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Search Engine Advertising and Contractual Strategies between

a Parent Firm and its Online Search Infomediary: A Game-

Theoretic Model

Siddharth Bhattacharya, Subodha Kumar, Sunil Wattal

Abstract

The widespread use of sponsored search advertising has led to firms using various kinds of

strategies such as referrals, poaching on competitors’ keywords, etc., to increase their

advertising effectiveness. One such trend that is growing rapidly in many sectors is the use of

online search infomediaries (to bring in additional demand) who advertise on the firm’s keyword

itself. However, in this unique context, what kind of contractual (pricing and advertising)

strategies work remains an open question. Using a game-theoretic model, we investigate what

pricing and advertising strategies between firms and their online search infomediaries maximize

profits under equilibrium, and how do traffic quality coming to either party, competition, and

current type, affect these equilibrium strategies. The results of the research will serve as

guidelines for firms to understand what decisions are optimal in various contexts and negotiate

their contracts accordingly.

Keywords: Game theory, affiliate marketing, contracting, advertising, traffic quality.

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1 Introduction

Over the years, there has been an emergence in the number of online infomediaries whose basic

objective is to offer consumers the opportunity to get price quotes, invoice prices as well as

product reviews and specifications from enrolled retailers. While on one hand they increase

consumer’s awareness, on the other hand, they provide sellers with information regarding

demand estimates/production requirements etc.(as shown in the Figure below). They help reduce

the search costs for consumers who can make more informed purchase decisions (Bakos 1997,

Brynjolfsson and Smith 2000).

Online Infomediary

These online infomediaries have moved from being mere information agents to becoming

contracted partners to focal firms that provide referral service to the focal firm’s website (an area

that has garnered increasing interest among researchers: (Chiou and Tucker 2012, Duan and

Zhang 2014, Ho et al. 2017). The focal firm utilizes these online infomediaries to generate

additional traffic by advertising the focal firm’s product. This increases the parent firm’s profit

which in turn pays the online infomediary (through fixed fee, commission etc) for its effort.

More recently, this phenomenon is being increasingly used to advertise on the search engine.

Here a focal firm utilizes an online infomediary to advertise on its keyword. For instance, we

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find that, in the auto industry, car dealers like “Audi” are utilizing search infomediaries like

“edmunds.com” or “cars.com” to generate additional traffic by advertising on Audi’s keywords

(as shown in Figure below). Similarly, big hotel chains like “Hyatt” often use search

infomediaries like “hotels.com” to do the same. In the search engine advertising context these

online infomediaries are often referred to as “Online Search Infomediaries” or OSIs. The focal

firm (referred to as “parent firm” hereafter) has the incentive to utilize these OSIs because it not

only gains from the additional traffic that the OSI generates (as the OSI advertises as well) but it

also benefits from the increased awareness of consumers which in turn leads to increase in

product sales. This is different from the context of online retailers(like Amazon.com, Flipkart

etc.) who are similarly employing third-party sellers to sell identical products on their platforms

and then paying them for their service. In the context of search engine(our focus area), on one

hand, parent firm and the OSIs compete directly with each other on search engine platforms; on

the other hand, it can coordinate its advertising strategies with OSI so as to mutually benefit at

the expense of other competitors. Moreover, there are multiple other factors like position at

which the ad is shown which also drives the actual demand, making the context unique.

An Example of OSI Marketing in Car Industry

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Once a consumer clicks on the OSI’s listing on the search engine he is redirected to the

OSI’s website(as shown in the following Figure the consumer clicks on cargurus.com which is

one of the many leading infomediaries in the car space). The OSI can decide whether to redirect

the consumer to the parent firm (by showing the listing of the focal firm as in the left of the

figure) or to a competing firm(by showing the listing of competing firm as shown in the right of

the figure).This makes it even more important for the parent firm to design its pricing and

advertising strategies to incentivize the OSI so that it behaves in the best interest of the parent.

Strategic behavior of OSI on consumer’s click

1.1 Research Questions and Contributions

The focus of our research is at the interplay of these factors and how that dictates firms’

strategies. We have two main forces that act simultaneously in determining the demand, which is

generated by the parent firm and OSI combined: first, advertising strategy between the firm and

the OSI, second, pricing strategy between the firm and the OSI. This leads us to ask the first

research question.

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(1) What pricing and advertising strategies should be followed between firm and OSI to

maximize profits?

Traditionally, firms have fixed contracts where they pay the OSI fixed payments per

month irrespective of the number of leads generated and lead quality. However, there is ample

scope for both the parent firm and the OSI to devise their strategies according to the quality of

leads coming to their websites and negotiate their strategies accordingly. This leads to our

second research question.

(2) How do the equilibrium pricing and advertising strategies change with the quality of traffic

coming to either party?

One may be tempted to think that as advertising effectiveness and quality of traffic

improve, so would the profit of the firm. However, in this research, we show that this might not

always be the case. In fact, the equilibrium profit depends on a combination of multiple factors:

the advertising level, the advertising expenditure done by the firm for both its own advertising

and for the OSI, the quality of traffic coming to each firm and its effect on demand etc. Some

factors dominate in certain situations over the other factors thereby dictating how the profit

reacts. For instance, our results show that when the ratio of advertising effectiveness is less than

a threshold, then the profit of the parent firm increases with the traffic quality as long as traffic

quality is greater than a threshold, while profit decreases with traffic quality as long as traffic

quality is below a threshold. This is especially important because firms can monitor the type of

traffic that the OSI is bringing in(for instance if edmunds.com is bringing in high quality leads

like Mercedes versus Kia then the parent firm might be willing to negotiate its contract with the

OSI). Thus, the results of the research will serve as guidelines for firms to understand what are

the equilibrium advertising and pricing decisions to maximize profit under various contexts.

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This research will also help find how these strategies would change depending on

whether the market is fixed or expands with advertising, competition from other firms and

customer heterogeneity (for instance, what stage of the funnel the user is in, etc.). Moreover, the

research will shed light on optimal budget allocation decisions for both firms and their OSIs.

Theoretically, this would be the first research (to our knowledge) to exhaustively look at pricing

and advertising strategies between the firm and its OSI in the online search advertising context

(Chiou and Tucker 2012, Chun et al. 2017).

1.2 Literature Review

Our study is closely related to three streams of literature: 1) affiliate marketing, 2) supply chain,

and 3) infomediaries. We discuss each of these briefly highlighting our contribution to each of

them.

Affiliate marketing has been growing in recent years with most big players in e-

commerce utilizing third-party sellers to generate additional demand. However, only a handful of

researchers have started investigating this phenomenon recently. Chiou and Tucker (2012) show

how the use of trademarks by OSI allows parent firm to decrease clicks on their paid search but

this gets outweighed by the increase in clicks on unpaid links. Ho et al. (2017) show how

cashback mechanisms may not be always useful for price discrimination and firms should try and

contract with multiple affiliates. Duan and Zhang (2014) show how search engine, social media,

and third-party referrals have differential relationship with sale measures. They show that while

search engine referrals explain most of the variance in sales, social media referrals explain most

of the variance in conversion. Though these researchers delve into diverse aspects of this

phenomenon, one of the important areas that remain unexplored is what contractual (pricing and

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advertising) strategies should be followed between the two parties and this is the main focus of

this paper. This is especially important because depending on the quality of traffic coming to

either party, they could negotiate to maximize their respective profits which would affect all

strategies under equilibrium conditions.

The supply chain literature has delved into the area of contract design. Different models

have been developed in this literature. They include: quantity flexibility (QF) contracts (Tsay

1999), backup agreements (Eppen and Iyer 1997), buy back or return policies (Emmons and

Gilbert 1998), incentive mechanisms (Lee and Whang 1999), revenue sharing (RS) contracts

(Cachon and Fisher 2000), allocation rules (Cachon and Lariviere 1999), and the quantity

discounts (Weng 1995). Most of these models address the problem of coordination in supply

chains and have not paid much attention to the lead generation mechanism which can affect the

bargaining power between the involved parties. Moreover, none of these are based in the online

search advertising context which is unique in terms of how ads are listed on the search engine

and the competitive landscape.

Finally, literature about online third-party intermediaries has shown that these reduce

search costs for consumers thus aiding them both in their information search and purchase

decisions (Bakos 1997, Brynjolfsson and Smith 2000). Ghose et al. (2007) show that digitally

enabled referral services can act as strategic tools for manufacturers to increase their channel

power and profits by diverting traffic from offline to online channels. Viswanathan et al. (2007)

show search infomediaries can act as a viable mechanism for market segmentation and price

discrimination. Similarly, other researchers (Bhargava and Choudhary 2004, Waldfogel and

Chen 2006) have explored other dimensions like increase in branded versus non-branded

searches in presence of infomediaries, or result of network effects on infomediaries strategies etc.

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Though these researchers have looked at diverse aspects of infomediaries, none of these have

focused on what should be the equilibrium pricing and advertising strategies between the two

parties, especially in the online search advertising context which is the focus of our research.

2 Model

We construct a game-theoretic model with two players: the firm and the OSI. The objective of

the game is for each player to maximize its own profit while optimizing their advertising and

pricing strategies respectively. Each player advertises on the search engine simultaneously

making it a simultaneous move game. We define the important components of the model next.

2.1 Demand

First, we look at the demand for both the parent and the OSI . The demand coming to each firm

is a function of two things: advertising level and advertising effectiveness of each firm (based on

the classic Lanchester model adopted by Case 1979). In this model, the market share of each firm

is affected by two factors: (i) positive direct response to its own advertising (given by the product

of firm’s advertising level and its advertising effectiveness, i.e., 𝐴 ∗ 𝑚1), and (ii) the loss of

existing customers due to advertising from firm 2 (given by the product of the second firm’s

advertising level and its advertising effectiveness, i.e., 𝐵 ∗ 𝑛1). For instance, for the parent firm,

demand is positively affected by the advertising done by itself but negatively affected by the

advertising done by the OSI . Similarly, for the OSI , its vice-versa (for detailed description of all

parameters, please refer to Table 1).

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Table 1: Parameter Descriptions

Parameter Description

r1 Quality of traffic coming to parent firm

r2 Quality of traffic coming to OSI

m1 Advertising effectiveness of parent firm for parent’s demand

n1 Advertising effectiveness of OSI for parent’s demand

m2 Advertising effectiveness of parent firm for OSI’s demand

n2 Advertising effectiveness of OSI for OSI ’s demand

A Advertising level of parent firm (Decision Variable)

B Advertising level of OSI (Decision Variable)

𝛼 Increment in parent firm’s advertising cost as OSI advertises

𝛽 Increment in OSI’s advertising cost as parent advertises

𝛿 Parent firm pays OSI proportional to the demand brought in by OSI

P Per unit payment made to OSI for demand it brings in (Decision Variable)

Thus, we can formulate the demand functions for each firm as follows:

Parent firm demand D1: 𝐴 ∗ 𝑚1 − 𝐵 ∗ 𝑛1

OSI demand D2: 𝐵 ∗ 𝑛2 − 𝐴 ∗ 𝑚2,

where A and B are the advertising level of parent firm and OSI, respectively; m1 and n1 are the

advertising effectiveness of parent firm and OSI, respectively, in the parent’s demand; and m2

and n2 are the advertising effectiveness of parent and OSI, respectively, in the OSI’s demand.

Knowing the demand, we can calculate the revenue received by the firm by multiplying

the demand by the quality of traffic coming to the firm. If r1 is the quality of traffic coming to

parent firm, then the revenue generated is given by:

Revenue generated by parent firm: 𝑟1 ∗ D1

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The parent firm pays the OSI proportional to the demand that the OSI brings in for the parent.

Thus, the additional revenue coming to parent due to OSI ’s advertising is given by

Revenue generated due to OSI ’s advertising: 𝛿 ∗ 𝑟2 ∗ D2,

where δ varies between 0 to 1 depending on the proportion of demand that OSI brings in for

parent.

2.2 Cost

The cost for each firm is a function of the advertising done by the firm. However, in the online

advertising context, the cost of each firm rises due to other player’s advertising. Further, it is

well established in the literature that cost is a nonlinear convex function of advertising (Yu et al.

2009). Thus, the cost function for the parent firm is given by:

Cost of parent C1: (𝐴2 + 𝛼 ∗ 𝐵2).

Similarly, the cost function for the OSI is given by:

Cost of OSI C2: (𝐴2 ∗ 𝛽 + 𝐵2),

where 𝛼 and 𝛽 capture how advertising of the parent and OSI s are dependent across both cost

functions, respectively.

2.3 Payment

As discussed earlier, the payment made to the OSI is proportional to the demand that the OSI is

bringing in for the parent. We formulate this as follows:

𝑝𝑎𝑦𝑚𝑒𝑛𝑡: 𝑝 ∗ 𝛿 ∗ r2 ∗ D2,

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where p is the per unit payment made to the OSI for the demand that it brings in.

2.4 Profit

Thus, having the revenue, cost, and payment, we now formulate the objective functions (profit)

of the parent and the OSI as follows. The objective function for parent firm is

𝑟1 ∗ D1 + 𝛿 ∗ 𝑟2 ∗ D2 − C1 − 𝑝𝑎𝑦𝑚𝑒𝑛𝑡.

The objective function for OSI is

𝑝𝑎𝑦𝑚𝑒𝑛𝑡 − C2.

Using the objective functions for both firms, we now solve the game to find the

equilibrium profit, advertising, and price. In the base model, for simplicity, we normalize the

advertising effectiveness of a firm on the demand of both firms to be the same, i.e., 𝑚1 = 𝑚2 =

𝑚 , and 𝑛1 = 𝑛2 = 𝑛 . This simplification still captures the reality well. However, for

completeness, later we discuss a more general case in Section 4. The solutions are given in the

following subsection. The proofs are given in the Appendix.

2.5 Results

Lemma 1:

a) The equilibrium profits of the parent and the OSI are given by:

Profit of parent 𝑄1∗ = −

𝒏𝟐(𝒏𝟐+𝒎𝟐𝜶)(𝒓𝟏−𝒓𝟐𝜹)𝟐

𝟒(𝒎𝟐−𝒏𝟐(𝟐+𝜶))

Profit of OSI 𝑄2∗ = −

𝑛2(−𝑛4+𝑚4(1+2𝛼)+𝑚2𝑛2(−2𝛼+(1+𝛼)2𝛽))(𝑟1−𝑟2𝛿)2

4(𝑚2−𝑛2(2+𝛼))2

b) The equilibrium advertising of the parent and the OSI s are given by:

Advertising of parent 𝐴∗ =1

2𝑚(𝑟1 + 𝛿(−𝑟2 +

−𝑚2𝑟1+𝑛2𝑟1+𝑚2𝑟2𝛿−𝑛2𝑟2𝛿

(𝑚2−2𝑛2−𝑛2𝛼)𝛿))

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Advertising of OSI 𝐵∗ =𝑛(−𝑚2𝑟1+𝑛2𝑟1+𝑚2𝑟2𝛿−𝑛2𝑟2𝛿)

2(𝑚2−2𝑛2−𝑛2𝛼)

c) The optimal price to be paid to OSI is given by:

𝑝∗ =−𝑚2𝑟1+𝑛2𝑟1+𝑚2𝑟2𝛿−𝑛2𝑟2𝛿

(𝑚2−2𝑛2−𝑛2𝛼)𝛿

Having found the optimal profit, advertising, and price, we next do sensitivity analysis

with respect to different parameters to get insight into how each would react with

increase/decrease in certain parameters. This is discussed in the next section.

3 Managerial Insights

3.1 Traffic Quality

In this section, we investigate how the optimal pricing and advertising strategies change with the

quality of traffic coming to each firm. Usually, firms have fixed contracts where they pay the

OSIs fixed payments per month irrespective of the number of leads generated and lead quality.

However, there is ample scope for both the parent and the OSI to devise their strategies

according to quality of leads coming to their websites and negotiate their contracts accordingly.

3.1.1 Traffic Quality of the Parent

The impact of the traffic quality of parent depends upon the level of relative effect of advertising

(i.e., 𝒎/𝒏). In the following proposition, we first consider a scenario where the relative effect of

advertising is low.

Proposition 1: When the relative effect of advertising is lower than a threshold (i.e., 𝒎/𝒏 < 𝟏),

with an increase in the quality of parent firm traffic (i.e., 𝒓𝟏 ), parent advertising always

increases, OSI advertising as well as the payment made to OSI always decreases, and the profit

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of the parent increases iff the traffic quality of parent firm is greater than a threshold (i.e., 𝒓𝟏 >

𝒓𝟐 ∗ 𝜹).

The results in Proposition 1 are illustrated in Figures 1-4. As the parent traffic quality

(i.e., r1) increases, there is a direct positive effect on the demand it gets. When the related effect

of advertising effectiveness is low, it could also be interpreted as the low level of the advertising

effectiveness of parent firm (for a given level of the advertising effectiveness of the OSI ).

Hence, in this scenario, the base advertising level of the parent firm is also low. Therefore, as

shown in Figure 1, with an increase in r1, the parent firm increases its advertising significantly to

benefit further from the improved traffic quality. Since both the advertising level and the quality

of its traffic increases for the parent firm (where the advertising increases significantly), it relies

less on the OSI by paying it less (as shown in Figure 3). As a result, the OSI advertises less with

an increase in r1 (as shown in Figure 2).

Figures 1-4: Illustration of the Results in Proposition 1 (when 𝒎/𝒏 < 𝟏)

r1→ r1 →

Figure 1: Parent advertising w r t r1 Figure 2: Affiliate advertising w r t r1

r1→ 𝛿 ∗ 𝑟2 r1 →

Figure 3: Payment w r t r1 Figure 4: Optimal Profit w r t r1

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In this case, there are three direct impacts of increase in r1 on the profit of the parent

firm: One is the positive effect of improved traffic quality on its demand, the second is the

negative effect of increased advertising expenditure, and the third is the positive effect of

reduced payment to the OSI . In addition to these, the demands from both the parent firm (i.e.,

𝐷1) and the OSI (i.e., 𝐷2) also change. As shown in Figure 4, when the quality of traffic is low

(i.e., 𝑟1 < 𝑟2 ∗ 𝛿), the negative effects dominate and hence the optimal profit decreases with an

increase in 𝑟1. However, when the quality of traffic is high (i.e., 𝑟1 > 𝑟2 ∗ 𝛿), the positive

effects dominate, and therefore the optimal profit starts increasing with 𝑟1. In the following

proposition, we now focus on a scenario where the relative effect of advertising (i.e., 𝑚/𝑛) is at

a moderate level.

Proposition 2: When the relative effect of advertising is moderate (i.e., 𝟏 ≤ 𝒎/𝒏 ≤ √𝟐 + 𝜶),

with an increase in the quality of parent firm traffic (i.e., 𝒓𝟏 ), parent advertising, OSI

advertising, and the payment made to OSI always increase, whereas the profit of the parent

increases iff the traffic quality of parent firm is greater than a threshold (i.e., 𝒓𝟏 > 𝒓𝟐 ∗ 𝜹).

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The results in Proposition 2 are illustrated in Figures 5-8. Similar to that in Proposition 1,

as the parent traffic quality (i.e., r1) increases, there is a direct positive effect on the demand it

gets. Therefore, as shown in Figure 5, with an increase in r1, the parent firm increases its

advertising to benefit further from the improved traffic quality. However, in this scenario, since

the relative effect of advertising is moderate, the base advertising level of the parent firm is not

very low, and therefore the advertising does not increase as significantly as that in Proposition 1.

Since the advertising cost doesn’t increase significantly for the parent firm with an increase in its

traffic quality, it is able to the pay a portion of extra revenue to the OSI . Therefore, as shown in

Figure 7, the payment increases with r1. As a result, the advertising of OSI also increases with

r1 (see Figure 6). Finally, similar to that in Proposition 1, profit of the parent increases with r1 if

and only if r1 is greater than a threshold (see Figure 8). The next proposition now considers a

scenario where the relative effect of advertising (i.e., 𝑚/𝑛) is high.

Figures 5-8 : When ratio of Advertising effectiveness is in the range 1<m/n <

√𝟐 + 𝜶 (corresponds to part b of PROPOSITION 1)

r1 → r1 →

Figure 5: Parent advertising w r t r1 Figure 6: Affiliate advertising w r t r1

r1→ 𝛿 ∗ 𝑟2 r1 →

Figure 7: Payment w r t r1 Figure 8: Optimal Profit w r t r1

Figures 5-8: Illustration of the Results in Proposition 2 (when 𝟏 ≤ 𝒎/𝒏 ≤ √𝟐 + 𝜶)

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Proposition 3: When the relative effect of advertising is high (i.e., 𝑚/𝑛 > √2 + 𝛼), with an

increase in the quality of parent firm traffic (i.e., 𝑟1), parent advertising, OSI advertising, and

the payment made to OSI always decrease, whereas the profit of the parent decreases iff the

traffic quality of parent firm is greater than a threshold (i.e., 𝑟1 > 𝑟2 ∗ 𝛿).

The results in Proposition 3 are illustrated in Figures 9-12. Unlike the earlier two

propositions, in this scenario, the relative effect of advertising is high, and therefore the base

advertising level of the parent firm is already at a high level. Therefore, as the parent traffic

quality (i.e., r1) increases, it reduces its advertising to maintain a similar demand level (see

Figure 9). In addition, it is able to reduce the payment to the OSI (see Figure 11). As a result, the

advertising of OSI also decreases with r1 (see Figure 10). The profit curve presented in Figure

12 is reverse of those in those in Figures 4 and 8, because, in this scenario, when the quality of

traffic is low (i.e., 𝑟1 < 𝑟2 ∗ 𝛿), the positive effects dominate and when the quality of traffic is

high (i.e., 𝑟1 > 𝑟2 ∗ 𝛿), the negative effects dominate.

Figures 9-12 : When ratio of Advertising effectiveness is high m/n >

√𝟐 + 𝜶 (corresponds to part c of PROPOSITION 1)

r1 → r1 →

Figure 9: Parent advertising w r t r1 Figure 10: Affiliate advertising w r t r1

r1 → 𝛿 ∗ 𝑟2 r1 →

Figure 11: Payment w r t r1 Figure 12: Optimal Profit w r t r1

Figures 9-12: Illustration of the Results in Proposition 3 (when 𝒎/𝒏 > √𝟐 + 𝜶)

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3.1.2 Traffic Quality of the OSI

As expected, with respect to the OSI traffic quality (i.e., 𝑟2), the equilibrium values of profit,

advertising, and payment vary exactly in the opposite manner as those with respect to 𝑟1. Hence,

we don’t discuss these results in detail.

3.2 Advertising Effectiveness

Advertising effectiveness is the another vital component driving demand (apart from traffic

quality). Even with the high-quality traffic, if the advertising effectiveness is not good, any

advertising strategy may not be effective. Thus, it’s crucial to see how the equilibrium values of

pricing, advertising, and profit change with advertising effectiveness.

Proposition 4: With an increase in the relative effect of advertising (i.e., 𝑚/𝑛 ),

a) When the traffic quality of parent firm is less than a threshold (i.e., 𝑟1 < 𝑟2 ∗ 𝛿),

parent advertising, OSI advertising, and the payment made to OSI decrease, whereas

the profit of the parent firm increases.

b) When the traffic quality of parent firm is greater than a threshold (i.e., 𝑟1 ≥ 𝑟2 ∗ 𝛿),

parent advertising, OSI advertising, the payment made to OSI , and the profit of the

parent firm all increase.

Let us first discuss part (a) of the proposition. When the traffic quality of the parent firm

is low, as its advertising effectiveness increases, it doesn’t gain much by advertising more.

Hence, in order to reduce cost, it reduces advertising as 𝑚/𝑛 increases. In response, the OSI

also reduces its advertising and therefore the payment to OSI also reduces. Because of the

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increase in advertising effectiveness and reduction in costs, the profit of the parent firm increases

with 𝑚/𝑛.

On the other hand, in part (b) of the proposition, since the traffic quality of the parent

firm is high, it gains by advertising more as its advertising effectiveness increases. In response,

the OSI also increases its advertising and therefore the payment to OSI also increases. In this

case, the benefit due to the increase in advertising effectiveness dominates the increase in costs,

and therefore the profit of the parent firm increases with 𝑚/𝑛.

4 Enhancements to fixed market model

4.1 Competition

In our base model, we have considered two players: the parent firm and the OSI . However, on

the search engine, it is realistic to assume the presence of a few more sponsored ads of other

parties that can have an indirect effect on the demand and cost of the parent firm and its OSI .

We capture this by changing our demand and cost functions in this section. In both the demand

functions of parent and OSI we introduce an additional term O capturing the competitors

advertising level and t which capturing the competitor’s advertising effectiveness. Thus our new

demand functions are given below.

Parent firm demand D1: A ∗ m − B ∗ n − O ∗ t

OSI demand D2: −A ∗ m + B ∗ n − O ∗ t

Revenue generated by parent firm: 𝑟1 ∗ D1

Revenue generated due to OSI ’s advertising: 𝛿 ∗ 𝑟2 ∗ D2

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where r1 is the traffic quality of parent and r2 is the traffic quality of OSI respectively as earlier.

The cost function also changes in presence of competition from a third player. We capture this

by introducing a term O2 ∗ γ since cost is a quadratic function of the advertising level of the

competing firm(similar to the cost function defined earlier). Our cost function is thus given by

Cost of parent C1: (𝐴2 + 𝛼 ∗ 𝐵2 + +O2 ∗ γ))

Cost of OSI C2: (𝐴2 ∗ 𝛽 + 𝐵2 + O2 ∗ γ)

𝑝𝑎𝑦𝑚𝑒𝑛𝑡: 𝑝 ∗ 𝛿 ∗ r2 ∗ D2,

Thus, having the revenue, cost, and payment, we now formulate the objective functions (profit)

of the parent and the OSI as follows. The objective function for parent firm is

𝑟1 ∗ D1 + 𝛿 ∗ 𝑟2 ∗ D2 − C1 − 𝑝𝑎𝑦𝑚𝑒𝑛𝑡.

The objective function for OSI is

𝑝𝑎𝑦𝑚𝑒𝑛𝑡 − C2.

Using the objective functions for both firms, we now solve the game to find the equilibrium

profit, advertising, and price(in a similar manner as in the fixed market case). The solutions are

given in the following subsection. Here the subscript for the optimal terms are named “c”

denoting that these solutions are for the competition case.

4.2 Results

Lemma 3:

a) The equilibrium profits of the parent and the OSI are given by:

Profit of parent 𝑄1𝑐

∗

=1

4(𝑚2 − 𝑛2(2 + 𝛼))(−4𝑂(𝑂𝑡2 + 𝑚2(2r1𝑡 + 𝑂𝛾)) − 𝑛4(r1 − r2𝛿)2 + 𝑛2(4𝑂2(2 + 𝛼)𝛾

− 𝑚2𝛼(r1 − r2𝛿)2 + 4𝑂𝑡(r1(3 + 𝛼) + r2(1 + 𝛼)𝛿)))

Profit of OSI 𝑄2𝑐

∗

20

= 1

4(−4𝑂2𝛾 −

1

(𝑚2 − 𝑛2(2 + 𝛼))2𝑚2(2𝑂𝑡 + 𝑛2(1 + 𝛼)(r1 − r2𝛿))(2𝑂𝑡(2 + 𝛽) + 2𝑚2(r1 − r2𝛿)

+ 𝑛2(−2 + 𝛽 + 𝛼𝛽)(r1 − r2𝛿)) +1

(𝑚2 − 𝑛2(2 + 𝛼))2(2𝑂𝑡 + (𝑚 − 𝑛)(𝑚 + 𝑛)(r1

− r2𝛿))(−2𝑛2𝑂𝑡(3 + 2𝛼) + 𝑛4(−r1 + r2𝛿) + 𝑚2(4𝑂𝑡 + 𝑛2(r1 − r2𝛿))))

b) The equilibrium advertising of the parent and the OSI s are given by:

Advertising of parent 𝐴𝑐∗ =

1

2𝑚(r1 + 𝛿(−r2 +

−𝑚2r1+𝑛2r1−2𝑂𝑡+𝑚2r2𝛿−𝑛2r2𝛿

(𝑚2−2𝑛2−𝑛2𝛼)𝛿))

Advertising of OSI 𝐵𝑐∗ =

𝑛(−𝑚2r1+𝑛2r1−2𝑂𝑡+𝑚2r2𝛿−𝑛2r2𝛿)

2(𝑚2−2𝑛2−𝑛2𝛼)

c) The optimal price to be paid to OSI is given by:

𝑝𝑐∗ =

−𝑚2r1+𝑛2r1−2𝑂𝑡+𝑚2r2𝛿−𝑛2r2𝛿

(𝑚2−2𝑛2−𝑛2𝛼)𝛿

Having found the optimal profit, advertising, and price, we next do sensitivity analysis

with respect to different parameters to get insight into how each would react with

increase/decrease in certain parameters. This is discussed in the next section.

4.3 Sensitivity analysis for Competition case

4.3.1 Traffic Quality of the Parent

The impact of the traffic quality of parent depends upon the level of relative effect of advertising

(i.e., 𝒎/𝒏). In the following proposition, we first consider a scenario where the relative effect of

advertising is low.

Proposition 5:In the competition case, when the relative effect of advertising is lower than a

threshold (i.e., 𝒎/𝒏 < 𝟏), with an increase in the quality of parent firm traffic (i.e., 𝒓𝟏), parent

advertising always increases, OSI advertising as well as the payment made to OSI always

decreases, and the profit of the parent increases iff the traffic quality of parent firm is greater

21

than a threshold (i.e., 𝒓𝟏 > 𝟐𝑶𝒕(−𝟐𝒎𝟐+𝒏𝟐(𝟑+𝜶))

𝒏𝟒+𝒎𝟐𝒏𝟐𝜶+ 𝐫𝟐𝜹) while it decreases if its lesser than this

threshold.

.Similar to the fixed market case, we find that even when we consider a third competitor

in the game the parent advertising, OSI advertising and payment made to OSI behave exactly

similarly as earlier. As the parent traffic quality (i.e., r1) increases, there is a direct positive

effect on the demand it gets. When the related effect of advertising effectiveness is low, it could

also be interpreted as the low level of the advertising effectiveness of parent firm (for a given

level of the advertising effectiveness of the OSI ). Hence, in this scenario, the base advertising

level of the parent firm is also low. Therefore, with an increase in r1, the parent firm increases its

advertising significantly to benefit from the improved traffic quality. The presence of a third

competitor only gives parent firm more incentive to advertise to benefit as much as possible from

the improved traffic quality. Thus, the parent’s advertising increases with r1. As both the

advertising level and the quality of its traffic increases for the parent firm, it can rely less on the

OSI by paying it less. Thus, the parent pays less to the OSI who in turn advertises less. As a

result, both the OSI advertising and payment made to OSI decreases with r1.

As for the parent firm profit, there are four direct impacts of increase in r1: One is the

positive effect of improved traffic quality on its demand, the second is the negative effect of

increased advertising expenditure, third is the positive effect of reduced payment to the OSI and

fourth is the negative effect of third party competitor’s advertising. In addition to these, the

demands from both the parent firm (i.e., 𝐷1) and the OSI (i.e., 𝐷2) also change. When the quality

of traffic is lower than a threshold (i.e., 𝑟1 < 2𝑂𝑡(−2𝑚2+𝑛2(3+𝛼))

𝑛4+𝑚2𝑛2𝛼+ r2𝛿), the negative effects

dominate and hence the optimal profit decreases with an increase in 𝑟1. However, when the

22

quality of traffic is higher than a threshold (i.e., 𝑟1 > 2𝑂𝑡(−2𝑚2+𝑛2(3+𝛼))

𝑛4+𝑚2𝑛2𝛼+ r2𝛿), the positive

effects dominate, and therefore the optimal profit starts increasing with 𝑟1.Thus under low level

of relative advertising effectiveness(𝑚/𝑛 <1) the profit behaves similarly as in the fixed market

case with only the threshold changing. In the next proposition, we now focus on a scenario where

the relative effect of advertising (i.e., 𝑚/𝑛) is at a moderate level.

Proposition 6: In the competition case, when the relative effect of advertising is moderate (i.e.,

𝟏 ≤ 𝒎/𝒏 ≤ √𝟐 + 𝜶), with an increase in the quality of parent firm traffic (i.e., 𝒓𝟏), parent

advertising, OSI advertising, and the payment made to OSI always increase, whereas the profit

of the parent decreases iff the traffic quality of parent firm is greater than a threshold (i.e., 𝒓𝟏 >

𝟐𝑶𝒕(−𝟐𝒎𝟐+𝒏𝟐(𝟑+𝜶))

𝒏𝟒+𝒎𝟐𝒏𝟐𝜶+ 𝐫𝟐𝜹) and increases if its less than a threshold.

Similar to that in Proposition 5, as the parent traffic quality (i.e., r1) increases, there is a

direct positive effect on the demand it gets. Therefore, , with an increase in r1, the parent firm

increases its advertising to benefit further from the improved traffic quality. However, in this

scenario, since the relative effect of advertising is moderate, the base advertising level of the

parent firm is not very low, and therefore the advertising cost does not increase as significantly

as that in Proposition 1. Since the advertising cost doesn’t increase significantly for the parent

firm with an increase in its traffic quality, it is able to the pay a portion of extra revenue to the

OSI . Therefore, the payment increases with r1. As a result, the advertising of OSI also increases

with r1. Profit of the parent on the other hand, increases with r1 if and only if r1 is lesser than a

threshold and increases if and only if r1 is greater than a threshold. This is reverse of the

previous proposition. Here, there are four direct impacts of increase in r1: One is the positive

23

effect of improved traffic quality on parent’s demand, the second is the positive effect of reduced

advertising expenditure, third is the negative effect of increased payment to the OSI and fourth

is the negative effect of third party competitor’s advertising. In addition to these, the demands

from both the parent firm (i.e., 𝐷1) and the OSI (i.e., 𝐷2) also change. When the quality of traffic

is lower than a threshold (i.e., 𝑟1 < 2𝑂𝑡(−2𝑚2+𝑛2(3+𝛼))

𝑛4+𝑚2𝑛2𝛼+ r2𝛿), the positive effect dominate and

hence the optimal profit increases with an increase in 𝑟1. However, when the quality of traffic is

higher than a threshold (i.e., 𝑟1 > 2𝑂𝑡(−2𝑚2+𝑛2(3+𝛼))

𝑛4+𝑚2𝑛2𝛼+ r2𝛿), the negative effect dominate, and

therefore the optimal profit starts decreasing with 𝑟1 .Thus under moderate level of relative

advertising effectiveness the profit behaves revrese as in the fixed market case. The next

proposition now considers a scenario where the relative effect of advertising (i.e., 𝑚/𝑛) is high.

Proposition 7: In the competition case, when the relative effect of advertising is high (i.e.,

𝒎/𝒏 > √𝟐 + 𝜶), with an increase in the quality of parent firm traffic (i.e., 𝒓𝟏 ), parent

advertising, OSI advertising, and the payment made to OSI always decrease, whereas the profit

of the parent decreases iff the traffic quality of parent firm is greater than a threshold (i.e., 𝒓𝟏 >

𝟐𝑶𝒕(−𝟐𝒎𝟐+𝒏𝟐(𝟑+𝜶))

𝒏𝟒+𝒎𝟐𝒏𝟐𝜶+ 𝐫𝟐𝜹) and increases if its less than this threshold.

Unlike the earlier two propositions, in this scenario, the relative effect of advertising is

high, and therefore the base advertising level of the parent firm is already at a high level.

Therefore, as the parent traffic quality (i.e., r1) increases, it reduces its advertising to maintain a

similar demand level. Thus, parent advertising decreases with r1 irrespective of presence of a

third competitor. In addition, it is able to reduce the payment to the OSI . As a result, the

advertising of OSI also decreases with r1. Profit of the parent as in the previous case(with

24

moderate level of advertising effectiveness), increases with r1 if and only if r1 is lesser than a

threshold and increases if and only if r1 is greater than a threshold. Here ,also there are four

direct impacts of increase in r1: One is the positive effect of improved traffic quality on parent’s

demand, the second is the positive effect of reduced advertising expenditure, third is the positive

effect of reduced payment to the OSI and fourth is the negative effect of third party competitor’s

advertising. In addition to these, the demands from both the parent firm (i.e., 𝐷1) and the OSI

(i.e., 𝐷2 ) also change. When the quality of traffic is lower than a threshold (i.e., 𝑟1 <

2𝑂𝑡(−2𝑚2+𝑛2(3+𝛼))

𝑛4+𝑚2𝑛2𝛼+ r2𝛿), the positive effect dominate and hence the optimal profit increases

with an increase in 𝑟1. However, when the quality of traffic is higher than a threshold (i.e., 𝑟1 >

2𝑂𝑡(−2𝑚2+𝑛2(3+𝛼))

𝑛4+𝑚2𝑛2𝛼+ r2𝛿), the negative effect dominate, and therefore the optimal profit starts

decreasing with 𝑟1.Thus under high level of relative advertising effectiveness the profit behaves

reverse as in the fixed market case.

4.3.2 Traffic Quality of the OSI

Proposition 8: When the relative effect of advertising is low (i.e., 𝑚

𝑛< 1), with an increase in

the quality of OSI traffic (i.e., 𝑟2), parent advertising always decreases, OSI advertising, and

the payment made to OSI always increases, whereas the profit of the parent decreases iff the

traffic quality of parent firm is lesser than a threshold (i.e., 𝑟2 <𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿)

and increases if its greater than this threshold.

As the OSI traffic quality (i.e., r2) increases, there is a direct positive effect on the

demand the OSI gets. When the related effect of advertising effectiveness is low, it could also be

interpreted as the low level of the advertising effectiveness of parent firm (for a given level of

the advertising effectiveness of the OSI ). Hence, in this scenario, the base advertising level of

25

the parent firm is also low. With an increase in r2, the parent firm has less incentive to advertise

itself since it cannot benefit much from its own advertising and instead pays OSI more to

capitalize on OSI ’s improved traffic quality(the presence of a third competitor has no bearing

on the parent’s strategy in this case) Thus, the parent’s advertising decreases with r1 while the

OSI ’s advertising decreases and so does the payment make to the OSI .

As for the parent firm profit, there are four direct impacts of increase in r2: One is the

positive effect of OSI ’s improved traffic quality on parent’s demand, the second is the positive

effect of redused advertising expenditure, third is the negative effect of increased payment to the

OSI and fourth is the negative effect of third party competitor’s advertising. In addition to these,

the demands from both the parent firm (i.e., 𝐷1) and the OSI (i.e., 𝐷2) also change. When the

quality of OSI traffic is lower than a threshold (i.e., 𝑟2 <𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿), the

negative effects dominate and hence the optimal profit decreases with an increase in 𝑟2 .

However, when the quality of traffic is higher than a threshold (i.e., 𝑟2 >

𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿), the positive effects dominate, and therefore the optimal profit

starts increasing with 𝑟2. In the next proposition, we now focus on a scenario where the relative

effect of advertising (i.e., 𝑚/𝑛) is at a moderate level.

Proposition 9: When the relative effect of advertising is moderate (i.e., 1 < 𝒎

𝒏< √𝟐 + 𝜶), with

an increase in the quality of OSI traffic (i.e., 𝑟2), parent advertising, OSI advertising, and the

payment made to OSI always decrease, whereas the profit of the parent decreases iff the traffic

quality of OSI is greater than a threshold (i.e., 𝒓𝟐 > 𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿) and

increases if its less than a threshold.

26

Here, with an increase in r2, the parent firm has no incentive to advertise itself (since

both the advertising level and advertising effectiveness are low) and thus the parent’s advertising

decreases with r1.As the quality of OSI traffic improves the OSI attracts higher demand itself

without much increase required in its advertising. Thus, parent doesn’t need to pay OSI to

increase advertising and so both OSI advertising and payment decrease with r2. Profit of the

parent on the other hand, increases with r1 if and only if r1 is lesser than a threshold and

decreases if and only if r1 is greater than a threshold. This is reverse of the previous proposition.

Here, there are four direct impacts of increase in r1: One is the positive effect of OSI ’s

improved traffic quality on parent’s demand, the second is the positive effect of reduced

advertising expenditure, third is the positive effect of reduced payment to the OSI and fourth is

the negative effect of third party competitor’s advertising. In addition to these, the demands from

both the parent firm (i.e., 𝐷1) and the OSI (i.e., 𝐷2) also change. When the quality of traffic is

lower than a threshold (i.e., 𝑟2 < 𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿), the positive effect dominate and

hence the optimal profit increases with an increase in 𝑟2. However, when the quality of traffic is

higher than a threshold (i.e., 𝑟2 > 𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿), the negative effect dominate,

and therefore the optimal profit starts decreasing with 𝑟2. The next proposition now considers a

scenario where the relative effect of advertising (i.e., 𝑚/𝑛) is high.

Proposition 10: When the relative effect of advertising is high (i.e., 𝑚/𝑛 > √2 + 𝛼), with an

increase in the quality of OSI traffic (i.e., 𝑟2), parent advertising, OSI advertising, and the

payment made to OSI always increases, whereas the profit of the parent decreases iff the traffic

quality of OSI is greater than a threshold (i.e., 𝒓𝟐 > 𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿) and

increases if its less than a threshold.

27

Unlike the earlier two propositions, in this scenario, the relative effect of advertising is

high, and therefore the base advertising level of the parent firm is already at a high level.

Therefore, since it can benefit from own advertising its advertising increases with r2.The OSI on

the other hand can benefit since its traffic quality is improving. Knowing this parent pays more

to OSI as it can bring in better demand for the parent as well. Thus, both the advertising of the

OSI and payment increase with OSI traffic quality (i.e., r2). Profit of the parent as in the

previous case(with moderate level of advertising effectiveness), increases with r1 if and only if

r1 is lesser than a threshold and increases if and only if r1 is greater than a threshold. Here ,also

there are four impacts of increase in r2: One is the positive indirect effect of improved traffic

quality on OSI ’s demand, the second is the negative effect of reduced advertising expenditure,

third is the negative effect of increased payment to the OSI and fourth is the negative effect of

third party competitor’s advertising. In addition to these, the demands from both the parent firm

(i.e., 𝐷1) and the OSI (i.e., 𝐷2) also change. When the quality of traffic is lower than a threshold

(i.e., 𝒓𝟐 < 𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿), the positive effect dominate and hence the optimal

profit increases with an increase in 𝑟1. However, when the quality of traffic is higher than a

threshold (i.e., 𝒓𝟐 > 𝑛2∗(r1)+𝑚2∗(r1)∗𝛼+2∗𝑂∗𝑡∗(1+𝛼)

(𝑛2+𝑚2∗𝛼)∗𝛿), the negative effect dominate, and therefore

the optimal profit starts decreasing with 𝑟1 .Thus under high level of relative advertising

effectiveness the profit behaves reverse as in the fixed market case.

4.3.4 Competitor’s Advertising level

Proposition 11: When the relative effect of advertising is lower than a threshold (i.e., 𝒎/𝒏 ≤

√𝟐 + 𝜶), with an increase in the advertising level of the competitor (i.e., 𝑶), parent advertising,

OSI advertising as well as the payment made to OSI always increase, and the profit of the parent

28

increases iff the advertising level of the competitor is lesser than a threshold (i.e., 𝑶 >

𝒕(−𝟐𝒎𝟐𝐫𝟏+𝒏𝟐(𝐫𝟏(𝟑+𝜶)+𝐫𝟐(𝟏+𝜶)𝜹))

𝟐(𝒕𝟐+(𝒎𝟐−𝒏𝟐(𝟐+𝜶))𝜸)) while it decreases if its greater than this threshold.

As the competitor’s advertising level (i.e., O) increases, there is a direct positive effect on

the demand the competitor gets. When the relative effect of advertising effectiveness is low, the

parent firm needs to advertises more to maintain the previous demand level in order to

counteract the lost demand to the competitor as the competitor’s advertising level(O) increases.

Hence, parent advertising increases with O. The parent also pays the OSI more so that the OSI in

turn increases its advertising to counteract the lost demand to the competitor. Thus, both payment

made to OSI and OSI advertising increases.

With an increase in advertising level of competitor, there is a negative effect of

competitor’s advertising on the demand that the parent gets, in order to counteract this negative

effect the parent firm increases its advertising level. This in turn has a positive effect on its

demand. Thus, when the competitor’s advertising level is lower than a threshold (i.e., 𝑂 <

𝑡(−2𝑚2r1+𝑛2(r1(3+𝛼)+r2(1+𝛼)𝛿))

2(𝑡2+(𝑚2−𝑛2(2+𝛼))𝛾)), the positive effect of the parent’s own advertising dominates

and thus the optimal profit increases with O. However, when the competitor’s advertising level is

higher than a threshold (i.e., 𝑂 > 𝑡(−2𝑚2r1+𝑛2(r1(3+𝛼)+r2(1+𝛼)𝛿))

2(𝑡2+(𝑚2−𝑛2(2+𝛼))𝛾))) , then due to increased

advertising level of competitor, the competitor can siphon off a major chunk of the demand.

Thus, the negative effects dominates, and therefore the optimal profit starts decreasing with O.

Proposition 12: When the relative effect of advertising is high (i.e., 𝒎/𝒏 > √𝟐 + 𝜶), with

increase in the advertising level of the competitor (i.e., 𝑶), parent advertising, OSI advertising,

and the payment made to OSI always decrease, whereas profit of the parent increases iff the

advertising level of the competitor is lesser than a threshold (i.e., 𝑶 >

29

𝒕(−𝟐𝒎𝟐𝐫𝟏+𝒏𝟐(𝐫𝟏(𝟑+𝜶)+𝐫𝟐(𝟏+𝜶)𝜹))

𝟐(𝒕𝟐+(𝒎𝟐−𝒏𝟐(𝟐+𝜶))𝜸)) while it decreases if its greater than this threshold.

Unlike the previous proposition, in this scenario, the relative effect of advertising is high,

and therefore the base advertising level of the parent firm is already at a high level. Therefore,

since the base advertising level is already high, the parent can decrease its own advertising which

decreases with O. Further, with high base advertising level the parent can reduce its payment to

OSI which in turn reduces its advertising. Thus, both the payment made to OSI and the OSI

advertising decrease with O.

With an increase in advertising level of competitor, there is a negative effect of

competitor’s advertising on the demand that the parent gets. However, since the base advertising

level of the parent is high it can still garner a major chunk of the demand. Thus, when the

competitor’s advertising level is lower than a threshold (i.e., 𝑂 <

𝑡(−2𝑚2r1+𝑛2(r1(3+𝛼)+r2(1+𝛼)𝛿))

2(𝑡2+(𝑚2−𝑛2(2+𝛼))𝛾)), the positive effect of the parent’s own advertising dominates

and thus the optimal profit increases with O. However, when the competitor’s advertising level is

higher than a threshold (i.e., 𝑂 > 𝑡(−2𝑚2r1+𝑛2(r1(3+𝛼)+r2(1+𝛼)𝛿))

2(𝑡2+(𝑚2−𝑛2(2+𝛼))𝛾))), due to increased advertising

level of competitor, the competitor can siphon off a major chunk of the demand. Thus, the

negative effects dominates, and therefore the optimal profit decreases with O.

4.4 Customer heterogeneity

Depending on the stage of the funnel the user is in (initial stage versus final stage), it is likely

that users would react differently to OSI advertising. Users in the initial stage of their decision-

making process are less likely to convert than users in final stages. This would affect pricing and

30

advertising strategies of all players. We intend to incorporate this into our model in the future

and find relevant strategies.

5 Conclusions

OSI marketing is becoming an increasingly important tool in the firm’s kitty to generate extra

traffic on their websites. This is evident from the fact that all the big e-commerce giants like

Amazon, eBay, Walmart, etc., are using thousands of third-party sellers to advertise their

products. Moreover, many such firms are utilizing such OSIs to advertise on the sponsored

search for their keywords to gain additional demand. Thus, in this context, it’s important to

reexamine traditional contractual decisions (such as pricing and advertising). This issue has been

considered important by several researchers and practitioners yet little formal analysis has been

done. Here, we try to address this important problem. Our analysis shows that traditional

business decisions can change dramatically depending on the traffic quality and level of

advertising effectiveness. Thus, firms should be careful in strategizing these decisions to avoid

spending wasteful amounts of money on advertising and payments to OSIs that do not bring in

substantial demand.

Theoretically, the study would contribute to three streams of literature: 1) affiliate marketing, 2)

supply chain, and 3) infomediaries. Although researchers in each of these areas have looked at

diverse strategies one of the important areas that has remained unexplored is what contractual

(pricing and advertising) strategies should be followed between the two parties in order to

maximize equilibrium profits and this is the focus of our paper. Practically, the insights from the

research will help managers strategize their advertising and pricing decisions when contracting

with search infomediaries rather than solely rely on the fixed payment model. It would also help

31

them adjust these strategies depending on competition from other players on the search engine,

customer type and number of infomediaries.

Appendix

Proofs for fixed market case

Sensitivity with r1 (parent traffic quality)

Price

𝐷[price, r1]

−𝑚2 + 𝑛2

(𝑚2 − 2𝑛2 − 𝑛2𝛼)𝛿

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1 , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

Parent firm advertising

𝐷[A1optimal, r1]//FullSimplify

−𝑚𝑛2(1 + 𝛼)

2(𝑚2 − 𝑛2(2 + 𝛼))

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

OSI Advertising

𝐷[B1optimal, r1]//FullSimplify

32

𝑛(−𝑚2 + 𝑛2)

2(𝑚2 − 2𝑛2 − 𝑛2𝛼)

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

Parent Profit

𝐷[profitoptimal, r1]//FullSimplify

{−𝑛2(𝑛2 + 𝑚2𝛼)(r1 − r2𝛿)

2(𝑚2 − 𝑛2(2 + 𝛼))}

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) 𝑎𝑛𝑑 𝑟1 > 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) 𝑎𝑛𝑑 𝑟1 < 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) 𝑎𝑛𝑑 𝑟1 > 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) 𝑎𝑛𝑑 𝑟1 < 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

Sensitivity with r2 (OSI traffic quality)

Parent firm advertising

𝐷[A1optimal, r2]//FullSimplify

𝑚𝑛2(1 + 𝛼)𝛿

2(𝑚2 − 𝑛2(2 + 𝛼))

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

OSI advertising

𝐷[B1optimal, r2]//FullSimplify

(𝑚 − 𝑛)𝑛(𝑚 + 𝑛)𝛿

2(𝑚2 − 𝑛2(2 + 𝛼))

33

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1 , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

Price

𝐷[price, r2]//FullSimplify

(𝑚 − 𝑛)(𝑚 + 𝑛)

𝑚2 − 𝑛2(2 + 𝛼)

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1 , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

Parent profit

𝐷[profitoptimal, r2]//FullSimplify

{𝑛2(𝑛2 + 𝑚2𝛼)𝛿(r1 − r2𝛿)

2(𝑚2 − 𝑛2(2 + 𝛼))}

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) 𝑎𝑛𝑑 𝑟1 > 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) 𝑎𝑛𝑑 𝑟1 < 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) 𝑎𝑛𝑑 𝑟1 > 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) 𝑎𝑛𝑑 𝑟1 < 𝑟2 ∗ 𝛿, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

Competition case

// sensitivity with O

Parent advertising

𝐷[A1optimal, 𝑂]//FullSimplify

34

−𝑚𝑡

𝑚2 − 𝑛2(2 + 𝛼)

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

OSI advertising

𝐷[B1optimal, 𝑂]//FullSimplify

𝑛𝑡

−𝑚2 + 𝑛2(2 + 𝛼)

Can be rewritten as after multiplying by negative sign on Numerator and denominator as

−𝑛𝑡

𝑚2 − 𝑛2(2 + 𝛼)

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

Price

𝐷[price, 𝑂]//FullSimplify

−2𝑡

(𝑚2 − 𝑛2(2 + 𝛼))𝛿

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

Profit

35

𝐷[profitoptimal, 𝑂]//FullSimplify

{−2𝑂𝑡2 − 2𝑚2(r1𝑡 + 𝑂𝛾) + 𝑛2(r1𝑡(3 + 𝛼) + 2𝑂(2 + 𝛼)𝛾 + r2𝑡(1 + 𝛼)𝛿)

𝑚2 − 𝑛2(2 + 𝛼)}

//solve for O to get a threshold

𝑂 →𝑡(−2𝑚2r1 + 𝑛2(r1(3 + 𝛼) + r2(1 + 𝛼)𝛿))

2(𝑡2 + (𝑚2 − 𝑛2(2 + 𝛼))𝛾)

threshold = t*((r1)*(-2*k^2) + ((r1)*(3+a) + (r2)*(s+a*s))) / 2*((t^2+(k^2-(2+a))*y) > 0,

k>0,t>0,r1>0,r2>0,0<s<1,0<y<1,0<a<1

//whenitssmallerthanthreshoildthenitspositivesoincreasingwith𝑂,/

/whenisgreaterthanthresholdthenitsnegetivesodecreasingwith𝑂.

𝐷[d1optimal, 𝑂]//FullSimplify

𝑡(−2𝑚2 + 𝑛2(3 + 𝛼))

𝑚2 − 𝑛2(2 + 𝛼)

𝐷[d2optimal, 𝑂]//FullSimplify

𝑛2𝑡(1 + 𝛼)

𝑚2 − 𝑛2(2 + 𝛼)

//sensitivity with respect to r1

𝐷[profitoptimal, r1]//FullSimplify

{−4𝑚2𝑂𝑡 + 𝑛4(−r1 + r2𝛿) + 𝑛2(2𝑂𝑡(3 + 𝛼) + 𝑚2𝛼(−r1 + r2𝛿))

2(𝑚2 − 𝑛2(2 + 𝛼))}

36

𝑟2 ∗ 𝑠 +2 ∗ 𝑂 ∗

𝑡

𝑛2∗ (−2𝑘2 + (3 + 𝑎))

1 + 𝑘2 ∗ 𝑎

((r2) *s )+ ((2 *((3 + a) - 2* k^2) *O* t)/((1 + a* k^2) *n^2)) < 0, k > 0 , t > 0, n>0,r2 >

0, 0 < O < 1, 0 < a < 1 , 0 < s < 1 , r2 > 0

{{r1 →2𝑂𝑡(−2𝑚2 + 𝑛2(3 + 𝛼))

𝑛4 + 𝑚2𝑛2𝛼+ r2𝛿}}

greaterthanthresholdthennegetivesodecreasingwithr1

lessthanthresholdthenpositivesoincreasingwithr1

Results:

• Approximate form

37

38

Results:

• Approximate form

Payment/price behaves exactly similarly

𝐷[price, r1]

−𝑚2 + 𝑛2

(𝑚2 − 2𝑛2 − 𝑛2𝛼)𝛿

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

39

𝑤ℎ𝑒𝑛𝑚

𝑛< 1 , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝐷[A1optimal, r1]//FullSimplify

−𝑚𝑛2(1 + 𝛼)

2(𝑚2 − 𝑛2(2 + 𝛼))

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

OSI advertising behaves exactly similarly.

𝐷[B1optimal, r1]//FullSimplify

𝑛(−𝑚2 + 𝑛2)

2(𝑚2 − 2𝑛2 − 𝑛2𝛼)

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1, 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝐷[d1optimal, r1]//FullSimplify

−𝑛2(𝑛2 + 𝑚2𝛼)

2(𝑚2 − 𝑛2(2 + 𝛼))

When m/n > sqrt(2+alfa) the parent firm demand decreases

When m/n < sqrt(2+alfa) the parent firm demand increases

𝐷[d2optimal, r1]//FullSimplify

𝑛2(𝑛2 + 𝑚2𝛼)

2(𝑚2 − 𝑛2(2 + 𝛼))

40

When m/n > sqrt(2+alfa) the OSI demand increases

When m/n < sqrt(2+alfa) the OSI demand decreases

------------------------------------------------------------------------------------------------------------

sensitivity with r2

Profit

𝑛2𝛿(2𝑂𝑡(1 + 𝛼) + 𝑛2(r1 − r2𝛿) + 𝑚2𝛼(r1 − r2𝛿))

2(𝑚2 − 𝑛2(2 + 𝛼))

{{r2 →𝑛2r1 + 𝑚2r1𝛼 + 2𝑂𝑡(1 + 𝛼)

(𝑛2 + 𝑚2𝛼)𝛿}}

This is always > 0

greater than threshold then negetive so decreasing with r2

less than threshold then positive so increasing with r2

𝐷[A1optimal, r2]//FullSimplify

𝑚𝑛2(1 + 𝛼)𝛿

2(𝑚2 − 𝑛2(2 + 𝛼))

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝐷[B1optimal, r2]//FullSimplify

(𝑚 − 𝑛)𝑛(𝑚 + 𝑛)𝛿

2(𝑚2 − 𝑛2(2 + 𝛼))

41

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1 , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝐷[price, r2]//FullSimplify

(𝑚 − 𝑛)(𝑚 + 𝑛)

𝑚2 − 𝑛2(2 + 𝛼)

𝑤ℎ𝑒𝑛𝑚

𝑛> √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 > 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

𝑤ℎ𝑒𝑛 1 <𝑚

𝑛< √(2+∝) , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 > 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 < 0

𝑤ℎ𝑒𝑛𝑚

𝑛< 1 , 𝐷𝑖𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 < 0, 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 0 , 𝐸𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 > 0

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