac.il.    Rica Gonen Department of Management and Economics, the Open University of Israel. E-mail: gonenr@openu.ac.il
Abstract

We consider mechanisms for markets that are double-sided and have players with multi-dimensional strategic spaces on at least one side. The players of the market are strategic, and act to optimize their own utilities. The mechanism designer, on the other hand, aims to optimize a social goal, i.e., the gain from trade. We focus on one example of this setting which is motivated by the foreseeable future form of online advertising.

Online advertising currently supports some of the most important Internet services, including: search, social media and user generated content sites. To overcome privacy concerns, it has been suggested to introduce user information markets through information brokers into the online advertising ecosystem. Such markets give users control over which data get shared in the online advertising exchange. We describe a model for the above foreseeable future form of online advertising, and design two mechanisms for the exchange of this model: a deterministic mechanism which is related to the vast literature on mechanism design through trade reduction and allows players with a multi-dimensional strategic space, and a randomized mechanism which can handle a more general version of the model.

Keywords: Mechanism design, double-sided market, multi-dimensional players, online advertising market

\DeclareCaptionType

mechanism[Mechanism][List of Mechanisms]

1 Introduction

Billions of transactions are carried out via exchanges at every given day, and the number of transactions and exchanges continues to grow as the need for competitiveness promotes adoption. The design of one-sided incentive compatible (truthful bidding) mechanisms for exchanges is relatively well understood. However, incentive compatible multi-sided mechanisms present a significantly more challenging problem as they introduce more sophisticated requirements such as budget balance.

More specifically, we are interested in designing exchanges (mechanisms) for multi-sided markets with strategic players. The players of the market are strategic, and act to optimize their own utilities. The mechanism designer, on the other hand, aims to optimize a social goal, i.e., the gain from trade (the difference between the total value of the sold goods for the buyers and the total costs of these goods for the sellers). The design of such mechanisms raises a few interesting questions. Can the mechanism maintain simultaneously different desirable economic properties such as: individual rationality (IR)—participants do not lose by participation, incentive compatibility (IC)—players are incentivized to report their true information to the mechanism and budget balance (BB)—the mechanism does not end up with a loss. Moreover, can the mechanism maintain these properties while suffering only a bounded loss compared to the optimal gain from trade? Finally, can this be done when all the players have a multi-dimensional strategic space?222We often refer to players with a multi-dimensional strategic space as multi-dimensional players.

The above questions can be studied in the context of many multi-sided markets. We focus on one such market, and leave the consideration of other multi-sided markets for future work. The market we consider is motivated by online advertising in its foreseeable future form. Online advertising currently supports some of the most important Internet services, including: search, social media and user generated content sites. However, the amount of information that companies collect about users increasingly creates privacy concerns in society as a whole, and even more so in the European society. In recent years EU regulators have actively been looking for solutions to guarantee that users’ privacy is preserved. In particular, the EU regulators have been looking for tools that enable the end user to configure their privacy settings so that only information allowed by the end-user is collected by online advertising platforms.

The market we study is induced by a new solution we suggest for the above privacy issue. In this solution mediators serve as the interface between end-users and the other players in the online advertising market. Each user informs his mediator of the attributes she is willing to reveal, and her cost, i.e., the compensation she requires for every ad she views. The mediator then tries to “sell” the user on the advertising market based only on the attributes she agreed to reveal, and, if successful, pays her the appropriate cost out of the amount he got from the sell.

As revealing more attributes is likely to result in a more profitable sale, our solution provides incentives for users to share their information with the advertising market while allowing users to retain control of the amount of information they would like to share. Notice that the fact that our solution motivates users to participate in the advertising market, and even to provide more precise information for targeting campaigns, means that our solution improves the efficiency of the advertising system and the digital economy as a whole (in addition to answering the privacy concerns discussed above). This is in sharp contrast to other natural approaches for dealing with privacy issues, such as cryptography based approaches, which reduce the amount of information available to the advertisers but give them nothing in return.

The advertising market induced by the above solution has mediators on one side, and advertisers on the other side. Each mediator has a set of users associated with him, and he is trying to assign these users to advertisers using the market. Each one of the users has a non-negative cost which she has to be paid if she is assigned to an advertiser. The mediators themselves have no cost of their own, however, each of them has to pay his users their cost if they are assigned to advertisers. Thus, the utility of a mediator is the amount paid to him minus the total cost of his users that are assigned. Finally, each advertiser has a positive capacity determining the number of users she is interested in targeting, and she gains a non-negative value from every one of the users assigned to her (as long as her capacity is not exhausted). Thus, the advertiser’s utility is her value multiplied by the number of users assigned to her (as long as this number does not exceed her capacity) minus her total payment.

A mechanism for the above market knows the mediators and the advertisers, but has no knowledge about their parameters or about the users. The objective of the mechanism is to find an assignment of users to advertisers that maximizes the gain from trade. In addition, the mechanism also decides how much to charge (pay) each advertiser (mediator). In order to achieve these goals, the mechanism receives reports from the advertisers and mediators. Each advertiser reports her capacity and value, and each mediator reports the number of his users and their costs. The mediators and advertisers are strategic, and thus, free to send incorrect reports. In other words, an advertiser may report incorrect capacity and value, and a mediator may report any subset of his users and associate an arbitrary cost with each user. We say that an advertiser is truthful if she reports correctly her capacity and value. A mediator is considered truthful if he reports to the mechanism his true number of users and the true costs of these users. Notice that we assume that the costs of the users are known to their corresponding mediators, i.e., the users are non-strategic. This assumption is reasonable given the high speed of the online advertising market, especially compared to the speed at which a private user can change her contract with her mediator.

To better understand the design challenge raised by this market, we observe that even if our setting is reduced to a single buyer-single seller exchange, still it is well known from [11] that maximizing gain from trade while maintaining individual rationality and incentive compatibility perforce to run into deficit (is not budget balanced). A well known circumvention of [11]’s impossibility is [10]’s trade reduction for the simple setting of double sided auctions. In [10]’s setting trade is conducted between multiple strategic sellers offering identical goods to multiple strategic buyers, where each seller is selling a single good and each buyer is interested in buying a single good. The result of [10] relaxes the requirement for optimal trade by means of a trade reduction. The trade reduction leads to an individually rational, incentive compatible and budget balance mechanism. Following [10]’s work several other mechanisms were designed using the technique of trade reduction. However, all the trade reduction mechanisms suggested in the literature to date allow only players with single dimensional strategic spaces.

1.1 Our Contribution

Given that existing trade reduction solutions do not apply in our setting, we describe new double-sided mechanisms able to handle mediators and advertisers with multi-dimensional strategic spaces. Our mechanisms guarantee desirable economic properties, and at the same time yield a gain from trade approximating the optimal gain from trade. If being truthful is a dominant strategy333Here and throughout the paper, a reference to domination of strategies should be understood as a reference to weak domination. We never refer to strong domination. for each advertiser and each mediator (regardless of other players’ strategies), then the mechanism is incentive compatible (IC); if no advertiser and no mediator have a negative utility by participating truthfully in the mechanism than it is individually rational (IR). Our objective is to construct mechanisms that are IC and IR.

We first study a special case of our setting where the advertisers’ capacities are publicly known (however, these capacities need not be all equal). The set of users of each mediator, on the other hand, remains unknown to the mechanism (i.e., the mechanism only learns about it through the mediator’s report). For this case we present a deterministic mechanism we term “Price by Removal Mechanism” (PRM) that works as follows: for every mediator find a threshold cost, and remove users of the mediator whose cost is above this threshold. Add a dummy advertiser with value that is the maximum threshold cost computed for the mediators and a capacity that is equal to the total number of users remaining. Assign the non-removed users to the advertisers using a VCG auction [15, 6, 9] in which the users are the goods and the bidders are the advertisers. Price the mediators according to their threshold cost, and price the advertisers according to the prices of the VCG auction describe above.

The method used to calculate the threshold costs of the above mechanism induce its properties. We prove that, for appropriately chosen threshold costs, the above mechanism is IC, IR, BB and provides a non-trivial approximation for the optimal gain from trade. More formally, if \tau is the size of the optimal trade, and \gamma is an upper bound, known to the mechanism, on the maximum capacity of any player (mediator or advertiser), then:

Theorem 1.1.

PRM is BB, IR, IC and \left(1-\frac{5\gamma}{\tau}\right)-competitive.

PRM generalizes the trade reduction ideas used so far in the literature for single dimensional strategic players, but is much more involved. Intuitively, PRM differs from previous ideas by the following observation. A trading set is the smallest set of players that is required for trade to occur. In the existing literature for single dimensional strategic players a trade reduction mechanism makes a binary decision regarding every trading set of the optimal trade, i.e., either the trading set is removed as a whole, or it is kept. On the other hand, dealing with multi-dimensional players requires PRM to remove only parts of some trading sets, and thus, requires it to make non-binary decisions.

Our deterministic mechanism PRM handles one type of multi-dimensional players (the mediators) and one type of single dimensional strategic space players (the advertisers). In order to enrich our strategic space even further, and allow advertisers to have multi-dimensional strategic spaces as well, we present also a randomized mechanism termed “Threshold by Partition Mechanism” (TPM). TPM applies to our general setting, i.e., we no longer assume that any capacity is known to the mechanism, and it works as follows: divide the set of mediators uniformly at random into to two sets (M_{1} and M_{2}) and divide the set of advertiser uniformly at random, as well, into two sets (A_{1} and A_{2}). Then use the optimal trade for M_{2} and A_{2} to produce threshold cost and threshold value that allow BB pricing as well as the needed reduction in trade for M_{1} and A_{1}. Analogously, use the optimal trade for M_{1} and A_{1} to produce threshold cost and threshold value that allow BB pricing as well as the needed reduction in trade for M_{2} and A_{2}.

The above description of TPM is not complete since the use of threshold cost and value from one pair (M_{i},A_{i}) to reduce the trade in the other pair might create an unbalanced reduction. To overcome this issue we create two random low priority sets: one of advertisers and the other of mediators. Then, whenever the reduction in trade is unbalanced, we remove additional low priority mediators or advertisers in order to restore balance (which can be done with high probability). The following theorem shows that the above mechanism is IC, IR, BB and provides a non-trivial approximation for the optimal gain from trade. The parameter \alpha is an upper bound, known to the mechanism, on the ratio between the maximum capacity of any player (mediator or advertiser) and the size of the optimal trade.444The parameters \gamma and \alpha both bound the maximum capacity of the players. Moreover, they are formally related by the formula \alpha=\gamma/\tau. We chose to formulate Theorems 1.1 and 1.2 in terms of the parameter that the mechanism corresponding to each theorem assumes access to.

Theorem 1.2.

TPM is BB, IR, IC and (1-28\sqrt[3]{\alpha}-20e^{-2/\sqrt[3]{\alpha}})-competitive.

We note that TPM is universally truthful, i.e., its IC property holds for every given choice of the random coins of the mechanism.

One drawback of our results is that the competitive ratios guaranteed by Theorems 1.1 and 1.2 are non-trivial only when no single advertiser or mediator has a large market power and the mechanism has access to a good bound on the maximum market power of any player. From a practical point of view we believe these assumption are both plausible. The number of agents using any given real life ad-exchange is usually very large, and the mechanism can use the large quantity of historical data available to it to estimate the bound it needs. From a more theoretical point of view, the impossibility result of [11] shows that no non-trivial competitive ratio can be achieved when one advertiser and one mediator control all the trade. This suggests, although we are unable to prove it formally, that the competitive ratio must deteriorate as a single advertiser or mediator gains more and more market power (i.e., when \gamma and \alpha increase).

1.2 Related Work

From a motivational point of view the most closely related literature to our work consists of works that involve mediators and online advertising markets, such as [1, 7, 14]. These works differ from ours in two crucial points. First, despite being motivated by the online display ads network exchange, the models studied by these works are actually auctions (i.e., one-sided mechanisms). Thus, they need not deal with the challenges and impossibility integrated by the double-sided structure of our market and the requirement to keep it from running into a deficit. Second, our focus is maximization of the gain from trade, unlike the above works which focus on revenue maximization.

Another related work involving both markets and mediators studies the phenomenon of markets in which individual buyers and sellers trade through intermediaries, who determine prices via strategic considerations [4]. An essential difference between the model of [4] and our model is that [4] does not assume private values for the players, and therefore, the impossibility of [11] does not apply in its model.

Last but not least is the literature on trade reduction and multi-sided markets. Deterministic mechanisms using trade reduction as a mean to achieve IC, IR and BB were described for various settings [10, 5, 2, 8, 12, 3]. Moreover, for a variant of the setting of [10, 3], [13] obtained a randomized mechanism achieving IC, IR and strong budget balance (i.e., it is BB and leaves no surplus for the market maker). The mutual grounds of all these settings is that all players participating in the trade have a single dimension strategic space. This idea was captured by [8] which provided a single trade reduction procedure applicable to all the above settings. In addition, [8] also defined a class of problems that can be solved by its suggested trade reduction procedure. Essentially this classification is based on partitioning the players participating in the trading set into equivalence classes.

As pointed out in the previous subsection, both our presented mechanisms extend significantly on the existing trade reduction literature. More specifically, even when all advertisers have known equal capacities (while mediators can still have a variable number of users), fitting our model into the classification of [8] still requires each mediator to have his own equivalence class (because a mediator with many users can always replace a mediator with a few users within a trading set, but the reverse is often not true). It follows that [8]’s trade reduction procedure might remove all the trade, and thus, achieves only a trivial gain from trade approximation.

2 Notation and Basic Observations

We begin this section with a more formal presentation of our model. Our model consists of a set P of users, a set M of mediators and a set A of advertisers. Each user p\in P has a non-negative cost c(p) which she has to be paid if she is assigned to an advertiser. The users are partitioned among the mediators, and we denote by P(m)\subseteq P the set of users associated with mediator m\in M (i.e., the sets \{P(m)\mid m\in M\} form a disjoint partition of P). The utility of a mediator m\in M is the amount he is paid minus the total cost he has to forward to his assigned users; hence, if x(p)\in\{0,1\} is an indicator for the event that user p\in P(m) is assigned and t is the payment received by m, then the utility of m is t-\sum_{p\in P(m)}x(p)\cdot c(p). Finally, each advertiser a\in A has a positive capacity u(a), and she gains a non-negative value v(a) from every one of the first u(a) users assigned to her; thus, if advertiser a is assigned n\leq u(a) users and has to pay t then her utility is n\cdot v(a)-t.

A mechanism for our model accepts reports from the advertisers and mediators, and based on these reports outputs an assignment of users to advertisers. In addition, the mechanism also decides how much to charge (pay) each advertiser (mediator). The objective of the mechanism is to output an assignment of users to advertisers that maximizes the gain from trade.

For ease of the presentation, it is useful to associate a set B(a) of u(a) slots with each advertiser a\in A. We then think of the users as assigned to slots instead of directly to advertisers. Formally, let B be the set of all slots (i.e., B=\bigcup_{a\in A}B(a)), then an assignment is a set S\subseteq B\times P in which no user or slot appears in more than one ordered pair. We say that an assignment S assigns a user p to slot b if (p,b)\in S. Similarly, we say that an assignment S assigns user p to advertiser a if there exists a slot b\in B(a) such that (p,b)\in S. It is also useful to define values for the slots. For every slot b of advertiser a, we define the value v(b) of b as equal to the value v(a) of a. Using this notation, the gain from trade of an assignment S can be stated as

\mathtt{GfT}(S)=\sum_{(p,b)\in S}[v(b)-c(p)]\enspace.

In addition to the above notation, we would like to define two additional shorthands that we use occasionally. Given a set A^{\prime}\subseteq A of advertisers, we denote by B(A^{\prime})=\bigcup_{a\in A^{\prime}}B(a) the set of slots belonging to advertisers of A^{\prime}. Similarly, given a set M^{\prime}\subseteq M of mediators, P(M^{\prime})=\bigcup_{m\in M^{\prime}}P(m) is the set of users associated with mediators of M^{\prime}.

2.1 Comparison of Costs and Values

The presentation of our mechanisms is simpler when the values of slots and the costs of users are all unique. Clearly, this is extremely unrealistic as all the slots of a given advertiser have the exact same value in our model. Thus, we simulate uniqueness using a tie-breaking rule. The rule we assume works as follows:

  • The mechanism chooses an arbitrary order \sigma on the mediators and advertisers. It is important that this order is chosen independently of the reports received by the mechanism. The mechanism then uses this order to break ties when comparing users to slots and when comparing between users (slots) associated with different mediators (advertisers). For example, when comparing the cost of user p with the value of a slot b, the mechanism breaks ties in favor of p if and only if the mediator of p appears earlier than the advertiser of b in \sigma.

  • We assume that the report of every mediator induces some order on the set of users of this mediators. The mechanism uses this order to break ties between the costs of users belonging to the same mediator.

  • Finally, since the slots of a given advertiser are all identical and non-strategic (recall that slots were introduced into the model just for the purpose of simplifying the presentation), any method can be used for tie-breaking between the slots of a given advertiser.

In the rest of this paper when costs/values are compared, unless it is explicitly specified that they are compared as numbers, the comparison is assumed to use the above tie breaking rule. Note that this assumption implies that two values (costs) are equal if and only if they belong to the same slot (user). We now prove a useful observation that follows from the way we defined the tie-breaking rule.

Observation 2.1.

When the slots are ordered in an increasing (or decreasing) value order, all the slots of a single advertiser are always consecutive.

Proof.

Let b_{1} and b_{2} be two slots of one advertiser, and let b be an arbitrary slot of another advertiser. It is enough to prove that v(b_{1})<v(b) implies v(b_{2})<v(b). The inequality v(b_{1})<v(b) can happen in two cases. If v(b_{1}) is smaller than v(b) as numbers, then we get v(b_{2})<v(b) since v(b_{1}) and v(b_{2}) are equal as numbers. Otherwise, if v(b_{1}) is equal to v(b) as numbers then the inequality v(b_{1})<v(b) implies that the advertiser of b_{1} (and b_{2}) appears later in \sigma than the advertiser of b, and thus, we also have v(b_{2})<v(b). ∎

2.2 Canonical Assignment

Given a set B^{\prime}\subseteq B of slots and a set P^{\prime}\subseteq P of users, the canonical assignment S_{c}(P^{\prime},B^{\prime}) is the assignment constructed by the following process. First, we order the slots of B^{\prime} in a decreasing value order b_{1},b_{2},\ldots,b_{|B^{\prime}|} and the users of P^{\prime} in an increasing cost order p_{1},p_{2},\ldots,p_{|P^{\prime}|}. Then, for every 1\leq i\leq\min\{|B^{\prime}|,|P^{\prime}|\}, the canonical assignment S_{c}(B^{\prime},P^{\prime}) assigns user p_{i} to slot b_{i} if and only if v(b_{i})>c(p_{i}). The canonical assignment is an important tool used frequently by the mechanisms we describe in the next sections. In some places we refer to the user or slot at location i of a canonical solution S_{c}(P^{\prime},B^{\prime}). By using this expression we mean user p_{i} or slot b_{i}, respectively. Additionally, the term |S_{c}(P,B)| is used very often in our proofs, and thus, it is useful to define the shorthand \tau=|S_{c}(P,B)|.

The following lemma shows that the above definition of \tau is consistent with the use of \tau in Section 1.1 as the size of the optimal trade.

Lemma 2.2.

The canonical assignment S_{c}(P^{\prime},B^{\prime}) maximizes \mathtt{GfT}(S_{c}(P^{\prime},B^{\prime})) among all the possible assignments of users of P^{\prime} to slots of B^{\prime}.

Proof.

For every 0\leq i\leq\min\{|B^{\prime}|,|P^{\prime}|\}, let S^{i}_{c}(P^{\prime},B^{\prime}) be the subset of S_{c}(P^{\prime},B^{\prime}) in which we keep only the assignments of the first i users. More formally,

S^{i}_{c}(P^{\prime},B^{\prime})=\{(p_{j},b_{j})\in S_{c}(P^{\prime},B^{\prime% })\mid 1\leq j\leq i\}\enspace.

Let i^{*} be the largest i such that S^{i}_{c}(P^{\prime},B^{\prime}) is contained in some assignment S^{*}\subseteq P^{\prime}\times B^{\prime} maximizing \mathtt{GfT}(S^{*}) among all the possible assignments of users of P^{\prime} to slots of B^{\prime}. Such an i^{*} clearly exists since S^{0}_{c}(P^{\prime},B^{\prime})=\varnothing is a subset of every feasible assignment. There are two cases to consider. The first case is when i^{*}=\min\{|B^{\prime}|,|P^{\prime}|\}. In this case the canonical assignment S_{c}(P^{\prime},B^{\prime}) is a subset of the optimal assignment S^{*}. Let us consider an arbitrary ordered pair (p_{j},b_{k})\in S^{*}\setminus S_{c}(P^{\prime},B^{\prime}), and let us assume, without loss of generality, that j\leq k. Since S^{*} is a legal assignment and (p_{j},b_{k}) belongs to S^{*} together with all the ordered pairs of S_{c}(P^{\prime},B^{\prime}), the ordered pair (p_{j},b_{j}) cannot belong to S_{c}(P^{\prime},B^{\prime}). By definition, this implies c(p_{j})>v(b_{j}), and thus, c(p_{j})\geq v(b_{j}) even when we compare them as numbers. Thus,

v(b_{k})-c(p_{j})\leq v(b_{j})-c(p_{j})\leq 0\enspace.

Summing the above inequality over all ordered pairs (p_{j},b_{k})\in S^{*}\setminus S_{c}(P^{\prime},B^{\prime}), we get:

\mathtt{GfT}(S_{c}(P^{\prime},B^{\prime}))=\mathtt{GfT}(S^{*})-\sum_{(p_{j},b_% {k})\in S^{*}\setminus S_{c}(P^{\prime},B^{\prime})}\mspace{-36.0mu }[v(b_{k})% -c(p_{j})]\geq\mathtt{GfT}(S^{*})\enspace,

which completes the proof of the lemma for the case i^{*}=\min\{|B^{\prime}|,|P^{\prime}|\}. The second case we need to consider is the case 0\leq i^{*}<\min\{|B^{\prime}|,|P^{\prime}|\}. In the rest of the proof we show that this case can never happen, which implies the lemma.

Assume towards a contradiction 0\leq i^{*}<\min\{|B^{\prime}|,|P^{\prime}|\}. Our objective is to show that there exists an assignment S^{\prime} obeying S^{i^{*}+1}_{c}(P^{\prime},B^{\prime})\subseteq S^{\prime} and \mathtt{GfT}(S^{\prime})\geq\mathtt{GfT}(S^{*}), which is a contradiction to the choice of i^{*}. By the choice of i^{*} we have S^{i^{*}}_{c}(P^{\prime},B^{\prime})\subseteq S^{*}, but S^{i^{*}+1}_{c}(P^{\prime},B^{\prime})\not\subseteq S^{*}, which can only happen when (p_{i^{*}+1},b_{i^{*}+1})\in S^{i^{*}+1}_{c}(P^{\prime},B^{\prime})\setminus S% ^{*}. There are now three cases that need to be considered:

  • If (p_{i^{*}+1},b_{i^{*}+1}) can be added to S^{*} without violating the feasibility, then we choose S^{\prime}=S^{*}\cup\{(p_{i^{*}+1},b_{i^{*}+1})\}. Clearly, S^{i^{*}+1}_{c}(P^{\prime},B^{\prime}) is a subset of S^{\prime} since S^{i^{*}+1}_{c}(P^{\prime},B^{\prime})=S^{i^{*}}_{c}(P^{\prime},B^{\prime})% \cup\{(p_{i^{*}+1},b_{i^{*}+1})\} and S^{i^{*}}_{c}(P^{\prime},B^{\prime}) is a subset of S^{*}. Additionally, since (p_{i^{*}+1},b_{i^{*}+1}) belongs to the canonical assignment, we know that v(b_{i^{*}+1})>c(p_{i^{*}+1}), and thus, v(b_{i^{*}+1})\geq c(p_{i^{*}+1}) also when we compare them as numbers. Thus,

    \mathtt{GfT}(S^{\prime})=\mathtt{GfT}(S^{*})+[v(b_{i^{*}+1})-c(p_{i^{*}+1})]% \geq\mathtt{GfT}(S^{*})\enspace.
  • When the previous case does not hold, there must be in S^{*} either an ordered pair containing p_{i^{*}+1} or an ordered pair containing b_{i^{*}+1}. In the current case we assume that S^{*} contains an ordered pair containing p_{i^{*}+1}, but does not contain an ordered pair containing b_{i^{*}+1}. The case that S^{*} contains both an ordered pair containing p_{i^{*}+1} and an ordered pair containing b_{i^{*}+1} is the next case we consider. Finally, the case that S^{*} contains an ordered pair containing b_{i^{*}+1}, but does not contain an ordered pair containing p_{i^{*}+1} is analogous to the current case, and is, thus, omitted.

    Let (p_{i^{*}+1},b) be the ordered pair of S^{*} that contains p_{i^{*}+1}. Since (p_{i^{*}+1},b_{i^{*}+1}) appears in the canonical solution, we must have c(p_{i^{*}+1})<v(b_{i^{*}+1}), and thus, c(p_{i})<v(b_{i}) for every 1\leq i\leq i^{*}. Hence, the assignment S^{i^{*}}_{c}(P^{\prime},B^{\prime}), which is contained in S^{*}, is equal to \{(p_{i},b_{i})\mid 1\leq i\leq i^{*}\}. This means that the slot b of the pair (p_{i^{*}+1},b)\in S^{*} cannot be one of the first i^{*} slots, and thus, obeys v(b)\leq v(b_{i^{*}+1}). We now consider the assignment S^{\prime}=S^{*}\cup\{(p_{i^{*}+1},b_{i^{*}+1})\}\setminus\{(p_{i^{*}+1},b)\}. Clearly this assignment contains S^{i^{*}+1}_{c}(P^{\prime},B^{\prime}) since S^{i^{*}+1}_{c}(P^{\prime},B^{\prime})=\{(p_{i},b_{i})\mid 1\leq i\leq i^{*}+1\} and S^{*} contains the assignment S^{i^{*}}_{c}(P^{\prime},B^{\prime})=\{(p_{i},b_{i})\mid 1\leq i\leq i^{*}\}. Additionally,

    \displaystyle\mathtt{GfT}(S^{\prime})= \displaystyle\mathtt{GfT}(S^{*})+[v(b_{i^{*}+1})-c(p_{i^{*}+1})]-[v(b)-c(p_{i^% {*}+1})]
    \displaystyle= \displaystyle\mathtt{GfT}(S^{*})+[v(b_{i^{*}+1})-v(b)]\geq\mathtt{GfT}(S^{*})\enspace.
  • The last case we need to consider is when S^{*} contains both an ordered pair (p_{i^{*}+1},b) containing user p_{i^{*}+1} and an ordered pair (p,b_{i^{*}+1}) containing slot b_{i^{*}+1}. In this case we choose S^{\prime}=S^{*}\cup\{(p_{i^{*}+1},b_{i^{*}+1}),(p,b)\}\setminus\{(p_{i^{*}+1}% ,b),(p,b_{i^{*}+1})\}. Since S^{i^{*}}_{c}(P^{\prime},B^{\prime})\subseteq S^{*} and S^{i^{*}}_{c}(P^{\prime},B^{\prime}) cannot contain (by definition) either the ordered pair (p,b_{i^{*}+1}) or the ordered pair (p_{i^{*}+1},b), we get that S^{\prime} contains S^{i^{*}+1}_{c}(P^{\prime},B^{\prime}). Additionally,

    \displaystyle\mathtt{GfT}(S^{\prime})= \displaystyle\mathtt{GfT}(S^{*})+[v(b_{i^{*}+1})-c(p_{i^{*}+1})]+[v(b)-c(p)]
    \displaystyle-[v(b)-c(p_{i^{*}+1})]-[v(b_{i^{*}+1})-c(p)]=\mathtt{GfT}(S^{*})% \enspace.\qed

3 Deterministic Mechanism

In this section we describe the deterministic mechanism “Price by Removal Mechanism” (PRM) for our model. Recall that PRM assumes public knowledge of the advertisers’ capacities. Accordingly, we assume throughout this section that the capacities of the advertisers are common knowledge (or that the advertisers are not strategic about them). We also assume that PRM has access to a value \gamma\geq 1 such that:

u(a)\leq\gamma\quad\forall\;a\in A\qquad\text{and}\qquad|P(m)|\leq\gamma\quad% \forall\;m\in M\enspace.

In other words, \gamma is an upper bound on how large can be the capacity of an advertiser or the number of users of a mediator. Informally, \gamma can be understood as a bound on the importance every single advertiser or mediator can have.

A description of PRM is given as Mechanism 3. Notice that Mechanism 3 often refers to parameters of the model that are not known to the mechanism (i.e., values of advertiser, the number of users of mediators and the costs of users). Whenever this happens, this should be understood as referring to the reported values of these parameters.

  {mechanism} Price by Removal Mechanism (PRM)  

  1. For every mediator m\in M, if the canonical assignment S_{c}(P\setminus P(m),B) is of size more than 4\gamma, denote by p_{m} the user at location |S_{c}(P\setminus P(m),B)|-4\gamma of the canonical assignment S_{c}(P\setminus P(m),B), and let c_{m} be the cost of p_{m}. Otherwise, set c_{m} to -\infty.

1

  1. For every mediator m\in M, let \hat{P}(m) be the set of users of mediator m whose cost is less than c_{m}. Intuitively, \hat{P}(m) is the set of users of mediator m that the mechanism tries to assign to advertisers.

  2. Assign the users of \bigcup_{m\in M}\hat{P}(m) to the advertisers using a VCG auction. More specifically, the users of \bigcup_{m\in M}\hat{P}(m) are the items sold in the auction, and the bidders are the advertisers of A plus a dummy advertiser a_{d} whose value and capacity are v(a_{d})=\max_{m\in M}c_{m} and u(a_{d})=\sum_{m\in M}|\hat{P}(m)|, respectively.

  3. Charge every non-dummy advertiser by the same amount she is charged (as a bidder) by the VCG auction.

For every user p assigned by the VCG auction, pay c_{m} to the mediator m of p.555Note that m is BB as he forwards to each of his assigned users her cost—which is less than c_{m}.  

Remark: It can be shown that the existence of the dummy advertiser never affects the behavior of Mechanism 3, and thus, one can safely omit it from the mechanism. Nevertheless, we keep this advertiser in the above description of the mechanism since its existence simplifies our proof that the mechanism is BB.

Let us recall our result regarding PRM.

Theorem 1.1.

PRM is BB, IR, IC and \left(1-\frac{5\gamma}{\tau}\right)-competitive.

PRM improves over the trade reduction procedure of [8] by guarantying a non-trivial competitive ratio even in the presence of players (mediators) with a multi-dimensional strategic space. However, the competitive ratio guaranteed by PRM is slightly worse than the competitive ratio guaranteed by the procedure of [8] when the players have single dimensional strategic spaces (which is 1-2/\tau). The next paragraph gives an intuitive explanation why it does not seem possible to improve the competitive ratio of PRM to match the competitive ratio of 1-2/\tau guaranteed by [8]’s procedure.

Observe that PRM needs to maintain IC for both mediators and advertisers. In order to maintain IC for the mediators the price for each mediator has to be computed while all his users are reduced from the trade. This is why the real reduction in trade might be larger by up to \gamma compared to the reduction 4\gamma explicitly stated by PRM, which explains the gap between the competitive ratio of 1-5\gamma/\tau and the explicit reduction of 4\gamma in the mechanism. It remains to understand why the explicit reduction is set to 4\gamma. The most significant difficulty in guaranteeing IC for the advertisers is that an advertiser might change her report in order to affect the costs \{c_{m}\}_{m\in M} of the mediators, and through them, manipulate the items offered in the VCG auction. PRM tackles this difficulty by guaranteeing that advertisers performing such manipulations are assigned no users. Since the users of mediator m are removed when c_{m} is calculated, securing this guarantee requires that advertisers having a slot in one of the last \gamma locations of the optimal trade are assigned no users. As an advertiser might have up to \gamma slots, this translates into a requirement that an advertiser whose earliest slot is in one of the last 2\gamma locations of the optimal trade is assigned no users. Moreover, to simplify the proof our analysis in fact requires that an advertiser whose earliest slot is in one of the last 3\gamma locations of the optimal trade is assigned no users. Taking into account, again, the fact that the users of mediator m are removed when c_{m} is calculated, guaranteeing the last property requires a trade reduction of 4\gamma. To summarize, maintaining the advertisers’ IC imposes an explicit trade reduction of 3\gamma that we present as 4\gamma for the sake of simplicity, and maintaining the mediators’ IC implies that the real trade reduction is larger by up to \gamma compared to the explicit one. Thus, a competitive ratio of 1-\frac{4\gamma}{\tau} seems to be inevitable in order to allow multi-dimensional strategic spaces.

In the rest of this section we prove Theorem 1.1. We begin with some basic properties of PRM. It is important to remember while reading the next proofs that the constructions of S_{c}(P,B) and S_{c}(P\setminus P(m),B) order the slots of B in the same order, and that the order of the users of P\setminus P(m) in the construction of S_{c}(P\setminus P(m),B) is obtained by removing the users of P(m) from the order used by the construction of S_{c}(P,B).

Observation 3.1.

For every mediator m\in M, |S_{c}(P\setminus P(m),B)|\leq\tau.

Proof.

Assume towards a contradiction that |S_{c}(P\setminus P(m),B)|>\tau, and let b^{\prime}_{\tau+1} and p^{\prime}_{\tau+1} be the slot and user at location \tau+1 of S_{c}(P\setminus P(m),B), respectively. Since p^{\prime}_{\tau+1} is assigned by S_{c}(P\setminus P(m),B) to b^{\prime}_{\tau+1}, it must be that v(b^{\prime}_{\tau+1})>c(p^{\prime}_{\tau+1}).

On the other hand, let b_{\tau+1} and p_{\tau+1} be the slot and user at location \tau+1 of S_{c}(P,B), respectively. Clearly b_{\tau+1} and b^{\prime}_{\tau+1} are exactly the same slot. Moreover, p^{\prime}_{\tau+1} is either equal to p_{\tau+1} or appears in S_{c}(P,B) in a larger location than p_{\tau+1}. Hence,

v(b_{\tau+1})=v(b^{\prime}_{\tau+1})>c(p^{\prime}_{\tau+1})\geq c(p_{\tau+1})\enspace,

which is a contradiction since p_{\tau+1} is not assigned to b_{\tau+1} by S_{c}(P,B). ∎

Using the last observation we can now prove the following lemma.

Lemma 3.2.

For every mediator m\in M the following is true:

  • When c_{m}\neq-\infty, p_{m} is assigned by the canonical assignment S_{c}(P,B).

  • All the users of \hat{P}_{m} are assigned by S_{c}(P,B).

Proof.

First, let us explain why the second claim we need to prove follows from the first one. If c_{m}=-\infty, then \hat{P}_{m}=\varnothing and the second claim is trivial. Otherwise, the first claim states that p_{m} is assigned by the canonical assignment S_{c}(P,B). By the definition of a canonical assignment, all the users having a lower cost than c_{m}=c(p_{m}) are assigned by S_{c}(P,B) as well. The second claim now follows since \hat{P}_{m} contains only users having a lower cost than c_{m}.

It remains to prove the first claim. The claim is trivial when c_{m}=-\infty, thus, we assume c_{m}\neq-\infty. Let F be the set of users that are assigned by S_{c}(P,B) but do not belong to P(m). Clearly the users of F take locations 1 to |F| in S_{c}(P\setminus P(m),B). On the other hand, we can lower bound the size of F by \tau-|P(m)|\geq\tau-\gamma. Thus, locations 1 to \tau-\gamma of S_{c}(P\setminus P(m),B) are all occupied by users assigned by S_{c}(P,B).

By Observation 3.1 and the fact that c_{m}\neq-\infty, p_{m} is chosen by PRM as the user located at location |S_{c}(P\setminus P(m),B)|-4\gamma\leq\tau-4\gamma of S_{c}(P\setminus P(m),B). This implies that p_{m} is assigned by S_{c}(P,B) since \tau-4\gamma\leq\tau-\gamma (and all the users at locations 1 to \tau-\gamma of S_{c}(P\setminus P(m),B) are assigned by S_{c}(P,B)). ∎

Corollary 3.3.

PRM is BB and assigns all the users of \bigcup_{m\in M}\hat{P}_{m}.

Proof.

Let M^{\prime} be the subset of mediators having a finite c_{m}. Formally, M^{\prime}=\{m\in M\mid c_{m}\neq-\infty\}. If M^{\prime}=\varnothing, then no users are assigned by PRM and \hat{P}_{m}=\varnothing for every mediator m\in M, which makes the corollary trivial. Hence, we assume from now on M^{\prime}\neq\varnothing. We define now p_{\tau} as the user at location \tau in S_{c}(P,B). Clearly, c(p_{\tau}) upper bounds the cost of any user assigned by S_{c}(P,M). On the other hand, by Lemma 3.2, for every m\in M^{\prime} the user p_{m} is assigned by S_{c}(P,B). Hence, c(p_{\tau})\geq\max_{m\in M^{\prime}}c(p_{m})=\max_{m\in M^{\prime}}c_{m}.

Let B^{\prime} be the set of slots assigned users by S_{c}(P,B), and let b_{\tau} be the slot at location \tau in S_{c}(P,B). Since p_{\tau} is assigned to b_{\tau} by S_{c}(P,B), we get: v(b_{\tau})>c(p_{\tau})\geq\max_{m\in M^{\prime}}c_{m}. Moreover, v(b_{\tau}) lower bounds the value of any slot in B^{\prime}, and thus, all the slots of B^{\prime} have values larger than \max_{m\in M^{\prime}}c_{m}. Combining this with the observation that |B^{\prime}|=\tau, we get that there exists a set A^{\prime}\subseteq A of advertisers with the following properties:

  • The advertisers of A^{\prime} all have values larger than \max_{m\in M^{\prime}}c_{m}=\max_{m\in M}c_{m}, and thus, larger than the value of the dummy advertiser a_{d}, and larger than the cost of any user in \bigcup_{m\in M^{\prime}}\hat{P}_{m}=\bigcup_{m\in M}\hat{P}_{m}.

  • The advertisers of A^{\prime} have a total capacity at least \tau\geq|\bigcup_{m\in M}\hat{P}_{m}|, where the inequality holds since Lemma 3.2 guarantees that the users of \bigcup_{m\in M}\hat{P}_{m} are all assigned by S_{c}(P,M).

The two above properties imply together that the VCG auction (and thus, also PRM) assigns all the users of \bigcup_{m\in M}\hat{P}_{m} to real advertisers. As no user is assigned to the dummy advertiser, and the dummy advertiser has a value of \max_{m\in M}c_{m}, the VCG auction charges the real advertisers at least \max_{m\in M}c_{m} for every user assigned to them. On the other hand, \max_{m\in M}c_{m} upper bounds the payment a mediator receives from PRM for a single assigned user, and thus, PRM is budget balanced. ∎

3.1 The Competitive Ratio of PRM

In this section we analyze the competitive ratio of PRM, and show that it is at least 1-5\gamma/\tau. If 5\gamma\geq\tau, then this is trivial. Thus, we assume throughout this section 5\gamma<\tau.

Observation 3.4.

Given a mediator m\in M, every user p\not\in P(m) which is assigned by the canonical assignment S_{c}(P,B) is also assigned by the canonical assignment S_{c}(P\setminus P(M),B).

Proof.

Let j^{\prime} and j be the locations of p in S_{c}(P\setminus P(m),B) and S_{c}(P,B), respectively. Clearly j^{\prime}\leq j, and thus, the slot at location j^{\prime} in S_{c}(P\setminus P(m),B) has a value larger or equal to the value of the slot at location j in S_{c}(P,B). This means that p must be assigned by S_{c}(P\setminus P(m),B) since it is assigned by S_{c}(P,B). ∎

Let us denote by c_{e} the cost of the user at location \tau-5\gamma of the canonical assignment S_{c}(P,B). Using the last observation we get:

Lemma 3.5.

For every mediator m\in M, c_{m}\geq c_{e}.

Proof.

Let L be the set of the users at locations \tau-5\gamma to \tau of S_{c}(P,B). Clearly all the users of L have a cost equal or larger than c_{e}. On the other hand, observe that by Observation 3.4 all the users of L\setminus P(m) are assigned by S_{c}(P\setminus P(m),B). Hence, the number of users of L that are assigned by S_{c}(P\setminus P(m),B) is at least:

|L\setminus P(m)|\geq|L|-|P(m)|\geq(5\gamma+1)-\gamma=4\gamma+1\enspace.

Recall that p_{m} is selected as the user at location |S_{c}(P\setminus P(m),B)|-4\gamma in S_{c}(P\setminus P(m),B). This, together with the observation that S_{c}(P,\setminus P(m),B) assigns at least 4\gamma+1 users of L, implies that one of the following must be true: either p_{m} is a user of L, or there exists a user of L that has a smaller location in S_{c}(P,\setminus P(m),B) than p_{m}, and thus, has a lower cost than p_{m}. In both cases the lemma follows.666In fact, it can be shown that the second case never happens.

Corollary 3.6.

PRM is \left(1-\frac{5\gamma}{\tau}\right)-competitive.

Proof.

Let \bar{P}_{e} be the set of users at locations 1 to \tau-5\gamma of the canonical assignment S_{c}(P,B). Consider an arbitrary user p\in\bar{P}_{e}, and let m be her mediator. Lemma 3.5 and the definition of \bar{P}_{e} imply together c(p)\leq c_{e}\leq c_{m}. On the other hand, c_{m}=c(p_{m}) is the cost of a user p_{m}\not\in P(m), and thus, we cannot have c(p)=c_{m} since the tie-breaking rule defined in Section 2.1 guarantees that no two users have the same cost. Hence, c(p)<c_{m}.

Recall that \hat{P}_{m} is defined as the set of all users of P(m) whose cost is less than c_{m}. Together with the previous observation, this gives \bar{P}_{e}\cap P(m)\subseteq\hat{P}_{m} for every mediator m\in M, and therefore, \bar{P}_{e}\subseteq\bigcup_{m\in M}\hat{P}_{m}. Notice now that Lemma 3.2 states that all the users of \bigcup_{m\in M}\hat{P}_{m} are assigned by PRM. Hence, in particular, the users of the subset \bar{P}_{e} are all assigned by PRM.

Denote by S the assignment produced by PRM. Then:

\displaystyle\mathtt{GfT}(S)= \displaystyle\sum_{(p,b)\in S}[v(b)-c(p)]=\sum_{(p,b)\in S}[v(b)-\max\nolimits% _{m\in M}c_{m}]+\sum_{(p,b)\in S}[\max\nolimits_{m\in M}c_{m}-c(p)] (1)
\displaystyle\geq \displaystyle\sum_{(p,b)\in S}[v(b)-\max\nolimits_{m\in M}c_{m}]+\sum_{p\in% \bar{P}_{e}}[\max\nolimits_{m\in M}c_{m}-c(p)]\enspace,

where the inequality holds by the above discussion and the observation that every user assigned by PRM belongs to \cup_{m\in M}\hat{P}_{m}, and thus, has a cost of at most \max_{m\in M}c_{m}.

The VCG auction assigns the users in a way that maximizes the total value of the advertisers from the assigned users. This means that the |S| users assigned by S are assigned to the slots at locations 1 to |S| of S_{c}(P,B). Let us denote this set of slots by B_{S}. Additionally, let us denote by \bar{B}_{e} the set of slots at locations 1 to |\bar{P}_{e}| in S_{c}(P,B). Then,

\sum_{(p,b)\in S}[v(b)-\max\nolimits_{m\in M}c_{m}]=\sum_{b\in B_{S}}[v(b)-% \max\nolimits_{m\in M}c_{m}]\geq\sum_{b\in\bar{B}_{e}}[v(b)-\max\nolimits_{m% \in M}c_{m}]\enspace,

where the inequality holds due to two observations: first \bar{B}_{e} is a subset of B_{S} since at least |\bar{B}_{e}|=|\bar{P}_{e}| users are assigned by S. Second, the dummy advertiser a_{d} has a value of \max_{m\in M}c_{m}, and thus, every slot of B_{S} has a value of at least \max_{m\in M}c_{m} since the VCG auction preferred assigning a user to this slot over assigning it to a_{d}.

Plugging the last inequality into Inequality (1) gives:

\displaystyle\mathtt{GfT}(S)\geq \displaystyle\sum_{b\in\bar{B}_{e}}[v(b)-\max\nolimits_{m\in M}c_{m}]+\sum_{p% \in\bar{P}_{e}}[\max\nolimits_{m\in M}c_{m}-c(p)]=\sum_{b\in\bar{B}_{e}}v(b)-% \sum_{p\in\bar{P}_{e}}c(p)
\displaystyle\geq \displaystyle\frac{\tau-5\gamma}{\tau}\cdot\sum_{(p,b)\in S_{c}(P,B)}\mspace{-% 18.0mu }v(b)-\frac{\tau-5\gamma}{\tau}\cdot\sum_{(p,b)\in S_{c}(P,B)}\mspace{-% 18.0mu }c(p)=\frac{\tau-5\gamma}{\tau}\cdot\mathtt{GfT}(S_{c}(P,B))\enspace,

where the second inequality holds since \bar{P}_{e} (\bar{B}_{e}) contains, by definition, the \tau-5\gamma users (slots) with the lowest costs (highest values) among the \tau users (slots) assigned by the canonical assignment S_{c}(P,B). The corollary now follows from the last inequality since S_{c}(P,B) is an assignment of users from P to slots of B maximizing the gain from trade. ∎

3.2 The Incentive Properties of PRM

In this section we prove the incentive parts of Theorem 1.1. We begin with a lemma analyzing the incentive properties of PRM for mediators.

Lemma 3.7.

For every mediator m, PRM is IR for m, and truthfulness is a dominant strategy for him.

Proof.

PRM calculates a threshold c_{m} based on the reports of advertisers and mediators other than m, and thus, c_{m} is independent of the report of m. For every user p\in P(m), if p’s cost is larger than c_{m}, then p is not added to \hat{P}_{m}, and thus, PRM does not assign p. On the other hand, if p’s cost is smaller than c_{m}, then p is added to \hat{P}_{m}, and by Lemma 3.2 she is assigned by PRM to some slot and m is paid c_{m}.

From the above description it is clear that when m is truthful, he is paid for each assigned user p\in P(m) at least the cost of the user. Thus, PRM is IR for m. To see why truthfulness is a dominant strategy for m, notice that the utility function of m is maximized when all the users of m whose cost is less than c_{m} are assigned, and no other user of m is assigned; which is exactly what PRM does when m is truthful. ∎

Proving the incentive properties of PRM for the advertisers is more involved. We start by proving that PRM is IR for advertisers.

Observation 3.8.

For every advertiser a, PRM is IR for a.

Proof.

PRM uses a VCG auction to assign users to advertisers, and to determine the cost each advertiser has to pay. Since VCG auctions are IR, the payment charged for every advertiser is upper bounded by the total value the advertiser gets from the users assigned to her. ∎

In the rest of the section we concentrate proving that truthfulness is a dominant strategy for the advertisers under PRM . Let us fix an advertiser a\in A. For technical reasons it is convenient to somewhat enhance the strategy space of a. Clearly, if we can show that truthfulness is a dominant strategy for a given the enhanced strategy space, then this is also true given the original strategy space.

Section 2.1 explains how ties between valuations of advertisers are broken using an order \sigma. In particular, when comparing the values of slots of two different advertisers with identical values, the values of the slots of the advertiser that appears earlier in \sigma are considered larger. We enhance the strategy space of a by allowing her to report both her value and her location in \sigma compared to the other advertisers and mediators. In other words, the relative order of the other advertisers and mediators in the order \sigma defining the tie-breaking rule is fixed, and a can choose to insert herself at any location within this order.

Given a possible strategy s for a, we denote by B_{s} the set of slots given that a uses this strategy, and by \ell(s) the lowest location in S_{c}(P,B_{s}) of a slot of a. Note that, since the slots of a appear sequentially in S_{c}(P,B_{s}), they must appear at locations \ell(s) to \ell(s)+u(a)-1 in S_{c}(P,B_{s}).

Observation 3.9.

For every two indices 1\leq i\leq j\leq|B|-(\gamma-1), if i+(\gamma-1)\leq j, then there exists a strategy s for advertiser a such that i\leq\ell(s)\leq j.

Proof.

\ell(s)-1 is equal to the total capacity of the advertisers appearing before the location set by the strategy s for a in the order \sigma. Since the capacity of every advertiser is at most \gamma, the values \ell(s) corresponding to two adjacent possible locations for a in \sigma are different by at most \gamma. Moreover, if a is inserted by s as the first advertiser in \sigma than \ell(s)=1, and if a is inserted by s as the last advertiser in \sigma than \ell(s)=|B|-(u(a)-1)\geq|B|-(\gamma-1). Hence, for every list of consecutive indices of length at least \gamma such that all the indices are between 1 and |B|-(\gamma-1), there must be a strategy s for a such that \ell(s) is within this list. ∎

The next claims analyze some possible strategies for a. Let \bar{s} be the strategy of a in which a reports a value of \infty, and let \bar{\tau} be the size of the canonical assignment S_{c}(P,B_{\bar{s}}).

Observation 3.10.

For every mediator m\in M and strategy s for a, |S_{c}(P\setminus P(m),B_{s})|\leq\bar{\tau}.

Proof.

Assume towards a contradiction that |S_{c}(P\setminus P(m),B_{s})|>\bar{\tau}, and let p_{\bar{\tau}+1} and b_{\bar{\tau}+1} be the user and slot, respectively, at location \bar{\tau}+1 in S_{c}(P\setminus P(m),B_{s}). By our assumption, p_{\bar{\tau}+1} is assigned to b_{\bar{\tau}+1}, and therefore, v(b_{\bar{\tau}+1})>c(p_{\bar{\tau}+1}).

Let us now define p^{\prime}_{\bar{\tau}+1} and b^{\prime}_{\bar{\tau}+1} to be the user and slot, respectively, at location \bar{\tau}+1 in S_{c}(P,B_{\bar{s}}). Clearly, p_{\bar{\tau}+1} appears at a location greater or equal to \bar{\tau}+1 in S_{c}(P,B_{\bar{s}}), hence, c(p_{\bar{\tau}+1})\geq c(p^{\prime}_{\bar{\tau}+1}). If b^{\prime}_{\bar{\tau}+1} is a slot of a, then by definition v(b^{\prime}_{\bar{\tau}+1})=\infty>v(b_{\bar{\tau}+1}). Otherwise, the location of b^{\prime}_{\bar{\tau}+1} in S_{c}(P\setminus P(m),B_{s}) must be smaller or equal to \bar{\tau}+1 (its location in S_{c}(P,B_{\bar{s}})), and thus, v(b^{\prime}_{\bar{\tau}+1})\geq v(b_{\bar{\tau}+1}).

Combining all the inequalities we proved gives: v(b^{\prime}_{\bar{\tau}+1})\geq v(b_{\bar{\tau}+1})>c(p_{\bar{\tau}+1})\geq c% (p^{\prime}_{\bar{\tau}+1}), which is a contradiction since b^{\prime}_{\bar{\tau}+1} and p^{\prime}_{\bar{\tau}+1} have the same location in S_{c}(P,B_{\bar{s}}), but p^{\prime}_{\bar{\tau}+1} is not assigned to b^{\prime}_{\bar{\tau}+1} by S_{c}(P,B_{\bar{s}}). ∎

Using the last observation we can now prove the following lemma, which considers strategies of a with a relatively large \ell(s).

Lemma 3.11.

Given that advertiser a uses a strategy s such that \ell(s)\geq\bar{\tau}-3\gamma, PRM assigns no users to a.

Proof.

Let c_{\bar{\tau}-3\gamma} be the cost of the user at location \bar{\tau}-3\gamma in S_{c}(P,B_{\bar{s}}) if \bar{\tau}>3\gamma, and -\infty otherwise. Our first objective is to show that c_{\bar{\tau}-3\gamma}\geq c_{m} for every mediator m\in M. Consider an arbitrary mediator m\in M. By Observation 3.10, |S_{c}(P\setminus P(m),B_{s})|\leq\bar{\tau}. There are now two cases to consider:

  • If |S_{c}(P\setminus P(m),B_{s})|\leq 4\gamma, then c_{m}=-\infty\leq c_{\bar{\tau}-3\gamma}.

  • If |S_{c}(P\setminus P(m),B_{s})|>4\gamma, then c_{m} is the cost of the user at location |S_{c}(P\setminus P(m),B_{s})|-4\gamma of S_{c}(P\setminus P(m),B_{s}). This user must appear in S_{c}(P,B_{\bar{s}}) either at location (|S_{c}(P\setminus P(m),B_{s})|-4\gamma)+\gamma\leq\bar{\tau}-3\gamma or at a smaller location. On the other hand, since \bar{\tau}\geq|S_{c}(P\setminus P(m),B_{s})|>4\gamma, c_{\bar{\tau}-3\gamma} is the cost of the user at location \bar{\tau}-3\gamma of S_{c}(P,B_{\bar{s}}), and thus, c_{\bar{\tau}-3\gamma}\geq c_{m}.

The fact that c_{\bar{\tau}-3\gamma}\geq\max_{m\in M}c_{m} implies that \bigcup_{m\in M}\hat{P}_{m} contains only users whose cost is smaller than c_{\bar{\tau}-3\gamma}, and there are only \bar{\tau}-3\gamma-1 such users. The VCG auction of PRM assigns the users of \bigcup_{m\in M}\hat{P}_{m} to the slots of B_{s} at the lowest locations, and thus, only slots at locations 1 to \bar{\tau}-3\gamma-1 have a potential to be assigned a user. Since we assume \ell(s)\geq\tau-3\gamma, no slot of a is among these slots, and thus, a is assigned no users by PRM. ∎

The following lemma considers strategies of a with a relatively small \ell(s).

Lemma 3.12.

Given a strategy s for advertiser a such that \ell(s)\leq\bar{\tau}-2\gamma and a mediator m\in M, |S_{c}(P\setminus P(m),B_{s})|=|S_{c}(P\setminus P(m),B_{\bar{s}})|.

Proof.

The order of the users is identical in both S_{c}(P\setminus P(m),B_{s}) and S_{c}(P\setminus P(m),B_{\bar{s}}). On the other hand, notice that \ell(s) must be at least 1, and thus, \bar{\tau}\geq 2\gamma+1 by the assumption \ell(s)\leq\bar{\tau}-2\gamma. Moreover, this assumption implies that all the slots of a appear at locations between 1 and \max\{\ell(s)+\gamma-1,\gamma\}\leq\max\{\bar{\tau}-\gamma-1,\gamma\}=\bar{% \tau}-\gamma-1 in both S_{c}(P\setminus P(m),B_{s}) and S_{c}(P\setminus P(m),B_{\bar{s}}). Hence, the sequences of the slots appearing in S_{c}(P\setminus P(m),B_{s}) and S_{c}(P\setminus P(m),B_{\bar{s}}) starting from location \bar{\tau}-\gamma are identical. This implies that for every \bar{\tau}-\gamma\leq i\leq\min\{|P\setminus P(m)|,|B|\}, S_{c}(P\setminus P(m),B_{s}) assigns its user at location i to its slot at location i if and only if S_{c}(P\setminus P(m),B_{\bar{s}}) assigns its user at location i to its slot at location i.

The above discussion implies that |S_{c}(P\setminus P(m),B_{s})|=|S_{c}(P\setminus P(m),B_{\bar{s}})| whenever it holds that:

|S_{c}(P\setminus P(m),B_{\bar{s}})|\geq\bar{\tau}-\gamma\enspace. (2)

Thus, the rest of the proof concentrate on proving Inequality (2). Let p_{\bar{\tau}} and s_{\bar{\tau}} be the user and slot at location \bar{\tau} of S_{c}(P,B_{\bar{s}}). Since \bar{\tau}=|S_{c}(P,B_{\bar{s}})| by definition, p_{\bar{\tau}} is assigned to s_{\bar{\tau}} by S_{c}(P,B_{\bar{s}}), and thus, v(s_{\bar{\tau}})>c(p_{\bar{\tau}}). Next, let p^{\prime}_{\bar{\tau}-\gamma} and s^{\prime}_{\bar{\tau}-\gamma} be the user and slot at location \bar{\tau}-\gamma of S_{c}(P\setminus P(m),B_{\bar{s}}). Clearly, p^{\prime}_{\bar{\tau}-\gamma} appears either at location \bar{\tau} or at a smaller location in S_{c}(P,B_{\bar{s}}), and thus, c(p^{\prime}_{\bar{\tau}-\gamma})\leq c(p_{\bar{\tau}}). Moreover, s^{\prime}_{\bar{\tau}-\gamma} appears at location \bar{\tau}-\gamma also in S_{c}(P,B_{\bar{s}}), and thus, v(s^{\prime}_{\bar{\tau}-\gamma})>v(s_{\bar{\tau}}). Combining all the above inequalities gives:

v(s^{\prime}_{\bar{\tau}-\gamma})>v(s_{\bar{\tau}})>c(p_{\bar{\tau}})\geq c(p^% {\prime}_{\bar{\tau}-\gamma})\enspace.

Hence, p^{\prime}_{\bar{\tau}-\gamma} is assigned to s^{\prime}_{\bar{\tau}-\gamma} by the canonical assignment S_{c}(P\setminus P(m),B_{\bar{s}}), which implies Inequality (2). ∎

Corollary 3.13.

For every mediator m\in M, c_{m} is identical given that advertiser a uses any strategy s obeying \ell(s)\leq\bar{\tau}-2\gamma.

Proof.

Consider two strategies s_{1} and s_{2} for a such that \ell(s_{1}),\ell(s_{2})\leq\bar{\tau}-2\gamma. Given that a uses strategy s_{1}, PRM selects c_{m} as the cost of the user at location |S_{c}(P\setminus P(m),B_{s_{1}})|-4\gamma in S_{c}(P\setminus P(m),B_{s_{1}}) or -\infty if |S_{c}(P\setminus P(m),B_{s_{1}})|\leq 4\gamma. Similarly, given that that a uses strategy s_{2}, PRM selects c_{m} as the cost of the user at location |S_{c}(P\setminus P(m),B_{s_{2}})|-4\gamma in S_{c}(P\setminus P(m),B_{s_{2}}) or -\infty if |S_{c}(P\setminus P(m),B_{s_{2}})|\leq 4\gamma. We claim that in both cases the same c_{m} is produced by PRM.

To see why this is the case observe that the two canonical assignments S_{c}(P\setminus P(m),B_{s_{1}}) and S_{c}(P\setminus P(m),B_{s_{2}}) share the order of the users, i.e., at a given location they both have the same user. Moreover, by Lemma 3.12:

|S_{c}(P\setminus P(m),B_{s_{1}})|=|S_{c}(P\setminus P(m),B_{\bar{s}})|=|S_{c}% (P\setminus P(m),B_{s_{2}})|\enspace.\qed

We are now ready to prove that truthfulness is a dominant strategy for a.

Lemma 3.14.

Given PRM, truthfulness is a dominant strategy for advertiser a.

Proof.

If \bar{\tau}\leq 3\gamma, then, by Lemma 3.11, PRM does not assign any users regardless of the strategy used by a, and thus, any strategy is a dominant strategy for a. Hence, we may assume \bar{\tau}>3\gamma for the remaining part of the proof. Let s_{t} be the truthful strategy of a, and let \hat{s} be any strategy such that \bar{\tau}-3\gamma\leq\ell(\hat{s})\leq\bar{\tau}-2\gamma (such a strategy exists by Observation 3.9). There are now two cases to consider, depending on the relationship between \ell(s_{t}) and \ell(\hat{s}).

Consider first the case \ell(s_{t})\leq\ell(\hat{s})\leq\bar{\tau}-2\gamma. By Corollary 3.13, every strategy s for a having \ell(s)\leq\bar{\tau}-2\gamma results in the same values c_{m} for all mediators, and thus, PRM ends up using exactly the same VCG auction for all these strategies (up to the bid of the bidder corresponding to a). Since VCG auctions are incentive compatible, this means that reporting the truth is a best response for a among all the possible strategies s for a obeying \ell(s)\leq\bar{\tau}-2\gamma. Moreover, truthfulness guarantees a non-negative utility for a since PRM is advertiser-side IR by Observation 3.8. Thus, being truthful dominates also strategies s for a having \ell(s)\geq\bar{\tau}-2\gamma\geq\bar{\tau}-3\gamma since such strategies result in 0 utility for a by Lemma 3.11.

Consider now the case \ell(s_{t})>\ell(\hat{s})\geq\bar{\tau}-3\gamma. By Lemma 3.11 a is assigned no users (and thus, gets 0 utility) when she uses her truthful strategy s_{t}. Let us observe what happens if advertiser a is modified so that her new truthful strategy is \hat{s}. By Lemma 3.11 the modified a also gets 0 utility when using her truthful strategy \hat{s}. On the other hand, the value of the modified advertiser for each user is at least as high as the value of the original advertiser, and thus, the modified advertiser gets at least as high utility as the original advertiser for every given outcome. Hence, to show that no strategy can give the original advertiser a positive utility, it is enough to show that the strategy maximizing the utility of the modified advertiser is her truthful strategy \hat{s}; which follows immediately from the first case (since the truthful strategy for the modified advertiser is \hat{s}, and \hat{s} obeys \ell(\hat{s})\leq\bar{\tau}-2\gamma). ∎

Corollary 3.15.

Given PRM, truthfulness is a dominant strategy for every advertiser.

Proof.

Follows from Lemma 3.14 since a is just an arbitrary fixed advertiser. ∎

4 Randomized Mechanism

In this section we describe the randomized mechanism “Threshold by Partition Mechanism” (TPM) for our model. Unlike the mechanism PRM from Section 3, TPM need not assume public knowledge about the advertisers’ capacities, i.e., the advertisers now have multi-dimensional strategy spaces. On the other hand, TPM assumes access to a value \alpha\in[|S_{c}(P,B)|^{-1},1] such that we are guaranteed that:

\frac{u(a)}{|S_{c}(P,B)|}\leq\alpha\quad\forall\;a\in A\qquad\text{and}\qquad% \frac{|P(m)|}{|S_{c}(P,B)|}\leq\alpha\quad\forall\;m\in M\enspace.

In other words, \alpha is an upper bound on how large can be the capacity of an advertiser or the number of users of a mediator compared to the size of the optimal assignment S_{c}(P,B). We remind the reader that \alpha is related to the value \gamma from Section 3 by the equation \alpha=\gamma/\tau, and thus, \alpha, like \gamma, can be informally understood as a bound on the importance of every single advertiser or mediator. It is important to note that \alpha is well-defined only when |S_{c}(P,B)|>0, and thus, we assume this inequality is true throughout the rest of the section.

A description of TPM is given as Mechanism 4. Notice that Mechanism 4 often refers to parameters of the model that are not known to the mechanism, such as the value of an advertiser or the number of users of a mediator. Whenever this happens, this should be understood as referring to the reported values of these parameters.

  {mechanism} Threshold by Partition Mechanism (TPM)  

  1. Let M_{L} be a set of mediators containing each mediator m\in M with probability \min\{17\sqrt[3]{\alpha},1\}, independently. Similarly, A_{L} is a set of advertisers containing each advertiser a\in A with probability \min\{17\sqrt[3]{\alpha},1\}, independently. Intuitively, the subscript L in M_{L} and A_{L} stands for “low priority”.

1

  1. Let \sigma_{A} be an arbitrary order over the advertisers that places the advertisers of A_{L} after all the other advertisers and is independent of the reports received by the mechanism. Similarly, \sigma_{M} is an arbitrary order over the mediators that places the mediators of M_{L} after all the other mediators and is independent of the reports received by the mechanism.

  2. Partition the mediators of M into two sets M_{1} and M_{2} by adding each mediator m\in M with probability 1/2, independently, to M_{1} and otherwise to M_{2}. Similarly, partition the advertisers of A into two sets A_{1} and A_{2} by adding each advertiser a\in A with probability 1/2, independently, to A_{1} and otherwise to A_{2}. The rest of the algorithm explains how to assign users of mediators from M_{1} to slots of advertisers from A_{1}, and how to charge advertisers of A_{1} and pay mediators of M_{1}. Analogous steps, which we omit, should be added for handling the advertisers of A_{2} and the mediators of M_{2}.

  3. Let \hat{p} and \hat{b} be the user and slot at location \lceil(1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2}))|\rceil of the canonical solution S_{c}(P(M_{2}),B(A_{2})). If (1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2}))|\leq 0, then the previous definition of \hat{p} and \hat{b} cannot be used. Instead define \hat{p} as a dummy user of cost -\infty and \hat{b} as a dummy slot of value \infty. Using \hat{p} and \hat{b} define now two sets

    \hat{P}=\{p\in P(M_{1})\mid c(p)<c(\hat{p})\}\qquad\text{and}\qquad\hat{B}=\{b% \in B(A_{1})\mid v(b)>v(\hat{b})\}\enspace.

    It is important to note that \hat{P} and \hat{B} are empty whenever \hat{p} and \hat{b} are dummy user and slot, respectively.

While there are unassigned users in \hat{P} and unassigned slots in \hat{B} do the following: Let m be the earliest mediator in \sigma_{M} having unassigned users in \hat{P}. Let a be the earliest advertiser in \sigma_{A} having unassigned slots in \hat{B}. Assign the unassigned user of \hat{P}\cap P(m) with the lowest cost to an arbitrary unassigned slot of \hat{B}\cap B(a), charge a payment of v(\hat{b}) to advertiser a and transfer a payment of c(\hat{p}) to mediator m.777Note that m is paid c(\hat{p}) for the assignment of each one of his users. Hence, m is always BB since the membership of p in \hat{P} implies c(p)<c(\hat{p}) (and c(p)\leq c(\hat{p}) when the costs are compared as numbers).  

Let us recall our result regarding TPM.

Theorem 1.2.

TPM is BB, IR, IC and (1-28\sqrt[3]{\alpha}-20e^{-2/\sqrt[3]{\alpha}})-competitive.

Intuitively, the analysis of TPM works by exploiting concentration results showing that the canonical assignments S_{c}(P(M_{1}),B(A_{1})) and S_{c}(P(M_{2}),B(A_{2})) are not very different from each other. This similarity allows us to use information from S_{c}(P(M_{2}),B(A_{2})) to set the payments charged to advertisers of B(A_{1}) and payed to mediators of P(M_{1}), and vice versa, while keeping a reasonable competitive ratio. The advantage of setting the payments this way is that it reduces the control players have on the payments they have to pay or are paid, which helps the mechanism to be IC.

In the rest of this section we prove Theorem 1.2. We start with the following observation.

Observation 4.1.

TPM is BB.

Proof.

We show that whenever TPM assigns a user p to a slot b, it charges the advertiser of b more than it pays the mediator of p. Consider an arbitrary ordered pair (p,b) from the assignment produced by TPM. We assume without loss of generality that p\in P(M_{1}); the other case is symmetric. The fact that p is assigned implies p\in\hat{P}, hence, \hat{P} is non-empty. Similarly, the fact that a user is assigned to b implies that \hat{B} is not empty.

Recall that the fact that \hat{P} and \hat{B} are not empty implies that \hat{p} and \hat{b} are not dummy user and slot, respectively. This means that \hat{p} and \hat{b} are matched by the canonical assignment S_{c}(P(M_{2}),B(A_{2})). Since a canonical assignment never assigns a user p^{\prime} to a slot b^{\prime} when c(p^{\prime})>v(b^{\prime}), we get c(\hat{p})<v(\hat{b}). The proof now completes by observing that the advertiser of b is charged v(\hat{b}) for the assignment of p to b, while the mediator of p is paid c(\hat{p}) for this assignment. ∎

4.1 The Incentive Properties of TPM

In this section we prove the incentive parts of Theorem 1.2. Specifically, we prove that TPM is IR and IC.

Lemma 4.2.

For every mediator m (advertiser a), TPM is IR for m (a), and truthfulness is a dominant strategy for him (her).

Proof.

We prove the lemma only for mediators. The proof for advertisers is analogous (with slots exchanging roles with users, v(\hat{b}) exchanging roles with c(\hat{p}), etc.), and thus, we omit it. Additionally, we assume without loss of generality that m\in M_{1} (the other case is symmetric).

Recall that TPM calculates a set \hat{P} of users, and each user p^{\prime}\in P(M_{1}) belongs to \hat{P} if and only if her cost is lower than some threshold c(\hat{p}). Additionally, note that TPM calculates the threshold c(\hat{p}) based on the reports of advertisers and mediators in A_{2} and M_{2}, respectively. Thus, m, which belongs to M_{1}, cannot affect this threshold. Similarly, m cannot affect the set of slots \hat{B}.

Whenever a user p\in P(m) is assigned to a slot the utility of m decreases by c(p) and increases by the payment m gets, which is c(\hat{p}). In other words, the utility of m changes by c(\hat{p})-c(p). When m is truthful this change is always non-negative since p can be assigned only when she belongs to \hat{P}, which implies that the cost of p is smaller than \hat{p}. This already proves that the utility of m is non-negative when he is truthful, and thus, TPM is IR for m.

We claim that there exists a value k which is independent of the report of m such that for any report of m the mechanism assigns the \min\{k,|\hat{P}\cap P(m)|\} users of m with the lowest reported costs. Before proving this claim, let us explain why the lemma follows from this claim. The above description shows that the utility of m changes by a c(\hat{p})-c(p) for every assigned user p\in P(m), thus, m wishes to assign as many as possible users having cost less than c(\hat{p}), and if he cannot assign all of them then he prefers to assign the users with the lowest costs. By reporting truthfully m guarantees that only users of cost less than c(\hat{p}) get to \hat{P}, and thus, has a chance to be assigned. Moreover, by the above claim the mechanism assigns the k users of \hat{P}\cap P(m) with the lowest costs (or all of them if |\hat{P}\cap P(m)|<k). Hence, the above claim indeed implies that truthfulness is a dominant strategy for m.

We are only left to prove the above claim. One can view TPM as considering the mediators according to the order \sigma_{M}. For every mediator of m^{\prime}\in M_{1} TPM assigns the users of \hat{P}\cap P(m^{\prime}) one by one (in an increasing reported costs order). Every assignment of a user “consumes” one unassigned slot of \hat{B}, and the assignment of users stops when all the users of \hat{P} are assigned, or there are no more unassigned slots in \hat{B}. This means that if TPM assigns users to all the slots of \hat{B} before considering m, then no user of m is assigned. The claim is true in this case with k=0. Otherwise, we choose k to be the number of unassigned \hat{B} slots immediately before TPM considers m. Notice that the report of m does not affect the behavior of TPM up to the point it considers m, and thus, k is independent of the report of m. If there are more than k users in \hat{P}\cap P(m), then only the k of them with the lowest costs are assigned before TPM consumes all the unassigned slots of \hat{B} and stops. Otherwise, if |\hat{P}\cap P(m)|\leq k then TPM manages to assigns all the users of \hat{P}\cap P(m) before it runs out of unassigned slots of \hat{B}. ∎

4.2 The Competitive Ratio of TPM

In this section we analyze the competitive ratio of TPM. Let us define \tilde{P} (\tilde{B}) as the set of the users (slots) at locations 1 to \lceil(1-11\sqrt[3]{\alpha})\tau\rceil of the canonical assignment S_{c}(P,B) (\tilde{P} and \tilde{B} are defined as the empty set when 1-11\sqrt[3]{\alpha}\leq 0). The following observation shows that most of the gain from trade of the canonical assignment S_{c}(P,B) comes from the users and slots of \tilde{P} and \tilde{B}, respectively. For convenience, let us denote by P_{o} the set of users that are assigned by S_{c}(P,B), and by B_{o} the set of slots that are assigned some user by S_{c}(P,B).

Observation 4.3.

\sum_{b\in\tilde{B}}v(b)-\sum_{p\in\tilde{P}}c(p)\geq(1-11\sqrt[3]{\alpha})% \cdot\mathtt{GfT}(S_{c}(P,B)).

Proof.

If 1-11\sqrt[3]{\alpha}\leq 0, then both \tilde{B} and \tilde{P} are empty, and the inequality that we need to prove holds since its left hand side is 0 and its right hand side is non-positive (recall that S_{c}(P,B) is an assignment of users from P to slots of B maximizing the gain from trade, and thus, its gain from trade is at least 0 since \mathtt{GfT}(\varnothing)=0). Thus, we may assume in the rest of the proof that 1-11\sqrt[3]{\alpha}>0.

Since \tilde{B} contains the \lceil(1-11\sqrt[3]{\alpha})\tau\rceil slots with the largest values among the slots of B_{o}, we get:

\sum_{b\in\tilde{B}}v(b)\geq\lceil(1-11\sqrt[3]{\alpha})\tau\rceil\cdot\frac{% \sum_{b\in B_{o}}v(b)}{\tau}\enspace.

Similarly, since \tilde{P} contains the \lceil(1-11\sqrt[3]{\alpha})\tau\rceil users with the lowest costs among the users of P_{o}, we get:

\sum_{p\in\tilde{A}}c(p)\leq\lceil(1-11\sqrt[3]{\alpha})\tau\rceil\cdot\frac{% \sum_{c\in P_{o}}c(p)}{\tau}\enspace.

Combining the two inequities gives:

\displaystyle\sum_{b\in\tilde{B}}v(b)-\sum_{p\in\tilde{P}}c(p)\geq \displaystyle\lceil(1-11\sqrt[3]{\alpha})\tau\rceil\cdot\frac{\left(\sum_{b\in B% _{o}}v(b)-\sum_{p\in P_{o}}c(p)\right)}{\tau}
\displaystyle= \displaystyle\lceil(1-11\sqrt[3]{\alpha})\tau\rceil\cdot\frac{\mathtt{GfT}(S_{% c}(P,B))}{\tau}\geq(1-11\sqrt[3]{\alpha})\cdot\mathtt{GfT}(S_{c}(P,B))\enspace.\qed

Observation 4.3 shows that one can prove a competitive ratio for TPM by relating the gain from trade of the assignment it produces to the gain from trade obtained by assigning the users of \tilde{P} to the slots \tilde{B}. The following lemma is a key lemma we use to relate the two gains.

Lemma 4.4.

There exists an event {\mathcal{E}}_{1} of probability at least 1-10e^{-2/\sqrt[3]{\alpha}} such that {\mathcal{E}}_{1} implies the following:

(i) \tilde{B}\cap B(A_{1})\subseteq\hat{B} (iii) |\hat{P}\setminus P(M_{L})|\leq|\hat{B}|
(ii) \tilde{P}\cap P(M_{1})\subseteq\hat{P} (iv) |\hat{B}\setminus B(A_{L})|\leq|\hat{P}|
(v) c(p)\leq\ell(P,B)\leq v(b) for every user p\in\hat{P} and slot b\in\hat{B}, where \ell(P,B) is a value which is independent of the random coins of TPM and obeys c(p)\leq\ell(P,B)\leq v(b) for every p\in P_{o} and b\in B_{o}.

Before proving Lemma 4.4, let us explain how the competitive ratio of TPM follows from it. Let \hat{S} be the assignment produced by TPM.

Lemma 4.5.

There exists an event {\mathcal{E}} of probability at least 1-20e^{-2/\sqrt[3]{\alpha}} such that {\mathcal{E}} implies:

\mathtt{GfT}(\hat{S})\geq\sum_{b\in\tilde{B}\setminus B(A_{L})}[v(b)-\ell(P,B)% ]+\sum_{p\in\tilde{P}\setminus P(M_{L})}[\ell(P,B)-c(p)]\enspace.
Proof.

Assume first that the event {\mathcal{E}}_{1} holds. Lemma 4.4 shows that given {\mathcal{E}}_{1} we have |\hat{P}\setminus P(M_{L})|\leq|\hat{B}|, hence, TPM assigns at least |\hat{P}\setminus P(M_{L})| users. Additionally, since TPM assigns all the users of \hat{P}\setminus P(M_{L}) before it starts assigning users of \hat{P}\cap P(M_{L}), we get that all the users of \hat{P}\setminus P(M_{L}) are assigned by \hat{S} given {\mathcal{E}}_{1}. On the other hand, Lemma 4.4 also shows that given {\mathcal{E}}_{1} all the users of \tilde{P}\cap P(M_{1}) belong to \hat{P}, and thus, the users of (\tilde{P}\cap P(M_{1}))\setminus P(M_{L}) are all assigned by \hat{S}. A similar argument shows that the slots of (\tilde{B}\cap B(A_{1}))\setminus B(A_{L}) are all assigned users by \hat{S} given {\mathcal{E}}_{1}. Finally, observe that {\mathcal{E}}_{1} also implies that c(p)\leq\ell(P,B)\leq v(b) for every pair (p,b)\in\hat{S}\cap(P(M_{1})\times B(A_{1})).

TPM assigns users of P(M_{2}) to slots of B(A_{2}) in an analogous way to the assignment of users of P(M_{1}) to slots of B(A_{1}). Thus, by symmetry, there must exist an event {\mathcal{E}}_{2} of probability at least 1-10e^{-2/\sqrt[3]{\alpha}} (the probability of {\mathcal{E}}_{1}) that guarantees the following three things.

  • The users of (\tilde{P}\cap P(M_{2}))\setminus P(M_{L}) are all assigned by \hat{S}.

  • The slots of (\tilde{B}\cap B(A_{2}))\setminus B(A_{L}) are all assigned users by \hat{S}.

  • c(p)\leq\ell(P,B)\leq v(b) for every pair (p,b)\in\hat{S}\cap(P(M_{2})\times B(A_{2})).

Let us now define {\mathcal{E}} as the event that both {\mathcal{E}}_{1} and {\mathcal{E}}_{2} happen together. By the union bound the probability of {\mathcal{E}} is at least 1-20e^{-2/\sqrt[3]{\alpha}} as promised. Additionally, combining the properties that follow from {\mathcal{E}}_{1} and {\mathcal{E}}_{2} by the above discussion, we get that {\mathcal{E}} implies the following properties.

  • The users of \tilde{P}\setminus P(M_{L}) are all assigned by \hat{S}.

  • The slots of \tilde{B}\setminus B(A_{L}) are all assigned users by \hat{S}.

  • c(p)\leq\ell(P,B)\leq v(b) for every pair (p,b)\in\hat{S}.

The last property holds since TPM assigns users of P(M_{1}) (P(M_{2})) only to slots of B(A_{1}) (B(A_{2})), and thus, (\hat{S}\cap(P(M_{2})\times B(A_{2})))\cup(\hat{S}\cap(P(M_{2})\times B(A_{2})% ))=\hat{S}.

In the rest of the proof we assume that {\mathcal{E}} happens. Consider an ordered pair (p,b)\in\hat{S}. The contribution of (p,b) to \mathtt{GfT}(\hat{S}) is:

v(b)-c(p)=[v(b)-\ell(P,B)]+[\ell(P,B)-c(p)]\enspace.

Notice that the two terms that appear in brackets on the right hand side of the last equation are both positive given {\mathcal{E}}. This allows us to lower bound the gain from trade of \hat{S} as follows:

\displaystyle\mathtt{GfT}(\hat{S})= \displaystyle\sum_{(p,b)\in\hat{S}}[v(b)-c(p)]=\sum_{(p,b)\in\hat{S}}\{[v(b)-% \ell(P,B)]+[\ell(P,B)-c(p)]\}
\displaystyle\geq \displaystyle\sum_{b\in\tilde{B}\setminus B(A_{L})}[v(b)-\ell(P,B)]+\sum_{p\in% \tilde{P}\setminus P(M_{L})}[\ell(P,B)-c(p)]\enspace.\qed
Corollary 4.6.

TPM is at least (1-28\sqrt[3]{\alpha}-20e^{-2/\sqrt[3]{\alpha}})-competitive.

Proof.

The corollary is trivial when 28\sqrt[3]{\alpha}+20e^{-2/\sqrt[3]{\alpha}}>1. Thus, we assume in this proof 28\sqrt[3]{\alpha}+20e^{-2/\sqrt[3]{\alpha}}\leq 1. Let \mathtt{Val}(M_{L},A_{L}) denote the expression:

\sum_{b\in\tilde{B}\setminus B(A_{L})}[v(b)-\ell(P,B)]+\sum_{p\in\tilde{P}% \setminus P(M_{L})}[\ell(P,B)-c(p)]\enspace.

Since every slot belongs to B(A_{L}) with probability 17\sqrt[3]{\alpha} and every user belongs to P(M_{L}) with the same probability, we get:

\displaystyle{\mathbb{E}}[\mathtt{Val}(M_{L},A_{L})]= \displaystyle(1-17\sqrt[3]{\alpha})\cdot\sum_{b\in\tilde{B}}[v(b)-\ell(P,B)]+(% 1-17\sqrt[3]{\alpha})\cdot\sum_{p\in\tilde{P}}[\ell(P,B)-c(b)]
\displaystyle= \displaystyle(1-17\sqrt[3]{\alpha})\cdot\mathtt{Val}(\varnothing,\varnothing)\enspace.

Additionally, the definition of \ell(P,B) guarantees that v(b)-\ell(P,B)\geq 0 and \ell(P,B)-c(p)\geq 0 for every b\in\tilde{B}\subseteq B_{o} and p\in\tilde{P}\subseteq P_{o}. Thus, \mathtt{Val}(M_{L},A_{L})\leq\mathtt{Val}(\varnothing,\varnothing) for every two sets M_{L}\subseteq M and A_{L}\subseteq A. Using Lemma 4.5 and the observation that TPM always produces assignments of non-negative gain from trade, we now get:

\displaystyle{\mathbb{E}}[\mathtt{GfT}(\hat{S})] \displaystyle=\Pr[{\mathcal{E}}]\cdot{\mathbb{E}}[\mathtt{GfT}(\hat{S})\mid{% \mathcal{E}}]+\Pr[\neg{\mathcal{E}}]\cdot{\mathbb{E}}[\mathtt{GfT}(\hat{S})% \mid\neg{\mathcal{E}}]
\displaystyle\geq \displaystyle\Pr[{\mathcal{E}}]\cdot{\mathbb{E}}[\mathtt{Val}(M_{L},A_{L})\mid% {\mathcal{E}}]={\mathbb{E}}[\mathtt{Val}(M_{L},A_{L})]-\Pr[\neg{\mathcal{E}}]% \cdot{\mathbb{E}}[\mathtt{Val}(M_{L},A_{L})\mid\neg{\mathcal{E}}]
\displaystyle\geq \displaystyle(1-17\sqrt[3]{\alpha})\cdot\mathtt{Val}(\varnothing,\varnothing)-% \Pr[\neg{\mathcal{E}}]\cdot\mathtt{Val}(\varnothing,\varnothing)=(1-17\sqrt[3]% {\alpha}-\Pr[\neg{\mathcal{E}}])\cdot\mathtt{Val}(\varnothing,\varnothing)\enspace. (3)

Recall that \Pr[\neg{\mathcal{E}}]\leq 20e^{-2/\sqrt[3]{\alpha}} by Lemma 4.5. Additionally, Observation 4.3 and the fact that |\tilde{P}|=|\tilde{B}| by definition imply together:

\displaystyle\mathtt{Val}(\varnothing,\varnothing)= \displaystyle\sum_{b\in\tilde{B}}[v(b)-\ell(P,B)]+\sum_{p\in\tilde{P}}[\ell(P,% B)-c(p)]
\displaystyle= \displaystyle\sum_{b\in\tilde{B}}v(b)-\sum_{p\in\tilde{P}}c(p)\geq(1-11\sqrt[3% ]{\alpha})\cdot\mathtt{GfT}(S_{c}(P,A))\enspace.

Plugging the last observations into Inequality 4.2 gives:

\displaystyle{\mathbb{E}}[\mathtt{GfT}(\hat{S})]\geq \displaystyle(1-17\sqrt[3]{\alpha}-\Pr[\neg{\mathcal{E}}])\cdot\mathtt{Val}(% \varnothing,\varnothing)
\displaystyle\geq \displaystyle(1-17\sqrt[3]{\alpha}-20e^{-2/\sqrt[3]{\alpha}})\cdot(1-11\sqrt[3% ]{\alpha})\cdot\mathtt{GfT}(S_{c}(P,A))
\displaystyle\geq \displaystyle(1-28\sqrt[3]{\alpha}-20e^{-2/\sqrt[3]{\alpha}})\cdot\mathtt{GfT}% (S_{c}(P,B))\enspace.

The corollary now follows by recalling that S_{c}(P,B) is the assignment of users from P to slots of B which maximizes the gain from trade. ∎

It remains now to prove Lemma 4.4. Let us begin with the following technical lemma.

Lemma 4.7.

Given a subset B^{\prime}\subseteq B_{o} and a probability q\in[0,1], let B^{\prime}[q] be a random subset of B^{\prime} constructed as follows: for every advertiser a\in A, independently, with probability q the slots of advertiser a that belong to B^{\prime} appear also in B^{\prime}[q]. Then, for every \beta\in(0,1]:

\Pr[||B^{\prime}[q]|-q\cdot|B^{\prime}||\geq\beta\tau]\leq 2e^{-2\beta^{2}/% \alpha}\enspace.

Similarly, given a subset P^{\prime}\subseteq P_{o} and a probability q\in[0,1], let P^{\prime}[q] be a random subset of P^{\prime} constructed as follows: for every mediator m\in M, independently, with probability q the users of mediator m that belong to P^{\prime} appear also in P^{\prime}[q]. Then, for every \beta\in(0,1]:

\Pr[||P^{\prime}[q]|-q\cdot|P^{\prime}||\geq\beta\tau]\leq 2e^{-2\beta^{2}/% \alpha}\enspace.
Proof.

We prove the first inequality; the second inequality is analogous. First, observe that the lemma is trivial when B^{\prime}=\varnothing since B^{\prime}=\varnothing implies ||B^{\prime}[q]|-q\cdot|B^{\prime}||=0<\beta\tau. Thus, we may assume in the rest of the proof B^{\prime}\neq\varnothing. For every advertiser a\in A, let X_{a} be an indicator for the event that slots of a appear in B^{\prime}[q]. Then:

|B^{\prime}[q]|=\sum_{a\in A}X_{a}\cdot|B^{\prime}\cap B(a)|\enspace.

The definition of \alpha implies |B(a)|\leq\alpha\tau for every advertiser a\in A, and thus, 0\leq|B^{\prime}\cap B(a)|\leq\alpha\tau. Hence, by Hoeffding’s inequality:

\displaystyle\Pr[||B^{\prime}[q]|-q\cdot|B^{\prime}||\geq\beta\tau]= \displaystyle\Pr[||B^{\prime}[q]|-{\mathbb{E}}[|B^{\prime}[q]|]|\geq\beta\tau]% \leq 2e^{-\frac{2(\beta\tau)^{2}}{\sum_{a\in A}|B^{\prime}\cap B(a)|^{2}}}
\displaystyle\leq \displaystyle 2e^{-\frac{2(\beta\tau)^{2}}{\alpha\tau\cdot\sum_{a\in A}|B^{% \prime}\cap B(a)|}}=2e^{-\frac{2\beta^{2}\tau}{\alpha\cdot|B^{\prime}|}}\leq 2% e^{-\frac{2\beta^{2}\tau}{\alpha\cdot|B_{o}|}}=2e^{-\frac{2\beta^{2}}{\alpha}}% \enspace.\qed
\Hy@raisedlink

Let {\mathcal{E}}^{\prime} be the event that the following inequalities are all true (at the same time):

(i) ||B_{o}\cap B(A_{2})|-|B_{o}|/2|\leq\sqrt[3]{\alpha}\cdot\tau (iii) ||\tilde{B}\cap B(A_{2})|-|\tilde{B}|/2|\leq\sqrt[3]{\alpha}\cdot\tau
(ii) ||P_{o}\cap P(M_{2})|-|P_{o}|/2|\leq\sqrt[3]{\alpha}\cdot\tau (iv) ||\tilde{P}\cap P(M_{2})|-|\tilde{P}|/2|\leq\sqrt[3]{\alpha}\cdot\tau
Observation 4.8.

\Pr[{\mathcal{E}}^{\prime}]\geq 1-8e^{-2/\sqrt[3]{\alpha}}.

Proof.

Observe that B_{o}\cap B(A_{2}), \tilde{B}\cap B(A_{2}), P_{o}\cap P(M_{2}) and \tilde{P}\cap P(M_{2}) have the same distributions as B_{o}[1/2], \tilde{B}[1/2], P_{o}[1/2] and \tilde{P}[1/2], respectively. Moreover, by definition \tilde{B}\subseteq B_{o} and \tilde{P}\subseteq P_{o}. Hence, by Lemma 4.7, each one of the four inequalities defining {\mathcal{E}}^{\prime} holds with probability at least 1-2e^{-2/\sqrt[3]{\alpha}}. The observation now follows by the union bound. ∎

Next, we need the following useful observation.

Observation 4.9.

It always holds that:

\min\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}\leq|S_{c}(P(M_{2}),B(A_{2}))% |\leq\max\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}\enspace.
Proof.

Let p_{\tau} and b_{\tau} be the user and slot at location \tau of S_{c}(P,B), respectively. The definition of a canonical assignment guarantees c(p_{\tau})<v(b_{\tau}). Additionally, the slots of B_{o} all appear in locations 1 to \tau of S_{c}(P,B), and thus, they all have values at least as large as v(b_{\tau}). Similarly, the users of P_{o} all have costs at most as large as c(p_{\tau}). Combining these observations, we get: c(p)\leq c(p_{\tau})<v(b_{\tau})\leq v(b) for every p\in P_{o} and b\in B_{o}.

The slots at locations 1 to |B_{o}\cap B(A_{2})| of S_{c}(P(M_{2}),B(A_{2})) all belong to B_{o} since B_{o} contains the \tau slots with the largest values. Similarly, the users at locations 1 to |P_{o}\cap P(M_{2})| belong to P_{o}. Combining both observations, we get that for every location 1\leq i\leq\min\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}, the user p^{\prime}_{i} at location i of S_{c}(P(M_{2}),B(A_{2})) and the slot b^{\prime}_{i} at this location belong to P_{o} and B_{o}, respectively, and thus, c(p^{\prime}_{i})<v(b^{\prime}_{i}). Hence, by the definition a canonical assignment, the pair (p^{\prime}_{i},b^{\prime}_{i}) belongs to S_{c}(P(M_{2}),B(A_{2})) for every 1\leq i\leq\min\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}; which completes the proof of the first inequality we need to prove.

Assume towards a contradiction that the second inequality we need to prove is wrong. In other words, we assume |S_{c}(P(M_{2}),B(A_{2}))|>\max\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}. Let j=\max\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}+1, and let p^{\prime}_{j} and b^{\prime}_{j} be the user and slot at location j of S_{c}(P(M_{2}),B(A_{2})), respectively. Our assumption implies that (p^{\prime}_{j},b^{\prime}_{j}) belongs to S_{c}(P(M_{2}),B(A_{2})), and thus, c(p^{\prime}_{j})<v(b^{\prime}_{j}). On the other hand, only the users at locations 1 to |P_{o}\cap P(M_{2})| of S_{c}(P(M_{2}),B(A_{2})) belong to P_{o}, hence, p^{\prime}_{j} does not belong to P_{o}. The user with the lowest cost that does not belong to P_{o} is the user p_{\tau+1} at location \tau+1 of S_{c}(P,B). Thus, we get: c(p^{\prime}_{j})\geq c(p_{\tau+1}). Similarly, we can also get v(b^{\prime}_{j})\leq v(b_{\tau+1}), where b_{\tau+1} is the slot at location \tau+1 of S_{c}(P,B). Combining the above inequalities gives:

c(p_{\tau+1})\leq c(p^{\prime}_{j})<v(p^{\prime}_{j})\leq v(b_{\tau+1})\enspace,

which contradicts the fact that p_{\tau+1} is not assigned to b_{\tau+1} by the canonical assignment S_{c}(P,B). ∎

The next few claims use the last observation to prove a few properties that hold given {\mathcal{E}}’.

Lemma 4.10.

The event {\mathcal{E}}^{\prime} implies: \hat{P}\subseteq P_{o} and \hat{B}\subseteq B_{o}.

Proof.

We prove the first inclusion. The other inclusion is analogous. Observation 4.9 and the definition of {\mathcal{E}}^{\prime} imply:

\displaystyle|S_{c}(P(M_{2}),B(A_{2}))|\leq \displaystyle\max\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}
\displaystyle\leq \displaystyle\max\{|P_{o}|/2+\sqrt[3]{\alpha}\cdot\tau,|B_{o}|/2+\sqrt[3]{% \alpha}\cdot\tau\}=(1/2+\sqrt[3]{\alpha})\tau\enspace.

Using the definition of {\mathcal{E}}^{\prime} again gives:

\displaystyle|P_{o}\cap P(M_{2})|\geq \displaystyle|P_{o}|/2-\sqrt[3]{\alpha}\cdot\tau=(1/2-\sqrt[3]{\alpha})\tau
\displaystyle\geq \displaystyle(1-4\sqrt[3]{\alpha})\cdot(1/2+\sqrt[3]{\alpha})\tau\geq(1-4\sqrt% [3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2}))|\enspace,

which implies, since |P_{o}\cap P(M_{2})| is integral,

|P_{o}\cap P(M_{2})|\geq\lceil(1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2% }))|\rceil\enspace. (4)

If \hat{p} is a dummy user then \hat{P} is empty, which makes the claim \hat{P}\subseteq P_{o} trivial. Thus, we may assume that \hat{p} is the user at location \lceil(1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2}))|\rceil of the canonical assignment S_{c}(P(M_{2}),B(A_{2})). Hence, Inequality (4) and the observation that the users of P_{o}\cap P(M_{2}) are the users with the lowest costs in P(M_{2}) imply together that \hat{p} belongs to the set P_{o}\cap P(M_{2})\subseteq P_{o}. On the other hand, P_{o} contains the \tau users with the lowest costs. Hence, every user with a cost lower than \hat{p} must be in P_{o} since \hat{p} is in P_{o}. The lemma now follows by observing that the definition of \hat{P} implies c(p)<c(\hat{p}) for every user p\in\hat{P}. ∎

Corollary 4.11.

There exists a value \ell(P,B) independent of the random coins of TPM such that:

  • c(p)\leq\ell(P,B)\leq v(b) for every user p\in P_{o} and slot b\in B_{o}

  • Whenever the event {\mathcal{E}}^{\prime} occurs, c(p)\leq\ell(P,B)\leq v(b) for every user p\in\hat{P} and slot b\in\hat{B}.

Proof.

Let \ell(P,B) be the value of the slot at location \tau of the canonical assignment S_{c}(P,B). Clearly, \ell(P,B) is independent of the random coins of TPM, as required. Additionally, for every slot b\in B_{o} it holds that v(b)\geq\ell(P,B) since b must be located at some location of S_{c}(P,B) between 1 and \tau. On the other hand, let p_{\tau} be the user at location \tau of S_{c}(P,B). Since the size of S_{c}(P,B) is \tau, p_{\tau} must be assigned to the slot at location \tau of S_{c}(P,B), which implies c(p_{\tau})\leq\ell(P,B).888In fact, we even have c(p_{\tau})<\ell(P,B) since the tie-breaking rule defined in Section 2.1 guarantees that the value of a slot is never equal to the cost of a user. Moreover, for every user p\in P_{o} it holds that c(p)\leq c(p_{\tau})\leq\ell(P,B) since p must be located at some location of S_{c}(P,B) between 1 and \tau.

The lemma now follows since Lemma 4.10 shows that the event {\mathcal{E}}^{\prime} implies that every user p\in\hat{P} belongs also to P_{o}, and every slot b\in\hat{B} belongs also to B_{o}. ∎

Lemma 4.12.

The event {\mathcal{E}}^{\prime} implies \tilde{P}\cap P(M_{1})\subseteq\hat{P} and \tilde{B}\cap B(A_{1})\subseteq\hat{B}.

Proof.

We prove the first inclusion. The other inclusion is analogous. The claim about \tilde{P}\cap P(M_{1}) is trivial when \tilde{P} is empty. Thus, we can assume throughout the proof that \tilde{P} is non-empty. Observation 4.9 and the definition of {\mathcal{E}}^{\prime} imply:

\displaystyle|S_{c}(P(M_{2}),B(A_{2}))|\geq \displaystyle\min\{|P_{o}\cap P(M_{2})|,|B_{o}\cap B(A_{2})|\}
\displaystyle\geq \displaystyle\min\{|P_{o}|/2-\sqrt[3]{\alpha}\cdot\tau,|B_{o}|/2-\sqrt[3]{% \alpha}\cdot\tau\}=(1/2-\sqrt[3]{\alpha})\tau\enspace.

Recall that \alpha\geq\tau^{-1}, and thus, \sqrt[3]{\alpha}\cdot\tau\geq 1. Using this inequality and the definition of {\mathcal{E}}^{\prime} again gives:

\displaystyle|\tilde{P}\cap P(M_{2})|\leq \displaystyle|\tilde{P}|/2+\sqrt[3]{\alpha}\cdot\tau=\lceil(1-11\sqrt[3]{% \alpha})\tau\rceil/2+\sqrt[3]{\alpha}\cdot\tau\leq(1/2-4\sqrt[3]{\alpha})\tau-% 1+\sqrt[3]{\alpha}\cdot\tau
\displaystyle= \displaystyle(1/2-3\sqrt[3]{\alpha})\tau-1\leq(1-4\sqrt[3]{\alpha})\cdot(1/2-% \sqrt[3]{\alpha})\tau-1
\displaystyle\leq \displaystyle(1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2}))|-1\enspace,

which implies, since |\tilde{P}\cap P(M_{2})| is integral,

|\tilde{P}\cap P(M_{2})|\leq\lceil(1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(% A_{2}))|\rceil-1<\lceil(1-4\sqrt[3]{\alpha})\cdot|S_{c}(P(M_{2}),B(A_{2}))|% \rceil\enspace. (5)

Inequality (5) and the observation that the users of \tilde{P}\cap P(M_{2}) are the users with the lowest costs in P(M_{2}) imply together that \hat{p} is a user of P(M_{2}) which does not belong to \tilde{P}\cap P(M_{2}), and therefore, does not belong to \tilde{P} either. On the other hand, \tilde{P} contains the \lceil(1-11\sqrt[3]{\alpha})\tau\rceil users with the lowest costs. Hence, every user p\in\tilde{P} has a cost smaller than c(\hat{p}) since \hat{p} does not belong to \tilde{P}. The lemma now follows by observing that the definition of \hat{P} implies that p\in\hat{P} for every user p\in P(M_{1}) obeying c(p)<c(\hat{p}). ∎

Corollary 4.13.

The event {\mathcal{E}}^{\prime} implies: -6.5\sqrt[3]{\alpha}\cdot\tau\leq|\hat{P}|-\tau/2\leq\sqrt[3]{\alpha}\cdot\tau and -6.5\sqrt[3]{\alpha}\cdot\tau\leq|\hat{B}|-\tau/2\leq\sqrt[3]{\alpha}\cdot\tau.

Proof.

We prove here only the bounds on the size of \hat{P}. The bounds on the size of \hat{B} are analogous. By Lemma 4.10, \hat{P}\subseteq P_{o}. On the other hand, by definition, \hat{P}\subseteq P(M_{1}). Thus, we get: \hat{P}\subseteq P_{o}\cap P(M_{1}). Combining this inclusion with the definition of {\mathcal{E}}^{\prime} gives:

\displaystyle|\hat{P}|\leq \displaystyle|P_{o}\cap P(M_{1})|=|P_{o}|-|P_{o}\cap P(M_{2})|
\displaystyle\leq \displaystyle|P_{o}|-[|P_{o}|/2-\sqrt[3]{\alpha}\cdot\tau]=|P_{o}|/2+\sqrt[3]{% \alpha}\cdot\tau=\tau/2+\sqrt[3]{\alpha}\cdot\tau\enspace.

On the other hand, by Lemma 4.12 and the definition of {\mathcal{E}}^{\prime},

\displaystyle|\hat{P}|\geq \displaystyle|\tilde{P}\cap P(M_{1})|=|\tilde{P}|-|\tilde{P}\cap P(M_{2})|\geq% |\tilde{P}|-[|\tilde{P}|/2+\sqrt[3]{\alpha}\cdot\tau]
\displaystyle= \displaystyle|\tilde{P}|/2-\sqrt[3]{\alpha}\cdot\tau\geq\lceil(1-11\sqrt[3]{% \alpha})\tau\rceil/2-\sqrt[3]{\alpha}\cdot\tau\geq\tau/2-6.5\sqrt[3]{\alpha}% \cdot\tau\enspace.\qed

We can now define the event {\mathcal{E}}_{1} referred to by Lemma 4.4. The event {\mathcal{E}}_{1} is the event that {\mathcal{E}}^{\prime} happens and in addition the following two inequalities also hold:

(i) |\hat{B}\setminus B(A_{L})|\leq|\hat{P}| (ii) |\hat{P}\setminus P(M_{L})|\leq|\hat{B}|

Next, let us prove Lemma 4.4. For ease of the reading, we first repeat the lemma itself.

Lemma 4.4.

There exists an event {\mathcal{E}}_{1} of probability at least 1-10e^{-2/\sqrt[3]{\alpha}} such that {\mathcal{E}}_{1} implies the following:

(i) \tilde{B}\cap B(A_{1})\subseteq\hat{B} (iii) |\hat{P}\setminus P(M_{L})|\leq|\hat{B}|
(ii) \tilde{P}\cap P(M_{1})\subseteq\hat{P} (iv) |\hat{B}\setminus B(A_{L})|\leq|\hat{P}|
(v) c(p)\leq\ell(P,B)\leq v(b) for every user p\in\hat{P} and slot b\in\hat{B}, where \ell(P,B) is a value which is independent of the random coins of TPM and obeys c(p)\leq\ell(P,B)\leq v(b) for every p\in P_{o} and b\in B_{o}.
Proof.

By definition, the event {\mathcal{E}}_{1} implies the inequalities: |\hat{B}\setminus B(A_{L})|\leq|\hat{P}| and |\hat{P}\setminus P(M_{L})|\leq|\hat{B}|. Additionally, {\mathcal{E}}_{1} implies the event {\mathcal{E}}^{\prime}, which, by Corollary 4.11 and Lemma 4.12, implies the other things that should follow from {\mathcal{E}}_{1} by the lemma. Hence, the only thing left to prove is that the probability of {\mathcal{E}}_{1} is at least 1-10e^{-2/\sqrt[3]{\alpha}}.

If 17\sqrt[3]{\alpha}\geq 1, then A_{L}=A and M_{L}=M, which implies that the two inequalities |\hat{B}\setminus B(A_{L})|\leq|\hat{P}| and |\hat{P}\setminus P(M_{L})|\leq|\hat{B}| are trivial. Hence, the events {\mathcal{E}}_{1} and {\mathcal{E}}^{\prime} are equivalent in this case, and thus, the probability of {\mathcal{E}}_{1} is at least 1-8e^{-2/\sqrt[3]{\alpha}} by Observation 4.8. Therefore, it is safe to assume in the rest of the proof that 17\sqrt[3]{\alpha}<1.

Our plan is to prove the inequality \Pr[{\mathcal{E}}_{1}\mid{\mathcal{E}}^{\prime}]\geq 1-2e^{-2/\sqrt[3]{\alpha}}. Notice that this inequality indeed implies the lemma since it implies:

\Pr[{\mathcal{E}}_{1}]=\Pr[{\mathcal{E}}^{\prime}]\cdot\Pr[{\mathcal{E}}_{1}% \mid{\mathcal{E}}^{\prime}]\geq(1-8e^{-2/\sqrt[3]{\alpha}})\cdot(1-2e^{-2/% \sqrt[3]{\alpha}})\geq 1-10e^{-2/\sqrt[3]{\alpha}}\enspace.

The event {\mathcal{E}}^{\prime} is fully determined by way TPM partitions M and A into M_{1},M_{2},A_{1} and A_{2}. Thus, it is enough to show that for every fixed partition for which the event {\mathcal{E}}^{\prime} holds, the event {\mathcal{E}}_{1} holds with probability at least 1-2e^{-2/\sqrt[3]{\alpha}}. Notice that the sets \hat{P} and \hat{B} become deterministic once we fix the partition of A and M. Hence, either |\hat{B}|\leq|\hat{P}|, which implies that the inequality |\hat{B}\setminus B(A_{L})|\leq|\hat{P}| holds regardless of the choice of A_{L}, or |\hat{P}|\leq|\hat{B}|, which implies that the inequality |\hat{P}\setminus P(M_{L})|\leq|\hat{B}| holds regardless of the choice of M_{L}. In both cases, all we need to show is that the other inequality holds with probability at least 1-2e^{-2/\sqrt[3]{\alpha}} over the random choice of A_{L} and M_{L}.

Let us assume, without loss of generality, that |\hat{B}|\leq|\hat{P}|. By the above discussion, all we need to prove is that \Pr[|\hat{P}\setminus P(M_{L})|\leq|\hat{B}|]\geq 1-2e^{-2/\sqrt[3]{\alpha}}, where the probability is over the random choice of M_{L}. By Corollary 4.13:

\displaystyle\Pr[|\hat{P}\setminus P(M_{L})|>|\hat{B}|]\leq \displaystyle\Pr[|\hat{P}\setminus P(M_{L})|>\tau/2-6.5\sqrt[3]{\alpha}\cdot\tau]
\displaystyle\leq \displaystyle\Pr[|\hat{P}\setminus P(M_{L})|>(1-17\sqrt[3]{\alpha})(\tau/2+% \sqrt[3]{\alpha}\cdot\tau)+\sqrt[3]{\alpha}\cdot\tau]
\displaystyle\leq \displaystyle\Pr[|\hat{P}\setminus P(M_{L})|>(1-17\sqrt[3]{\alpha})\cdot|\hat{% P}|+\sqrt[3]{\alpha}\cdot\tau]\enspace.

Notice now that \hat{P}\setminus P(M_{L}) has the same distribution as \hat{P}[1-17\sqrt[3]{\alpha}]. Hence, by Lemma 4.7:

\displaystyle\Pr[|\hat{P}\setminus P(M_{L})|>|\hat{B}|]\leq \displaystyle\Pr[|\hat{P}\setminus P(M_{L})|>(1-17\sqrt[3]{\alpha})\cdot|\hat{% P}|+\sqrt[3]{\alpha}\cdot\tau]
\displaystyle\leq \displaystyle\Pr[||\hat{P}[1-17\sqrt[3]{\alpha}]|-(1-17\sqrt[3]{\alpha})\cdot|% \hat{P}||>\sqrt[3]{\alpha}\cdot\tau]\leq 2e^{-2/\sqrt[3]{\alpha}}\enspace.\qed

5 Conclusions

We considered mechanisms for double-sided markets that interact with strategic players where at least one side of the market has players with multi-dimensional strategic spaces. In particular, we explored the question of how many sides of the market can have players with multi-dimensional strategic spaces, while still allowing for mechanisms that both maintain the desired basic economic properties (individual rationality, incentive compatibility and budget balance) and suffer only a (small) bounded loss compared to the socially optimal outcome. To answer this question we presented two mechanisms. One mechanism that is deterministic and allows one side to have players with multi-dimensional strategic spaces, and another mechanism that is randomized and allows two sides to have players with multi-dimensional strategic spaces.

Our mechanisms significantly extend the literature on trade reduction—a technique used to achieve the basic economic properties of individual rationality, incentive compatibility and budget balance in a multi-sided market. While all the previous trade reduction solutions dealt with players having single dimensional strategic spaces, our deterministic algorithm performs a non-binary trade reduction which leads to the first trade reduction solution applying to multi-dimensional strategic spaces. As given, this deterministic trade reduction solution deals with two sides: advertisers and mediators; however, we believe that our ideas are more general, and we currently work on extending the mechanism in order to handle more market sides.

From a more practical point of view, our study is motivated by a foreseeable future form of online advertising in which users are incentivized to share their information via participation in a mediated online advertising exchange. Our model captures some of the challenges introduced by such exchanges, but does not encompass their online nature. In this sense our results in this paper can be seen as a step towards the study of a more realistic model which also captures the online nature of ad exchanges.

References

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