Strategic pricing model based on genetic algorithm: The case of electronic publishing market

Abstract

The electronic-book (e-book) is one of the new technological changes that has significantly influenced the publishing industry in the last century. This has forced publishers to reconsider their distribution channels, since the Internet has provided a new means with which to serve readers. In this paper, a strategic market analysis is proposed from the perspective of a traditional publisher that needs to decide whether to switch to e-publishing business. The analysis model determines the publishing market equilibrium in three different market scenarios. Besides, it shows the impact of readers’ choices and price sensitivities on the publishers’ profits. The proposed decision support model has its basis on game theory and it is built in an oligopoly setting to reflect the severe market competition. The readers’ utilities and demands are modeled using the multinomial logit (MNL) model. Although the first scenario possesses a global optimum solution, in the remaining two market scenarios genetic algorithms (GAs) are used in order to find the solutions of the oligopolies. Numerical applications reveal the industry equilibrium point, where the sum of the profits of all publishers in the market is maximum.

Keywords

Pricing decision game theory competitive market model genetic algorithm channel management electronic book dual channel

1 Introduction

Traditionally, the term “publishing” refers to the distribution of printed works such as books and newspapers. With the advent of digital information systems and the Internet, the scope of publishing has expanded to include electronic resources, such as the electronic versions of books and periodicals, as well as micropublishing, websites, blogs or video games. In this work, an electronic book (e-book) will be considered as the digital version of a traditional printed book (p-book) to be read digitally on a personal computer, handheld computer, PDA or a dedicated e-book reader [1, 2]. Although a group of very traditional readers refuses to read the books on electronic environment, e-books have many advantages over p-books. First of all, e-books are quickly downloadable from the Internet at lower prices. A reader can carry more than one book, even a library, with him. Moreover, it is easy to delete the book after reading it. As there are not any print or stock related costs, the e-book is environment friendly and the cost of an e-book is lower. These advantages have directed publishers to seriously consider e-publishing. Since April of 2011, e-book sales have outsold printed books on Amazon.com [3] and continue to grow throughout the marketplace [4]. It has been estimated that e-book sales will account for 50% of the publishing industry’s sales by 2020 and 90% by 2030 [2]. Like all the other new technologies, e-publishing can provide significant economic and social benefits only if it becomes widely available, in other words, it is commercially successful.Ojala, (1998) stated that ‘The greatest enigma in the online world remains pricing and it’s amazing that, after a quarter-century of information being sold online, no one has a definitive pricing model [5].’ In this regard, investigating the economic issues that arise due to the presence of multiple publishers is an important research area.

In this paper, a strategic marketing analysis framework in a publishing market is proposed in the presence of multiple competing publishers. The proposed publishing market consists of p-publishers (publishers of p-books) that try to decide on whether or not switch to e-publishing (publishing e-books). The proposed decision support model addresses the following research question: ‘How much should a p or e-publisher prices books in different competitive market scenario?’ The model computes the unit prices, and accordingly the profits of the publishers in three different market scenarios. Since the publishers need to make their decision in an uncertain market environment, the customer utility and demand are modeled using the multinomial logit (MNL) model. For each scenario, a non-cooperative pricing game is built, whose players are the publishers. Solving the game, the mutual best response strategies that determine the equilibrium point(s) are studied.

The rest of the paper is organized as follows.Section 2 discusses the related work in the literature and their differences from this one. Section 3 gives brief introductions to the two main methodologies that constitute the framework. In Section 4, the formulation of the model, possible market scenarios and their demonstrative examples are given in detail. The results are discussed at the end of the Section 4. Finally, conclusions and future work are given in Section 5.

2 Related work

In literature, the GA has been applied to a variety of field, frequently as a part of decision support systems. A comprehensive representation of GA, including encoding, adaptation and genetic optimizations, can be found as a sourcebook [6]. The authors provides with an in-depth coverage of several problems, such as reliability design, scheduling and network design routing. In one of the recent works, the authors used a hybrid system by evolving a case-based reasoning with GA for wholesaler’s returning book forecasting [7]. In literature, it is possible to encounter the use of GA into the game theory-based models. These works are from various research areas. Riechmann, (2001) has shown that economic learning via GA can be described as a specific form of an evolutionary game [8]. In his paper, he pointed out that GA learning results in a series of near Nash equilibrium (NE). In another research, the authors discussed a new evolutionary strategy for the multiple objective design optimization of internal aerodynamic shape [9]. They claimed that game theory replaces a global optimization problem by a non-cooperative game based on NE with several players solving local constrained sub-optimization tasks. The authors stated that game theory is not only the primary method for the formal modeling of interactions between individual, but it also underlies how biologists think about social interactions on an intuitive level [10]. They used GA as an alternative method of searching evolutionary stable sets in a well-studied game of biological communication.

There are several research in the literature on the adoption of e-books and e-publishing; however this paper’s concern is a more specific area. The concentration is on the economics and management aspects of e-publishing. The most relevant research are as follows: Jiang and Katsamakas, (2010) examined how the entry of an e-book seller affects strategic interaction in the book markets and impacts sellers or consumers [11]. Their work is a good example of the application of game theory in analyzing the market asymmetries. The research of Hua et al. (2011) has the same research question as the one in this paper [2]. In their work, the authors derived the conditions under which a publisher should sell only p-books, only e-books, and both of them simultaneously. They used the newsvendor model to analyze demand behavior; whereas the demand was modeled using MNL model in this paper. As the demand varies linearly on offered price, they determined the closed-form expression of optimum prices. In this paper, the proposed model considers the pricing strategies of competitors when defining unit prices; hence it necessitates solving the complicated problems using nonlinear techniques. Bernstein et al. (2010) used the MNL model for the equilibrium analysis of retailers [12]. They differentiated retailers’ choices as bricks-and-mortar and clicks-and-mortar, which represent the traditional retail channel and Internet channel, respectively. Their study has some common grounds with the one in this paper, since they analyze the supply chain channel structure choice in an oligopoly setting. In fact, their idea of switching from bricks-and-mortar to clicks-and-mortar is adapted to e-publishing business. A set of mathematical models are examined and different pricing and launch strategies of e-books are compared in [13]. This article has the similar research question, but the authors have not considered the competitors’ pricing strategies. They conducted sensitivity analysis to investigate the impacts of different copyright, launch modes and channels distribution on the publisher’s pricing options.

3 Techniques used in the proposed framework

3.1 Game theory

Game theory has been recognized as a cornerstoneof micro-economics that can be applied to analyze problems with conflicting objectives and interactivedecision makers [14]. It provides us well defined equilibrium criteria to measure game optimality under various scenarios [15]. The ability of game theory to model individual, independent decision makers whose actions potentially affect all other decision makers makes game theory particularly attractive to analyze the performance of such types of situations.

A Nash equilibrium is a strategy profile where no player can improve its utility with a unilateral deviation [16, 17]. It corresponds to the steady-state of the game and is predicted as the most probable outcome of the game [18]. The game structures in which the players do not have the option of planning as a group in advance of choosing their actions are called non-cooperative games. It does not mean that the players cannot cooperate, but any cooperation must be self-enforcing.

3.2 Genetic algorithm

The genetic algorithm is a search heuristic that is inspired from the process of Darwinian evolutionary ideas of natural selection and genetic [19, 20]. It has been recognized as a general search strategy and an optimization method. GA begins with a set of k randomly generated states/chromosomes which is called the population. Each chromosome, or each individual in the population, is represented as a string over a finite alphabet (most commonly, a string of 0 s and 1 s) and each chromosome encodes a solution of the problem. New generations are produced using a fitness function that is an evaluation function that the chromosomes are rated in each state. Better state returns higher fitness function values and the objective is to convey more successful/ fitter chromosomes to new generations. The fitness is used to judge how much a solution is near the optimum solution. Two genetic operators, crossover and mutation, are utilized to produce a new generation. For each new population (solution), a pair of parent solutions is selected for breeding a new generation. Crossover may be considered as the main engine for exploration in the GA. The idea behind crossover is that the new chromosome may be better than both of the parents if it takes the best characteristics from each of the parents. If all generations are produced by using the crossover operator, there may be the same value for the same gene in all the chromosomes. Mutation operator is used to create some genetic diversity by altering one or more gene values in a chromosome from its initial state. The objective of mutation operator is to prevent remaining into a local optimum point.

4 The proposed framework

An n-firm oligopoly setting is considered to study the structure of the game [17]. In the proposed game, the publishers that sell their books through a retail store (p-publishers) want to reach more reader by publishingtheir books on an Internet channel (e-publishers) (Fig. 1). The e-publisher _i will continue to sell its books on the retail stores; hence it has to define two different prices: a unit retail price p _i for its traditional channel and a unit online price $p_{e i}$ for its Internet channel. The p-publisher_i needs to define only its unit retail price p _i. As the stocking and maintenance costs of an e-book are assumed to be lower than the ones of a traditional book, the following assumption on the prices is set: p _ei ≤ p _i. A = {1, 2, … , n} ∪ A ₀ denotes the set of publishers. A ₀ represents the non-purchasingalternative.

4.1 Customer utility model

A reader is assumed to derive a different utility when obtaining the book from a retailer’s physical store (alternative i) than obtaining it in an electronic form (alternative ei). Furthermore, a reader is assumed to have a no-purchase alternative (A ₀). In other words, if s/ he does not like any offer, s/ he will not buy any book. Then, the set of alternatives is A ^p = {1, 2, … , n} ∪ A ₀ when all the publishers sell from their retail stores, while it is A ^E = {1, e1, 2, e2, … , n, en} ∪ A ₀ when all publishers sell both from their retail stores and their online stores. It is also possible to have a case with k e-publishers and (n-k) p-publishers, then the set of alternatives is A ^EP (k) = {1, e1, … , k, ek, k + 1, k + 2 … , n}.

The customer utility is modeled using the multinomial-logit model (MNL). The MNL model is one of the random-utility models that are based on a probabilistic model of individual customer utility [21]. The random utility models assume that a firm has only probabilistic information on the utility function of any given customer, and this can be modeled by assuming that customers’ utilities for alternatives are themselves random variables.

For the MNL model, the probability that an alternative i is chosen from a set A ⊆ {1, 2, … , n} that contains i is given by: $P_{i} (A) = e^{\frac{u_{i}}{μ}} \sum_{j = 1}^{n} e^{\frac{u_{j}}{μ}}$ (1) where P _i (A) is the probability that a customer selects alternative i from a subset A of alternatives, u _i is the utility that customer has for alternative i, μ is a scale parameter.

The logit demand function is based on the MNL model. The no-purchase alternative of a customer is considered with utility U ₀. It is common to model u _i as a linear function of several known attributes, including the price. Assuming the representative component of utility u _i is linear in price and interpreting the choice probabilities as fractions of a population of customers of size M lead to the class of logit-demand functions [21].

In the multiple-product case, by considering each user of the same type (b _i = b), market size (M), price (p) and coefficient of the price sensitivity (b), the demand function of publisher _i is given by:

$\begin{matrix} d_{i} (p_{1}, p_{2}, \dots, p_{n}) \\ = M \frac{e^{- {bp}_{i}}}{1 + \sum_{j = 1}^{n} e^{- {bp}_{j}}}, i = 1, \dots n \end{matrix}$ (2)

The MNL probability that a customer chooses product j as a function of the vector of prices p = {p ₁, p ₂, …, p _n} is then given by: ${Prob}_{j} (p) = \frac{e^{- {bp}_{j}}}{1 + \sum_{i = 1}^{n} e^{- {bp}_{i}}}$ (3)

4.2 Choice of channel structure

In the proposed strategic analysis, each p-publisher that has an incentive to publish its books on an electronic environment, in other words that has an incentive to move to e-publishing business, faces three marketing scenarios:

P-P competition: All publishers in the market are p-publishers.

E-P competition: Some publishers remain asp-publishers, but the rest moves to e-publishing.

E-E competition: All publishers in the market are e-publishers.

The list of notation for the equilibrium analysis under different scenarios is given as:

$p_{i}^{PP}$ : Eq. price of p-publisher_i under P-P competition

$p_{i}^{EP}$ : Eq. price of p-publisher_i under E-P competition

$p_{ei}^{EP}$ : Eq. price of e-publisher_i under E-P competition

$p_{i}^{EE}$ : Eq. price of p-publisher_i under E-E competition

$p_{ei}^{EE}$ : Eq. price of e-publisher_i under E-E competition

$Π_{i}^{PP}$ : Eq. profit of p-publisher_i under P-P competition

$Π_{i}^{EP}$ : Eq. profit of p-publisher_i under E-P competition

$Π_{ei}^{EP}$ : Eq. profit of e-publisher_i under E-P competition

$Π_{ei}^{EE}$ : Eq. profit of e-publisher_iunder E-E competition

4.2.1 Scenario I: P-publisher vs. p-publisher (P-P Competition)

In this scenario, the set of alternatives for readers is A ^P = {1, 2, …, n} ∪ A ₀. All p-publishers simultaneously set their prices. Each p-publisher’s aim is to define its optimum price $(p_{i}^{PP})$ in the given market environment that maximizes its profit:

$\begin{matrix} max_{p_{i}^{PP}} (Π_{i}^{PP}) = max_{p_{i}^{PP}} (p_{i}^{PP} - c_{i}) (M \frac{e^{- {bp}_{i}^{PP}}}{1 + \sum_{t = 1}^{n} e^{- {bp}_{t}^{PP}}}) \\ s . t . p_{i}^{PP} - c_{i} \geq 0, \forall i \\ p_{i}^{PP} \geq 0, \forall i \end{matrix}$ (4) where b is the price elasticity and c _i is the unit cost of p - publisher _i. The probability that p - publisher _i with the price $p_{i}^{PP}$ is chosen by a customer is given as: ${Prob}_{i}^{PP} = \frac{e^{- b_{p_{i}}^{PP}}}{1 + \sum_{t = 1}^{n} e^{- {bp}_{t}^{PP}}}$ (5)

The first order condition of the choice probability with respect to its price $(\partial {Prob}_{i}^{PP} / \partial p_{i}^{PP})$ is negative, which means that the price increase of ap-publisher _i reduces its own demand; whereas the first order condition with respect to its competitor’s price $(\partial {Prob}_{i}^{PP} / \partial p_{t}^{PP})$ , i ≠ t is positive, which means that the price increase of the competitor’s price increase the demand of p-publisher _i.

As p-publishers determine their prices simultaneously, they need to consider the competition, i.e. the prices offered by other p-publishers in their market. The problem is modeled as a game where the players are the p-publishers, the strategies of the players are their offered unit prices and the payoffs of the players are their profit functions. Solving such a game means predicting the strategy of the publisher. One can see that if the strategies from the players are mutual best responses to each other, no player would have to deviate from the given strategies and the game would reach a steady state. Such a point is called the NE point of the game [16]. In the game, p-publishers determine their prices independently and the information is strictly limited to local information. Hence, the game has a non-cooperative setup.

Global optimality conditions are used in order to analyze the existence and the uniqueness of the equilibrium point. A convex optimization problem is defined as a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing [22]. With a convex objective and a convex feasible region, there can be only one optimum solution, which is globally optimum. Furthermore, a function f is concave if – f is convex. The constraints in the proposed problem (4) are linear, so they are convexes. As a firststep, the profit maximizing objective function of a p-publisher is converted to a convex minimization problem form by taking its opposite: $min_{p_{i}^{PP}} (- Π_{i}^{PP})$ . With the constraint $p_{i}^{PP} \geq c_{i}$ , we found that $\frac{\partial^{2} (- Π_{i}^{PP})}{\partial (p_{i}^{PP})^{2}} \geq 0$ . Therefore, the vector $p = [p_{1}^{PP}, p_{2}^{PP}, \dots, p_{n}^{PP}]$ denotes the solution (the NE) of this gamewith: $p_{i}^{PP} = {BR}_{i} (p_{- i}^{PP})$ (6) where $p_{- i}^{PP}$ represents the vector of best responses of all p - publisher _t, t ≠ i. The NE is the point that solves the set of equations: $\frac{\partial Π_{i}^{PP}}{\partial p_{i}^{PP}} = 0, \forall i$ . It is also the global optimum of the given problem.

This scenario is demonstrated on a simple but representative example with two p-publishers in the market. The model parameters are set as in Table 1. The target customer group is assumed to be consist of M = 100 readers. The unit costs of two p-publishers are assumed to be the same and equal to 1. The price sensitivities in the readers’ demand functions are differentiated in order to analyze the impact of readers’ price sensitivities on the publishers’ price determination. For the first scenario, the equilibrium price values and related demands and profits are given in Table 2.

As the price sensitivity of the customers of p - publisher ₂ is set higher than the one of p - publisher ₁, p - publisher ₂ offers a lower price ( $p_{2}^{PP} = 1.7123$ ) in the equilibrium. In Fig. 2a, the price sensitivity b ₁ is fixed and b ₂ is increased gradually. With the increase of the price sensitivity of its readers, p - publisher ₂ is obliged to decrease its offered price ( $p_{2}^{PP}$ ). P - publisher ₂ resists until b ₂ = 2.1; after that $p_{2}^{PP}$ increases dramatically to 23.68, which can be interpreted as p - publisher ₂ is retreated from the competition. The variations of the offered prices have direct impact on the profits of p-publishers (Fig. 2b).

In Fig. 3a and b, the profits of two p-publishers and the reaction of p - publisher ₁ are analyzed in respect to p - publisher ₂’s price. During the simulations, the $p_{2}^{PP}$ is increased incrementally and every time, the proposed model calculates the optimum price of p-publisher ₁ and accordingly the profits of both p-publishers. In Fig. 4, the point that maximizes the total profit of two p-publishers represents the equilibrium point where $p_{1}^{PP} = 2.1123$ and $p_{2}^{PP} = 1.7123$ . As p - publisher ₂ increases its price, its competitor p - publisher ₂ finds the chance to increase its price.

4.2.2 Scenario II: E-publisher vs. p-publisher (E-P Competition)

In the second scenario, the first k (1 ≤ k ≤ n) publishers are assumed to be move on e-publishing, whereas the remaining n - k publishers are stayed as p-publisher. The set of alternatives for consumers is A ^EP (k) = A ^P ∪ {e1, e2, … , ek}.

In this scenario, p - publisher _i determines only one price ( $p_{i}^{EP}$ ), but e - publisher _i determines both a price for its traditional channel ( $p_{i}^{EP}$ ) and a price for its Internet channel ( $p_{ei}^{EP}$ ). The p-publisher’s price is assumed to be influenced from other p-publisher’s prices, whereas the e-publisher’s price is influenced from both other e-publishers’ prices and from the price of its own traditional channel. In other words, if e - publisher _i increases its Internet channel price ( $p_{ei}^{EP}$ ), the demand to its traditional channel increases. From this point of view, two channels of an e-publisher can be considered as ‘competing’ [12]. This property is also visible from the own and cross-partial derivatives of the choice probabilities. The first order condition of the choice probability of p - publisher _i with respect to its traditional channel price $(\frac{\partial {Prob}_{i}^{EP}}{\partial p_{i}^{EP}})$ is proved to be negative, which means that the price increase of p - publisher _i’s traditional channel price reduces the demand to its traditional channel. The first order condition with respect to its Internet channel price $(\frac{\partial {Prob}_{i}^{EP}}{\partial p_{ei}^{EP}})$ is proved to be positive, which means that the price increase of its Internet channel price increases the demand to its traditional channel. It is possible to make similar interpretations for the negativity of $\frac{\partial {Prob}_{ei}^{EP}}{\partial p_{ei}^{EP}}$ , and the negativity of $\frac{\partial {Prob}_{ei}^{EP}}{\partial p_{i}^{EP}}$ . Both type of publishers’ aim is to define their optimum prices ( $p_{i}^{EP} and p_{ei}^{EP}$ ) that maximize their profits. P - publisher _j wants to maximize its profit:

$\begin{matrix} max_{p_{j}^{EP}} (Π_{j}^{EP}) \\ = max_{p_{j}^{EP}} (p_{j}^{EP} - c_{j}) (M . \frac{e^{- b . p_{j}^{EP}}}{1 + \sum_{t = k + 1}^{n} e^{- b . p_{j}^{EP}}}) \\ s . t . p_{j}^{EP} - c_{j} \geq 0, j = k + 1, k + 2, \dots, n \\ p_{j}^{EP} \geq 0, j = k + 1, k + 2, \dots, n \end{matrix}$ (7)

On the other hand, the objective of e-publisher_i is to choose $p_{i}^{EP}$ and $p_{ei}^{EP}$ that maximizes its own profit:

$\begin{matrix} max_{p_{ei}^{EP}} (Π_{ei}^{EP}) = max_{p_{ei}^{EP}} (p_{i}^{EP} - c_{i}) \\ (M . \frac{e^{- b . p_{i}^{EP}}}{1 + \sum_{t = 1}^{n} e^{- b . p_{t}^{EP}} + e^{- ϑ b . p_{ei}^{EP}}}) \\ + (p_{ei}^{EP} - c_{ei}) (M . \frac{e^{- ϑ b . p_{ei}^{EP}}}{1 + \sum_{t = 1}^{k} e^{- ϑ b . p_{et}^{EP} + e^{- b . p_{i}^{EP}}}}) \\ s . t . P_{i}^{EP} - c_{i} \geq 0, \forall i \\ P_{ei}^{EP} - c_{ei} \geq 0, \forall i \\ P_{i}^{EP} \geq 0, P_{ei}^{EP} \geq 0, \forall i \end{matrix}$ (8)

The first term of the profit function belongs to the profit earned from p-publishing, while the second term belongs to the profit earned from e-publishing. In the proposed model, the coefficient ϑ ≥ 1 is inserted to the demand function because an e-reader’s sensitivity to price is assumed to be higher than the one of a p-reader. Similar to the previous scenario, all the publishers are assumed to determine their unit prices simultaneously. The problem is modeled as a non-cooperative game. The vector $p = [p_{1}^{EP}, p_{e 1}^{EP}, \dots, p_{k}^{EP}, p_{ek}^{EP}, p_{k + 1}^{EP}, p_{k + 2}^{EP}, \dots, p_{n}^{EP}]$ denotes the solution (the NE) of this game for: $p_{i}^{EP} = {BR}_{i} (p_{- 1}^{EP}) and p_{ei}^{EP} = {BR}_{i} (p_{i}^{EP}, p_{- ei}^{EP})$ (9) where $p_{- i}^{EP}$ represents the vector of best responses of all p-publisher_t, t ≠ i, and $p_{- ei}^{EP}$ represents the vector of best responses of all e-publisher_t, t ≠ i. The NE is the point that solves the set of equations: $\frac{\partial Π_{j}^{EP}}{\partial p_{j}^{EP}} = \frac{\partial Π_{ei}^{EP}}{\partial p_{ei}^{EP}} = 0, \forall i, j .$ (10)

In this setting, it is not possible to derive closed-form expressions for the equilibrium prices, demands and profits. The convexity of the maximization problem in this scenario cannot be demonstrated. Furthermore, the increase of the number of publishers in the problem increases its complexity. Therefore, the GA is used as a computing technique to find solutions. Due to the probabilistic nature of the solution, the GA does not guarantee optimality even when it may be reached. However, they are likely to be close to the global optimum [23]. The general structure of the GA is summarized inFig. 4.

Chromosome representation: A chromosome represents a solution to the problem and is encoded as a vector of random keys. In a direct representation, a chromosome represents a solution of the original problem (genotype), whereas in an indirect representation a chromosome does not represent a solution of the original problem, it needs special procedures (phenotype) [24]. A GA spends most of its time on evaluating these solutions which increases the importance of chromosome representation. For the given problem, the chromosomes are directly represented and built as real-valued genes [25, 26].

The decision variables in the given problem are the unit prices offered by publishers. The issue that necessitates some engineering in chromosome representation and crossover technique is the fact that a p-publisher needs to determine only one type of price, whereas an e-publisher needs to determine two types of prices. In Fig. 5, a chromosome that composes of two types of gene is generated as a demonstrative purpose. Each gene in the chromosome is assumed to compose of two parts: The first for p-publishing price and the second for e-publishing price. Since a p-publisher does not offer any e-publishing price, the second value of p-publisher’s gene is defined as zero.

Fitness function: Given a particular chromosome, the fitness function returns a single numerical value, which is supposed to be proportional to the utility or the ability of this chromosome. Each publisher in this scenario tries to maximize its own profit. The higher the profit, the higher the fitness value should be. Therefore, the profit functions can be considered as the fitness functions. Each gene in a chromosome If belongs to a publisher whose type is known a priori.

The gene belongs to a p-publisher, its profit value is calculated using Equation (7), whereas if it belongs to an e-publisher, its profit value is calculated using Equation (8). Then, fitness value of a chromosome is the sum of the profit values of each gene.

Population size: It is obvious that smaller population sizes result in shorter CPU time, but larger population sizes explore enough of the solution space and consistently find good solutions. In the GA implementation, two different population sizes are chosen (100, 150) after realizing some investigations on the solution quality and computation time.

Initial population generation: The starting population of chromosomes of the appropriate length is randomly generated.

Parent selection: Selection process is defined as selecting the chromosomes on which to base the next generation. It is a random process with priority being given to chromosomes that are most fit. The Roulette Wheel selection technique, also called stochastic sampling with replacement, is used [27]. In this technique, the chromosomes are mapped to contiguous segments of a line where each chromosome’s segment is proportional in size to its fitness value [28]. A random number is generated and the chromosome whose segment spans the random number is selected. This procedure is repeated until the mating population is obtained.

Crossover: The crossover is responsible for the information exchange between mating chromosomes and the convergence speed of the GA. It is to form new chromosomes that inherit segments of information stored in parent chromosomes. The probability that the crossover operation is applied to a particular chromosome during a generation is defined as the crossover rate. For this research, the one-point crossover is used, the crossover rate is set to 0.9 and the crossover operator generates two offsprings from each pair. An example of one-point crossover for this scenario is illustrated as follows:

Two offsprings created from Chromosome ₁ and Chromosome₂:

Mutation: The mutation is a genetic operator that alters one or more gene values in a chromosome from its initial state. This allows having completely new chromosomes in the population with whom it is possible to arrive at better solutions. This operator helps to prevent the population from stagnating at any local optimum. The mutation is carried out according to the mutation probability, which is set to 0.01 for this research. The mutation operator is defined as decreasing 50% of a randomly selected gene value.

Replacement: Combining the newly reproduced population with the initial one, the population size is doubled at the end of each crossover process. For replacement, the parents and their offsprings are sorted according to their fitness values and the best half of these chromosomes are carried to the next generation.

Stopping conditions: The stopping criterion determines when the genetic process stops evolving. For the GA, one of the most frequently used stopping criteria is the specification of a maximum number of generations. In the GA implementation of the scenario, the maximum number of generations is defined as 250. Another frequently used stopping criterion is to terminate the evaluation if there is not any change to the population’s best fitness values for a specified number of generations. This number is set to 50 in this scenariosetting.

For each population size, the GA is run 50 times and the best result is chosen from these 50 results. The second scenario is demonstrated on an example with e-publishe r ₁ and p-publisher ₂ in the market. The model parameters are set as in Table 3. The representative chromosome with the first gene belonging to e-publishe r ₁ and the second gene belonging to p-publisher ₂ for this case is as follows:

For this scenario, the equilibrium price values and related demands and profits are obtained as in Table 4. The results at the equilibrium point confirm that the e-publishing price is lower than the p-publishing prices because of lower publishing costs. E-publishe r ₁ reaches bigger market share, which is proportional to its total profit, since it offers two different publishing channels for different preferences.

In Fig. 6a and b, the profits of two types of publishers and their offered prices are analyzed in respect to the price sensitivity of customers of e-publishe r ₁. It is obvious that, as the price sensitivity of e-reader increases, e-publishe r ₁ is obliged to decrease its offered price ( $p_{e 2}^{EP}$ ) to retain its e-readers. Besides, the p-publishing price of e-publishe r ₁ shows a decreasing trend, since it tries to compensate its demand loss by p-publishing. E-publishe r ₁ resists until ϕ _e1. b ₁ = 2.3; after that point its price increases dramatically which can be interpreted as e-publishe r ₁ is retreated from the competition. In the market there is only one e-publisher, so its e-publishing price decrease does not influence dramatically the demand to p-publisher ₂. Therefore, the profit of e-publishe r ₁ shows a decreasing trend, whereas the one of p-publisher ₂ remains nearly constant.

In Fig. 7a and b, the profits of two p-publishers and the reaction of p-publisher ₁ are analyzed in respect to p-publishe r ₂’s price. For each value of $p_{2}^{EP}$ , the proposed model calculates the equilibrium prices and accordingly the profits of two publishers. The equilibrium point where two publishers’ profits are maximized is shown in Fig. 7b.

Figure 8a and b are drawn to show the robustness of the GA. Figure 8a illustrates how best and mean fitness values changes in respect to generations. The algorithm stops at iteration 232, since it finds the same value at the last 50 iterations. Figure 8b shows how the distances between chromosomes change in respect to generations. It is possible to observe the convergence of the chromosomes to the optimum solution.

4.2.3 Scenario III: E-publisher vs. e-publisher (E-E Competition)

In the last scenario, it is assumed that all n publishers in the market adopt e-publishing. The set of alternatives is then A ^EE = {1, e1, 2, e2, …, n, en} ∪ A ₀. All e-publishers determine two prices: A price for their traditional channel ( $p_{i}^{EE}$ ) and a price for their Internet channel ( $p_{ei}^{EE}$ ). Two channels of each e-publisher can be considered as ‘competing’, just like in the previous scenario.

Each e-publisher’s aim is to define its optimum prices ( $p_{i}^{EE}$ and $p_{ei}^{EE}$ ) in the given market environment that maximizes its own profit:

$\begin{matrix} max_{p_{i}^{EE}, p_{ei}^{EE}} (Π_{i}^{EE}) = max_{p_{i}^{EE}, p_{ei}^{EE}} (p_{i}^{EE} - c_{i}) \\ (M . \frac{e^{- b . p_{i}^{EE}}}{1 + \sum_{t = 1}^{n} e^{- b . p_{t}^{EE}} + e^{- ϑ b . p_{ei}^{EE}}}) \\ + (p_{ei}^{EE} - c_{ei}) (M . \frac{e^{- ϑ b . p_{ei}^{EE}}}{1 + \sum_{t = 1}^{n} e^{- ϑ b . p_{et}^{EE}} + e^{- b . p_{i}^{EE}}}) \\ s . t . p_{i}^{EE} - c_{i} \geq 0, \forall i \\ p_{ei}^{EE} - c_{ei} \geq 0, \forall i \\ p_{i}^{EE} \geq 0, p_{ei}^{EE} \geq 0, \forall i \end{matrix}$ (11)

The problem in the third scenario is again modeled as a non-cooperative game. Now, the vector $p = [p_{1}^{EE}, p_{e 1}^{EE}, \dots, p_{k}^{EE}, p_{ek}^{EE}, \dots, p_{n}^{EE}, p_{en}^{EE}]$ denotes the solution (the NE) of this game for: $p_{i}^{EE} = {BR}_{i} (p_{- i}^{EE}, p_{ei}^{EE}) and p_{ei}^{EE} = {BR}_{i} (p_{i}^{EE}, p_{- ei}^{EE})$ (12) where $p_{- i}^{EE}$ represents the best responses ofp-publishing prices of all e-publisher_t, t ≠ i, and $P_{- ei}^{EE}$ represents the vector of best responses of e-publishing prices of all e-publisher_t, t ≠ i. The NE is the point that solves the set of equations:

$\frac{\partial Π_{i}^{EE}}{\partial p_{i}^{EE}} = \frac{\partial Π_{ei}^{EE}}{\partial p_{ei}^{EE}} = 0, \forall i$ . Since the convexity of the maximization problem cannot be demonstrated, the sub-optimal solutions are again computed using the GA (Fig. 4).

The GA settings for the second scenario are valid for this third scenario also. In this scenario, all the publishers in the market are e-publishers. Hence, the chromosomes for e-publishers are defined as in Fig. 9. For this case, the fitness value of a chromosome is the sum of the profit values of each gene which are calculated using Equation (11). It is assumed that there are two e-publishers (e-publishe r ₁ and e-publisher ₂) in the third scenario (Table 5). The equilibrium price values and related demands and profits are given in Table 6.

Both e-publishing and p-publishing prices ofe-publishe r ₂ are lower than the ones of e-publisher ₁. The reason is that both types of readers (e-readers andp-readers) of e-publishe r ₂ are more sensitive to price than the readers of e-publisher ₁. E-publishe r ₂ is obliged to hold its prices down in order to grab more readers. In Fig. 10a and b, the profits of e-publishers and their offered prices are analyzed in respect to the price sensitivity of customers of e-publishe r ₂. It is obvious that, as the price sensitivity of e-reader increases, e-publishe r ₂ is obliged to decrease its offered price ( $p_{e 2}^{EE}$ ) to retain its e-readers.

Besides, the p-publishing price of e-publishe r ₂ shows a decreasing trend, since it tries to compensate its demand loss by p-publishing. E-publishe r ₂ resists until ϕ_e2 . b ₂ = 2.1; after that point its price increases dramatically which can be interpreted as e-publishe r ₂ is retreated from the competition. In Fig. 11a and b, the profits of two e-publishers and the reaction of e_publishe r ₂ are analyzed in respect to the e_publisher ₁’s p-publishing price ( $p_{e 1}^{EE}$ ). The equilibrium point where two e-publishers’ profits are maximized is shown in Fig. 11b. Figure 12a and b are to demonstrate the robustness of GA.

4.2.4 Discussion

Based on the numerical results of three demonstrative examples, it is possible to make some observations. The scenario where the sum of the profits of all the publishers in the market is maximum is the third one: the E-E competition. The E-E competition can be interpreted as the industry equilibrium. The results show that the coefficient of price sensitivity (b) is the fundamental factor that determines both types of price levels. As the price sensitivity of readers increases, publishers are obliged to decrease their prices in order to maintain more reader. In this sense, it is the reader that determines the price level. In the case e-publisher ₁ is alone in the market (E-P competition), its demand and accordingly its profit are higher than in the competitive market (E-E competition). In the case there is at least onee-publisher in the market, the p-publisher loses profit. The p-publisher cannot reduce price like an e-publisher because of its higher maintenance and stocking costs. An e-publisher’s profit reaches its maximum level, if its competitor is a p-publisher. Therefore, the most profitable decision for the p-publisher is entering toe-publishing business in a market environment where the competitors seem to remain as p-publishers.

5 Conclusions

The principle objective of this work is to come up with a strategic analysis framework which would be valid for different types of decision environments in different sectors by making slight modifications. In this paper, the concentration is on the publishing industry, especially on a p-publisher that tries to decide on whether to enter the e-publishing business. However, the same decision support framework can help an existing firm when changing its business, or an existing firm to determine new prices, or a new entrant firm to determine prices.

Showing the effectiveness of a local search algorithm to find the equilibrium points of the quite complex games is the other contribution of this work. Especially for the second and third scenario, the complexity of the problem increases with the number of publishers in the market. Furthermore, closed-form expressions for the equilibrium points have not been demonstrated. The GA has been used as a computing technique to solve these problems. The results have revealed that the price sensitivity of readers is the most fundamental parameter that affects both the choice of the reader and the price of the book. Thehigher the reader’s price sensitivity is, the more the publisher tends to sell e-book to the reader.

Going forward, the equilibrium points may be found by using other soft computing algorithms and different algorithms can be compared in terms of obtained values and the execution time. Since the games are played offline in the proposed framework, the execution times are not questioned. The price sensitivity coefficient has a direct and profound impact on the equilibrium results, a future work can concentrate on determining it in the most efficient way.

In the proposed decision support model, the stocking and maintenance costs of an e-book are assumed to be lower than the ones of a traditional book, and hence the unit price of an e-book is assumed to be equal or lower than the one of a p-book. One of the two e-publishers whom this work has been presented to has commented on the cost structures. They both have agreed on the stocking and maintenance costs, but one of them has commented that, sometimes the initial cost of creating an e-book can exceed the one of a traditional book. Going forward, this difference between initial costs may be integrated to the model, instead of assuming them equal.

References

C.C. Miller and J. Bowman, E-books outsell print books at Amazon. The New York Times (2011) Available at: http://www.nytimes.com/2011/05/20/technology/20amazon.html

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