Capacity constraints and the inevitability of mediators in adword auctions
Abstract
One natural constraint in the sponsored search advertising framework arises from the fact that there is a limit on the number of available slots, especially for the popular keywords, and as a result, a significant pool of advertisers are left out. We study the emergence of diversification in the adword market triggered by such capacity constraints in the sense that new market mechanisms, as well as, new forprofit agents are likely to emerge to combat or to make profit from the opportunities created by shortages in adspace inventory. We propose a model where the additional capacity is provided by forprofit agents (or, mediators), who compete for slots in the original auction, draw traffic, and run their own subauctions. The quality of the additional capacity provided by a mediator is measured by its fitness factor. We compute revenues and payoffs for all the different parties at a symmetric Nash equilibrium (SNE) when the mediatorbased model is operated by a mechanism currently being used by Google and Yahoo!, and then compare these numbers with those obtained at a corresponding SNE for the same mechanism, but without any mediators involved in the auctions. Such calculations allow us to determine the value of the additional capacity. Our results show that the revenue of the auctioneer, as well as the social value (i.e. efficiency ), always increase when mediators are involved; moreover even the payoffs of all the bidders will increase if the mediator has a high enough fitness. Thus, our analysis indicates that there are significant opportunities for diversification in the internet economy and we should expect it to continue to develop richer structure, with room for different types of agents and mechanisms to coexist.
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1 Introduction
Sponsored search advertising is a significant growth market and is witnessing rapid growth and evolution. The analysis of the underlying models has so far primarily focused on the scenario, where advertisers/bidders interact directly with the auctioneers, i.e., the Search Engines and publishers. However, the market is already witnessing the spontaneous emergence of several categories of companies who are trying to mediate or facilitate the auction process. For example, a number of different AdNetworks have started proliferating, and so have companies who specialize in reselling ad inventories. Hence, there is a need for analyzing the impact of such incentive driven and forprofit agents, especially as they become more sophisticated in playing the game. In the present work, our focus is on the emergence of market mechanisms and forprofit agents motivated by capacity constraint inherent to the present models.
For instance, one natural constraint comes from the fact that there is a limit on the number of slots available for putting ads, especially for the popular keywords, and a significant pool of advertisers are left out due to this capacity constraint. We ask whether there are sustainable market constructs and mechanisms, where new players interact with the existing auction mechanisms to increase the overall capacity. In particular, leadgeneration companies who bid for keywords, draw traffic from search pages and then redirect such traffic to service/product providers, have spontaneously emerged. However, the incentive and equilibria properties of paidsearch auctions in the presence of such profitdriven players have not been explored. We investigate key questions, including what happens to the overall revenue of the auctioneers when such mediators participate, what is the payoff of a mediator and how does it dependent on her quality, how are the payoffs of the bidders affected, and is there an overall value that is generated by such mechanisms.
Formally, in the current models, there are slots to be allocated among () bidders (i.e. the advertisers). A bidder has a true valuation (known only to the bidder ) for the specific keyword and she bids . The expected click through rate (CTR) of an ad put by bidder when allocated slot has the form i.e. separable in to a position effect and an advertiser effect. ’s can be interpreted as the probability that an ad will be noticed when put in slot and it is assumed that . can be interpreted as the probability that an ad put by bidder will be clicked on if noticed and is refered as the relevance of bidder . The payoff/utility of bidder when given slot at a price of per click is given by and they are assumed to be rational agents trying to maximize their payoffs. As of now, Google as well as Yahoo! uses schemes closely modeled as RBR(rank by revenue) with GSP(generalized second pricing). The bidders are ranked according to and the slots are allocated as per this ranks. For simplicity of notation, assume that the th bidder is the one allocated slot according to this ranking rule, then is charged an amount equal to . Formal analysis of such sponsored search advertising model has been done extensively in recent years, from algorithmic as well as from game theoretic perspective[2, 6, 3, 1, 7, 4, 5].
In the following section, we propose and study a model wherein the additional capacity is provided by a forprofit agent who competes for a slot in the original auction, draws traffic and runs its own subauction for the added slots. We discuss the cost or the value of capacity by analyzing the change in the revenues due to added capacity as compared to the ones without added capacity.
2 The Model
In this section, we discuss our model motivated by the capacity constraint, which can be formally described as follows:

Primary Auction (auction) : Mediators participate in the original auction run by the search engine (called auction) and compete with advertisers for slots (called primary slots). For the th agent (an advertiser or a mediator), let and denote her true valuation and the bid for the auction respectively. Further, let us denote by where is the relevance score of th agent for auction. Let there are mediators and there indices are respectively.

Secondary auctions (auctions):

Secondary slots: Suppose that in the primary auction, the slots assigned to the mediators are respectively, then effectively, the additional slots are obtained by forking these primary slots in to additional slots respectively, where for all . By forking we mean the following: on the associated landing page the mediator puts some information relevant to the specific keyword associated with the auction along with the space for additional slots. Let us call these additional slots as secondary slots.

Properties of secondary slots and fitness of the mediators: For the th mediator, there will be a probability associated with her ad to be clicked if noticed, which is actually her relevence score and the position based CTRs might actually improve say by a factor of . This means that the position based CTR for the th secondary slot of th mediator in modeled as for and otherwise. Therefore, we can define a fitness for the th mediator, which is equal to . Thus corresponding to the th primary slot (the one being forked by the th mediator), the effective position based CTR for the th secondary slot obtained is where
(1) Note that , however could be greater than .

auctions: Mediators run their individual subauctions (called auctions) for the secondary slots provided by them. For an advertiser there is another type of valuations and bids, the ones associated with auctions. For the th agent, let and denote her true valuation and the bid for the auction of th mediator respectively. In general, the two types of valuations or bids corresponding to auction and the auctions might differ a lot. We also assume that and whenever is a mediator. Further, for the advertisers who do not participate in one auction (auction or auction), the corresponding true valuation and the bid are assumed to be zero. Also, for notational convenience let us denote by where is the relevance score of th agent for the auction of th mediator.

Payment models for auctions: Mediators could sell their secondary slots by impression (PPM), by payperclick (PPC) or payperconversion(PPA). In the following analysis, we consider PPC.


Freedom of participation: Advertisers are free to bid for primary as well as secondary slots.

True valuations of the mediators: The true valuation of the mediators are derived from the expected revenue (total payments from advertisers) they obtain from the corresponding auctions^{1}^{1}1This way of deriving the true valuation for the mediator is reasonable for the mediator can participate in the auction several times and run her corresponding auction and can estimate the revenue she is deriving from the auction. ex ante.
3 Bid Profiles at SNE
For simplicity, let us assume participation of a single mediator and the analysis involving several mediators can be done in a similar fashion. For notational convenience let
The auction as well as the auction is done via RBR with GSP, i.e. the mechanism currently being used by Google and Yahoo!, and the solution concept we use is Symmetric Nash Equilibria(SNE)[2, 7]. Suppose the allocations for the auction and auction are and respectively. Then the payoff of the th agent from the combined auction (auction and auction together) is
where
From the mathematical structure of payoffs and strategies available to the bidders wherein two different uncorrelated values can be reported as bids in the two types of auctions independently of each other^{2}^{2}2This assumption was motivated by some empirical examples from Google Adword^{3}^{3}footnotemark: 3., it is clear that the equilibrium of the combined auction game is the one obtained from the equilibria of the auction game and the auction game each played in isolation. In particular at SNE[2, 7],
and
which implies that (see Eq. (1))
where
is the true valuation of the mediator multiplied by her relevance score as per our definition^{1}^{1}footnotemark: 1, which is the expected revenue she derives from her auction ex ante given a slot in the auction and therefore the mediator’s payoff at SNE is
4 Revenue of the Auctioneer
In this section, we discuss the change in the revenue of the auctioneer due to the involvement of the mediator. The revenue of the auctioneer with the participation of the mediator is
and similarly, the revenue of the auctioneer without the participation of the mediator is
Therefore,
Thus revenue of the auctioneer always increases by the involvement of the mediator. As we can note from the above expression, smaller the better the improvement in the revenue of the auctioneer. To ensure a smaller value of , the mediator’s valuation which is the expected payments that she obtains from the auction should be better, therefore fitness factor should be very good. There is another way to improve her true valuation. The mediator could actually run many subauctions related to the specific keyword in question. This can be done as follows: besides providing the additional slots on the landing page, the information section of the page could contain links to other pages wherein further additional slots associated with a related keyword could be provided^{3}^{3}3For example, the keyword “personal loans” or “easy loans” and the mediator “personalloans.com”.. With this variation of the model, a better value of could possibly be ensured leading to a winwin situation for everyone.
Theorem 1
Increasing the capacity via mediator improves the revenue of auctioneer.
5 Efficiency
Now let us turn our attention to the change in the efficiency and as we will prove below, the efficiency always improves by the participation of the mediator.
Theorem 2
Increasing the capacity via mediator improves the efficiency.
6 Advertisers’ Payoffs
Clearly, for the newly accommodated advertisers, that is the ones who lost in the auction but win a slot in auction, the payoffs increase from zero to a postitive number. Now let us see where do these improvements in the revenue of the auctioneer, in payoffs of newly accommodated advertisers, and in the efficiency come from? Only thing left to look at is the change in the payoffs for the advertisers who originally won in the auction, that is the winners when there was no mediator. The new payoff for th ranked advertiser in auction is
where
is her payoff from the auction. Also, for , her payoff when there was no mediator is
Similarly, for , her payoff when there was no mediator is
Therefore, in general we have,
Thus, for the th ranked winning advertiser from the auction without mediation, the revenue from the auction decreases by and she faces a loss unless compensated for by her payoffs in auction. Further, this payoff loss will be visible only to the advertisers who joined the auction game before the mediator and they are likely to participate in the auction so as to make up for this loss. Thus, via the mediator, a part of the payoffs of the originally winning advertisers essentially gets distributed among the newly accommodated advertisers. However, when the mediator’s fitness factor is very good, it might be a winwin situation for everyone. Depending on how good the fitness factor is, sometimes the payoff from the auction might be enough to compensate for any loss by accommodating new advertisers. Let us consider an extreme situation when and . The gain in payoff for the advertiser is
Therefore as long as
the advertiser faces no net loss in payoff and might actually gain.
7 Concluding Remarks
In the present work, we have studied the emergence of diversification in the adword market triggered by the inherent capacity constraint. We proposed and analyzed a model where additional capacity is created by a forprofit agent who compete for a slot in the original auction, draws traffic and runs its own subauction. Our study potentially indicate a fold diversification in the adword market in terms of (i) the emergence of new market mechanisms, (ii) emergence of new forprofit agents, and (iii) involvement of a wider pool of advertisers. Therefore, we should expect the internet economy to continue to develop richer structure, with room for different types of agents and mechanisms to coexist. In particular, capacity constraints motivates the study of yet another model where the additional capacity is created by the search engine itself, essentially acting as a mediator itself and running a single combined auction. This study will be presented in an extended version of the present work.
References
 [1] G. Aggarwal, A. Goel, R. Motwani, Truthful Auctions for Pricing Search Keywords, EC 2006.
 [2] B. Edelman, M. Ostrovsky, M. Schwarz, Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords, American Economic Review 2007.
 [3] S. Lahaie, An Analysis of Alternative Slot Auction Designs for Sponsored Search, EC 2006.
 [4] S. Lahaie, D. Pennock, Revenue Analysis of a Family of Ranking Rules for Keyword Auctions, EC 2007.
 [5] M. Mahdian, H. Nazerzadeh, A. Saberi, Allocating online advertisement space with unreliable estimates, EC 2007
 [6] A. Mehta, A. Saberi, U. Vazirani, V. Vazirani, AdWords and generalized online matching, FOCS 2005.
 [7] H. Varian, Position Auctions, To appear in International Journal of Industrial Organization.