Optimal Mobile App Advertising Keyword Auction Model with Variable Costs

2013 8th International Conference on Communications and Networking in China (CHINACOM) Optimal Mobile App Advertising Keyword Auction Model with Variable Costs Meng Wang, Zhide Chen, Li Xu and Lei Shao School of Mathematics and Computer Sciences,Fujian Normal University Key Lab of Network Security and cryptography, Fujian Province Fuzhou, China Email: wangmengfjnu@gmail.com, zhidechen@fjnu.edu.cn, xuli@fjnu.edu.cn, 308449646@qq.com Abstract With the rapid development of mobile App advertising, how to allocate App advertising more reasonable is now need to be studied. This paper proposes an App advertising auction mechanism based on optimal keyword auctions. First, we analyze the development trend of the App advertising and obtain the result that the core competitiveness of mobile advertising lies in the precise. Next, we describe the App advertising auction mechanism in detail. The advertising click-through rate only depends on the location of the premise and per-click cost. In considering the premise of taking the variable costs of App advertising slots into account, we put forward optimal keyword auctions model. A mathematical analysis is used to get the expected revenues of advertisers, advertising alliance platform and App developers. Finally, we use the expected revenue expressions to analyze the feasibility properties of the model and give corresponding proof. According to the results of our analysis, we can better guide App advertising and maximum revenue of any equilibrium of this dynamic auction. In order to closer to the actual situation, our model also takes the individual rationality, incentive compatibility properties into account to prove the feasibility of the mechanism. Keywords App Advertising, Keyword Auctions, VCG Mechanism, GSP Mechanism. I. Introduction With the open of mobile App platform, App advertising is showing a huge market. Compared with the age of the Internet ads that appear on the PC screen, the mobile applicaiton advertising through the intelligent mobile terminal further precision on the road. His delivery can be accurate to follow your geographical location, time, type, operating system, price and brand. At this stage, the core competitiveness of mobile advertising lies in the precise. Here we require App is an abbreviation of the mobile Application. Now the methods of App advertising can be mainly divided into two types. The first one is mobile manufacturers using their own brand and put the advertisement in the App store for free download. By using these applications, most users can better understand the brand. In general, the propaganda effect is very obvious. The other type of method is implanting the application of mobile ads into hot or their related products to achieve the purpose of marketing. Although in this method, there is no direct way to release their own brand of propaganda effect directly. But with wide adaptability and also assured accuracy, it was favored by most of the advertisers. Perhaps see a huge role in mobile applications, mobile App advertising, at home and abroad have been born many mobile application advertising platform, including meters ads and the recent percentage Communications L-Sense mobile Internet advertising platform. They all sought through advertising matching system to put the ads into the most suitable place. Therefore, the App store and application developers are actively exploring mobile advertising. Between App advertising and application, how to reasonable delivery has become a problem worthy of study. Researches on keyword auction are mainly focused on the design of keyword auctions mechanism. The general situation is based on the search engine website, such as Yahoo and Google. The representative work is Varian to Google. Take Yahoo qualifying auction as an example, analysis of the equilibrium of the game 19]. Edelman discussed the generalized second price (GSP) auction mechanism balanced existence and honest reporting of the dominant bidding strategy under the premise of the existence of equilibrium 4]. Ming proposed the mechanism for qualifying auction forward-looking Nash equilibrium and prove that the Nash equilibrium does not always guarantee the auction is honest 3]. Rong Wenjin et al discussed how to set the optimal reserve price of keyword auctions and put forward the corresponding calculation method 20]. About the optimal keyword auctions mechanism design, Garg & Narahari directly apply Myerson s results and formulas to design the optimal keywords auctions 14]. Jun Li also studied the optimal keyword auctions mechanism based on the variable costs 13]. In this paper, we will research App advertising and propose the App advertising model. Section II simply describes the keyword auctions and App advertising auction. Section III gives a model of keyword auctions and gives a few assumptions to simplify the model. Section IV analyze the model, it can be divided into three parts and the first part gives the analysis of each participant revenues, the second part gives analysis of the feasibility of keyword auctions mechanism, the third part analyzed the properties of the model. Section V concludes the paper. A. Keyword Auctions II. Background Keyword auctions is a form of auction theory abstracted from the practice of search engine in the recent ten years. Here we will give a brief description of how the keyword auctions operate. When internet users enter the keyword they 166 978-1-4799-1406-7 2013 IEEE

want to search in a search engine, he ll get to a page containing the results of the most relevant links with keywords and sponsored links (paid advertising). These ads can be significant and practical search results for us to distinguish, and different search to generate different sponsored links. Advertisers set advertising according to the search words. For example, a travel agents purchase the keywords Maldives, every travel agency link will appear on the search results page when users search for the word, when the user clicks on a sponsored link, he is taken to the advertiser s website and then the search engine of the internet let users bring his page and got its pay. The number of search engine advertising slots that can provide is certain. In a search results page, the value of positions is different, at the top of the page, the possibility of click is obviously larger than that at the bottom of the page. Therefore, the search engines need a mechanism to allocate advertising slots to advertisers, the auction is naturally a good choice. Keyword auctions mechanism is used to allocate the slots for advertisers and to determine how much advertisers need to pay. It consists of two main rules: payment rules and allocation rules. Payment rules following two mechanisms: the generalized first price auction (GFP) and generalized second price auction (GSP). For GFP mechanism, advertisers pay their own price. For GSP mechanism, advertisers pay the highest bid of other advertisers. At present, the widely used mechanism of search engines are built on the basis of GSP. In the two payment rules, advertisers pay nothing when get nothing. Allocation rule also has two types: one is the sort of direct, namely advertising sequence according to bid ranking. The other type is a comprehensive ranking method, namely considering advertisers bids and other factors, such as expected click rate and the correlation of the keywords. In practice, we often use the Comprehensive Ranking, such as Baidu bidding ranking advertisements. We can known that in keyword auctions, the advertiser s bid reflects the private information of the advertiser click valuation unit. Sort way determines the expected distribution of the allocation of advertising resources and expected to pay for the advertisers. The final user click advertising resources to achieve the allocation and the search engine to obtain payment. B. Mobile App Advertising Around the world, there are over 650,000 mobile Internet applications and more than ten billion times downloads. Users on the mobile Internet and mobile Internet terminal behavior habits in gradual training, more and more time is spent by the user on it. The prospects seem good, in fact, there are some problems. Although Crazy Bird created astonishing record, but this is just a case, How to get the maximum revenue became a urgent problem to solve. According to the survey that the time user spent on mobile Internet accounts for 10% of the total time online, but mobile advertising accounted for the entire online advertising compared to only 0.5% to 1%. Scale translated over 500 billion online advertising and mobile advertising market size from 250 million to 500 million yuan. Mobile advertising value have not broke. App advertising can be built in tens of thousands of quantity and is collected in the App. According to the population (the user behavior database), the application content type, mobile phone brand models, operating system platform, mobile phone price, regional, delivery time, telecom operators and redirect dimension. It will target audience who may be interested in advertising or the audience of people interested in the App content matching degree high advertising to the front of the user mobile phone screen. We can realize the accurate transmission of the user s mobile phone. We use the keyword auctions mechanism model to achieve the allocation of App advertising. However, in the traditional keyword auctions mechanism, the assumption of setting the cost of keyword advertising slots is not necessarily reasonable. First of all, we are very clear that the revenue is zero if the App does not display ads. Whether display advertising or not is a trade-off between users and advertisers. The App without advertising can attract users more and advertising with App may have a negative impact on the user experience, resulting in the loss of brand value in the application, we call it variable cost that App developers can provide a bit of a variable keyword advertising cost. III. Model This model defines the relationship between the three, advertisers, advertising alliance platform and App developers. Advertisers in accordance with the actual App advertising effects to pay the reasonable cost of advertising to the App developers. App developers select the appropriate advertisers through advertising platform and display advertising to increase revenue, while a large number of mobile advertising sales cost will be saved. It is easy to put click quantity into revenue. Advertising alliance platform match through its own advertising auction by connecting the upstream and downstream of the App developers and advertisers, provide a platform to promote efficiently for advertisers, small and medium-sized App developer. In advertising alliance platform, an App corresponding to an App advertising slot, which is based on a combination of factors to carry out. We use the keyword auctions model to classify the nature of the Apps, also classified advertisers demands. We assume that advertisers are taking neither risk nor a conservative attitude. We can say the advertisers are risk-neutral. We suppose there are N risk-neutral advertisers to compete K App advertising slots. Keyword auctions use the show amount of payment mode. App downloads and user stickiness are two key factors for the improvement of the show amount. Advertising in the App, users have the probability of running App and see the advertisement, we call it the display rate. α i j indicates the display rate of advertiser i s ad on App advertising slot j. We also can set the rank of App downloads. The more popular the Application is, the more users, and the corresponding higher display rate. So we define display rate meet the decreasing nature: The more popular the advertising in the Application, the higher the display rate. So what we represent the Application pop coefficient is r i, α i j is based on r i after sorting the results. α i j is a function of r i. We have α i1 > α i2 > α i3 > α i4 > α i5...... > α ik, i 1, 2, 3,..., N (1) 167

In the App market platform, there is statistical ranking App downloads, and ranking data corresponding download variable is represented by d i. We have r i εd i. ε is a constant and a positive number. We also have α i j λr i. λ is a constant and a positive number. When the user runs the App and see the ad. It can bring advertisers a certain value of v in average. It is private information for advertisers. Assuming that the advertiser i s valuation ] v i have a distribution function g i ( ) on an interval v, v. It has a positive continuously differentiable density function g i ( ). The valuations between various advertisers are independent. Distribution function G i ( ) is that we all know and meet the hazard rate monotonically non-decreasing conditions. is monotonically decreasing. g i ( ) 1 G( ) i This model have certain personal characteristics estimated with the auction price. T is v, v ] N represents the valuation of the advertisers slots. v (v 1, v 2, v 3,..., v N ) and b (b 1, b 2, b 3,..., b N ) respectively denote the valuation vector and bid vector of N advertisers. v i (v 1, v 2, v 3,..., v i 1, v i+1,..., v N ) and b i (b 1, b 2, b 3,..., b i 1, b i+1,..., b N ), respectively denote valuation vectors and bid vector of other N 1 advertisers except advertiser i. The advertiser s bid b i is a function of its own valuation v i, that is b i β i (v i ), i 1, 2, 3,..., N. Where β i ( ) is also known as the strategy of advertiser i. App developers provide variable costs of K advertising, c (c 1, c 2, c 3,..., c K ) is the cost of display a App advertising once. The cost in addition to the display rate was a single display cost corresponding. To research keyword auctions mechanism, we only consider direct real mechanism. The bid space is T. A keyword auctions mechanism (P, M) consists of two main rules: (a) Allocation rule: Keyword auctions allocation rules P is a map from advertisers bid space T to {0, 1} NK, that is P {0, 1} NK. Specifically, P(b) {P i j (b)}, P i j (b) 1 represents advertiser i from advertising alliance won the slot of App j. We set only one slot on each App, and each advertiser can get at most one slot in the same period of time. Thus P i j (b) 1, j 1, 2,..., K (2) P i j (b) 1, i 1, 2,..., N (3) j1 (b) Payment rule: Keyword auctions payment rules M is a map from advertisers bid space T to R NK, that is M : T R NK. Specifically, M(b) {M i j (b)}, M i j (b) represents the show price paid by the advertiser i for getting the slot in the Application j when the bid price vector is b. IV. A. Participants Revenues Analysis Advertisers revenues: We assume that other advertisers take their true valuation except advertiser i, that is b i v i. Hence we define the revenue of the advertisers: t i (b i v) v i M i j (b i, v i )]α i j P i j (b i, v i ) (4) Among them: j1 v i M i j (b i, v i )]λεr i P i j (b i, v i ) j1 P i (b i, v i ) M i (b i, v i ) v i P i (b i, v i ) M i (b i, v i ) λεr i P i j (b i, v i ) (5) j1 λεr i P i j (b i, v i )M i j (b i, v i ) (6) j1 The expected revenues of the advertisers are: ξ i (b i v i ) E v i t i (b i v)] (7) E v i v i P i (b i, v i ) M i (b i, v i )] v i P i (b i ) M i (b i ) E v i ( ) represents taking expectations for a distribution of the other N-1 advertisers valuations. P i (b i ) E v i P i (b i, v i )] (8) M i (b i ) E v i M i (b i, v i )] (9) They respectively denote excepted CTR and expect to pay of advertiser i when the bid is b i. App developer s expected revenue: App developers can have more than one App in the auction, and also can be only one. When the the advertisers bid with true valuation, the expected revenue of App developer community is equal to the expected revenue of advertisers R minus retention valuation of these advertising slots of App developers VC. ξ 0 R VC E vi M i (v i )] E v P i j (v)c j ] j1 When App developers only have one advertising slot, the revenue is: ξ 0 R VC E vi M i (v i )] E v P i j (v)c j ] E v ( ) indicates taking expectations for the N advertisers valuation. E vi ( ) indicates taking expectations for valuation of the advertiser i. Advertising alliance revenue: The expected revenue of advertising alliance is equal to the product of advertisement display, click rate and the number of single click charges. ξ m E vi M i j (v)p i j (v)c j ] (10) j1 j1 168

B. Feasibility of Keyword Auctions Mechanism Definition 1 Individual Rationality. Mobile App advertising keyword auctions mechanism is individually rational, and the revenue is not a negative, we have ξ i (v i v i ) 0, i 1, 2, 3,..., N (11) Definition 2 Incentive Compatibility. Mobile App advertising keyword auctions mechanism is incentive compatibility that meet: ξ i (v i v i ) ξ i (b i v i ) (12) b i belongs to the valuation space. i 1, 2, 3,..., N. Definition 3 Feasible Mechanism. When the mechanism satisfies individual rationality, incentive compatibility, as well as the allocation of feasibility, we can say the mobile App advertising Keyword auctions is feasible. Definition 4 Sale Mechanism. Optimal keyword auctions meet the equilibrium strategy: (1) All advertisers bid according to the strategy β i ( ). That is, for all i, we have β i β. (2) When the other advertisers bid according to the strategy b i β(v i ), any real value x for advertiser i s optimal reaction is showing their true valuations, the bid should be b β(v). Hence, independent and identically distributed private valuation v, equilibrium strategies β i ( ) and above constitute the initial sale of mobile App advertising keyword auctions mechanism. Lemma 1 The keyword auctions mechanism is feasible, if and only if the following conditions hold (i) (ii) ξ i (v i v i ) ξ i (v v) + v i Pi ( j) j ]. P i ( ) is monotonically non-decreasing. (iii) ξ i (v v) 0, i 1, 2, 3 N. (iv) P i j (b) 1, j 1, 2, K and (v) P i j (b) 1, i 1, 2, N. j1 β(v) meet the sale mechanism. Proof: (Sufficiency) Condition (iv) is clear from the definition of allocation rule. As ξ i (v i v i ) ξ i (v v)+ v i Pi ( j) j ] ξ i (v v) 0 satisfies the principle of individual rationality. According to Eq.(7). We have: ξ i (b i v i ) E v i v i P i (b i, v i ) M i (b i, v i )] (13) E v i b i P i (b i, v i ) M i (b i, v i ) + (v i b i )P i (b i, v i )] ξ i (b i b i ) + (v i b i ) P i (b i ) By the conditions (i) and (ii), then ξ i (v i v i ) ξ i (b i v i ) ξ i (v i v i ) ξ i (b i b i ) (v i b i ) P i (b i ) vi b i P i (b i )dt (v i b i ) P i (b i ) vi b i P i (t) P i (b i ) ] dt 0 Therefore, it satisfies the principle of incentive compatibility. It actual meet sale mechanism by knowing the income. When the conditions above are met, we can say the mechanism (i)-(v) is feasible. (Necessity) According to the definitions of individual rationality, feasibility of equilibrium strategies and the allocation of feasibility, conditions (iii), (iv), (v) were set up. It can be obtained by incentive compatibility of condition Eq.(12) and Eq.(13) that: (v i b i ) P i (b i ) ξ i (v i v i ) ξ i (b i b i ) (v i b i ) P i (v i ) By b i v i, we know that P i ( ) is monotonically nondecreasing. ξ i (v i v i ) is a monotonically non-decreasing continuous. So (i) and (ii) were set up. Lemma 2 Under the feasible mechanism, we can have the expected revenue of App developers ξ 0 E v λεr i P i j (v)v i 1 F i(v i ) c j ] f i (v i ) α i j ξ i (v v) j1 Proof: We can have the expected payoff of advertiser i under the feasible mechanism M i (v i ) v i P i (v i ) ξ i (v i v i ) (14) v i v i P i (v i ) ξ i (v v) P i ( j) j By Eq.(14), Eq.(8) and Eq.(5), we have R E vi M i (v) ] v v i P i (v i ) f i (v i )dv i v vi P i (t)dt f i (v i )dv i ξ i (v v) v v i P i (v i ) f i (v i )dv i v 1 F i (v i )] P i (v i )dv i ξ i (v v) v v i 1 F ] i(v i ) E v i P i (v i, v i )] f i (v i )dv i f i (v i ) ξ i (v v) E v λεr i P i j (v) v i 1 F ] i(v i ) f j1 i (v i ) ξ i (v v) Then by the above revenues ξ 0 R VC N E vi M i (v i )] E v K P i j (v)c j ], we can get the result. j1 169

C. Revenue Maximization Optimal mobile App advertising keyword auctions mechanism: Theorem 1 (Revenue maximization theorem) The keyword auctions feasible mechanism is determined by the expected revenue of (ξ i (v v), i 1, 2,..., N), App developers expected revenue and advertising the allocation rules and valuation limit on (v). Proof: According to Lemma 2, a feasible keyword auctions mechanism if optimal, must make P i j ( ) and {ξ i (v v)} maximum ξ. So the problem of optimal keyword auctions mechanism can be attributed to ξ i (v v) 0 and under the condition of (ii) and (iv) maximum (OP): (OP) E v j1 λεr i P i j (v) v i 1 F i(v i ) c ] j f i (v i ) α i j (15) Theorem 2 (Optimal keyword auctions mechanism properties) Keyword auctions mechanism (P, M) is optimal when the following conditions are met. (vi) ξ i (v v) 0, i 1, 2 N. (vii) Allocation rules P condition (ii) and (iv) to maximize (OP). Proof: It is difficult for condition (i) to determine a mechanism is feasible directly, the following theorem and Lemma 1 can solve this problem, and design an optimal keyword auctions. Theorem 3 If the allocation rules P satisfy the condition (vii), we define payment rules M as follows: b i (viii) M i (b i, b i ) b i P i (b i, b i ) Pi ( j, b i ) j ] (16) In this equation, i 1,2, N. Then we can say keyword auctions mechanism (P, M) is an optimal mechanism. Proof: First, we can prove that the real bid is a Nash equilibrium. Assuming that other advertising in addition to advertiser i outside the main according to their true valuation, that is b i v i. The advertiser i quotation obey balancing strategy. With Eq.(16), when b i, then seek advertiser i bid expected payoff is j b i M i (b i ) b i P i (b i ) Pi ( j) j ] Thus, the expected revenue of the advertiser i is ξ i (b i v i ) v i P i (b i ) M i (b i ) (17) v i P i (b i ) b i P bi i (b i ) P i ( j) j ] bi (v i b i ) P i (b i ) + P i ( j) j ] v i jb i v i b i P i (b i ) j ] + P i ( j) j ] v i P i ( j) j ] + P i (b i ) j P i ( j) j ] v i P i ( j) j ] According to (ii), the above equation, if and only if equality was b i v i. In other words, advertiser i will be his true valuation v i. Then the excepted payoff of advertiser i is. jb i v i M i (v i ) v i P i (v i ) Pi ( j) j ] So by Eq.(7), we have ξ i (v v) v i P i (v i ) M i (v i ) v i Pi ( j) j ] We get ξ i (v v) 0, i 1, 2, N. Therefore, the mechanism satisfies the condition of (i) and (iii) of Lemma 1. Under the condition of (ii) and (iv) of P, we can maximum (OP). Hence it satisfies the condition (ii) and (iv) in Lemma 1. Thus it is a feasible keyword auctions mechanism. This feasible mechanism also meets the condition (vi) and (vii) of Theorem 2. Then we can say the mechanism is optimal. V. Conclusion In this paper, we propose the mobile advertising auction model based on the optimal keyword auctions. Analysis and characterization of the model under the assumption that consider the App developers have their variable cost. We reached the keyword auctions revenue maximization theorem and given the properties of the optimal keyword auctions mechanism. The advertising click-through rate only depends on the location of the premise and per-click costs. We analyzed the expected revenues of advertisers, advertising alliance platform and App developers, get the corresponding expected revenue expressions. Now, many businesses already open mobile platform, such as Google has been fully open App platform, and tens of millions of investment construction of this platform. As Mobile App become more and more popular, App advertising needs a continuous improvement auction mechanism to adapt the change. At this stage, we need to combined theory with the actual constantly. When considering the influence of advertisers and advertising for click-through rates, CTR separation 170

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