Advertising Space Exchange in a Network using Market Equilibrium Algorithms Amin Saberi

Alexander D. Shkolnik

Stanford University Stanford, CA 94304

Stanford University Stanford, CA 94304

[email protected] ABSTRACT We present a prototype of an online advertisement space exchange platform that enables its participants to advertise on each others’ websites and simulates a virtual exchange economy. Our main contribution is a system design that effectively and practically realizes the exchange based on a competitive equilibrium computed from the elicited preferences of the advertisers. The current prototype is used by 741 bloggers registered in various communities including Academia, Cuisine, Life and Parenting. The platform circulated roughly 2, 000 advertisements at any given time serving nearly 80, 000 impressions per day. The current paper discusses the implementation of the prototype and its performance in practice. We also present observations about the exchange market dynamics, the structure of the underlying network and its effects on the distribution of prices, wealth and income.

Categories and Subject Descriptors J.4 [Computer Applications]: Social And Behavioral Sciences—economics

General Terms Economics, Market Equilibrium

Keywords Online advertising, market mechanisms, market equilibrium

1.

INTRODUCTION

We present an online advertisement exchange platform, named Adoptic, that enables its participants to advertise on each others’ websites. In this platform, participants can advertise in the network for free as long as they provide their advertising space in return. Such a trade oriented approach motivates a virtual exchange economy for online advertising which treats an advertiser as a trader and advertising space as a good.

[email protected] Our main contribution is the design of a system that effectively realizes a fair exchange based on the computed equilibrium prices and allocations. The main components of this system are as follows: 1. A simple interface to allow users to create and manage advertisements promoting content of their blog, website or social network profile. The interface also elicits the preferences of an advertiser using a social graph of fans, favorites and communities. These preferences are translated into a utility function by a market mechanism that assigns a value to each combination of advertising space alternatives. 2. A market mechanism for computing the competitive equilibrium prices in polynomial time. The mechanism solves a series of market equilibrium problems to keep the trade efficient. The allocations of each advertisement maximize the utilities subject to a spending budget. The mechanism also allows the advertisers to save their virtual currency and spend it at a later date. 3. An advertisement server that coordinates the exchange dictated by equilibrium allocations in real-time. The advertisers pay the publishers with virtual credit for impressions at equilibrium prices. The Ad Server also keeps track of and executes all virtual credit transactions. The current systems is fully implemented and is used by over 700 bloggers. These bloggers form an explicit social advertising network of fans, favorites and communities covering a broad range of blogging topics. There are, in all, thirty four communities which were introduced as the user base grew. Roughly one third of the communities were requested by the users themselves. The behavior of these groups yielded a complex network over which the exchange market operates. Several communities, like Parenting and Life exhibited rapid growth as invitations were sent by the already registered members. Participation was the highest among these very social users. They created advertisements most frequently, joined more communities than other users and represented the most interconnected areas of Adoptic’s social network. Other communities like Academia, Science and Art were slow in growth but maintained high levels of activity. Privacy was a bigger concern for these users as they were more cautious about what to advertise on their blogs. As a result, they spent more time refining their favorite blog lists and were reticent to join numerous communities.

The outline of the paper is as follows. In §3 we mathematically formulate advertisement space exchange and design a market mechanism to model it a series of market equilibrium problems. In §4 we provide an in depth description of the prototype including the user interface, the Ad Server and the algorithm used to compute the market equilibria. In §5 we confirm our implementation in practice and provide some insight into the dynamics of the exchange market over a real-world data set.

2.

RELATED WORK:

Most of the current literature on online advertising focuses on sponsored search [9] and similar markets in which advertisers and publishers are regarded as separate participants. The main difference in our platform is that it creates an advertising network which provides opportunities and value for free. For example, a blogger who has a website with a typically small amount of traffic would get very small returns from AdSense. Advertising using AdWords or AdSense in order to increasing the traffic to a small blog is also costly. One approach for mutual advertising is ”Link exchange”, a termed coined after a company of the same name [11]. The concept consists of organizing a network of websites each of whom agrees to refer each other to their visitors by displaying a url link of other websites who are participating in the exchange. The main difference of link exchange with Adoptic is its one-to-one exchange rate for all advertisements. Link exchange also does not give much control to advertisers over where their advertisements will be placed. Adoptic allows bloggers to advertise on their friends’ websites, websites with similar content or the websites of influential bloggers. The algorithm for computing the prices builds on the literature in economic theory on competitive equilibria as well as more recent results on the design of efficient algorithms for computing prices [6, 5, 3]. In the current application, users are protected from most market aspects such as prices, allocations and budgets. Yet, the underlying market mechanism ensures a fair allocation of advertisement space that encourages sharing of space and higher utilization. In this respect, Adoptic is similar to BitTorrent and other marketbased peer-to-peer systems [4].

3.

FORMULATION

We consider some number of participants on the internet each of whom can act as both a publisher and an advertiser. A publisher owns a website which serves impressions realized by some incoming rate of its visitors. An advertiser can create advertisements which, in general, can be any type of digital content. Each advertiser can also be assumed to have some set of preferences that dictate where to publish his/her advertisements. The general problem is then to determine how to allocate advertisements to websites and price each impression in a fashion that will satisfy all parties.

3.1

Exchange Economy & Equilibria

We consider a pure exchange economy with a set of m traders endowed with n divisible goods. Each trader proceeds by selling his/her goods at market prices and buying a new bundle of goods that is more preferred. Let π ∈
vector of market prices with a real nonnegative πj recording the price for good j. Let x ∈ 0, that is, each trader is endowed with at least some amount of a good. Each trader i evaluates a utility function ui :
(x∗i1 , . . . , x∗in )

j

satisfies (1) and is a maximizer of ui , i.e.

ui (x∗i1 , . . . , x∗in )

≥ ui (xi1 , ..., xin ) .

2. For each good j the market clears, X ∗ X xij = wij . i

(2)

i

The equilibrium (x∗ , π ∗ ) has been shown to exist under various sets of assumptions with the most celebrated result formulated by Arrow and Debreu [2, 1] in an even more general case of economies with production.

3.2

Advertisement Space Exchange

We formalize an online advertisement exchange as a set A of advertisements and a set S of websites along with a function β : A → S which assigns each advertisement to a website. This models a ”belongs to” relationship in which the publisher, an owner of a website, can also act as an advertiser and generate advertisements based on the website’s content. This is especially appropriate in the context of online advertisement for blogs (scenarios where the advertiser does not necessarily own a website are covered in §3.5). We also define a function α : A × S → R+ which assigns a nonnegative real number αij to indicate an advertiser’s desire, or preference level, to publish advertisement i on website j. Furthermore, each website j has some expected number of visitors ηj per interval of time T . When a visit to website j occurs, an advertisement i is delivered to be shown to the visitor, also called an impression. The advertiser then pays the publisher some price for the service rendered. In order to be able to deliver advertisements in real-time we keep an allocation probability matrix P (t) which assigns a probability pij (t) of showing advertisement i on website P j at time t. This matrix satisfies i pij (t) = 1. In order to accommodate for changing user preferences as well as variation in website traffic over time, the matrix P (t) is updated at times tn = nT for n = 0, 1, 2, . . . , where each interval is referred to as a trade round. During the nth trade round the allocation probability matrix has a constant value P (tn ) and each website j maintains a constant

price πjn . Each website also keeps a running virtual credit balance bj (t). When website j is visited the probability distribution {pij (tn ) : i = 1, 2, ..., m} is sampled to select an advertisement i to be shown. Letting i be the chosen advertisement belonging to website j (i.e. j = β(i)), an impression at time t served by website k triggers a transaction [bk (t) ← bk (t) + πkn , bj (t) ← bj (t) − πkn ] in which the advertiser pays the publisher his/her price for the acquired impression.

3.3

scale the prices such that the average price per impression π ¯ n during trade round n, defined in (5), is unit. P j πj ηj (5) π ¯= P j ηj Such a scaling does not change equilibrium prices since in (1), the budget constraint, prices can be scaled without altering equilibrium conditions. We chose π ¯ = 1 simply due to a pleasant interpretation.

Market Mechanism Formulation

We now establish a procedure to convert advertisement space exchange (A, S, β, α) into an exchange economy. The variables of the derived market mechanism are xij , the number of times advertisement i is to be published on website j, and πj , the price paid to the publisher by the advertiser. Consider a trade round beginning at tn when the expected number of visitors during the next T units of time to website j is ηˆj and the number of advertisements belonging to website j is mj . Letting γj ∈ [0, 1] and ρi ∈ (0, 1) perform the following steps,

1. For each website j, add a market good in amount ηˆj . 2. For each advertisement i, add a market trader. 3. Assign initial endowments of market goods as, „ « 8 < 1 ηˆ + γ bj (tn ) for j = β(i) j j n−1 πj . wij = mj : 0 otherwise 4. Aggegate preferences αij into utility functions, !1/ρi X ρi αij xij ui (xi1 , . . . , xin ) = .

(3)

(4)

j

Step 1 models the websites as goods with supply of each equal to the expected number of visitors in the upcoming trade round n. In steps 2 and 3 an advertisement is modeled as a trader with an initial endowment being a fraction of the number of impressions available on the website it belongs to. Thus each traders sells some amount of advertising space and buys advertising space brought to market by other traders. The term γj bj (tn )/πjn−1 is added to adjust for accumulated wealth b(tn ) due to excessive selling of impressions in prior trade rounds. Such scenarios occur due to the fact that the ηˆj ’s are estimated values. Parameters γj are adjusted to ensure that each initial endowment is nonnegative and controls how quickly accumulated balance will be spent. The utility functions formed in step 4 are a common and well studied CES family of utilities chosen for their ability to model a wide range of trader preferences. The result is a market equilibrium problem with some equilibrium prices π n and allocations xn at time tn . We solve a series of these problems in time, each at the beginning of a trade round, to ensure that the system remains efficient as preferences and supplies change over time. In order to deliver advertisements in real-time we recover the probability matrix P (tn ) by setting pij (tn ) = xn ηjn and ij /ˆ

3.4

Market Model Parameters

In §3.3, we’ve left several parameters and estimates up to the choice of implementation. Our prototype currently employs trade rounds of duration T set to one hour. Currently estimates ηˆj for each website j and each of the 24 daily trade rounds are computed as averages over three previous days. Parameters ρi are related to the elasticity of substitution as 1/(1 − ρi ) [9]. We restrict the range of ρ to (0, 1) to ensure convergence to equilibrium as explained in §4.2. This restricted range however still allows for modeling a variety of trader preferences from perfect substitutes (ρ ↑ 1) to CobbDouglass utilities (ρ ↓ 0) in which there is a perfect balance between substitutability and complementarity. Our prototype currently uses values of ρi set to 1/2 for each trader, a choice made to simplify the closed form subproblem solutions in Algorithm 1 line 6. The choices of γj ’s are adjusted based on the advertiser’s selection of advertising strategy (aggressive, conservative, etc...). These variables also help regulate how quickly a market returns from a shock (see §5.2). In a pure exchage discussed upto now the balance bj (t) of website j is zero in expectation. A more general model should accommodate participants who have little or no adverting space to sell. In such a model we define bj (t) as a variable to record the wealth of website j at time t. In the next section we extend the current model to one where advertisers can save and spend money according to their choice of parameter γj .

3.5

Monetary Endowments Extension

Adding money to our model gives some desired flexibility as advertisers can spend without necessarily owning adverting space to sell. Money can be added to the model by treating it as another good or by modeling a Fisher market. In the latter case we adjust our model by adding a special good representing money indicated by a zero index and having a fixed price π0 = 1. To modify the previous model we only have to define a new utility vi as in (6) and assign some monetary endowment wi0 ≥ 0 to each trader. 1

vi (xi0 , ..., xin ) = αi0 xσi0i + ui (xi1 , ..., xin ) ρi

(6)

The parameter σi ∈ (0, 1) is related to elasticity of substitution of money in the same way as ρi . The addition of a good with fixed price π0 = 1 also removes the requirement of scaling the prices such that π ¯ = 1, as the prices are set relative to π0 automatically by the budget constraint (1). In the case of the Fisher market, the mechanism differs in that traders do not have initial endowments of goods but instead come to market with some initial endowment of money wi0 . Goods are bought from the market directly with the

budget constraint taking the form in (7). X πj xij ≤ wi0 .

(7)

In the context of advertisement exchange, wi0 is some fraction of the surplus from the wealth bj (t). In order to have a stable economy over times tn , that is, bj (t) is bounded for each website over time, the initial monetary endowments wi0 for each upcoming trade round n are assigned as follows, ! ηj πjn−1 bj (tn ) 1 + γj n−1 (8) wi0 = mj π ¯jn−1 πj where πjn−1 and π ¯ n−1 are values from the previous trade round and γj ∈ (0, 1) is the advertiser spending rate variable that may need adjustment to keep monetary endowments of each trader positive. This process works by approximating the pure exchange economy as traders sell goods at prices assigned during the previous trade round and are also able to use wealth accumulated by the balance bj (t).

4.

PROTOTYPE

We now describe the implementation of the prototype including the user interface, the Ad Server which delivers advertisements in real-time and the algorithm implemented to compute market equilibria. We conclude with an overview of the performance in practice.

4.1

Web Application

The user interface component of the prototype is a web application implemented in Ruby on Rails and running a MySQL database. The sign up process is open by invitation to anyone with a blog and requires pasting a HTML/Javascript code in a place within the blog HTML where the user would like to display advertising.

4.1.1

User Interface to Elicit Preferences

The user interface, shown in Appendix B, allows users to create and manage their advertisements. An RSS reader is provided to help users create the advertisements based on the content of their website (blog). The interface also contains social networking features including communities, fans and favorites. A user may register a website for any number of communities each representing a topic of interest (e.g. art, politics, news, etc.) and each advertisement must be assigned to a community. An advertisement circulates only in the community to which it is assigned to ensure relevancy. In addition, the advertiser may chose any number of favorite websites on which to advertise on more frequently. As a result each user has a list of favorites and a list of fans. These concepts generalize to settings where a user may just be an advertiser, in which case the list of fans is empty, and where the user may only chose to publish, in which case the list of favorites is empty. Community, fan and favorite concepts induce a social graph of advertisers and publishers. We use the notation j ∼C k to indicate that website j and website k are both registered for community C. We write j → k to indicate that website j is in the list of fans of website k and k is in the list of favorites of website j. Thus the number of fans of a website j is an indegree and the number of favorites is an outdegree in the social graph. The distributions of these quantities can be found in Figure 2.

In §3.3 we described how to formulate advertisement space exchange as a market equilibrium problem. In step 4 of this procedure preference coefficients αij were aggregated into a CES utility function. We can now state how to compute the preference matrix α based on the social graph described above. Given advertiser A with advertisement i and publisher B with website j, the desirability (or preference) of the advertiser to display advertisement i on website j is a computed as a function of whether A is a fan of B and the number of communities A and B have in common. This function is given in (9). X αij = I({β(i) ∼C j}) + R I({β(i) → j}) (9) C

where I is the indicator function, β(·) is defined in §3.2 and parameter R is set to model the relative desirability between community members and fan/favorite relationships. In the current version of the prototype R is set to the value 3. One can conceive of may ways to elicit preferences that are much more complicated than the form in (9). For example, favorites of a website may be ranked yielding a weighted social graph. In addition one can devise many ways to elicit preferences implicitly. This method would usually require a web crawler to analyze links in between websites and search and match websites based on keywords. These extensions however do not add any significant difficulty to the implementation.

4.1.2

Delivering Advertisements

To coordinate the exchange we deliver advertisements in real time based on the probability matrix P (t) established in §3.3. Each website visit executes the HTML/Javascript code and contacts the prototype’s Ad Server which uploads the advertisement to the website. A screenshot of a sample advertisement being displayed on a blogger’s website is shown in Appendix A. For every impression served, the Ad Server issues a balance transaction between the advertiser and the publisher and records on which websites each of the advertisements was shown and how much was paid to the publisher.

4.2

Tâtonnement Algorithm

In this section we describe an algorithm used to compute a pure exchange economy equilibrium with CES utilities. For the sake of completeness, we start by describing classic results on tˆ atonnement procedure and it is applications. We also discuss implementation and performance issues.

4.2.1

Description of Algorithm

We model every advertiser as a price taking agent maximizing the following constrained maximization problem. !1 ρi X ρi for i = 1, 2, . . . , m max α x ij ij n n xij ∈R+ ,πj ∈R+

subject to

j

X j

πj xij =

X

πj wij

j

The utility of the advertiser is assumed to be CES. We employ CES utility functions for ρi ∈ (0, 1). This is essential for achieving Weak Gross Substitutability (WGS) which has

several desirable properties including convergence, convexity and in most cases uniqueness [3]. It is classically known that the tˆ atonnement, an iterative procedure of price adjustment based on the excess demand vector globally converges to equilibrium. There are a series of recent results deriving algorithms that converge to the equilibria in polynomial time. Two examples are [3, 6]. We use a particular variation that updates the prices multiplicatively and therefore is closer to [6]. For the sake of completeness, we have included the description of the algorithm. Let n be the number of goods and m be the number of traders, each with scaled utility ui . The utilities are scaled in order to obtain a transformed market where Wj = 1 for all j, that is, the supply of each good is unit. Line 6 is the price update used based on a time step of ∆t = 21 . On line 6, a constrained maximization subproblem is solved for each trader i separately. Closed form solutions for such subproblems with CES utility functions may be derived using the Lagrange multiplier method. Convergence is considered to be reached when the supply and demand for each good are within a tolerance  > 0 of each other. We note that when this tolerance is reached and the subproblem on line 6 is solved with sufficient accuracy the result (x, π) satisfies a (1 + )-approximate equilibrium. are P P That is, the utilities w for x ≤ (1 + ) within (1 P + ) of the maximum, ij ij i i P all j and j xij πj ≤ (1 + ) i wij πj for all i. Algorithm 1 Tˆ atonnement for pure exchange Require:  > 0 1: πj ← 1 for j = 1, 2, . . . , n 2: repeat 3: X ← 0n 4: for i = P 1 to m do 5: si ← j πj wij 6: xi ← argmax{ui (y) | y ∈
4.2.2

Implementation and Performance

The algorithm is implemented in C with a Ruby API layer to improve integration with the rest of the prototype. The evident O(mn) memory requirement for Algorithm 1 is bypassed due to the fact that the endowments w and the preferences α are sparse. The allocations xi for each trader P are not stored since only the aggregate demands X = xi are important for the computation. We can therefore replace, in Algorithm 1, all xi by one storage vector y and prior to outputting the solution (x, π), solve m subproblems from line 6, one for each xi . A naive implementation of the algorithm will take O(mn) time for each iteration. This is because it is expected that some of the communities will have a large number of participants which will make the trade graph dense. One way for avoiding this problem is to define a good for each commu-

nity. The good and the demand of each individual can be converted to the good and demand of the communities to which he or she belongs. This is reminiscent of economies with production in a very special case. Using such a sparse implementation, along with observing that price updates can be implemented in parallel, we observe that each iteration of the algorithm can be implemented in time proportional to the maximum degree in the graph. There are theoretical results that show that the number of iterations needed for convergence is at most linear. We have observed that the actual number of iterations needed in practice remains constant for our choice of parameters. Finally, in this application we can achieve much faster convergence rates by setting the initial prices to the ones computed in the previous trade round. In practice, when prices are updated hourly, this results in only one or two iterations until convergence is reached.

Figure 1: Algorithm Performance. Figure 1 suggests that for our choice of graphs, the number of iterations needed to converge to a solution remains pretty much constant as the size of the market grows. These results were obtained by generating random markets of various sizes resembling properties of the real markets observed in practice. The social graph (see §4.1.1) of fans (indegree) and favorites (outdegree) was generated using a power law 1 . Each degree density ∝ k2 √ website was assumed to have registered for four out of b nc communities selected uniformly at random. The expected number of visitors to each website is modeled as a Poisson random variable with parameter 10. The market size is given by n websites where each website is assumed to own one advertisement assigned randomly to one of the four communities of the website. While the latter assumption is not realistic, it is convenient for presenting running times since m = n. Our current implementation is not parallel because of the small size of our current network. The simulations were per-

formed on an Intel Core 2 Duo T7200 2 GHz processor with 2GB memory. The current implementation takes about 149 seconds on a graph with 20,000 traders and 20,000 goods.

set in §5.1. In §5.2 we present some observations that give insight into the dynamics of the exchange market.

5.1 Finally, we note that due to the sparse nature of preferences in the application markets that are very large (n, m > 100, 000) can be partitioned into smaller markets based on community memberships and resulting prices in separate markets can be related based on the total traffic (impressions served) in each exchange market.

4.3

Practical Considerations

There are many practical considerations when a system like the presented prototype is put into practice. In any market mechanism fraudulent or strategic behaviour of users is a potential concern. In the context of the current prototype fraudulent or strategic behaviour would aim to acquire wealth which may be spent on undeserved advertising space. By design, the only participation required of each user is a statement of their preferences. As preferences dictate expenditure of wealth, little gained benefit can be expected. Users can still act strategically by using information regarding market structure and supplies of goods available. However, the market mechanism maintains the system in a competitive equilibrium, and presence of strategic users described reduces options for strategic behavior with their numbers. In addition, the fact the supplies ηj are not provided by the users but are measured by the market mechanism precludes other forms of strategic behaviour. The fact that the market operates at a Walrasian equilibrium also prevents coalitions (a subset of participants who strategically reach an agreement in order to gain a mutual benefit). This fact comes out of standard microeconomic theory and states that any Walrasian equilibrium has the core property [10]. It is also useful to note that users who attempt to create fake fan websites will gain little or no benefit. We demonstrate in §5 that fans need to have considerable income to alter equilibrium prices. Impression fraud presents a great risk in the implementation of our prototype. This occurs when a user simulates fake visits to his/her website to gain significant income unfairly. The current prototype detects fake visits based on website traffic from the same IP and warns of possible fraudulent behaviour. If the attacker is able to produce visits from multiple IPs, detection becomes very difficult if not impossible. However, given that this behaviour take place in a market, there is incentive for honest participants to report such cases of fraudulent activity.

5.

RESULTS

The results and their discussion are based on a real world data set collected from the interaction of users with the web application interface as well as the records of impressions by the Ad Server. The users consist of a group of bloggers invited by email to form a community and bloggers who were invited by those already in the system. These users were given the option of joining communities and provided with searching and browsing capabilities to select favorite websites on which to advertise (see §4 for description of the user interface). We describe the profile of the analyzed data

Analyzed Data

The empirical results of this section are obtained from data collected over a five day period during the first week of September 2008. This set of data comprises 120 trade rounds, or hours, of continuous advertisement space exchange with a market equilibrium problem solved at the start of each trade round. The analyzed data set consists of 741 websites, 34 communities and roughly 2, 000 advertisements circulating at any given time. Of the 741 websites, roughly 40% consistently generated new advertisements. Each advertisement was removed from circulation after about a month to prevent outdated content

5.1.1

Statistics of the Data Set

Figure 2 presents a statistical summary of the data set. Panel (a) is a histogram of a representative number of advertisements circulating in the communities. The average number of advertisements per community was roughly 55 with a maximum of 550 advertisements in one of the communities not shown due to scale. Panel (b) is a histogram of a representative number of advertisements belonging to an advertiser. The average number of advertisements created by those who consistently generated new advertisements was roughly 7 with a few advertisers circulating up to 50 advertisements at a time. Panel (c) describes websites populations within the communities. The average number of communities per website was 4 with a mean website population of 61.5 per community and a maximum population of 235. The social graph represented by the fans and favorites described in §4 had the expected power law distribution ∝ k1α as shown by the log plot in panels (d) and (e) with exponent (slope) α = 1.66 for the fans and 1.57 for the favorites. Finally, as shown in panel (f ) an average website serves an average 6.6 impressions per hour reaching average peak levels during daytime of 37 impressions served per hour. The densities of five day daily and peak time averages are presented.

5.1.2

Distribution of Equilibrium Prices

Figure 3 describes how prices distribute in the exchange network. Panel (a) shows a histogram of average market prices πj over the five day period where the average price per impression, given by (5), was π ¯ = 1 and the maximum impression price was 6.87. This plot represents a typical price distribution with communities in the network. Panel (b) shows prices on a loglog plot with R >> 1 in (9) to ensure that the effects of communities on prices are negligible. We observe a power law distribution of prices in our network. In the panel, both the number of fans (indegree) and (scaled) prices are shown to have mirroring distributions. This confirms a frequent findings in markets where the exchange takes place over edges in a graph, referred to as graphical economics [7, 8]. In such markets price distributions may follow the degree distributions of the underlying exchange graph.

5.2

Discussion

The forces that govern the market dynamics in our framework arise due to website traffic and the structure of the social graph. We study their interplay to shed light on the operation of the market in practice.

Figure 2: Statistics of collected data set.

Figure 3: Prices on a power law degree network.

Figure 4: Impact of sudden traffic shock on neighborhood.

5.2.1

Shocks in the Network:

It is not uncommon for a blog post to become virally popular and bring large surges of website traffic. In Figure 4 we observe such a traffic shock which causes the publisher to earn more income than is typical. In this case we’ve considered a blog which advertises primarily in the Academia community. While this website has a very moderate traffic of a dozen visits per day, during the first week of September 2008, the author published a blog post which won a lot of attention. In Panel (b) we show the spike in the number of impressions served occurring in the middle of the week. In Panel (a), we show two credit balance (wealth) curves over time. The top curve shows how the wealth of the blogger who authored the post, b(t), incurs a sudden jump. The lower curve shows the average wealth of the fans of this blog which turns negative during the traffic shock. This would cause some of these advertisers to freeze spending in order to avoid going further into debt as guaranteed by the market mechanism described in §3.3.

much control is retained by the market mechanism. For example, the introduction of communities to the network mitigates these effects by ensuring that wealth distributes itself more evenly. This may not be desirable for the advertiser since he/she relinquishes some control over advertisement placement. In addition, the advertiser selected parameter γ (rate of spending accumulated wealth) may be used to avoid or risk the negative effects of a shock. For example a high value of γ will spend any excess wealth quickly leaving the advertiser at a risk of going into debt during a shock.

5.2.2

Location in the Network:

The position in the network can yield great benefits from the market to the advertiser. This benefit can be measured in term of acquired impressions or rather income in the market. Income is a better measurement since the advertiser gets to select how to spend it in our prototype. We compute the income of a website as the number of impressions served (per hour) times the price per impression.

Recovery from a shock: The system seems to return to normal operation after roughly 40 trade rounds. This property of the market is discussed in §3.3 when introducing the parameter γ, set by the advertiser to control how quickly accumulated balance is spent. In this particular example, γ = 0.25 dictating that one quarter of the accumulated balance be spent on advertising during each trade round. The fans of the blog seem to have largely recovered but this may not always be the case. It may often take more time since wealth distributes itself primarily based on the directed edges of the social graph. The social graph can be highly asymmetric (i.e. fans of a blog may not always be in its favorites list) and the credit lost by the fans of the blog may be carried to a different part of the network.

Figure 5 shows a data plot where each data point represents a website. Coordinate x is the average number of impressions served per hour and coordinate y represents the average number of impressions per hour acquired by the website’s advertisements. The color-map represents the price per impression on each of the websites with the darker green colors representing very low impression prices. We can observe that the higher priced websites are concentrated in the region {(x, y) : x  y} confirming that higher priced website are rewarded with more acquired impressions. The size of each data point is proportional to the number of fans of the website (popularity in the network). The fact that the data points are curved reflects the presence of a significant number of publishers who do not consistently advertise.

Practical Implications: Mitigating these affects is a tradeoff between how much control is given to the user and how

Network position and price: While we know that a greater number of impressions served and a greater popular-

implementation. The authors also greatly appreciate the work of Carl Jackson and Nitin Verma on the web application.

7.

Figure 5: Average number of impressions served vs acquired (per hour) for each website. ity should drive prices lower and higher respectively, there are clear examples to the contrary. Consider the data point labeled C in the plot which has a low popularity and a supply comparable to many low priced websites. Regardless, the price of an impression on website C is very high. This result confirms the fact that the population which makes up the fans of a publisher is a very important factor in determining the price. We now extend the example just provided by considering websites marked A, B and C in Figure 5. Real case study: Blogs A and B were some of the earliest to join and are located far apart in the network with no communities in common. Blog B publishes advertisements primarily in the Academia community while A advertises mostly in the Parenting community. Since both A and B got an early start and have interesting blogs, over time they have gained a high popularity in the network with A having 99 fans and B having 205 fans. Both acquire a comparable number of impressions for advertising and are serving impressions priced at 1.27 for A and 4.11 for B. Website C, who also advertises in the Parenting community, joined the network within weeks before this data was collected. Because of this C has only 2 fans, B being one of them. Similar data points suggest, based on the number fans, that C is at a disadvantage. Yet an impression on this blog costs 3.45 which is unusual with such a high popularity and supply of impressions. Looking at the fans of C however, we see see that they are relatively influential in the network. That is, their average income is 15.31. In comparison, blogs A and B have fans whose average incomes are 7.80 and 9.39 respectively.

6.

ACKNOWLEDGMENTS

The authors would like to thank Mehdi Yahyanejad for numerous helpful discussions, insights and help with algorithm

REFERENCES

[1] K. Arrow, H. Block, and L. Hurwicz. On the stability of the competitive equilibrium, ii. Econometrica 27, 1959. [2] K. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econometrica 22, 1954. [3] B. Codenotti, B. McCune, S. Penumatcha, and K. Varadarajan. Market equilibrium via exess demand function. STOC, 2005. [4] B. Cohen. Incentives build robustness in bittorrent. Workshop on Economics of Peer-to-Peer Systems, 2003. [5] N. Devanur, C. Papadimitriou, A. Saberi, and V. Vazirani. Market equilibrium via a primal-dual-type algorithm. FOCS, 2002. [6] L. Fleischer, R. Garg, S. Kapoor, R. Khandekar, and A. Saberi. A fast and simple algorithm for computing market equilibria. WINE, 2008. [7] S. Kakade, M. Kearns, and L. Ortiz. Graphical economics. COLT, 2004. [8] S. Kakade, M. Kearns, L. Ortiz, R. Pemantle, and S. Suri. Economic properties of social networks. NIPS, 2004. [9] S. Lahaie, D. Pennock, and A. Saberi. Algorithmic Game Theory: Sponsored Search Auctions. Cambridge University Press, 2007. [10] A. Mas-Colell, M. Whinston, and J. Green. Microeconomic Theory. Oxford University Press, 1995. [11] Wikipedia. Link exchange. http://en.wikipedia.org.

APPENDIX Appendix A gives a snapshop of a sample advertisement as it appears on a blog. Appendix B is a screenshot of the user interface described in §4. For more screenshots please visit http://www.stanford.edu/~saberi/wwwmarket/ .

A.

SCREENSHOT OF ADVERTISEMENT

B.

SCREENSHOT OF USER INTERFACE