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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.18 No.4 pp.776-793
DOI : https://doi.org/10.7232/iems.2019.18.4.776

# A Study on Pricing under a Coopetitive and Duopolistic Market of Vertically Differentiated Products

Sina Keyhanian, Abbas Ahmadi*, Behrooz Karimi
Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran
July 15, 2018 April 12, 2019 September 23, 2019

## ABSTRACT

We consider two firms introducing vertically differentiated products to a market ruled by the consumers’ willingnessto-pay. If consumers behave indifferently to all the products, they would always choose the product which offers highest net utility. The paper considers a more realistic case in which high utility products do not necessarily satisfy the consumers who are sensitive to price enhancements. They are eager to analyze whether choosing a new entrant offering higher quality products is worth the price. In such a setup, competing firms offering vertically differentiated products could employ cooperative bundles. By introducing new products in stages, we show that the market gets divided based on consumers’ willingness to pay more for an expensive product. We have employed computational method on a newly built coopetition model. Our results show that cross bundling can improve both sellers’ profits, also leading to a boost in the entrant’s position in the market. In fact, the applied strategy of simultaneous cooperation and competition in this paper shows that, coopetition drives consumers’ attention to purchase cross-bundles instead of individual products or direct bundles, which in the end benefits both firms. Also it is shown that previous solutions are not necessarily optimal for duopolies practicing coopetition.

## 1. INTRODUCTION

In the modern economic world, numerous firms choose to partake not only the competitive acts but also the cooperative actions with their rivals. Many studies, for instance, the research by Okura and Kasuga (2007) have vehemently described the phenomenon which includes both the competitive and cooperative acts simultaneously in the name of coopetition. The recent century has also seen the numerous and rapid emergence of diverse literature defining the coopetition, game theories and models of the actual market scenario in different countries and industries. It is, however, right to note that most of these sources have majorly focused on defining the market and economic features and models of the public and private industries (Brandenburger and Nalebuff, 2002).

Although there is only a handful of research studies on the coopetition especially in the duopoly market, most global industries have demonstrated its features and impacts. For instance, the agreement between the Amazon and the Apple towards the end of 2007 (Stadtler and Van Wassenhove, 2016). This coopetition partnership aimed at making Amazon become the Apple iPad’s Frenemy. It involved the introduction of a new application in the Apple’s iPad’s phone known as the Kindle e-reading. Immediately after the Apple launched its iPad, it started distributing the Amazon’s e-books via the app. This coopetition model benefited both parties as the Amazon secured a broader market for the Kindle application while Apple emerged as the adequate content provider.

Another recent case of coopetition partnership was between the Toyota and BMW (Stadtler and Van Wassenhove, 2016). The two firms entered into a contract to introduce their joint expertise in the hybrid and carbon fiber technologies. According to the partnership, the two companies, BMW and Toyota were still to compete with each other in the automobile production. However, they will not go alone in the enhancement and development of their production technology. Another significant coopetition pact in the same field regards to Renault and Nissan (Kraus et al., 2018). The two firms triggered a contract to enable them to produce vehicles through the “Synergies” technology. However, the agreement allowed them to sell the same product although with a slight adjustment in the shared market.

An alternative technique for price discrimination, first suggested by Stigler (1968) and analyzed further by Adams and Yellen (1976), is for the decision maker to package two or more products in bundles rather than selling them separately. Bundling is an effective marketing strategy that can be used by firms seeking to maximize sales or order values and consequently profits. In the context of vertically differentiated products, bundling provides firms with the opportunity of offering the best product mix at prices that maximize profits while at the same time maximizing consumer utility. Bundling is most effective in situations whereby a firm sells different products, usually components which can be jointly packaged and sold as a single bundle. The firm in this case sells the bundle at a price which is lower than the sum of its individual components.

Whether as individual components or bundles, consumers assess the value of a commodity based on price and quality (Debo et al., 2005). Product value consequently determines their willingness to pay (WTP) for the commodity in question. However in a monopoly set up, firms will realize profits using product components or bundles that offer the highest net utility. In market set ups where consumers perceive quality based on price, consumers will be more likely to pay for commodities with a higher price up to a certain limit known as the consumer expectancy threshold (CETH). Upon reaching the CETH limit, consumers will reconsider decisions on what product to buy.

In a monopoly market set up, Adams and Yellen (1976) recommend cross bundling as an alternative for price discrimination in order to maximize profits. With different market players, competing firms can decide to cross bundle their products or services. This paper considers the possibility of cross bundling competitors’ components. Individually, these components are of a lower quality compared to when offered as part of a bundle. However, it is critical to note that coopetition is not only an effective cost reduction tool but also quality improvement.

In order to define adequate coopetition marketing strategy, this study process will, therefore, enhance a game theory and mathematical model which would introduce efficient cooperation in the duopolistic market. To improve its success by introducing a reliable model, the article will analyze the previous literature concerning the topic and compare it with the current marketing phenomenon to introduce adequate breed which would produce success to the firms willing to enter into pacts with their competing rivals in the modern market.

The main contributions of this paper are listed as follows: First, the willingness of consumers to pay more for a higher quality product is modelled and the preliminaries are studied leading to the definition of switching willingness- to-pay value. Second, considering introducing products in both cases of sequential and simultaneous and modeling their corresponding settings. The problem structures leads to handling both blue oceans (with only one winner) and red oceans (with more than one winner and hence the challenge to share the revenue). Third, useful managerial insights are extracted from numerical examples which can be helpful for decision makers seeking to perform better technologically and enter markets containing stable incumbents. The insights show that coopetition agreements, not only can benefit the new entrant to enhance her position in the market but also can increase the whole market’s profit within the same market size by beneficial employment of price discrimination.

The rest of the paper is organized as follows. Next section provides a brief review of previous studies on bundling strategies especially when it comes to vertically differentiable products. Section 3 discusses the model and how modelling the consumer behavior can affect the decisions on pricing for the new entrant regarding the incumbent characteristics. Section 4 provides some numerical experiments along with analytics which provides insights on the subject. Eventually, the achievements are summarized in section 5.

## 2. LITERATURE REVIEW

Many research studies have increasingly emerged in the current century concerning the prowess of the coopetition marketing. However, it is proper to note that although the term coopetition has been in existence since the late ancient civilization, it became increasingly and widely used in the mid-20th century as revealed by Dagnino (2009). Around this time, the concept was commonly used in the academic institutions, especially in the research collaboration. The more prominent multinational firms later also adopted the model but applied it quietly. Currently, however, the concept has significantly evolved, and its principles have been enhanced to make a great sense to the organizations and businesses of varied kinds (Dumez and Jeunemaitre, 2006). In the modern networked global universe, it has become significantly easier to implement the model considering the emergence of technological and sophisticated business models including various types of contractual agreements.

According to Matsumura and Kanda (2005), alliance literature has indicated that the value creation is dependent on the utilization and pooling of the valuable resources that industries often acquire through diverse ways which involve interfirm partnership. Traditionally, such alliances or collaboration have been viewed under assumptions such as heterogeneous customers, suppliers and other partnership groups (Chin et al., 2008). In the 1980s, coopetition was mainly perceived as a price discrimination strategy, but recently the mechanism has widely been accepted by the policymakers across USA, Europe and Asia (Cachon and Lariviere, 2005).

In fact, the strategy often adds value not only to the consumers but also to the involved firms especially if it results in the improvements of the existing services and products along with creating new ones. The cooperation is further essential as it also increases the competition among the rivalry networks. While most researches and literature have determined its success among the involved parties, some have, however, defied the findings, stating that it is a risky and staggering relationship that often results into a consequential failure and a detrimental “race” and performance of the alliance (Cachon and Lariviere, 2005). Some of the initial studies have determined that coopetition usually matters to the involved firms and the performance of the partnership for better and for worse. Most of these studies were, however, conducted in specific firms which could single-handedly explain the phenomenon. Other research studies, for instance, Bollapragada and Mallik (2008) have also revealed the evidence that industry only matters if the firms benefit from the cooperation.

Cooperation has also been defined under the concept of collaboration in supply chain models. An economic impact analysis has been recently studied for various design variables of collaborations such as information sharing (Simatupang and Sridharan, 2018). The new entrant and incumbent structure of the market can also be seen as a Stackelberg game model in which the leader (incumbent) has more power than the follower (new entrant), see (Golmohammadi et al., 2016) for a two echelon supply chain modeling of this settings.

All in all, some studies have indicated that coopetition is not necessarily a success model especially in sectors with inadequate intensive knowledge such as manufacturing (Banciu et al., 2010). In light of the elaborated issues above, it could, therefore, be stated that the firms’ success in the coopetition highly depends on the type and specific factors which favor the cooperation in addition to external factors such as the economic and industrial context which the coopetition was embedded in.

Bundling, i.e., the sale of two or more products for one combined price, is an effective strategy for firms to increase profits. For example, a travel agency could offer a vacation package which consists of combining flight, hotel, rental car and a media service package bundles. Previously, bundling strategies was common among horizontally differentiated products. The bundle involved combining products of distinctive characteristics into one package. Nowadays, it is common for firms to bundle vertically differentiated products. For example, Tsiros and Heilman (2005) explain that some supermarkets in China offer bundles consisting of a carton of soon-to-expire milk and a carton of milk with a longer shelf life. They demonstrate that by pricing the bundle at a lower amount compared to the regular price of two components, pricesensitive customers are encouraged to buy more which leads to a suitable sell out of the soon-to-expire products. However, recent researches in the context of duopoly markets have provided different insights. Ginevičius and Krivka (2008) provide a brief review on the game theoretic models employed in this context. There are examples for both pricing models in decentralized supply chains without bundling strategies (Li et al., 2018), and for pricing models with bundling strategies that have considered different game theoretic conditions along with centralized settings, (Giri et al., 2017). (Li et al., 2018) consider both price and quality in their model within a two-echelon supply chain. They also discuss the impacts of bargaining power and decision structures on the final price and quality outcomes. (Giri et al., 2017), however consider duopolistic suppliers selling complementary products to common retailer under assumption of sequential movements. They show that the supply chain is better off selling bundles than introducing the products individually to the market. Also Pan and Zhou (2017) discuss sequential moves under the assumption of market power in supply chains employing bundling. They show that the optimal decision in such a setting is affected by two factors; product complementarity and profitability.

In a two-stage channel coopetition, Shao and Li (2019) consider two types of product strategy in terms of quality. Considering a new entrant versus an incumbent supplier, they determine that the retailer is better off with the bundling strategy. In this case, bundling encourages the new supplier of the complementary component to introduce his lower quality components. Further, the bundling strategy does not affect the product strategy of the incumbent supplier. Shugan et al. (2016) were stimulated by the fact that similar industries bundle ancillary services differently. They demonstrate that product lines such as airlines and amusement parks with low core differentiation routinely bundle high-end product lines and vice versa. Ginevičius and Krivka (2008) investigate game theoretic models which help to estimate market equilibrium of a duopoly. They emphasize the role of the game theory in analysing players’ behaviour in the market. This is rooted in a deep investment in product differentiation as well as the first and second mover advantage. These two concepts are highly critical in our research model.

Definitions of the relationships between players vary in different game theory models. Banciu et al. (2010) define a median level of retailers who use the space from a shopping mall leased to them to attract their customers. The main decision maker is the shopping mall owner who earns profit from the retailers’ sales. They discuss the optimal order of vertically differentiated quality ratings associated with a variety of bundling strategies. However, their research considers a direct relationship between the firms (sellers) and the customers (buyers). To capture arbitrary profits earned by sellers, Banciu and Ødegaard (2016) used the copula theory alongside the concept of price independence. This scenario is however impossible if components of the product bundle are considered independent. The research also considers the concept of customer behaviour, especially under the competitive market set up. Coopetition creates a new market space in which a portion of the customers buys the product bundle that maximizes their needs regardless of the brand. This is common when the customer population is price sensitive. A model of bundling monopoly was proposed by Spulber (2017) in which merging policies are also of concern. He considered suppliers’ population is constituted of some input suppliers (IS) and some bundled input monopolists (BIM). He finally shows that his results are in contrary with the Cournot effect.

Most of the existing work on vertically differentiated products assume that consumers purchase at most one component (see for example Mussa and Rosen, 1978;Moorthy, 1984;Bhargava and Choudhary, 2001;Pan and Honhon, 2012). Honhon and Pan (2017) believe that in practice, consumers may buy more than one component for some product categories. They refer to the concept of consumers buying possibly more than one components as their basket shopping behaviour.

In this paper we consider two firms that simultaneously compete and cooperate on the product bundles that they present to the market. The goods in this case are vertically differentiated and are ranked according to their individual quality ratings. Product value is determined by the customers who consequently define the market segmentation. The assumption of consumers willing to pay more for higher quality products have also been discussed in the literature in several ways. Some view higher quality as being a green product such as Wei et al. (2018) who discuss whether consumers find green products worth more price than the normal products they use daily. Some other view higher quality as more healthy agricultural products. Loureiro et al. (2002) approach consumers’ willingness to pay more for eco-labeled apples where there is a concern for the usage of pesticides in the relevant market. Numerous similar cases, both old and new researches, can also be found in literature, some of which are Laroche et al. (2001), Raje et al. (2002), Frash et al. (2015) and Meyer et al. (2018), Morales and Higuchi (2018). We, however, have applied this assumption in a general manner which has led to the definition of customer expectancy threshold (CETH) and switching willingness to pay (SWTP). Our research employs the optimization approach to find the optimal price vectors and quality orderings in the market scenario. Clearly, the focus in the optimal price vector would be the alteration and structure of competitive and cooperative bundle prices. The simultaneous occurrence of these two leads to the coopetitive space. Several studies have developed optimization approaches to design and price the bundles. Hanson and Martin (1990) formulate the bundling problem as a mixed integer programming model to determine the optimal bundle prices that will maximize profit given the number of consumer segments and reservation price. On the other hand, Hitt and Chen (2005) and Wu et al. (2008) use the non-linear programming approach and customized bundle pricing strategy for information goods. In their settings, prices only depend on the number of components in a bundle.

## 3. PROPOSED MODEL OF COMPETITION AND COOPERATION

For the beginning let’s try the case in which a normal product (not considering the generation here) is already in the market and the premium product is presented. Let p and Q show price and quality respectively with indices N (Normal) and P (Premium). Based on Figure 1 the consumers with valuation $0 ≤ θ < p N Q N$ choose to buy nothing and $p N Q N ≤ θ < p P − p N Q P − Q N$ choose to buy the normal product. The consumers are assumed to be highly alert to squandering. This means they only tend to buy a more expensive product if it offers them a certain minimum amount of utility. Thus, for the rest of the interval we assume that the consumer will only switch from the normal product to the new entrant’s premium product only if the added utility is at least greater than the simple quality versus price proportionality: $Q P − Q N p P − p N$. In other words, we should have:

$( Q P − Q N ) θ − ( p P − p N ) ≥ Q P − Q N p P − p N$
(1)

We call the right-hand side of the above inequality, customer expectancy threshold (CETH). By substituting $Q P − Q N = y$ and $p P − p N = x$ we get:

$y θ − x ≥ y x ⇒ θ − x y ≥ 1 x ⇒ θ ≥ 1 x + x y = 1 p P − p N + p P − p N Q P − Q N .$
(2)

The right hand side of inequality (2) is the minimum switching willingness to pay (SWTP) which triggers the consumers with higher WTP than this value to purchase the higher quality product. The SWTP can be calculated in two manners: 1) a consecutive manner, in which products are ordered by the quality ratings and each product’s SWTP is calculated regarding to the previous product, 2) a pairwise manner, in which there is a set of SWTPs which is calculated for each product regarding to each of the lower quality products. In Figures 1-3 we use the consecutive manner to avoid sophistication in depiction and provide an easy to understand description of how to market gets divided. While in the main analysis of the model we use the pairwise SWTP calculations to capture all the responses of the market to each of the products. Figures 2

Lemma 1. With the consumers being alert of squandering, to catch the highest share of the market the new entrant should present its product to the market with a price level of$p P ∗ = Q P × p N Q N$.

Proof. Based on Figure 1, the premium player should move its utility function to its left as much as possible to increase her share of the market. The most shift it can make is until after the incumbent player’s proportion of price and quality (the P/Q ratio).■

If we consider the perception of consumers regarding to ratios between quality and price, the above attempt would seem relevant especially when the consumers are sensitive to paying more. The same P/Q ratio will indeed assure them of the fairness of the new entrant in pricing her new product. Which leaves the rest of the purchasing decision up to whether the further utility is worth it or not.

Now, under the settings of Lemma 1, the consumers with valuations $p P − p N Q P − Q N ≤ θ < p P − p N Q P − Q N + 1 p P − p N$ will choose to stay on their old choice and the new entrant will be able to capture only the consumers with valuations greater than or equal to $p P − p N Q P − Q N + 1 p P − p N$. This result clearly leaves a smaller share of the market to capture for the new entrant. However, it also accounts for the vivid fact that a new entrant would not be able to catch a sufficient share of the market at the beginning. The new entrant might still desire to sacrifice price and decrease the P/Q ratio as much as possible to capture more market share, however such an action may interfere with having a rational profit making strategy.

Preposition 1. Under same P/Q ratios, if the new entrant sets her price higher than$p N + 1 / ( 1 − p Q )$and lower than$p N + 2 / ( 1 − p Q )$then, (1) a positive share of the market is guaranteed for the new entrant and, (2) the incumbent’s market share exceeds the new entrant’s market share.

Proof. Since $p P Q P = p N Q N = p Q$, also $p P − p N Q P − Q N = p Q$ and therefore, for the new entrant to have a positive share of the market we should have $1 − p Q − 1 p P − p N > 0$ which con cludes the lower bound of the preposition. And for the incumbent’s market share to be superior, we should simply have $1 p P − p N > 1 − p Q − 1 p P − p N$ which concludes the up per bound.■

The above proposition follows with the theory that for the new entrant to capture more market share than the incumbent, she can set the price (and correspondingly the quality) as high as possible. To avoid this non-realistic case, we shall assume that the quality (and correspondingly the price) of the new product is limited to an upper bound due to technological restrictions.

Lemma 2. If the qualities and prices of both prod ucts are represented as the equations $Q P = Q N + ( z + 1 ) 2 z$ and $p P = p N + z + a$ respectively, where a is a real constant and z∈R+, then the premium player will cap ture a positive share of the market if and only if $1 a − 2 < z$.

Proof I. For the premium product seller to have a positive share of the market we should have $1 x + x v < 1 ⇒ y + x 2 − x y x y < 0$.

Knowing that x, y > 0 we get $y + x 2 − x y < 0 ⇒ x 2 < y ( x − 1 ) ⇒ x 2 x − 1 < y$ for x>1. Substituting z = x−1 we get $( z + 1 ) 2 z < z + a$ which implies $1 a − 2 < z$ for a > 2 .■

Proof II. Using the fact that based on Lemma 1 the P/Q ratios should be equal, we have $p N Q N = p P Q P = p N + z ( z + a ) Q N + ( z + 1 ) 2$ which implies $p N Q N = z ( z + a ) ( z + 1 ) 2$ and since $p N Q N < 1$ we conclude that $a z < 2 z + 1 or 1 / ( a − 2 ) < z$. ■

The profit function of premium product seller would be as follows:

$π 2 = p P ( 1 − p P − p N Q P − Q N − 1 p P − p N )$
(3)

By substituting premium price with its optimal value in Lemma 1 we get:

$π 2 = Q P p N Q N ( 1 − p N Q P Q N − 1 Q P − Q N − 1 p N ( Q P Q N − 1 ) )$
(4)

Knowing that $p P Q P = p N Q N$ the optimal value for the premium price is calculated as follows:

$π 2 = p P ( 1 − p P Q P − Q P p P 1 Q P − Q N ) ⇒ π 2 = p P − p P 2 Q P − Q P Q P − Q N ⇒ ∂ π 2 ∂ p P = 1 − 2 p P Q P = 0 ⇒ p P ∗ = Q P 2$
(5)

The optimal profit value is $π 2 ( p P ∗ ) = Q P [ 1 4 − 1 Q P − Q N ]$. Off course this can be true only if the incumbent player (N) has set her price half of her product or service quality. In other words, equation (5) is just revealing the strategy of incumbent player through which the new entrant would gain the biggest possible profit.

If we consider maximizing the sum of all players’ profits the objective function would be $max p N , p P ≥ 0 π = ∑ S ∈ { N , P } p S Q S$. The demands are clearly expressed based on Figure 2 which leads us to maximizing the following function:

$π = p N ( 1 p P − p N ) + p P ( 1 − p Q − 1 p P − p N ) = p N − p P p P − p N + p P − p P 2 Q P = − 1 + p P − p P 2 Q P , ∂ π ∂ p P = 1 − 2 p P Q P = 0 ⇒ p P ∗ = Q P 2$
(6)

Which also implies that $p N ∗ = Q N 2$. So, when considering the maximization of both profits the resulting prices would end up halving their corresponding qualities.

Let’s assume that the sellers can cross bundle their products (CB = {N, P}) under an alliance, and if the cross bundle is sold each will get a proportion of the profit corresponding to its own product’s quality; e.g. the premium player receives $Q P / ( Q N + Q P )$ times the profit. Like the entrance of the premium product, the introduction of a new product to the market is to be considered in stages. That means after the market has reached a stable state in its previous version (where there were only two competing components; normal and premium product), now a new entrant; the cross bundle, can be introduced to the market.

The sellers are clearly forced to price the bundle lower than the sum of the components’ prices. If the bundle price is set higher than that then the consumers would buy the components separately. Notice that the net utility lines in Figures 2 and 3 are in fact congruent due to the same P/Q ration, but are shown in a separate manner to ease the understanding of how the market is divided based on competition among consecutive offered product packages.

Lemma 3. If the product bundle’s price is less than the sum of the components’ prices then the bundle’s quality will be valued less than the sum of the qualities of the components.

Proof. To catch the most profit for the new entrant, due to Lemma 1 we should have $p B Q B = p N Q N = p P Q P$ which is equivalent to $p N + p P Q N + Q P = p B Q B$ or $Q B Q N + Q P = p B p N + p P$ which implies that if we set $p B ≤ p N + p P$ we get $Q B ≤ Q N + Q P$.■

As much as this lemma can be understood in terms of value perception it can also be formulated as realworld instances. For instance, in the cable TV industry the normal product may include some prevalent local channels and a few international channels, while the premium product can include those international channels along with some premium channels. This way the bundle would include some common channels among both of its components which leads to a lower size than the sum of its components sizes.

Along with its entrance to the market the cross bundle captures a share of each of the products’ markets by the inverse differential of prices due to Lemma 1. The winning end of normal product is always lower than the winning end of cross bundle since pB > pP. But the winning end of premium product is not always lower than the cross bundle. For the single premium product to have a positive share we should have $1 p P − p N < 1 p B − p P$ which implies that $p B < 2 p P − p N$ and by Lemma 3 we get $p B < min { 2 p P − p N , p N + p P }$. Now two states can be considered:

State I. $p B < min { 2 p P − p N , p N + p P }$; the single premium product will have non-zero demand.

In this state, the profit of premium player would be:

$π 2 = p B ( 1 p P − p N − 1 p B − p N ) Q P Q N + Q P + p P ( 1 p B − p P − 1 p P − p N ) + p B ( 1 − 1 p B − p N − p Q ) Q P Q P + Q N = ( 1 p B − p P − 1 p P − p N ) ( p P − p B Q P Q N + Q P ) + p B Q P Q N + Q P ( 1 − 1 p B − p N − p Q )$
(7)

State II.$p B ≥ 2 p P − p N$; the single premium product’s demand will be zero.

$π 2 = p B ( 1 − 1 p B − p N − p Q ) Q P Q N + Q P$
(8)

The second player’s profit will be greater in state 1 than state 2 when $p P − p B Q P Q N + Q P > 0 or p B < p P ( Q N + Q P ) Q P$. Therefore, the second player would be better off if the cross-bundle price is set lower than $min { p P ( Q N + Q P ) / Q P , 2 p P − p N , p N + p P }$.

### 3.1 Incorporating Willingness to Pay

In a realistic case, consumer’s willingness to pay, al so plays a key role in defining the consumer expectancy threshold. Here we use $CETH = ( 1 − θ 2 ) Q P − Q N p P − p N$. To justify using the expression $( 1 − θ 2 )$, remember that consumers with lower willingness to pay are less willing to pay more for an expensive product. Therefore, this coefficient boosts the threshold in a way that consumers with low valuations would not tend to buy the premium product but as the valuation increases the consumers are likely to spend more for the premium product. Notice that, applying a linear modification, (1−θ), would result in losing the strict descending behaviour of the willingness.

Lemma 4. With$CETH = ( 1 − θ 2 ) Q P − Q N p P − p N$, only the con sumers with SWTP equal to$− x y + x 2 y 2 + 4 y x 2 + 4 y 2 2 y$will buy the premium product. In other words, the premium sellers profit would be$p P × y ( 2 − x ) + x 2 y 2 + 4 y x 2 + 4 y 2 2 y$.

Proof. Rewriting the conditions, we get:

$( Q P − Q N ) θ − ( p P − p N ) ≥ ( 1 − θ 2 ) Q P − Q N p P − p N ⇒ y θ − x ≥ ( 1 − θ 2 ) y x ⇒ y θ 2 + x y θ − x 2 − y ≥ 0$
(9)

Because of the positive coefficient of the first statement of quadratic function in left hand side (call it g(θ)), it forms a parabola with a negative global minimum:

$a r g m i n θ ∈ [ 0 , 1 ] = − x y y ⇒ g ( − x ) = − x 2 − y − x 2 y 4 < 0$
(10)

Therefore, we can conclude that the parabola hits positive x-axis by its second/bigger root which is: $SWTP = θ ∗ = − x y + x 2 y 2 + 4 y x 2 + 4 y 2 2 y$ and from this point ahead it will have a positive value; the premium player profits only in this interval. ■

### 3.2 Coopetition: Simultaneous Competition and Cooperation

Some studies have considered coopetition as the presence of both competition and cooperation together in discrete and separate periods of time. In those settings, based on management decisions, the firms are competing in some periods and cooperating in the others. The dynamically discrete nature of coopetition carries useful insights on how sequential decisions can affect both the market and the firms’ outcomes. In this paper we apply simultaneous coopetition which focuses on how the outcomes are spread across the market, assuming that cooperation and competition are happening within the same discrete period of time by displaying their concurrent nature in defining the offered product packages.

To model this situation, we should use a specified demand function which is affected by both products’ performances in the market and their effect on each other’s shares. Here we model the problem under the assumption in which we consider two sellers each providing two vertically differentiated products. The first seller has a high marginal production cost but owns a bigger share of the market. However, her product has been in the market for a long time and a meaningful portion of the consumers would react to a new product which could meet their needs. On the other hand, the second seller provides a better-quality product with lower marginal cost (due to having although small but a more talented expert team). The summary of notations used to model the problem can be found in Table 1. Let’s call the first seller’s product variants normal (NI, NII) and the second seller’s products premium (PI, PII). Certain characteristics of our study are similar to some previous researches (see Banciu et al., 2010, Honhon and Pan, 2016;Zhang and Frazier, 2011). However, in this article we consider two generations of products for each of the sellers. The reason of this doubling is mainly because of the cross-bundling assumption in the problem. If there were only one generation of products then the cross-bundle (being the only possible bundle in that case) would be already a cooperation which would not leave any chance to compete on bundles in the problem.

Demand structure is assumed to be handled by a uniform distribution of consumers’ willingness-to-pay (Debo et al., 2005) in which the demand can be calculated by using a cumulative distribution function based on the market segmentation. Consider a uniform distribution $f ( T ) ∈ [ 0 , 1 ]$ in which the quality rating follows up a subsequent ordering property based on which the bundles always provide a better-quality rating than the components alone (see Figure 4).

All the products presented into the market can be represented by a set multiplication such as $S ∈ { N , P } × { I , I I , B }$. In this case, the profit function can be expressed as in equation (10). Equations (11) and (12) represent the price arbitrage constraints which guarantee that the bundle price would adhere a sub-additive behaviour in terms of the prices for each of the components in the bundle (Banciu et al., 2010).

Since coopetition should be meaningful among various products, we assume that the sellers compete on products with the same generation (as the first or second generation or a bundle). According to Vives (2001), the demand functions are separated based on the fact that each demand substitutability can be measured in terms of its generation peer. If we show the quality ratings of two peer products by Q and Q for the lower and higher qualified products respectively, the substitutability ratio is defined by $Q ¯ Q _ 2 + Q ¯ 2 and Q _ Q _ 2 + Q ¯ 2$ respectively. Since the higher qualified product (the premium) is in a smaller segment of the market, the higher its prices get the more consumers it loses to the other product. The model is formulated as follows:

$max p { N I , P I , N I I , P I I , N B , P B } ≥ 0 π = ∑ S ∈ { N , P } × { I , I I , B } p S Q S$
(11)

Subject to.

$p N B ≤ p N I + p N I I$
(12)

$p P B ≤ p P I + p P I I$
(13)

$Q N I = d N I p P I − p N I$
(14)

$Q P I = d P I p N I − p P I$
(15)

$Q N I I = d N I I p P I I − p N I I$
(16)

$Q P I I = d P I I p N I I − p P I I$
(17)

$Q N B = d N B p P B − p N B$
(18)

$Q P B = d P B p N B − p P B$
(19)

We define the demand function simply as $Q N I = d N I p p I − p N I$. The appellation of the demand substitutability is now clear; other than incorporating the effect of each product corresponding to its nature, the denominators facilitate further calculations i.e. when summing up the demands of a specific generation’s products.

$Q N I + Q P I = ( β α 2 + β 2 p P I − p N I ) + ( α α 2 + β 2 p N I − p P I )$
(20)

Based on lemma 1, we can rewrite prices as $p N I = p Q α$ and $p P I = p Q β$, and calculate each generation market sizes as follows:

$Q N + Q P = ( β 2 α 2 + β 2 p Q − p N ) + ( α 2 α 2 + β 2 p Q − p P ) = p Q − p N − p P$
(21)

Figure 5 shows the combination of the products in market when there is a normal ordering based on quality levels. Consider the order of two products are switched. This change can have profound consequences on the whole concept of the problem. In a way that if we still assume the separation of coopetition for product peers, an overlapping situation will occur in the coopetition in which an outsider will affect the consumers who will be attracted by the specific product peers (see Figure 6). This is inevitable since we are using uniform distribution.

Rewriting the expressions again we find out that this order switching, mostly affects the market size and therefore the demand behaviour on the relevant product peer. Figure 7 shows the overlapping that can occur due to order switching.

The new arrangement adds a shift to the consumer valuation as follows:

$α ( T 5 ⊕ − κ I ) − p N I = β ( T 5 ⊕ + κ I ) − p P I ⇒ ( β − α ) T 5 ⊕ = κ I ( − α − β ) + p P I − p N I ⇒ T 5 ⊕ = p P I − p N I β − α − κ I β + α β − α .$
(22)

## 4. NUMERICAL RESULTS

For start, simply let the initial market include a normal player presenting a product (NI) with quality 10 at price 2.6 i.e. $p / Q = 0.26$. For a new entrant to present a product (PI) with quality of 35 the best possible price then would be 9.1. The second generation of these products (NII and PII) are introduced at qualities of 47 and 55, following the same P/Q ratio and priced at 12.22 and 14.3, respectively. The corresponding zone detachments based on the winnings of products would be at points 0.2600, 0.4138, 0.5805 and 0.7408 for products NI, PI, NII and PII respectively.

### Example (1). Case I: Competitive

In the competitive case, we assume that there are no cooperative agreements among the incumbent and the new entrant. Therefore, the bundles CPII and CNII will not be present at the market. Using the SWTP formula for the case where P/Q ratios are equal, which is $p / Q + 1 / ( p ¯ − p _ ) ( p _ < p ¯ )$ , Table 3 shows the pairwise SWTP calculated for each product or bundle in correspondence to every single product or bundle with lower quality than it.

For instance, the SWTP value regarding NII and PI, 0.5805, means any consumer with WTP higher than 0.5805 will choose to purchase NII over PI. Also notice that the SWTP values higher than 1 can be assumed as 1 without loss of generality. And SWTP values lower than 0.26 (the P/Q ratio of this example) represent consumers who purchase nothing. Considering these modifications, the SWTP values leave us with 14 distinct intervals to analyse the competitive ocean, corresponding winner(s) and their shares, within each. Table 4 shows market’s status in each of these intervals.

Intervals consisting of more than one winner represent a red ocean with higher competitive intensity. In such a situation the profits are divided among the winners through relational to their valued qualities. For instance, in (0.741, 0.809), profits through selling PII and NB’s would be 0.3667 and 0.320, respectively. Finally, Table 5 shows the profits and captured market size by each of the products or bundles.

### Example (1). Case II: Coopetitive

In the coopetitive case, as discussed in Section 3, cooperative-bundles or cross-bundles (CPII and CNII) are introduced to the market under a revenue sharing contractual agreement between the incumbent and the new entrant. Under the agreed coopetition, the pairwise SWTP values of Table 3 will be updated as shown in Table 6. Notice that the SWTP values related to the products introduced before than the cross bundles have not changed.

The new SWTP values this time, lead to 20 distinct intervals to analyse. The optimal status of the market for those intervals is also shown in Table 7.

Finally, Table 8 shows the updated profits and captured market size by each of the products or bundles under the coopetition agreement.

The results show that applying the coopetition via introduction of cross-bundles to the market changes the market’s profit structure substantially. Although the size of the market which contains consumers who purchase a product does not change (0.74=1–0.26), but the captured market by the incumbent decreases from 0.4183 to 0.3890, and the captured market by the entrant increases from 0.3217 to 0.3510. Meanwhile, both the incumbent and the new entrant’s profit increase from 4.3320 and 4.2429, to 6.0233 and 5.8973, respectively (see Figure 8). Therefore, the whole market’s profit increases from 8.5750 in the competition case to 11.9206 in the coopetition case. Notice that the new entrant’s profit remains lower than the incumbent’s profit.

Eventually, we can extract the following insights for a highly sub-additive market in which the products maintain the same P/Q ratio. Notice that the sub-additive attribute of the market is in fact the due to the sensitivity of consumers to pay more for expensive products.

Insights:

1. Introduction of cooperative (cross) bundles will force the incumbent's and the new entrant's direct bundles out of the market completely.

2. By introducing the cooperative bundles, within the same market size, the whole market's profit increases significantly. The cooperative bundle including its cheaper product from the new entrant is more likely to attract consumers than the other. However, the incumbent's profit remains higher than of the new entrant.

3. The individual products’ profit making decreases significantly following the introduction of cooperative bundles, leading to the idea that cooperation within a competitive market between two firms, most likely drives the consumers' attention to the cooperative offers.

4. Interestingly, coopetition, which invites the rivals to cooperate, in contrary leads to an increase in the intensity of competition in the market (the red oceans).

Now consider another sub-additive situation where the bundles quality rating equals $( Q 1 + Q 2 ) a$ such that a < 1. Let a=11/12 and the price and quality of products along with the cooperative bundles be as shown in Table 9. The assortment of products is composed as shown in Figure 9

The profit for each firm can be calculated based on the sharing rule for cooperative bundles and the fact that the market can be divided equally among equivalent products with same maximal utility. These calculations have been performed in Table 10 for each sub-interval. Also, it has been emphasized that some sub-intervals experience a red ocean situation (in this case more than one equivalent product).

Until the valuation (0.567, 0.599], the maximal utility is possessed by product PI, but in the next interval; (0.599, 0.650] a competitive situation occurs. To find the maximal utility we should investigate the intersection of utility functions for the equivalent products in this interval. Since the initial value of net utility for product PII is lower than product PI, we are sure that its utility stays lower until the end of the interval. Therefore, it suffices to just consider NII, PI and CNII (see Figure 10).

The solution offered by Honhon and Pan (2017) maybe optimal for monopoly cases but when considering duopoly, it can represent a very low profit solution, see Table 11 and Table 12. Since upon this type of pricing in this specific example, the only bundle with the highest utility would be CNII, the profit will be shared on the relevant market size based on the corresponding quality ratings respectively. Notice that this kind of growth in the utility of products can be easily investigated under the distance between quality rating and price (QS - pS); clearly the higher this distance the higher utility the consumer would achieve. The profit has significantly diminished under a highly red ocean. This example acts as a contradicting assumption which shows that under duopoly the mentioned solution cannot be optimal.

Finding the exact optimal price which maximizes the profit function in this case is a hard problem but we can catch some insight about the quadratic behaviour of the profit by solving some examples. Consider the normal seller selling a product with quality level 10 at price 3. A new entrant wants to join the market with its product bearing quality level 15 and the consumers consider their valuation instincts in choosing whether to go for the more expensive product or not. By using Lemma 3, we find out that the premium seller can range his price from 3 to 8 and achieve a maximum profit somewhere between (an approximate price level of 5.27), see Figure 11.

Before getting into cooperative bundling, using Lemma 3 now consider all the four products (both generations of each seller’s products) are presented to the market. Let the quality levels of products $N I , P I , N I I , P I I$ be 10, 15, 20 and 25 with prices $3.00,$5.27, $7.93 and$12.20, respectively. Then, the relevant $θ ∗$ values for the pairs $( N I , P I ) , ( P I , N I I ) and ( N I I , P I I )$ would be 0.686, 0.715 and 0.898, respectively. Because the intersection of the utility function of product $NI$ with the x-axis is 0.334, this combination let’s all the products to have a share in the market, Figure 12.

Now consider another example in which the quality levels of products $N I , P I , N I I , P I I$ be 9, 12, 15 and 18 with prices $3.00,$4.00, $5.00 and$6.00, respectively (Figure 13). Here, the proportionality of price to quality has been intentionally set the same which would lead to a simultaneous intersection of utility functions on the x-axis therefore, a same θ value for all consecutive pairs. Now, based on lemma 3, consumers with valuations (0.334, 0.758) will choose to buy NI and consumers with greater valuation than 0.758 will buy product PII; a perfect opportunity to stimulate the consumers to buy the bundle. Knowing this and the rules of profit sharing in cooperative bundling, we get a total of $1.2749 profit for normal seller and a$1.4502 profit for premium seller.

Consider the bundle (NI, PII) which the consumer values sub-additively at level of $22.7 = ( 9 + 18 ) 18 / 19$. It can be shown that at price $8.13 (which is even more than the one third of the valued quality level), we get a value of 0.695 for θ (a sooner triumph than the other products) with$1.9117 profit for normal seller (a 50% increase) and a \$1.6502 profit for premium seller (a 13% increase). Again, the situation is better for the normal seller.

Conjecture. The premium player can only have greater size of the market by bundling, if the consumer values the bundle in a way that the difference with its previous winning product is more than the difference of the winning products quality from its pair (4.700155>3) - also try for QB=21 (for which the difference equals 3 which is the same as the previous one) some small values plus 7 would get a little bit bigger market size. Hence, the rule goes greater and equal. As you can see the bundling can improve both players’ profits.

Now for the actual model, for an unconstrained version of the problem; ignoring capacity and market size boundaries, we can evaluate the intersection of all derivatives to find an optimal vector of prices. Apart from possible zero values for the denominators, there is always a unique solution for optimal prices. The results as shown in Table 13, interestingly alter based on the quality ratings being sub-additive and super-additive.

The random numerical results show that as the quality rating of the bundles increase, the prices of the components increase and relatively at a same reverse behaviour the prices of the bundles decrease. The parameters of the numerical examples were set randomly using uniform distributions and among them only the ones that occur around intersections and reveal insights on the change of behaviour both on market and consumers’ side were reflected. The increase in the component prices will decrease the net utility of the customers who are consuming those products which is clear since the consumers would rather buy the higher quality goods now, see Figure 14.

## 5. CONCLUSION

Being the best in the market is not expected to be a perpetual situation. Soon or later a new entrant will likely emerge in the market with the advantage of innovative ideas and higher quality products. The higher quality offering, targets consumer’s taste of utility. However, providing a better quality product will inevitably lead to setting higher prices. In this paper, we studied simultaneous cooperation and coopetition for markets sensitive to price and utility. Consumers were considered rigorous on spending their money on new products priced at a higher level. Due to conflicting settings in the market, the competitors were left with no choice than to consider cooperation alongside of the other options. The findings were remarkable and indicated that cooperative bundling can benefit both the new entrant and the incumbent. The result surprisingly remained the same for when the consumers assumed the bundle value under sub-additive quality ratings. The consumer expectancy threshold level was introduced to model the sensitivity of consumers' willingness-to-pay on the higher priced product. The threshold implied that consumers expect a surplus on the additional money they are willing to spend. This was also reflected in a quadratic behaviour of the new entrant’s profit. The behavior also reflected that the optimal share of the market may not be achieved by the price at which the optimum profit is gathered. Our assumptions helped configure a new type of market in which sellers together can present their products’ bundles directly and indirectly (cooperative/cross bundles).

Our findings showed that coopetition can drive consumers’ attention to purchase cross-bundles instead of individual products or direct bundles. The advantage of the new entrant also led to the fact that this coopetition is indeed in the benefit of both sides. Since our model’s structure was considered in stages, there can be many interesting ways for further developments of the model to be accomplished. Therefore, we encourage researchers to generalize the model and study other aspects. Some recommendations are as follows: 1) Cooperation was considered only through cooperative bundles. Revenue sharing contracts can also be coordinated among the sellers for better profit and market share achievements. Also information sharing is an available strategy. The normal seller can use the premium seller’s knowledge on more efficient production processes. Meanwhile, as the paper’s results show, the new entrant will also enjoy better market position brought by the cooperative activities. 2) Different market power structures can influence the way the players would behave.

## Figure

New-entrant and the different segmentations that are formed by the coopetition.

Divided market following the introduction of the premium product.

Divided market following introduction of the cross bundle to the previous stable market.

Market segmentation for two sellers’ product components and bundles.

Coopetition parameter shows the interval in which there exist potential consumers who are indifferent to the relevant product peers.

Switching quality rating orders can lead to overlapping coopetition.

Potential consumers that are indifferent to different product peers will experience an overlap with the other sets when order switching occurs.

Comparing the outcome and status of the incumbent and the new entrant for both competitive and coopetitive cases.

Numerical example 2 product assortments in the unit interval.

Intersection of utilities for equivalent products in numerical example 1; a subinterval of (0.599,0.650] in which the intersections occur.

Captured share of market and gained profit by premium seller on different values of her price

An example in which all the products gain a positive share of the market.

An example with same proportionality of price and quality for all products.

Comparing components and bundle pricing behaviour when quality ratings of bundles increase.

## Table

Nomenclature

Numerical example 1 prices and quality ratings of products and bundles

Numerical example 1’s competitive case pairwise Switching Willingness to Pay (SWTP)

Numerical example 1’s competitive case optimal market status; ocean, size and winners within distinct intervals

Numerical example 1’s competitive case optimal profits and captured market by each product

Numerical example 1’s coopetitive case pairwise switching willingness to pay (SWTP)

Numerical example 1’s coopetitive case optimal market status; ocean, size and winners within distinct intervals

Numerical example 1’s coopetitive case optimal profits and captured market by each product

Numerical example 2 prices and quality ratings

Numerical example 2 prices and quality ratings

Numerical example 3 price and quality: applying Honhon and Pan (2017) solution; when prices halve quality ratings

Numerical example 3 profits: applying Honhon and Pan (2017) solution; when prices halve quality ratings

Optimal price vectors for different quality ratings under various scenarios of additivity

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