# Coordination of a Decentralized System with a Mobile Platform and an App by a Bilateral Advertising Contract

Authors: Jinjie Zhu; Shulin Liu
Journal: Discrete Dynamics in Nature and Society
 (2022)

---

## Abstract

A previous study investigates the advertising strategies of the platform and App by assuming that the platform’s advertisement will increase the number of the App’s users, but the App’s advertisement will not increase the number of the platform’s users; and the platform overcharges the App and takes the advertising fee as a source of its revenue. However, the existing users of the App may recommend the App to the users who have not used it. As a result, the App’s advertisement may increase the number of the platform’s users. Additionally, it is not reasonable that the platform takes the advertising fee of the App as a source of its revenue. These motivate us to reanalyze the previous work. This paper reanalyzes the previous work by additionally assuming that the App’s advertisement will increase the number of the platform’s users and assuming that the platform receives its revenue from its users and shares a proportion of the App’s sales revenue and the App receives its revenues from its users and a proportion of its advertising cost subsidized by the platform. We find that these new assumptions have some significant effects on the previous results. We use dynamical optimization approaches to analyze a decentralized system and find the two parties’ optimal advertisement efforts and proportions. To achieve the efficiency of the integrated system that is proved to be more efficient than the decentralized system, we design a bilateral advertising contact for the decentralized system and show that there exists a unique contact that can coordinate the decentralized system. We find that both parties are better off under some mild conditions and the proportion that the platform bears the App’s advertising cost becomes greater with the contract than without the contract. We have gained some managerial insights.


---

## Body

## 1. Introduction

Mobile Apps have become an indispensable part of the daily life of users worldwide in the era of mobile Internet. In 2018, the global mobile App revenue has reached $92.1 billion and continues to increase closely to hundreds of billions of dollars (Source:https://newzoo.com/insights/articles/newzoos-2018-global-mobile-market-report-insights-into-the-worlds-3-billion-smartphone-users/.). The huge success of Apps cannot be achieved without two major mobile platforms, i.e., Apple’s iOS and Google’s Android OS. As a two-sided market connecting users and Apps, the revenue of the platform comes not only from the payment of users to enjoy the functional services provided by the platform but also more importantly from the share of Apps’ sales revenue. For example, Apple charges 30% share of sales revenue for all published Apps in iOS App Store (Source: https://developer.apple.com/programs/; https://developer.apple.com/in-app-purchase/.). In addition, the platform owner is willing to provide an advertising subsidy for the App, leading to greater profits for both of them [1].Both platforms and Apps are constantly facing new challenges and opportunities in occupying market share. [2] consider a decentralized system consisting of two parties: the platform owner and app developer, where the two parties determine their equilibrium advertising efforts and the optimal trajectories of the numbers of users independently by maximizing their separate objectives. Then, they consider an integrated system with the two parties being integrated as one party, where the two parties determine their equilibrium advertising efforts and the optimal trajectories of the numbers of users by maximizing the sum of their objectives. After that, they design a bilateral participation contract to coordinate the decentralized and integrated system. They numerically demonstrate that both the platform owner and app developer can be better off with the contract than without it.This paper finds that some of the main assumptions and results in the study by Wang et al. [2] are problematic. First, they assume that the platform’s advertisement will positively increase the number of the App’s users, but the App’s advertisement will not increase the number of the platform’s users. According to Pedersen and Nysveen [3], the existing users of the App may recommend the App to the users who have not used it yet, i.e., the App’s unaware users. This recommendation effect will lead the App’s unaware users to have to register as the platform’s users before downloading the desired Apps [4]. Thus, the App’s advertisement may increase the number of the platform’s users. This is ignored in the study by Wang et al. [2] and is captured by a nonnegative parameter κ in our model. When κ = 0, our model reduces to their case, i.e., only positive effect of the platform’s advertisement on the number of the App’s users is considered. Second, they assume that the platform owner overcharges the App and takes the advertising fee as a source of its revenue in some cases. Clearly, this assumption is not reasonable. The reason is that the platform’s revenue should partially come from the App’s revenue, but not from the App’s cost. Third, they assume that a loss of existing users of the platform (App) negatively influences its users’ growth and is captured by the same decay coefficient δ. We think the influences may be the same or different. Fourth, we find that they present wrong expressions of the optimal trajectories of the numbers of platform users and app users for the integrated and decentralized systems (see their Propositions 2 and 4). Thus, their subsequent results and interpretations associated with these expressions are problematic. Finally, as in the study by Wang et al. [2], it is of significant importance to coordinate the two decentralized systems and the integrated system in the sense that the respective optimal advertising efforts of the platform and the App in the new decentralized system are equal to their respective optimal advertising efforts in the integrated system, and the sum of optimal objectives of the platform and the App in the new decentralized system is equal to the optimal objective in the integrated system. Thus, the efficiency is improved with coordination.This paper aims at reanalyzing the work of Wang et al. [2] by considering the four aspects mentioned above. Our main contributions are shown as follows.Some of our results for the integrated system, the decentralized system, and the decentralized system with the bilateral participation contract generalize theirs. Specifically, our Propositions1, 3, and 7 for three systems, respectively, degenerate into their Propositions 1, 3, and 5 when κ = 0 and δP = δA = δ. However, some of our results for the integrated and decentralized system do not degenerate into theirs: our Propositions 2(i) and 4(i) do not degenerate into their Propositions 2 and 4 because their Propositions 2 and 4 present wrong expressions of the optimal trajectories of the number of platform users and app users for the integrated and decentralized systems. Thus, their subsequent results and interpretations based on their Propositions 2 and 4 are problematic.We utilize the phase diagram to show that the unique steady-state of the numbers of platform users and App users for the integrated and decentralized systems are all stable nodes and this makes sense in reality. In contrast, they are silent on this.Finally, we theoretically compare the optimal values of the objective functions for the decentralized systems without the contract and with the contract. Our results are totally different from the related results of Wang et al. [2] and improve their results.The remainder of this paper is organized as follows. Section2 reviews related literature. Section 3 establishes a model of an integrated system and a decentralized system consisting of one platform and one App. Section 4 analyzes the two systems. Section 5 designs a bilateral advertising contract to coordinate the decentralized system with the integrated system. Section 6 concludes by discussing the major results and gaining some managerial insights. All proofs are provided in .


## 2. Literature Review

This paper will contribute to two streams of literature on cooperative advertising and App advertising.Mathematical modeling of cooperative advertising has always been a focus of operations research. Traditionally, vertical cooperative advertising in the supply chain describes a financial agreement where the manufacturers will share a certain percentage of the retailers’ advertising costs [5]. Specifically, the manufacturers set some guidelines for advertising, and the advertising preparation and planning are basically done by the retailers. Then, the retailers can require the manufacturers to provide certain advertising subsidies under the previously determined agreements [6, 7]. Cooperative advertising is therefore essentially an advertising contract or financial agreement about sharing the associated costs [8, 9]. Manufacturers and retailers can have such a cooperative relationship because manufacturers’ advertising focuses on brand image, while retailers’ advertising focuses on sales [7, 10, 11]. To sum up, cooperative advertising can stimulate the retailers’ investment in advertising, so as to achieve the impact of advertising on sales that the manufacturers expect [12]. In addition, the revenue sharing contract and two-part-tariff contract are also introduced to coordinate the supply chain [13, 14]. However, a few studies have proposed bilateral advertising contracts in which retailers may also support their manufacturers’ advertising programs [15–17].Dynamic cooperative advertising can involve a time dimension to characterize the time dependence of advertising decisions. In dynamic models, the crucial state variables, such as brand goodwill, sales, and market share, are usually assumed to change over time, as described by differential equations. In addition, the control variables of the model are not only the advertising effort but also the price, product quality, inventory, and so on [18]. Jørgensen et al. [19] considered the long and short-term effects of advertising on demand and band goodwill. He et al. [20] extended their work to a stochastic dynamic sales model based on the decisions of advertising and prices. De Giovanni [21] introduced the manufacturers’ quality improvement efforts, while Zhang et al. [16] considered the impact of advertising on the consumers’ reference prices.Different from the dynamic advertising in the supply chain, in our model setting, the numbers of the platform users and App users are two state variables, which are affected by their respective advertising efforts. Additionally, the platform’s user growth is also affected by the App’s user base and vice versa. Our model captures a scenario where the platform is willing to provide advertising subsidies to the App given that a revenue-sharing contract exists between the platform and the App. This is in line with the actual economic environment.The other stream of our research is related to App advertising. For the theoretical modeling of advertising, Zhang et al. [22] modeled the strategic selection of targeted advertising and mass advertising. Sun et al. [23] studied the optimal sequence of fading advertising shown to App users by considering the sojourn and exposure effects. Choi et al. [24] investigated the search-based advertising auction problem, and Yuan et al. [25] added the quality score effect to that issue. Guo et al. [26] conducted an economic analysis of the reward advertising mechanism in the game Apps. However, the abovementioned studies only focus on the App developers’ advertising strategies and do not concern the platform’s interaction.In the context of a revenue sharing contract between the platform and the App, Avinadav et al. [27] and Avinadav et al. [28] study the optimal prices and qualities of Apps under three different risk attitudes of developers. Chen et al. [29] analyzed the optimal strategies for developers to provide paid or free Apps in single and two platforms. However, the advertising decisions are not the concern of the above research. Hao et al. [1] established a bilateral market model to determine the platform owner’s optimal advertising revenue sharing contract. They found that it is the best way to subsidize the App through advertising channels, which will generate greater profits for both the platform and the App. However, their model setting is static.For the dynamic advertising model setting, most of the studies are influenced by Kumar and Sethi [30], regarding the dynamic pricing and advertising strategies of web content. However, relatively little research has been done on the dynamic advertising modeling of Apps. Ji et al. [4] studied the joint decisions of the platform and App on the advertising investment and in-App advertising adoption. They showed that revenue sharing and advertising cost sharing should be adopted to coordinate the mobile application system.Wang et al. [2] explored the dynamic optimal advertising strategies when the platform and the App play multiple roles as sellers, ad publishers, and advertisers. This paper builds on their work, but differs from theirs in three aspects, as follows:(1)
The dynamics of the number of platform users and App users are differentThey assume that the number of App’s users will not affect the number of the platform’s users. Instead, according to Pedersen and Nysveen [3], the recommendation effect will be generated when the App has a large user base. Thus, the current users of the App may recommend the unaware users to register on the platform. Therefore, the App’s user base may contribute positively to the platform’s user growth to some extent. Therefore, the number of the App’s users will positively affect the number of the platform’s users, which is considered and captured by a parameter κ in the current paper.(2)
They assume that a loss of existing users of both the platform and App influences their user growth negatively and is captured by the same decay coefficientδ. We assume that the loss should be captured by two different or same decay coefficients of δP and δA.Whenκ = 0 and δP = δA = δ, our motion equations of the numbers of the platform and App reduce to theirs (see our (4) and (5) and their (2) and (3)).(3)
The flows of revenues and expenditures between the platform and App are differentWe assume that the platform is willing to undertake a proportionϕ (t) of the App’s advertising cost CA (t) (0 ≤ ϕ (t) ≤ 1), as the advertising subsidy. The platform undertakes a part of the App’ advertising cost (i.e., ϕ (t) CA (t)) when 0 < ϕ (t) < 1 and does not undertake any of CA (t) (or equivalently, the App undertakes the whole advertising cost of CA (t)) when ϕ (t) = 0. In contrast, they define ϕ (t) as the platform owner’s charge on the app developer’s payment rate. By their equation (4), the platform undertakes a part of the App’s advertising cost (i.e., (1 − ϕ (t)) CA (t)) when 0 ≤ ϕ (t) < 1, does not undertake any of CA (t) (or equivalently, the App pays the whole advertising cost of CA (t)) when ϕ (t) = 1, and overcharges the App and takes the advertising fee as a source of revenue (i.e., (ϕ (t) − 1) CA (t)) when ϕ (t) > 1.The abovementioned comparison shows that ourϕ (t) (0 ≤ ϕ (t) ≤ 1) has the same meanings as their 1 − ϕ (t) (0 ≤ ϕ (t) ≤ 1). Clearly, our meaning of ϕ (t) is more easily understandable. Their assumption of ϕ (t) > 1 does not make sense and is not reasonable: it is well known that the platform’s revenue should partially come from the App’s revenue, but not from the App’s cost; they assume that the platform takes the advertising fee of the App as a source of revenue and that the revenue is (ϕ (t) − 1) CA (t), which is unreasonable. Accordingly, the flows of revenues and expenditures between the platform and App are different from theirs (see our (2) and (3) and their (4) and (5)). Additionally, the total instantaneous profit of the integrated system is the same as theirs (see our (6) and their (6)).Finally, this paper is mostly related to the literature on the coordination of advertising efforts that analyzes the proposed supply chain (SC) using the decentralized models, centralized models, and coordination models. For example, Johari and Hosseini-Motlagh [31] studied a SC with one manufacturer and two retailers facing deterministic effort-dependent market demands, where the supplier shares a fraction of retailers’ advertising expenditures and incurs national advertising expenditures. The cooperation is designed by the Nash bargaining model. Ebrahimi et al. [32] investigated the coordination of promotional effort and replenishment decisions in a two-echelon SC including a single supplier and a single retailer, where a stochastic demand influenced by the retailer’s promotional effort is assumed and the supplier does not share any fraction of retailers’ advertising expenditures. A coordination model is proposed based on a delay in payment contract, where the supplier offers a delay in payment to the retailer as long as the retailer selects the globally optimal solution, which is obtained in the centralized model. Hosseini-Motlagh et al. [33] investigated a competitive SC consisting of one supplier and two retailers, where only the retailers invest in advertising and incur advertising expenditures, the supplier shares a fraction of the retailers’ advertising expenditures, and the demand is stochastic and dependent on the effort. The coordination contract is designed in such a way that the supplier considers the trade-off between reducing lead time and paying tax on carbon emissions while providing enough incentives for the competitive retailers.In contrast, this paper is different from the above three papers in several aspects. First, our decentralized models, centralized models, and coordination models are dynamical, and theirs are static. Second, our demands (the numbers of the users of the platform and App) follow a system of differential equations and do not depend on the efforts, and their demands are stochastic or deterministic and depend on the efforts. Last, our meaning of coordination is different from theirs. By the coordination, we mean the contract designed into the decentralized system in which the respective optimal advertising efforts of the platform and the App are equal to their respective optimal advertising efforts in the integrated system. With the coordination, we expect that the aggregate optimal objective value of the integrated system is equal to that of the decentralized system with the contract.


## 3. Two Systems

### 3.1. Costs and Revenues

We consider a system consisting of one platform and one App and respective users. The platform owner grants the App access to the platform, and the users download the desired App only through the platform. For convenience, we refer to the platform owner as “she” and the App developer as “he,” respectively. All users are divided into two groups: the platform users and the App users. At timet, there are x (t) users who purchase the related products of the platform, such as value-added services, accessories, and music. Meanwhile, there are y (t) users who purchase the related products of the App, including paid downloads or in-App purchases.Denote the advertising efforts of the platform and App byuP (t) and uA (t), respectively. Exerting advertising efforts will incur the costs. We assume that the costs of advertising efforts of the platform and App take quadratic functions as in the studies by Wang et al. [2]; Zhang et al. [16]; Chintagunta and Jain [34]; and Sethi [35]:(1)CPt=12uP2tandCAt=12uA2t,which implies an increasing marginal advertising cost.The App exerts advertising effort not only to increase his number of users but also indirectly benefits the platform through the revenue sharing contract. Similar to the studies by Hao et al. [1] and Wang et al. [2], we assume that the platform is willing to undertake a proportion ϕ (t) of the App’s advertising cost CA (t) (0 ≤ ϕ (t) ≤ 1), as the advertising subsidy, in order to stimulate the App to increase his advertising effort. Accordingly, the total cost of the platform is CP (t) + CA (t) ϕ (t).The platform and the App provide separate services for their users and then earn sales revenue from their users, respectively. Since the platform provides a user download environment for the App, we assume that the platform will share a proportionλ of App’s sales revenue (0 ≤ λ ≤ 1) when the purchase occurs in the App. Let pP and pA denote the marginal sales revenue of the platform and the App, respectively. Then, the App receives a total revenue of pAy (t), among which λpAy (t) is paid to the platform. Thus, the platform receives a total revenue of pPx (t) + λpAy (t) and the App’ total revenue is the sum of the sale net revenue of (1 − λ) pAy (t) and the advertising subsidy from the platform of CA (t) ϕ (t).Accordingly, the instantaneous profits at timet of the platform and the App are given as follows: (Our two profit functions are different from (4) and (5) assumed by Wang et al. [2]. The differences are presented in the Literature section in a detail, where we also justify why our assumption is reasonable and understandable and theirs are not.)(2)πPt=pPxt+λpAyt−12uP2t−12uA2tφt,(3)πAt=1−λpAyt+12uA2tφt−12uA2t.We summarize the revenue and the expenditure flows of the platform and the App, as shown in Figure1, where inflows indicate revenues and outflows mean expenditures.Figure 1
Revenue and expenditure flows.The motion equations of the numbers of both the platform and App are assumed as follows: (Our two motion equations are generalized, but different from Wang et al. [2] (see their (2) and (3)). The coefficients of x (t) and y (t) are identical in the study by Wang et al. [2] (δP = δA) and may be different in our model. They assume that the App can only advertise within the platform and thus ignore the impact of the App’s user base on the platform’s user growth, i.e., κ = 0. According to Ji et al. [4], the App’s user base could also contribute to the platform’s user growth, which is captured in our model by a parameter κ. When κ = 0, our two equations reduced to theirs.)(4)x′t=uPt+κyt−δPxt,x0=x0>0,(5)y′t=γuAt+βxt−δAyt,y0=y0>0,where the parameters of α, γ, and β have the same meanings as in the (2) and (3) of Wang et al. [2]; δP and δA have the similar meaning as δ in their (2). We summarize the meanings of all these parameters as follows: (i) The parameters α, γ, κ, β, δP, and δA are positive. (ii) The constant α (γ) measures the advertising effectiveness of the platform (App) on her (his) user growth. (iii) The parameter κ characterizes the indirect positive impact of the App’s user base (y (t)) on the platform’s user growth x′ (t). The reason is that the recommendation effect will be generated when the App has a large user base [2]. It means that the App’s existing users will make recommendations to the users who have not yet downloaded the App, namely, the App’s unaware users. This leads to the App’s unaware users having to register as the platform’s users before downloading the desired App. Therefore, the App’s user base contributes positively to the platform’s user growth to some extent. (iv) The parameter β characterizes the indirect positive impact of platform’s user base (x (t)) on the App’s user growth y′ (t). The reason is that the larger the user base of a platform, the more likely the App will be downloaded when the developer advertises it. Thus, the platform’s user base will generate a potential market effect for the App [33]. (v) Some of the platform’s users may leave the platform for various reasons, such as a lack of good user experience and functional innovation. This is captured by the decay coefficient δP. Similarly, this phenomenon may occur to the App’s user and is characterized by the decay coefficient δA.In reality, the number of platform users and App users is finite. On the one hand, the contributions to increases in the platform users and App users are characterized byκ and β, respectively. On the other hand, the contributions to decreases in the platform users and App users are characterized by δP and δA, respectively. Thus, it is reasonable to assume δPδA > κβ to guarantee the finite numbers of platform users and App users. Otherwise, the number of platform users and App users will approach infinite as t ⟶ +∞. Now, we consider two specific systems based on the above assumptions. Later on, we will find that this assumption plays a crucial role in our analysis.


### 3.2. Integrated System

Suppose that the platform and the App are vertically integrated into one company, which is considered an integrated system. Thus, the total instantaneous profit of the integrated system (denoted byπI (t)) is the sum of the profits of the platform and the App. Using (2) and (3), we have(6)πIt=πPt+πAt=pPxt+pAyt−12uP2t−12uA2t.We consider an infinite horizon with a positive discount rateρ. The integrated system’s objective is to maximize the present value of (6) over [0, ∞) by choosing the advertising efforts of the platform and App subject to (4) and (5), i.e. (This is the same as (6) in Wang et al. [2]),(7)VI∗x0,y0=maxuPt≥0,uAt≥0VI=∫0∞e−ρtpPxt+pAyt−12uP2t−12uA2tdt,s.t. (4) and (5).


### 3.3. Decentralized System

In the decentralized system, the platform and the App separately maximize the present values of their own profits over [0, ∞) by choosing their own advertising efforts. We model the decentralized system as a Stackelberg game structure. The platform, as a leader, first determines the advertising effortuP (x, y) and the App’s advertising subsidy rate ϕ (x, y). After the platform’s actions, the App as a follower, then determines the advertising effort uA (x, y).Therefore, the platform’s optimization problem is(8)VDP∗x0,y0=maxuPt≥0,φt≥0VDP=∫0∞e−ρtpPxt+λpAyt−12uP2t−12φtuA2tdt,s.t. (4) and (5). The App’s optimization problem is(9)VDA∗x0,y0=maxuAt≥0VDA=∫0∞e−ρt1−λpAyt−12uA2t+12φtuA2tdt,s.t. (4) and (5).For simplicity, in the rest of this paper, we suppresst in time-dependent variables. We summarize all notations in Table 1.Table 1
Summary of notation.
tTimet, t ≥ 0Control variablesuP (t)Advertising effort by the platform at timetuA (t)Advertising effort by the App at timetϕ (t)App’s advertising subsidy rate set by the platform at timetState variablesx (t)The number of platform users at timety (t)The number of App users at timetParametersαThe platform’s advertising effectiveness on her user growthγThe App’s advertising effectiveness on his user growthβThe platform’s user base effectiveness on the App’s user growthκThe App’s user base effectiveness on the platform’s user growthδP, δAUser growth decay parameter of the platform and the App, respectivelyρDiscount rateλThe sharing rate of the App’s sales revenue set by the platformpP, pAMarginal sales revenue of the platform and the App, respectivelyπP (t), πA (t), πI (t)Instantaneous profits of the platform, the App, and the integrated systemVIValue function of the integrated systemVDPValue function of the platform in the decentralized systemVDAValue function of the app in the decentralized system


## 3.1. Costs and Revenues

We consider a system consisting of one platform and one App and respective users. The platform owner grants the App access to the platform, and the users download the desired App only through the platform. For convenience, we refer to the platform owner as “she” and the App developer as “he,” respectively. All users are divided into two groups: the platform users and the App users. At timet, there are x (t) users who purchase the related products of the platform, such as value-added services, accessories, and music. Meanwhile, there are y (t) users who purchase the related products of the App, including paid downloads or in-App purchases.Denote the advertising efforts of the platform and App byuP (t) and uA (t), respectively. Exerting advertising efforts will incur the costs. We assume that the costs of advertising efforts of the platform and App take quadratic functions as in the studies by Wang et al. [2]; Zhang et al. [16]; Chintagunta and Jain [34]; and Sethi [35]:(1)CPt=12uP2tandCAt=12uA2t,which implies an increasing marginal advertising cost.The App exerts advertising effort not only to increase his number of users but also indirectly benefits the platform through the revenue sharing contract. Similar to the studies by Hao et al. [1] and Wang et al. [2], we assume that the platform is willing to undertake a proportion ϕ (t) of the App’s advertising cost CA (t) (0 ≤ ϕ (t) ≤ 1), as the advertising subsidy, in order to stimulate the App to increase his advertising effort. Accordingly, the total cost of the platform is CP (t) + CA (t) ϕ (t).The platform and the App provide separate services for their users and then earn sales revenue from their users, respectively. Since the platform provides a user download environment for the App, we assume that the platform will share a proportionλ of App’s sales revenue (0 ≤ λ ≤ 1) when the purchase occurs in the App. Let pP and pA denote the marginal sales revenue of the platform and the App, respectively. Then, the App receives a total revenue of pAy (t), among which λpAy (t) is paid to the platform. Thus, the platform receives a total revenue of pPx (t) + λpAy (t) and the App’ total revenue is the sum of the sale net revenue of (1 − λ) pAy (t) and the advertising subsidy from the platform of CA (t) ϕ (t).Accordingly, the instantaneous profits at timet of the platform and the App are given as follows: (Our two profit functions are different from (4) and (5) assumed by Wang et al. [2]. The differences are presented in the Literature section in a detail, where we also justify why our assumption is reasonable and understandable and theirs are not.)(2)πPt=pPxt+λpAyt−12uP2t−12uA2tφt,(3)πAt=1−λpAyt+12uA2tφt−12uA2t.We summarize the revenue and the expenditure flows of the platform and the App, as shown in Figure1, where inflows indicate revenues and outflows mean expenditures.Figure 1
Revenue and expenditure flows.The motion equations of the numbers of both the platform and App are assumed as follows: (Our two motion equations are generalized, but different from Wang et al. [2] (see their (2) and (3)). The coefficients of x (t) and y (t) are identical in the study by Wang et al. [2] (δP = δA) and may be different in our model. They assume that the App can only advertise within the platform and thus ignore the impact of the App’s user base on the platform’s user growth, i.e., κ = 0. According to Ji et al. [4], the App’s user base could also contribute to the platform’s user growth, which is captured in our model by a parameter κ. When κ = 0, our two equations reduced to theirs.)(4)x′t=uPt+κyt−δPxt,x0=x0>0,(5)y′t=γuAt+βxt−δAyt,y0=y0>0,where the parameters of α, γ, and β have the same meanings as in the (2) and (3) of Wang et al. [2]; δP and δA have the similar meaning as δ in their (2). We summarize the meanings of all these parameters as follows: (i) The parameters α, γ, κ, β, δP, and δA are positive. (ii) The constant α (γ) measures the advertising effectiveness of the platform (App) on her (his) user growth. (iii) The parameter κ characterizes the indirect positive impact of the App’s user base (y (t)) on the platform’s user growth x′ (t). The reason is that the recommendation effect will be generated when the App has a large user base [2]. It means that the App’s existing users will make recommendations to the users who have not yet downloaded the App, namely, the App’s unaware users. This leads to the App’s unaware users having to register as the platform’s users before downloading the desired App. Therefore, the App’s user base contributes positively to the platform’s user growth to some extent. (iv) The parameter β characterizes the indirect positive impact of platform’s user base (x (t)) on the App’s user growth y′ (t). The reason is that the larger the user base of a platform, the more likely the App will be downloaded when the developer advertises it. Thus, the platform’s user base will generate a potential market effect for the App [33]. (v) Some of the platform’s users may leave the platform for various reasons, such as a lack of good user experience and functional innovation. This is captured by the decay coefficient δP. Similarly, this phenomenon may occur to the App’s user and is characterized by the decay coefficient δA.In reality, the number of platform users and App users is finite. On the one hand, the contributions to increases in the platform users and App users are characterized byκ and β, respectively. On the other hand, the contributions to decreases in the platform users and App users are characterized by δP and δA, respectively. Thus, it is reasonable to assume δPδA > κβ to guarantee the finite numbers of platform users and App users. Otherwise, the number of platform users and App users will approach infinite as t ⟶ +∞. Now, we consider two specific systems based on the above assumptions. Later on, we will find that this assumption plays a crucial role in our analysis.


## 3.2. Integrated System

Suppose that the platform and the App are vertically integrated into one company, which is considered an integrated system. Thus, the total instantaneous profit of the integrated system (denoted byπI (t)) is the sum of the profits of the platform and the App. Using (2) and (3), we have(6)πIt=πPt+πAt=pPxt+pAyt−12uP2t−12uA2t.We consider an infinite horizon with a positive discount rateρ. The integrated system’s objective is to maximize the present value of (6) over [0, ∞) by choosing the advertising efforts of the platform and App subject to (4) and (5), i.e. (This is the same as (6) in Wang et al. [2]),(7)VI∗x0,y0=maxuPt≥0,uAt≥0VI=∫0∞e−ρtpPxt+pAyt−12uP2t−12uA2tdt,s.t. (4) and (5).


## 3.3. Decentralized System

In the decentralized system, the platform and the App separately maximize the present values of their own profits over [0, ∞) by choosing their own advertising efforts. We model the decentralized system as a Stackelberg game structure. The platform, as a leader, first determines the advertising effortuP (x, y) and the App’s advertising subsidy rate ϕ (x, y). After the platform’s actions, the App as a follower, then determines the advertising effort uA (x, y).Therefore, the platform’s optimization problem is(8)VDP∗x0,y0=maxuPt≥0,φt≥0VDP=∫0∞e−ρtpPxt+λpAyt−12uP2t−12φtuA2tdt,s.t. (4) and (5). The App’s optimization problem is(9)VDA∗x0,y0=maxuAt≥0VDA=∫0∞e−ρt1−λpAyt−12uA2t+12φtuA2tdt,s.t. (4) and (5).For simplicity, in the rest of this paper, we suppresst in time-dependent variables. We summarize all notations in Table 1.Table 1
Summary of notation.
tTimet, t ≥ 0Control variablesuP (t)Advertising effort by the platform at timetuA (t)Advertising effort by the App at timetϕ (t)App’s advertising subsidy rate set by the platform at timetState variablesx (t)The number of platform users at timety (t)The number of App users at timetParametersαThe platform’s advertising effectiveness on her user growthγThe App’s advertising effectiveness on his user growthβThe platform’s user base effectiveness on the App’s user growthκThe App’s user base effectiveness on the platform’s user growthδP, δAUser growth decay parameter of the platform and the App, respectivelyρDiscount rateλThe sharing rate of the App’s sales revenue set by the platformpP, pAMarginal sales revenue of the platform and the App, respectivelyπP (t), πA (t), πI (t)Instantaneous profits of the platform, the App, and the integrated systemVIValue function of the integrated systemVDPValue function of the platform in the decentralized systemVDAValue function of the app in the decentralized system


## 4. Analysis of Two Systems

### 4.1. Integrated System

Proposition 1.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that the optimal advertising efforts of the platform and App are positive.). Then, in the integrated system, the respective optimal advertising efforts of the platform and the App are (When κ = 0 and δP = δA = δ, our two (10) and (11) reduce to (11) and (12) in Wang et al. [2], respectively.)(10)uIP∗=αρ+δApP+βpAρ+δPρ+δA−κβ,(11)uIA∗=γρ+δPpA+κpPρ+δPρ+δA−κβ.
By (10), the optimal platform’s advertising effort uIP∗ increases with the effectiveness of the platform’s advertising effort (α) and the effectiveness of the App’s user base on the platform’s user growth (κ), as well as the platform’s marginal sales revenue pP. Additionally, uIP∗ increases with the App’s marginal sales revenue pA, and the effectiveness of the platform’s user base on the App’s user growth β (Apparently, ∂uIP∗/∂κ > 0. Using δPδA − κβ > 0, we have ∂uIP∗/∂β=αpAρ+δPρ+δA−κβ+κρ+δApP+βpA/ρ+δPρ+δA−κβ2>0). Similarly, by (11), the App’s optimal advertising effort uIA∗ increases with the parameters γ, β, and pA and also increases with the parameters pP and κ (Similar to the above proof of ∂uIP∗/∂β > 0, we have ∂uIA∗/∂κ > 0.). Moreover, by (10) and (11), the higher the user growth decay rate (δP and δA) and/or discount rate (ρ), the weaker the optimal advertising efforts (uIP∗ and uIA∗).Proposition 2.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that xISS and yISS are positive.). Then, in the integrated system, the following are noted:(i)
The optimal trajectories of the numbers of platform users and App users are given by (If one substitutes (11) and (12) in Wang et al. [2] into their (2) and (3) and solves the resulting simultaneous differential equation, one can find that the solution is different from (13) and (14) in Proposition 2 of Wang et al. [2]. In other words, they present incorrect expressions of the solutions. Thus, their subsequent analysis based on the solution is problematic.)(12)x∗t=Aeq1t+Beq2t+xISS,(13)y∗t=Ceq1t+Deq2t+yISS,where(14)xISS=αδAuIP∗+γκuIA∗δPδA−κβ=pPα2δAρ+δA+γ2κ2+pAα2βδA+γ2κρ+δPδPδA−κβρ+δPρ+δA−κβ,(15)yISS=αβuIP∗+γδPuIA∗δPδA−κβ=pAγ2δPρ+δP+α2β2+pPγ2κδP+α2βρ+δAδPδA−κβρ+δPρ+δA−κβ,Δ=δP−δA2+4κβ,q1=−δP+δA+Δ2,q2=−δP+δA−Δ2,A=Δ−δP−δAx0−xISS+2κy0−yISS2Δ,B=Δ+δP−δAx0−xISS−2κy0−yISS2Δ,C=Δ+δP−δAy0−yISS+2βx0−xISS2Δ,D=Δ−δP−δAy0−yISS−2βx0−xISS2Δ.(ii)
There exists a unique steady-state of the number of platform users and App users (xISS, yISS) that is a stable node(iii)
The optimal present value of the total profit, i.e., the optimal objective value is(16)VI∗x0,y0=uIP∗2+uIA∗22ρ+uIP∗αx0+uIA∗γy0.The part (ii) of Proposition2 shows that the optimal trajectories of the number of platform users and App users converge to the steady-state that is the stable node E (xISS, yISS). This can be proved by using the phase diagram analysis as follows. By δPδA − κβ > 0, we have δP/κ > β/δA > 0, implying that the slope of the demarcation curve x′t=0 is higher than the slope of the demarcation curve y′t=0. The two demarcation curves (straight lines), intersecting at point E (xISS, yISS), divide the phase space into four distinct regions, labeled I through IV as shown in Figure 2. Point E represents the intertemporal equilibrium of the integrated system. Since ∂x′/∂x=−δP<0 and ∂y′/∂y=−δA<0, we have the four groups of xy-arrowheads. Then, we can draw a family of streamlines, or trajectories, to portray the dynamics of the integrated system from any conceivable initial point. As shown in Figure 2, all the streamlines associated with point E flow noncyclically toward it. Thus, point E (xISS, yISS) is a stable node.By (14) and (15), xISS increases with the parameters α, κ (Since ∂uIP∗/∂κ > 0 and ∂uIA∗/∂κ > 0 implied by Proposition 1, by (14), and δPδA − κβ > 0, we h000ave ∂xISS/∂κ=αδA∂uIP∗/∂κ+γuIA∗+γκ∂uIA∗/∂κ/δPδA−κβ+βαδAuIP∗+γκuIA∗/δPδA−κβ2>0) and pP, while yISS increases with the parameters γ, β (Similar to the above proof of ∂xISS∗/∂κ > 0, we have ∂yISS/∂β > 0.), and pA. Furthermore, both xISS and yISS decrease in δP, δA, and ρ. (Since δPδA − κβ > 0 and both uIP∗ and uIA∗ decrease in ρ, δP, and δA, we have ∂xISS/∂δA=αδA∂uIP∗/∂δA+γκ∂uIA∗/∂δAδPδA−κβ−καβuIP∗+γδPuIA∗/δPδA−κβ2<0,∂yISS/∂δA=αβ∂uIP∗/∂δA+γδP∂uIA∗/∂δAδPδA−κβ−δPαβuIP∗+γδPuIA∗/δPδA−κβ2<0. Similarly, we can prove that xISS and yISS both decrease in δP and ρ.) By (16), VI∗ is positively related to the initial number of platform users and App users, i.e., x0 and y0. Additionally, VI∗ increases with α, γ, β, κ, pP, and pA, while decreases in ρ, δP, and δA (Since uIP∗ (uIA∗) increases in α (γ), pP, pA, β, and κ decrease in ρ, δP, and δA, we can prove VI∗ increases in α, γ, pP, pA, β, and κ, decreases in ρ, δP, and δA by (10), (11), and (16).Figure 2
The phase diagram.


### 4.2. Decentralized System

Proposition 3.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that uDP∗ and uDA∗ are positive.). Then, in the decentralized system, the optimal advertising effort of the platform is as follows:(17)uDP∗=αρ+δApP+λβ​pAρ+δPρ+δA−κβ,and the optimal App’s advertising subsidy rate set by the platform for the App is as follows:(18)φD∗=3λ−1ρ+δPpA+2κpP1+λρ+δPpA+2κpP.Furthermore, the optimal advertising effort of App is as follows:(19)uDA∗=γ1+λρ+δPpA+2κpP2ρ+δPρ+δA−κβ.Next, we perform some comparative static analyses of the results of Proposition3, leading to some useful managerial insights.By (17), the platform’s optimal advertising effort uDP∗ increases with the App’s marginal sales revenue pA and the effectiveness β of the platform’s user base on the App’s user growth. Clearly, ∂uDP∗/∂λ > 0, implying that the more the platform shares the App’s sales revenue, the more effort the platform spends on advertising. Additionally, uDP∗ increases with α, κ, and pP and decreases in ρ, δP, and δA. By (18), we have ∂ϕD∗/∂κ > 0, ∂ϕD∗/∂λ > 0, ∂ϕD∗/∂pP > 0, and ∂ϕ∗/∂pA < 0. Thus, the optimal App’s advertising subsidy rate set by the platform ϕD∗ increases with the effectiveness of the App’s user base on the platform’s user growth (κ), the sharing rate of the App’s sales revenue set by the platform (λ), and the platform’s marginal sales revenue (pP), but decreases with the App’s marginal sales revenue (pA).Remark 1.
To guarantee the required condition 0 ≤ϕD∗ ≤ 1, the inequality of 1/3 − 2κpP/[3 (ρ + δP) pA] ≤ λ ≤ 1 must hold because these two inequalities are equivalent by (18). It is assumed that the proportion λ that the platform shares from the App’s sales revenue is between 0 and 1. Thus, to be meaningful economically, we must add an assumption of 1/3 − 2κpP/[3 (ρ + δP) pA] ≥ 0 (i.e., pA ≥ (2κpP)/(ρ + δP)), implying that the platform shares at least a proportion 1/3 − 2κpP/[3 (ρ + δP) pA] of the App’s sales revenue. Under this assumption, we can get some managerial insights as follows:(i)
Clearly,λ = 1 is equivalent to ϕD∗ = 1 by (18). This implies that the platform shares all the sales revenue of the App if and only if it undertakes all the advertising costs of the App.(ii)
SinceϕD∗ = 0 is equivalent to λ = 1/3 − 2κpP/[3 (ρ + δP) pA], the platform will not carry any advertising cost of App if and only if it shares a proportion 1/3 − 2κpP/[3 (ρ + δP) pA] of the App’s sales revenue(iii)
Assume thatκ = 0. Then, 0 ≤ ϕD∗ ≤ 1 is equivalent to 1/3 ≤ λ ≤ 1 by (18). This means that when the App’s users have no impact on the platform’s user growth, the platform will share at least a 1/3 proportion of the App’s sales revenue. Additionally, with κ = 0, our decentralized system reduces Wang et al. [2], and ϕD∗ = (3λ − 1)/(1 + λ) > 0 by (18), implying that ϕD∗ only depends on λ and increases with λ. The latter result is contrary to theirs (see their (18)) and is in line with our intuition mentioned above.(iv)
Given that the proportion that the platform shares the App’s sales revenue is between the open interval of (1/3 − 2κpP/[3 (ρ + δP) pA], 1), the platform will carry a proportion ϕD∗ > 0 of App’s advertising costs.
The App’s optimal advertising effortuDA∗ increases with the parameters κ, pP, and λ that have positive effects on the platform’s revenue. The reason here is different from that of the integrated system. It is because ϕD∗ increases with the parameters κ, pP, and λ. This means that the more revenue the platform earns, the more advertising subsidies she provides to the App, which incentivizes the App to increase his advertising effort. Therefore, the revenue sharing rate will not be a burden for the App to determine his advertising effort. Additionally, uDA∗ increases with γ, β, and pA and decreases in ρ, δP, and δA.Proposition 4.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that xDSS and yDSS are positive.). Then, in the decentralized system:(i)
The optimal trajectories of the number of platform users and App users are(20)x∗t=A¯eq1t+B¯eq2t+xDSS,(21)y∗t=C¯eq1t+D¯eq2t+yDSS,whereA¯=Δ−δP−δAx0−xDSS+2κy0−yDSS/2Δ,B¯=Δ+δP−δAx0−xDSS−2κy0−yDSS/2Δ,C¯=Δ+δP−δAy0−yDSS+2βx0−xDSS/2Δ,D¯=Δ−δP−δAy0−yDSS−2βx0−xDSS/2Δ.(22)xDSS=αδAuDP∗+γκuDA∗δPδA−κβ=2pPα2δAρ+δA+γ2κ2+pA2λα2βδA+γ2κρ+δP1+λ2δPδA−κβρ+δPρ+δA−κβ,(23)yDSS=αβuDP∗+γδPuDA∗δPδA−κβ=2pPγ2κδP+α2βρ+δA+pAγ2δPρ+δP1+λ+2λα2β22δPδA−κβρ+δPρ+δA−κβ.(ii)
There exists a unique steady-state of the number of platform users and App users (xDSS, yDSS) that is a stable node(iii)
The optimal present values of the profits of the platform and the App are given by(24)VDP∗x0,y0=uDP∗2+uDA∗22ρ+ρ+δApP+λβpAx0+κpP+ρ+δPλpAy0ρ+δPρ+δA−κβ,(25)VDA∗x0,y0=uIP∗−uDP∗uDP∗+uIA∗−uDA∗uDA∗ρ+1−λpAρ+δPρ+δA−κββx0+ρ+δPy0.Furthermore, the sum ofVDP∗x0,y0 and VDP∗x0,y0 (denoted by VD∗) is given by(26)VD∗≡VDP∗x0,y0+VDA∗x0,y0=uDP∗2uIP∗−uDP∗+uDA∗2uIA∗−uDA∗2ρ+uIP∗αx0+uIA∗γy0.Corollary 1.
Suppose thatδPδA − κβ > 0. Then, in the decentralized system:(i)
∂VDP∗/∂λ > 0(ii)
Ifλ ≥ 1/2, then ∂VDA∗/∂λ < 0(iii)
∂VD∗/∂λ > 0Part (i) of Corollary1 shows that a higher sharing rate of App’s sales revenue leads to more revenue for the platform. Part (ii) shows that if λ ≥ 1/2, a higher sharing rate of App’s sales revenue definitely results in less revenue for the App, regardless of the effects of the other parameters on VDA∗. Thus, the revenue sharing rate should be a crucial reference for App developers to select platforms. Part (iii) shows that the revenue sharing rate has a positive effect on the optimal aggregate present values of profits of the decentralized system (VD∗). In conclusion, it is necessary to set a reasonable revenue sharing rate to maintain a mutually beneficial relationship between the platform and the App.Proposition 5.
Suppose thatδPδA − κβ > 0. Then, we have(i)
uDP∗ ≤ uIP∗ and uDA∗ ≤ uIA∗(ii)
xDSS ≤ xISS and yDSS ≤ yISS(iii)
VD∗ (≡ VDP∗ + VDA∗) ≤ VI∗, where VI∗ and VD∗ are given by (16) and (26), respectively.Proposition5 shows that the optimal advertising efforts of the platform and App and the steady-state of their user numbers in the decentralized system are always less than or equal to the corresponding ones in the integrated system. Moreover, the optimal aggregate present value of profits in the decentralized system is always less than or equal to the optimal present value of total profit in the integrated system, implying that the decentralized system is less efficient than the integrated system as expected.


## 4.1. Integrated System

Proposition 1.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that the optimal advertising efforts of the platform and App are positive.). Then, in the integrated system, the respective optimal advertising efforts of the platform and the App are (When κ = 0 and δP = δA = δ, our two (10) and (11) reduce to (11) and (12) in Wang et al. [2], respectively.)(10)uIP∗=αρ+δApP+βpAρ+δPρ+δA−κβ,(11)uIA∗=γρ+δPpA+κpPρ+δPρ+δA−κβ.
By (10), the optimal platform’s advertising effort uIP∗ increases with the effectiveness of the platform’s advertising effort (α) and the effectiveness of the App’s user base on the platform’s user growth (κ), as well as the platform’s marginal sales revenue pP. Additionally, uIP∗ increases with the App’s marginal sales revenue pA, and the effectiveness of the platform’s user base on the App’s user growth β (Apparently, ∂uIP∗/∂κ > 0. Using δPδA − κβ > 0, we have ∂uIP∗/∂β=αpAρ+δPρ+δA−κβ+κρ+δApP+βpA/ρ+δPρ+δA−κβ2>0). Similarly, by (11), the App’s optimal advertising effort uIA∗ increases with the parameters γ, β, and pA and also increases with the parameters pP and κ (Similar to the above proof of ∂uIP∗/∂β > 0, we have ∂uIA∗/∂κ > 0.). Moreover, by (10) and (11), the higher the user growth decay rate (δP and δA) and/or discount rate (ρ), the weaker the optimal advertising efforts (uIP∗ and uIA∗).Proposition 2.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that xISS and yISS are positive.). Then, in the integrated system, the following are noted:(i)
The optimal trajectories of the numbers of platform users and App users are given by (If one substitutes (11) and (12) in Wang et al. [2] into their (2) and (3) and solves the resulting simultaneous differential equation, one can find that the solution is different from (13) and (14) in Proposition 2 of Wang et al. [2]. In other words, they present incorrect expressions of the solutions. Thus, their subsequent analysis based on the solution is problematic.)(12)x∗t=Aeq1t+Beq2t+xISS,(13)y∗t=Ceq1t+Deq2t+yISS,where(14)xISS=αδAuIP∗+γκuIA∗δPδA−κβ=pPα2δAρ+δA+γ2κ2+pAα2βδA+γ2κρ+δPδPδA−κβρ+δPρ+δA−κβ,(15)yISS=αβuIP∗+γδPuIA∗δPδA−κβ=pAγ2δPρ+δP+α2β2+pPγ2κδP+α2βρ+δAδPδA−κβρ+δPρ+δA−κβ,Δ=δP−δA2+4κβ,q1=−δP+δA+Δ2,q2=−δP+δA−Δ2,A=Δ−δP−δAx0−xISS+2κy0−yISS2Δ,B=Δ+δP−δAx0−xISS−2κy0−yISS2Δ,C=Δ+δP−δAy0−yISS+2βx0−xISS2Δ,D=Δ−δP−δAy0−yISS−2βx0−xISS2Δ.(ii)
There exists a unique steady-state of the number of platform users and App users (xISS, yISS) that is a stable node(iii)
The optimal present value of the total profit, i.e., the optimal objective value is(16)VI∗x0,y0=uIP∗2+uIA∗22ρ+uIP∗αx0+uIA∗γy0.The part (ii) of Proposition2 shows that the optimal trajectories of the number of platform users and App users converge to the steady-state that is the stable node E (xISS, yISS). This can be proved by using the phase diagram analysis as follows. By δPδA − κβ > 0, we have δP/κ > β/δA > 0, implying that the slope of the demarcation curve x′t=0 is higher than the slope of the demarcation curve y′t=0. The two demarcation curves (straight lines), intersecting at point E (xISS, yISS), divide the phase space into four distinct regions, labeled I through IV as shown in Figure 2. Point E represents the intertemporal equilibrium of the integrated system. Since ∂x′/∂x=−δP<0 and ∂y′/∂y=−δA<0, we have the four groups of xy-arrowheads. Then, we can draw a family of streamlines, or trajectories, to portray the dynamics of the integrated system from any conceivable initial point. As shown in Figure 2, all the streamlines associated with point E flow noncyclically toward it. Thus, point E (xISS, yISS) is a stable node.By (14) and (15), xISS increases with the parameters α, κ (Since ∂uIP∗/∂κ > 0 and ∂uIA∗/∂κ > 0 implied by Proposition 1, by (14), and δPδA − κβ > 0, we h000ave ∂xISS/∂κ=αδA∂uIP∗/∂κ+γuIA∗+γκ∂uIA∗/∂κ/δPδA−κβ+βαδAuIP∗+γκuIA∗/δPδA−κβ2>0) and pP, while yISS increases with the parameters γ, β (Similar to the above proof of ∂xISS∗/∂κ > 0, we have ∂yISS/∂β > 0.), and pA. Furthermore, both xISS and yISS decrease in δP, δA, and ρ. (Since δPδA − κβ > 0 and both uIP∗ and uIA∗ decrease in ρ, δP, and δA, we have ∂xISS/∂δA=αδA∂uIP∗/∂δA+γκ∂uIA∗/∂δAδPδA−κβ−καβuIP∗+γδPuIA∗/δPδA−κβ2<0,∂yISS/∂δA=αβ∂uIP∗/∂δA+γδP∂uIA∗/∂δAδPδA−κβ−δPαβuIP∗+γδPuIA∗/δPδA−κβ2<0. Similarly, we can prove that xISS and yISS both decrease in δP and ρ.) By (16), VI∗ is positively related to the initial number of platform users and App users, i.e., x0 and y0. Additionally, VI∗ increases with α, γ, β, κ, pP, and pA, while decreases in ρ, δP, and δA (Since uIP∗ (uIA∗) increases in α (γ), pP, pA, β, and κ decrease in ρ, δP, and δA, we can prove VI∗ increases in α, γ, pP, pA, β, and κ, decreases in ρ, δP, and δA by (10), (11), and (16).Figure 2
The phase diagram.


## 4.2. Decentralized System

Proposition 3.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that uDP∗ and uDA∗ are positive.). Then, in the decentralized system, the optimal advertising effort of the platform is as follows:(17)uDP∗=αρ+δApP+λβ​pAρ+δPρ+δA−κβ,and the optimal App’s advertising subsidy rate set by the platform for the App is as follows:(18)φD∗=3λ−1ρ+δPpA+2κpP1+λρ+δPpA+2κpP.Furthermore, the optimal advertising effort of App is as follows:(19)uDA∗=γ1+λρ+δPpA+2κpP2ρ+δPρ+δA−κβ.Next, we perform some comparative static analyses of the results of Proposition3, leading to some useful managerial insights.By (17), the platform’s optimal advertising effort uDP∗ increases with the App’s marginal sales revenue pA and the effectiveness β of the platform’s user base on the App’s user growth. Clearly, ∂uDP∗/∂λ > 0, implying that the more the platform shares the App’s sales revenue, the more effort the platform spends on advertising. Additionally, uDP∗ increases with α, κ, and pP and decreases in ρ, δP, and δA. By (18), we have ∂ϕD∗/∂κ > 0, ∂ϕD∗/∂λ > 0, ∂ϕD∗/∂pP > 0, and ∂ϕ∗/∂pA < 0. Thus, the optimal App’s advertising subsidy rate set by the platform ϕD∗ increases with the effectiveness of the App’s user base on the platform’s user growth (κ), the sharing rate of the App’s sales revenue set by the platform (λ), and the platform’s marginal sales revenue (pP), but decreases with the App’s marginal sales revenue (pA).Remark 1.
To guarantee the required condition 0 ≤ϕD∗ ≤ 1, the inequality of 1/3 − 2κpP/[3 (ρ + δP) pA] ≤ λ ≤ 1 must hold because these two inequalities are equivalent by (18). It is assumed that the proportion λ that the platform shares from the App’s sales revenue is between 0 and 1. Thus, to be meaningful economically, we must add an assumption of 1/3 − 2κpP/[3 (ρ + δP) pA] ≥ 0 (i.e., pA ≥ (2κpP)/(ρ + δP)), implying that the platform shares at least a proportion 1/3 − 2κpP/[3 (ρ + δP) pA] of the App’s sales revenue. Under this assumption, we can get some managerial insights as follows:(i)
Clearly,λ = 1 is equivalent to ϕD∗ = 1 by (18). This implies that the platform shares all the sales revenue of the App if and only if it undertakes all the advertising costs of the App.(ii)
SinceϕD∗ = 0 is equivalent to λ = 1/3 − 2κpP/[3 (ρ + δP) pA], the platform will not carry any advertising cost of App if and only if it shares a proportion 1/3 − 2κpP/[3 (ρ + δP) pA] of the App’s sales revenue(iii)
Assume thatκ = 0. Then, 0 ≤ ϕD∗ ≤ 1 is equivalent to 1/3 ≤ λ ≤ 1 by (18). This means that when the App’s users have no impact on the platform’s user growth, the platform will share at least a 1/3 proportion of the App’s sales revenue. Additionally, with κ = 0, our decentralized system reduces Wang et al. [2], and ϕD∗ = (3λ − 1)/(1 + λ) > 0 by (18), implying that ϕD∗ only depends on λ and increases with λ. The latter result is contrary to theirs (see their (18)) and is in line with our intuition mentioned above.(iv)
Given that the proportion that the platform shares the App’s sales revenue is between the open interval of (1/3 − 2κpP/[3 (ρ + δP) pA], 1), the platform will carry a proportion ϕD∗ > 0 of App’s advertising costs.
The App’s optimal advertising effortuDA∗ increases with the parameters κ, pP, and λ that have positive effects on the platform’s revenue. The reason here is different from that of the integrated system. It is because ϕD∗ increases with the parameters κ, pP, and λ. This means that the more revenue the platform earns, the more advertising subsidies she provides to the App, which incentivizes the App to increase his advertising effort. Therefore, the revenue sharing rate will not be a burden for the App to determine his advertising effort. Additionally, uDA∗ increases with γ, β, and pA and decreases in ρ, δP, and δA.Proposition 4.
Suppose thatδPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that xDSS and yDSS are positive.). Then, in the decentralized system:(i)
The optimal trajectories of the number of platform users and App users are(20)x∗t=A¯eq1t+B¯eq2t+xDSS,(21)y∗t=C¯eq1t+D¯eq2t+yDSS,whereA¯=Δ−δP−δAx0−xDSS+2κy0−yDSS/2Δ,B¯=Δ+δP−δAx0−xDSS−2κy0−yDSS/2Δ,C¯=Δ+δP−δAy0−yDSS+2βx0−xDSS/2Δ,D¯=Δ−δP−δAy0−yDSS−2βx0−xDSS/2Δ.(22)xDSS=αδAuDP∗+γκuDA∗δPδA−κβ=2pPα2δAρ+δA+γ2κ2+pA2λα2βδA+γ2κρ+δP1+λ2δPδA−κβρ+δPρ+δA−κβ,(23)yDSS=αβuDP∗+γδPuDA∗δPδA−κβ=2pPγ2κδP+α2βρ+δA+pAγ2δPρ+δP1+λ+2λα2β22δPδA−κβρ+δPρ+δA−κβ.(ii)
There exists a unique steady-state of the number of platform users and App users (xDSS, yDSS) that is a stable node(iii)
The optimal present values of the profits of the platform and the App are given by(24)VDP∗x0,y0=uDP∗2+uDA∗22ρ+ρ+δApP+λβpAx0+κpP+ρ+δPλpAy0ρ+δPρ+δA−κβ,(25)VDA∗x0,y0=uIP∗−uDP∗uDP∗+uIA∗−uDA∗uDA∗ρ+1−λpAρ+δPρ+δA−κββx0+ρ+δPy0.Furthermore, the sum ofVDP∗x0,y0 and VDP∗x0,y0 (denoted by VD∗) is given by(26)VD∗≡VDP∗x0,y0+VDA∗x0,y0=uDP∗2uIP∗−uDP∗+uDA∗2uIA∗−uDA∗2ρ+uIP∗αx0+uIA∗γy0.Corollary 1.
Suppose thatδPδA − κβ > 0. Then, in the decentralized system:(i)
∂VDP∗/∂λ > 0(ii)
Ifλ ≥ 1/2, then ∂VDA∗/∂λ < 0(iii)
∂VD∗/∂λ > 0Part (i) of Corollary1 shows that a higher sharing rate of App’s sales revenue leads to more revenue for the platform. Part (ii) shows that if λ ≥ 1/2, a higher sharing rate of App’s sales revenue definitely results in less revenue for the App, regardless of the effects of the other parameters on VDA∗. Thus, the revenue sharing rate should be a crucial reference for App developers to select platforms. Part (iii) shows that the revenue sharing rate has a positive effect on the optimal aggregate present values of profits of the decentralized system (VD∗). In conclusion, it is necessary to set a reasonable revenue sharing rate to maintain a mutually beneficial relationship between the platform and the App.Proposition 5.
Suppose thatδPδA − κβ > 0. Then, we have(i)
uDP∗ ≤ uIP∗ and uDA∗ ≤ uIA∗(ii)
xDSS ≤ xISS and yDSS ≤ yISS(iii)
VD∗ (≡ VDP∗ + VDA∗) ≤ VI∗, where VI∗ and VD∗ are given by (16) and (26), respectively.Proposition5 shows that the optimal advertising efforts of the platform and App and the steady-state of their user numbers in the decentralized system are always less than or equal to the corresponding ones in the integrated system. Moreover, the optimal aggregate present value of profits in the decentralized system is always less than or equal to the optimal present value of total profit in the integrated system, implying that the decentralized system is less efficient than the integrated system as expected.


## 5. The Decentralized System with a Bilateral Advertising Contract

In this section, we design a bilateral advertising contract between the platform and App to improve the efficiency of the original decentralized system and expect the sum of the optimal present values of profits of the platform and the App of the decentralized system with the contract to be equal to the optimal present value of total profit of the integrated system (VI∗). In other words, the contract is designed into the original decentralized system in order to implement the optimal objective value of the integrated system.By introducing the bilateral advertising contract into the original decentralized system, we mean that the platform bears a proportionϕ of the App’s advertising cost while the App needs to bear a proportion φ of the platform’s advertising cost. We denote the bilateral advertising contract as an ordered pair (ϕ, φ) with 0 ≤ ϕ ≤ 1 and 0 ≤ φ ≤ 1. Under the contract, we have two new decentralized systems for which the optimization problems of the platform and the App, respectively, become(27)VDP∗x0,y0;φ,ϕ=maxuPt≥0VDP=∫0∞e−ρtpPxt+λpAyt−121−ϕuP2t−12φuA2tdt,s.t. (4) and (5),(28)VDA∗x0,y0;φ,ϕ=maxuAt≥0VDA=∫0∞e−ρt1−λpAyt−121−φuA2t−12ϕuP2tdt,s.t. (4) and (5).Remark 2.
ϕ in (27) and (28) is an undetermined parameter, while ϕ in (8) is a control variable.
Separately solving (27) and (28), we have the optimal advertising efforts of the platform and the App, denoted by uDP∗ (ϕ, φ) and uDA∗ (ϕ, φ), respectively. We need to determine a particular pair (ϕ∗, φ∗) to coordinate the new decentralized system with the integrated system, where by coordination, we mean that the respective optimal advertising efforts of the platform and the App in the new decentralized system are equal to their respective optimal advertising efforts in the integrated system, i.e., uDP∗ (ϕ∗, φ∗) = uIP∗ and uDA∗ (ϕ∗, φ∗) = uIA∗. With the coordination, we expect that VDP∗ (x0, y0; ϕ∗, φ∗) + VDA∗ (x0, y0; ϕ∗, φ∗) = VI∗ (x0, y0). The following three propositions present the main results.Proposition 6.
Consider the decentralized system with a bilateral advertising contract of (ϕ, φ). Suppose that δPδA − κβ > 0 (The condition δPδA − κβ > 0 guarantees that uDP∗ (ϕ, φ) and uDA∗ (ϕ, φ) are positive.). Then, the optimal advertising efforts of the platform and the App are both constants, i.e.,(29)uDP∗ϕ,φ=α1−φρ+δApP+λβpAρ+δPρ+δA−κβ,(30)uDA∗ϕ,φ=γ1−ϕ1−λρ+δPpAρ+δPρ+δA−κβ.Proposition 7.
The new decentralized system with the contract of (ϕ∗, φ∗) is coordinated with the integrated system, i.e., uDP∗(ϕ∗, φ∗) = uIP∗ and uDA∗ (ϕ∗, φ∗) = uIA∗, where uIP∗ and uIA∗ are given by (10) and (11), and (ϕ∗, φ∗) is given as(31)ϕ∗=λρ+δPpA+κpPρ+δPpA+κpP,(32)φ∗=1−λβpAρ+δApP+βpA.Next, we can gain some managerial insights from Proposition7. Clearly, by (31), we have ∂ϕ∗/∂κ > 0, ∂ϕ∗/∂λ > 0, ∂ϕ∗/∂pP > 0, and ∂ϕ∗/∂pA < 0. These results associated with the bilateral advertising contract are similar to those for the decentralized system without a bilateral advertising contract. Therefore, they have similar insights and are robust.By (32), we have ∂φ∗/∂λ < 0, ∂φ∗/∂pA > 0, and ∂φ∗/∂pP < 0. These three inequalities have some managerial insights. The optimal proportion (φ∗) of the platform’s advertising costs that the App is willing to bear decreases with the proportion (λ) of the App’s sales revenue that the platform shares, increases with the marginal sales revenue of the App (pA), and decreases with the marginal sales revenue of the platform (pP).By comparing (31) with (18), we find that the optimal proportion of the App’s advertising cost that the platform bears becomes smaller in the original decentralized system (without the contract) than in the new decentralized system with the contract because λρ+δPpA+κpP/ρ+δPpA+κpP−3λ−1ρ+δPpA+2κpP/1+λρ+δPpA+2κpP=λ−12ρ+δP2pA2/ρ+δPpA+κpP1+λρ+δPpA+2κpP≥0. This is in line with intuition. Since the contract also requires the App to bear a proportion of the platform’s advertising costs, the platform bears a higher proportion of the App’s advertising costs compared to the original decentralized system. Furthermore, when κ = 0 in (31), we have ϕ∗ = λ. Namely, when the App’s users have no impact on the platform’s user growth, the optimal proportion that the platform shares from the App’s sales revenue is exactly equal to the proportion of the App’s advertising cost that it bears. This result is partially the same as Wang et al. [2] (see their (24)).Proposition 8.
Suppose thatδPδA − κβ > 0. Then, under the unique bilateral advertising contract (ϕ∗, φ∗), the sum of VDP∗ (x0, y0; ϕ∗, φ∗) and VDA∗ (x0, y0; ϕ∗, φ∗) is equal to the optimal present value of the aggregate profit of the integrated system, VI∗ (x0, y0), i.e.,(33)VDP∗x0,y0;ϕ∗,φ∗+VDA∗x0,y0;ϕ∗,φ∗=VI∗x0,y0,where(34)VDP∗x0,y0;ϕ∗,φ∗=uDP∗uIP∗+2uDA∗−uIA∗uIA∗2ρ+ρ+δApP+λβpAx0+κpP+ρ+δPλpAy0ρ+δPρ+δA−κβ,(35)VDA∗x0,y0;ϕ∗,φ∗=uIP∗−uDP∗uIP∗+2uIA∗−uDA∗uIA∗2ρ+1−λpAρ+δPρ+δA−κββx0+ρ+δPy0,uIP∗ and uIA∗ are given by (10) and (11), uDP∗ and uDA∗ are given by (17) and (19).Propositions7 and 8 show that, as expected, the new decentralized system under the contract (ϕ∗, φ∗) not only coordinates with the integrated system (i.e., uDP∗ (ϕ∗, φ∗) = uIP∗ and uDA∗ (ϕ∗, φ∗) = uIA∗) but also implements the aggregate optimal objective value of the integrated system (see (33)).Corollary 2.
Suppose thatδPδA − κβ > 0. Then, ∂VDP∗ (x0, y0; ϕ∗, φ∗)/∂λ > 0 and ∂VDA∗ (x0, y0; ϕ∗, φ∗)/∂λ < 0.Compared to Corollary1 without the bilateral advertising contract, under the unique bilateral advertising contract (ϕ∗, φ∗), VDP∗ (ϕ∗, φ∗) for the platform still increases with the revenue sharing rate λ. However, VDA∗ (ϕ∗, φ∗) for the App always decreases in the revenue sharing rate λ.Proposition 9.
Suppose thatδPδA − κβ > 0. Then,(i)
Suppose thatλ ≤ 1 − 4α2β2/[4α2β2 + γ2 (ρ + δP)2] (This condition is applied to guarantee the nonnegativity of the right-hand side in (36).) holds. Then, the necessary and sufficient condition for VDP∗ (x0, y0, ϕ∗, φ∗) ≥ VDP∗ (x0, y0) is as follows:(36)uIA∗−uDA∗2≤uIP∗−uDP∗uDP∗orpPpA≥1−λγ2ρ+δP2−4λα2β24α2βρ+δA.(ii)
Suppose thatλ ≤ 1 − 2α2β2/[4α2β2 + γ2 (ρ + δP)2](This condition is applied to guarantee the nonnegativity of the right-hand side in (37).) holds. Then, the necessary and sufficient condition for VDA∗ (x0, y0, ϕ∗, φ∗) ≥ VDA∗ (x0, y0) is as follows:(37)uIA∗−uDA∗2≥uIP∗−uDP∗uDP∗−uIP∗2orpPpA≤1−λγ2ρ+δP2−4λ−2α2β22α2βρ+δA.(iii)
Suppose thatλ ≤ 1 − 4α2β2/[4α2β2 + γ2 (ρ + δP)2] holds. Then, the necessary and sufficient condition for VDP∗ (x0, y0; ϕ∗, φ∗) ≥ VDP∗ (x0, y0) and VDA∗ (x0, y0; ϕ∗, φ∗) ≤ VDA∗ (x0, y0) is as follows:(38)pPpA≥1−λγ2ρ+δP2−4λ−2α2β22α2βρ+δA.(iv)
Suppose thatλ ≤ 1 − 4α2β2/[4α2β2 + γ2 (ρ + δP)2]. Then, the necessary and sufficient condition for VDP∗ (x0, y0; ϕ∗, φ∗) ≤ VDP∗ (x0, y0) and VDA∗ (x0, y0; ϕ∗, φ∗) ≥ VDA∗ (x0, y0) is as follows:(39)pPpA≤1−λγ2ρ+δP2−4λα2β24α2βρ+δA.(v)
Suppose thatλ ≤ 1 − 4α2β2/[4α2β2 + γ2 (ρ + δP)2]. Then, the necessary and sufficient condition for VDP∗ (x0, y0; ϕ∗, φ∗) ≥ VDP∗ (x0, y0) and VDA∗ (x0, y0; ϕ∗, φ∗) ≥ VDA∗ (x0, y0) is as follows:(40)1−λγ2ρ+δP2−4λα2β24α2βρ+δA≤pPpA≤1−λγ2ρ+δP2−4λ−2α2β22α2βρ+δA.(vi)
VDP∗ (x0, y0; ϕ∗, φ∗) ≤ VDP∗ (x0, y0) and VDA∗ (x0, y0; ϕ∗, φ∗) ≤ VDA∗ (x0, y0) do not hold simultaneouslyThe first two parts of Proposition9 identify the respective conditions under which the platform and the App are individually better off with the contract (ϕ∗, φ∗) than without the contract. Parts (iii) and (iv) show that it was possible that, under some conditions, one is better off and the other is worse off. Part (v) shows how to identify the conditions under which both parties are better off—it is a win-win situation all around. Finally, part (vi) shows that it was impossible that both parties are worse off with the contract than without it. In other words, at least one of the two parties will be better off with the contract.The conditions (38)∼(40) in Proposition 9 are mutually exclusive. It is of significance to check whether these conditions are satisfied. Next, we use some numerical examples to verify that each of the above conditions can be satisfied by taking different ranges of the revenue sharing rate λ and fixing the other parameters. Thus, Proposition 9 is valid.Example 1.
The condition (38) can be satisfied. Let α = γ = 5, pP = 20, pA = 15, β = 0.2, κ = 0.1, ρ = 0.1, δP = 0.3, δA = 0.2, x0 = 50, and y0 = 30. Solving (38) for λ yields(41)λ≥2α2β2+γ2ρ+δP2pA−2α2βρ+δApP4α2β2+γ2ρ+δP2pA.
Substituting all the given parameters into the above equation, we haveλ ≥ 0.25. Thus, the condition (38) is satisfied for the given parameters when λ ≥ 0.25.Example 2.
The condition (39) can be satisfied. Let α = 5, γ = 10, pP = 20, pA = 15, β = 0.2, κ = 0.1, ρ = 0.1, δP = 0.3, δA = 0.2, x0 = 50, and y0 = 30. Solving (39) for λ yields(42)λ≤γ2ρ+δP2pA−4α2βρ+δApP4α2β2+γ2ρ+δP2pA.
Substituting all the given parameters into the above equation, we haveλ ≤ 0.4. Thus, the condition (39) is satisfied for the given parameters when λ ≤ 0.4.Example 3.
The condition (40) can be satisfied. Let α = 5, γ = 10, pP = 20, pA = 15, β = 0.2, κ = 0.1, ρ = 0.1, δP = 0.3, δA = 0.2, x0 = 50, and y0 = 30. Solving (40) for λ yields(43)γ2ρ+δP2pA−4α2βρ+δApP4α2β2+γ2ρ+δP2pA≤λ≤2α2β2+γ2ρ+δP2pA−2α2βρ+δApP4α2β2+γ2ρ+δP2pA.
Substituting all the given parameters into the above equation, we have 0.4 ≤λ ≤ 0.7. Thus, the condition (40) is satisfied for the given parameters when 0.4 ≤ λ ≤ 0.7.


## 6. Conclusion

This paper reanalyzes the work of Wang et al. [2] by improving some of their assumptions. In particular, we assume that (i) the number of the App’s users may increase the number of the platform’s users; (ii) a loss of existing users of the platform (App) negatively influences its users’ growth differently or equally; and (iii) the platform owner cannot overcharge the app developer and take the advertising fee as a source of revenue, but charge the App’s revenue as a source of revenue. Our models and analytical results are different from or are extensions of the work of Wang et al. [2].We reanalyze the two revised systems (the decentralized system and the integrated system with the platform and App) and design the bilateral advertising contract to coordinate the decentralized system with the integrated system.First, for the two respective revised systems, we have obtained the two parties’ equilibrium advertising efforts and the optimal trajectories of the numbers of users, and we have shown that there exist the unique steady-states of user numbers for the two parties that are shown as the stable nodes by using the phase diagram analysis. In particular, for the decentralized system, we have found the optimal advertising subsidy rate that is set by the platform for the App.Second, to achieve the efficiency of the integrated system that is proved to be more efficient than the decentralized system, we design a bilateral advertising contact for the decentralized system where we additionally assume that the App bears a proportion of the platform’s advertising cost. We show that there exists a unique bilateral advertising contact that can coordinate the decentralized system with the integrated system and simultaneously implement the optimal aggregate objective value of the integrated system.Finally, our analytical results may have some interesting theoretical findings and managerial insights for practice as follows .(i)
In the decentralized system, the more the platform shares the App’s sales revenue or the more the platform’s marginal sales revenue, the more the platform should undertake the App’s advertising cost. However, the more the App’s marginal sales revenue, the less the platform should undertake the App’s advertising cost. These results are irrespective of whether the bilateral advertising contact is incorporated into the decentralized system or not and thus are robust.(ii)
To guarantee that there exists the optimal advertising subsidy rate in the decentralized system without the contract, the proportion of the App’s sales revenue that the platform shares is at least 1/3(iii)
The platform shares all the sales revenue of the App if and only if it undertakes all the advertising costs of the App in the decentralized system without the contract(iv)
When the App’s users have no impact on the platform’s user growth, as assumed in the study by Wang et al. [2], the platform will share at least a 1/3 proportion of the App’s sales revenue in the decentralized system without the contract in order to guarantee the existence of the optimal advertising subsidy rate, and the optimal advertising subsidy rate is different from the study by Wang et al. [2].(vi)
The optimal advertising subsidy rate becomes smaller in the original decentralized system (without the contract) than in the decentralized system with the contract(vii)
The optimal present value of the platform always increases with the revenue sharing rate in both the original decentralized system (without the contract) and the decentralized system with the contract. As the revenue sharing rate increases, the optimal present value of the App decreases in the original decentralized system if the revenue sharing rate is greater than 1/2, and always decreases in the decentralized system with the contract.(viii)
The unique bilateral advertising contact guarantees that at least one of the platforms and the App can benefit from it, compared to the original decentralized system. Furthermore, when the ratio of the platform’s marginal sales revenue to the App’s marginal sales revenue is within a particular interval, then both the platform and the App can get more benefits from the bilateral advertising contract than from the original decentralized system, achieving a win-win situation.Our analytical results, including some managerial insights, come from the assumptions that the costs of advertising efforts of the platform and App take quadratic functions and/or there is only one App. What if we assume the other forms of functions, such as cubic functions and several Apps. Can we design a different bilateral participation contract between the platform and the App to coordinate the decentralized system? For example, by the different contract, we mean that the platform shares a proportion of the App’s sales revenue while the App shares a proportion of the platform’s sales revenue. Can we compare these two contracts? [36].



---

*Source: 6468725-2022-10-10.xml*