Computing https://doi.org/10.1007/s00607-018-0642-5

On understanding price-QoS war for competitive market and confused consumers Driss Ait Omar1 · M’Hamed Outanoute2 · Mohamed Baslam1 · Mohamed Fakir1 · Belaid Bouikhalene3 Received: 21 October 2017 / Accepted: 20 June 2018 © Springer-Verlag GmbH Austria, part of Springer Nature 2018

Abstract How will bounded rationality influence telecommunication network fluctuations? Recently, there has been an increased research interest in telecommunication network pricing, which leads to many proposals for new pricing schemes motivated by different objectives namely: to maximize service provider’s revenue, to guarantee fairness among users and to satisfy quality of service (QoS) requirements for differentiated network services. In the present paper, we consider a system with N rational service providers (SPs) that offer homogeneous telecommunication services to bounded rational costumers. All SPs offer the same services and seek to persuade more customers in the same system, we model this conflict as a noncooperative game. On the one hand, each SP decide his policies of price and QoS in order to maximize his profit. One the other hand, we assume that the customers are boundedly rational and make their subscription decisions probabilistically, according to Luce choice probabilities. Furthermore, the customers decide to which SP to subscribe, each one may migrate to another SP or alternatively switch to “no subscription state” depending on the observed price/QoS. In this work, we have proved through a detailed analysis the existence and uniqueness of Nash equilibrium. We evaluate the impact of user’s bounded rationality on the equilibrium of game. Using the price of anarchy, we examine the performance and efficiency of equilibrium. We have shown that the S Ps have an interest in confusing customers, which means more than the customers are irrational, the S Ps earn more. Keywords Pricing · QoS · Bounded rationality · Nash equilibrium · Luce choice probabilities Mathematics Subject Classification 93A30 · 90B18 · 91B06

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1 Introduction Bounded rationality is a concept proposed by Simon [30] which has called into question human rationality. Rationality is limited because there are limits to our capacity for reflection and the available information. When one speaks of a bounded rationality and the systematic customers’ confusion, one is strongly in a competitive environment between the actors which belong to it. In the case of telecommunication networks, service providers follow this kind of publication strategies totally or partially confused. Currently, the games theory is widely used to analyze the behavior of customers and service providers in the telecommunication networks. The competition in terms of prices and QoS among SPs entails the formation of non-cooperative games. In this paper, we consider that the market consists of N SPs and boundedly rational customers. In order to optimize revenues, each rational SP decide best responses of its price pi and its QoS qi . As SPs share the same market, the behavior of each depends on those of his opponents and those of customers. In this work, we present a model to calculate a single criteria Nash equilibrium (price) for several SPs, taking into account Bertrand’s model in game theory, which implies that players (S Ps) only choose prices. Our model is mainly inspired from the paper [3], where the authors have considered the rationality of SPs and have constructed a Markov model formalizing the behavior of customers depending on the strategic actions of the SPs to study a non-cooperative game for pricing and QoS as a decision parameters. But in our case, we will present that the demand for a SP is related to the size of the market (number of consumers) and the probability of choosing it to make the model of the application non-linear and take into account the degree of irrationality of users (λ). Rationality implies that each customer has to reason to increase his own payoff. In other words, he possesses all the information on the market that allows him to rationalize his choice, so he is able to maximize his own gain. von Neumann and Morgenstren [33] justified the idea of maximizing the expected utility hypothesis in their work. In the real world, the assumption of “full rationality” almost never holds. This real world “Bounded rationality” is one of the major impediments to applying conventional game theory in the real world. Bounded rationality of consumers means that rather than choosing the optimal action, customers choose an action that gives them a higher payoff with higher probability. In particular, we adopt the Luce model of probabilistic choice, that determines the probability with which a customer will decide to subscribe with a given S Pi. We consider also that the bounded rationality of consumers can be measured by a degree of irrationality that can be expressed by, λ, with λ = 0 corresponding to the rational behavior and λ → 1 corresponding to the totally random choice. To clarify the purpose of our research, this work aims to: – Extend the model of user gain proposed in the basic work [2] so that it takes the QoS parameter into equilibrium. – Reformulate the model using the Kleinrock’s [14] function to study the Price-QoS Nash Equilibrium. – Analyze the impact of the customers’ bounded rationality on the Price-QoS equilibrium.

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The instead of this paper is organized as follows:In Sect. 2, we quote the notable related works. In Sect. 3, we present the problem formulation by modeling of customer behavior using the Luce probabilistic model and we describe the utility model of S Ps and the gain of users in the telecommunication networks. Next, we present the theoretical analysis of the non-cooperative game considered in this study and the method used to learn Nash Equilibrium point respectively in the Sects. 4 and 5. Finally, we give a numerical analysis obtained on the models proposed in this work to validate what was obtained in the theoretical analysis in Sect. 6 and we conclude this paper with perspectives in Sect. 7.

2 Related work In related works, the concept of bounded rationality has attracted the attention of many researchers. In the work [27], the authors focused on the different models existing in the literature of customers’ bounded rationality. In this book [4], the author discussed the models of bounded rationality, also he presented the design problem of the agents’ choice that is governed by a probabilistic model and based his modeling by the Luce model. Furthermore in the book [17], the authors discussed the concept of bounded rationality, showing that irrational decision-making can be explained if one takes into account that human capacities to process information are limited. Thus, they used and improved heuristic techniques to predict the quality of the decision by formulating and solving optimization problems. In the paper [12], the authors were able to present some interesting results on customer confusion (Confusing consumers to make them irrational) with real price of the services offered by the S Ps and its impact on the dynamics of the telecommunication networks. In the work [9,20] the authors proposed additional pricing models in which consumers have imperfect price information and have to pay other fees to search for complete information that makes them rational. They concluded that as long as uncertainty is strong, the decision to acquire information becomes a strategic complement. Bounded rationality of the players (SPs) have been studied in many researches. The authors of papers [1,6,10,34] proposed that partial information can be modeled as marginal profiles. Players decide their strategy according to their respective marginal profit. In the paper [3] the authors modeled the behavior of customers towards the telecommunication networks and the bounded rationality of the service providers. The migration or the dynamic behavior of customers in the field of telecommunication networks is formalized in the form of a chain of markov. In this dynamic system, the authors have demonstrated theoretically and numerically the stability of the equilibrium between the players who are the S Ps. But the concept of bounded rationality on customers are little studied especially in telecommunication networks. The authors of the article [5] considered a Bertrand oligopoly model by modeling the rationality of consumers who make their purchasing decisions in a probabilistic way. According to the Luce model, this business model is aimed at companies of all kinds; that is more general. In the article [7], the authors proposed a model of sales prices by taking into account the way the company follows to present its prices to consumers and it is this way that confuses them. The authors of the article [22] were able to introduce the

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concept of confusion of customers in the telecommunication networks and proposed a competitive price model between two rational S Ps. The model proposed in this work is based on the price parameter without using the other pole of the strategies of S Ps which is QoS quality of service. The authors investigated the impact of customer confusion on a market of two S Ps. In our work we will extend the study by proposing a new competitive model between N S Ps that take their strategies according to price and quality of service QoS. Our model will be based on a new formulation of the demand which is made nonlinear and we will adapt the probabilistic model of Luce used in the article [5] for the case of the telecommunication networks.

3 Problem formulation 3.1 Customer behavior model The modeling of customer behavior is a very importanttask when one is studying economically a market. In the telecommunication networks, the authors Maillé et al. [18] modeled this behavior in the form of a Markov chain. In this study, we use the Luce model to mathematise the discrete choice of clients by exploiting the softmax function or the normalized exponential function [8], as in the article [25]. 3.1.1 The Luce model The Luce model is a first probabilistic choice model that incorporates boundedly rational choice of customers [13–26,33]. By using this model, customers can select the SP that maximize their payoff with higher probabilities, but not necessarily the best response with probability one. More precisely, the choice probabilities for specific strategies are proportional to the expected payoffs associated with such strategies. We augment this framework by including a free parameter λ that determines a degree of customers irrationality. When customers faced with a choice among different alternatives i ∈ {1 . . . N }, the perfectly rational decision maker always chooses the most preferred option(s) i ∈ {arg maxi (u i )}, where u i is a profit of users who have chosen SP i. In contrast, to capture bounded rationality, we assume that customers choose the alternative i ∈ {1 . . . N } with probability, as in [31], is given by: exp(u i /λ) ρi =  N j=1 exp(u j /λ)

(1)

where λ ∈ [0, 1], is a degree of irrationality of customers to the telecommunication networks. As λ increases (approximates 1), the customerr is less likely to choose the offer (the S P) with the highest expectation and he will not explore other offers from other S Ps. So in this case, the choice is random and it is said that the customer choice is irrational. But, If λ decreases (approaches 0), the customer is likely to make a rational choice.

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3.1.2 Utility and behaviors models We consider that the utility u i of customers that are subscribed with S Pi is a function that depends on the strategies price pi and QoS qi of SP i: u i ( pi , qi ) = vi (qi ) − pi

(2)

where vi (qi ) are a revenues of customers that are subscribed with SP i. We assume that the customers never decide to subscribe with more than one SP, i.e. each consumer has (N + 1) different choices, so, the probability that customers will decide to subscribe with SP i is:   exp vi (qiλ)− pi   ρi (p, q) = (3)  v (q )− p 1 + Nj=1 exp j jλ j 3.2 Utility model We consider a population of n customers, therefore the expected demand of S Pi is given by, nρi (p, q). So, the utility function of S Pi , is exactly the difference between its revenues, nρi (p, q) pi , and the fee paid to buy a given amount of bandwidth μi : Πi (p, q) = nρi pi − Fi (ρi , qi ) = nρi pi − ϑi μi (n, ρi , qi )

(4)

where ϑi is the price of unit of bandwidth and μi (ρi , qi ) is the amount of bandwidth required by SP i to guarantee the promised QoS qi , which has the following form: μi (n, ρi , qi ) = (nρi )gi (qi ) + h i (qi )

(5)

where n is a number of customers in the market and gi (qi ) and h i (qi ) are positive increasing functions. The profile function of SP i becomes:   n exp vi (qiλ)− pi  ( pi − ϑi gi (qi )) − ϑi h i (qi ) .  Πi (p, q) =  v (q )− p 1 + Nj=1 exp j jλ j

(6)

3.3 The profit of users in telecomunication The real gain of users in the telecommunication networks is normally not only dependant on his subscription in S Pi , but also secondarily on the strategies of others S P−i (opponents of S Pi ). We propose in this section the modeling of this profit by adding the constraint on the QoS to the first model proposed in the paper [2].

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The revenue of user u if he chooses the subscription at S Pi is: Riu ( p ∗ , q ∗ ) = u i ( pi , qi ) −

N 

β uj [( pi∗ − p ∗j ) + (qi∗ − q ∗j )]

(7)

j=1, j=i

where – β uj ∈ [0, 1] ∀ j ∈ [1, N ] and j = i is the sensitivity of the user u at the motivating strategies of adversaries of its S Pi . If β uj → 0 then the user u is faithful to his S Pi, but if β uj → 1 then the user u is totally attracted by the offer of the operator S P j and in this case we are not talking about the fidelity of u. – p ∗ is the vector of Nash Equilibrium Prices. – q ∗ is the vector of Nash Equilibrium QoS. Then, the profit (welfare) of user u in the telecommunication networks is the accumulate of his revenue in all S Ps, it is presented in this equation: G u ( p∗ , q ∗ ) =

N 

Riu ( p ∗ , q ∗ )

(8)

i=1

4 Analysis of the non-cooperative game In the telecommunication networks, S Ps can compete on several variables. They can, for example, they can compete based on their choices of prices and QoS. However, we consider a system with N rational SPs who decide their price and QoS strategies in order to maximize their individual utility/payoff Πi (.). These SPs are selfish, hence they do not cooperate with each other to manage their policies. But, they have all current informations on all the strategies of their opponents. Then, the game studied in this work are a non-cooperative game and at complete information. The noncooperative game between rational S Ps is formulated as follows: Let G = [N , {Pi , Q i } , {Πi (.)}] denote the non-cooperative price QoS game (NPQG), where N = {1, . . . , N} is the index set identifying the S Ps, Pi is the price strategy set of S Pi , Q i is the QoS strategy set of S Pi , and Πi (.) is the utility function of S Pi which is defined in Eq. 6. We assume that the strategy spaces Pi and Q i of each S Pi are compact and convex sets with maximum and minimum constraints,  for  any given S Pi we consider strategy spaces the closed intervals Pi = pi , pi and   Q i = q i , q i . Let the price vector p = ( p1 , . . . , p N )T ∈ P N = P1 × P2 × · · · × PN , QoS vector q = (q1 , . . . , q N )T ∈ Q N = Q 1 × Q 2 × · · · × Q N . In order to maximize their utilities, each S Pi decides a price pi and QoS qi . Formally, the problem can be expressed as: max

pi ∈Pi ,qsi ∈Q i

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Πi (p, q), ∀i ∈ N .

On understanding price-QoS war for competitive market and…

4.1 Fixed QoS game The most basic and fundamental competition pertains to pricing choices. we consider in this section, the Bertrand Model that examines the interdependence between rivals decisions in terms of pricing decisions. In this section, we present the utility for all the S P s that offer homogeneous services to customers, as well as the gain of customers in the telecommunication networks. In this part, we consider that SPs have fixed their QoS, q, at some predetermined ¯ and we consider only the price game. The utility function (6) become: point, q,   n exp v¯i −λ pi  ( pi − ϑi g¯i ) − ϑi h¯ i .  (9) Πi ( pi , p−i ) = N v¯ j − p j 1 + j=1 exp λ where v(q¯i ) = v¯i , g¯i = gi (q¯i ) and h¯ i = h i (q¯i ) are positive real constants. An important derivative proprety of relation (3) is that: ∂ρi ρi (1 − ρi ) , =− ∂ pi λ Proof



∂ρi = ∂ pi

∂

  v¯ − p exp i λ i  v¯ − p   1+ Nj=1 exp j λ j ∂ pi      ∂ 1+ N exp v¯ j − p j    N j=1 v¯ j − p j λ v¯i − pi 1 + j=1 exp − exp λ ∂ pi λ  2  N v¯ j − p j 1 + j=1 exp λ

  v¯ − p ∂ exp i λ i ∂ pi

=











2 v¯ − p    exp i λ i N v¯ j − p j 1 + j=1 exp + λ λ λ =    2 N v¯ j − p j 1 + j=1 exp λ   2   exp v¯i −λ pi exp v¯i −λ pi  +   =−  2   v¯ − p  v¯ − p λ 1 + Nj=1 exp j λ j λ 1 + Nj=1 exp j λ j

− exp

=−

v¯i − pi λ

ρ2 ρi (1 − ρi ) ρi + i =− λ λ λ  

therefor, the profit derivative for SP i is: ∂Πi ( pi , p−i ) ρi (1 − ρi ) + nρi , ∀i = 1 . . . N = −n ( pi − ϑi g¯i ) ∂ pi λ

(10)

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with second derivative ∂ 2 Πi ( pi , p−i ) ρi (1 − ρi ) ρi (1 − ρi )(2ρi − 1) = −n ( pi − ϑi g¯i ) − 2n 2 2 λ λ ∂ pi

(11)

Definition 1 A price vector p∗ = ( p1∗ , . . . , p ∗N ) is the Nash Equilibrium price of the game G = [N , {Pi , Q i }, {Πi (.)}] if: ∀(i, pi ) ∈ (N , Pi ), Πi ( pi∗ , p∗−i ) ≥ Πi ( pi , p∗−i ) ∗ , p∗ , . . . , p∗ ) with p∗−i = ( p1∗ , . . . , pi−1 i+1 N

Theorem 1 The game G = [N , {Pi , Q¯ i }, {Πi (.)}] admits a Nash Equilibrium price . (existence of NE) if and only if pi > ϑi g¯i − 2ρ2λ i −1 Proof To prove existence, we note that each S Ps strategy space Pi is defined by all prices in the closed interval bounded by the minimum and maximum prices. Thus, the joint strategy space P a non-empty, convex, and compact subset of the Euclidean space R N . In addition, the utility functions are concave with respect to prices as can be seen from the second derivative test. Equation 11 represent the second derivative of the utility function 9 relative to the price. The condition for this function as strictly concave is: ∂ 2 Πi ( pi , p−i ) < 0, ∂ pi2

ρi (1 − ρi ) ρi (1 − ρi )(2ρi − 1) < 0, − 2n 2 λ λ 2λ Finally pi > ϑi g¯i − , 2ρi − 1

then

− n ( pi − ϑi g¯i )

which ensures existence of a Nash equilibrium.

 

Theorem 2 The game G = [N , {Pi , Q¯ i }, {Πi (.)}] admits a unique Nash Equilibrium price if and only if pi > ϑi g¯i − 2(2ρ3λi −1) . Proof To prove the uniqueness of a Nash Equilibrium, we use the most common method to show uniqueness is the following condition of Rosen [28]. Moulin [23],(see, for example, [21]) which resumed by the following proposition that holds for a concave game [16]: If a concave game satisfies the dominance solvability condition: −



 ∂ 2 Ui (p, qs )

∂ 2 Ui (p, qs )



−

∂p ∂p ≥ 0 ∂ pi2 i j j, j=i

then the game G admits a unique NE.

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(12)

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We have ∂ 2 Πi = ∂ pi ∂ p j



 n n ( pi − ϑi g¯i )  2 2ρ − ρ i + ρi ρ j , i λ2 λ

so: 

∂ 2 Πi

∂ p ∂ p j=i

i



 n



n ( pi − ϑi g¯i )  2

=

2ρi − ρi + ρi

ρj



λ2 λ j j=i



n ( pi − ϑi g¯i ) n

ρi (1 − ρi ), =

− 1) + (2ρ i λ2 λ

(13)

from (11) and (13) we have:



 ∂ 2 Πi 

∂ 2 Πi

n ( pi − ϑi g¯i ) (2ρi − 1) 2n = ρi (1 − ρi ) − + −



2 ∂ pi ∂ p j λ2 λ ∂ pi j=i



n ( pi − ϑi g¯i ) n

+

(2ρi − 1) +

, 2 λ λ  if

n( pi −ϑi g¯i ) λ2

(2ρi − 1) +

n λ

 ≥ 0 , i.e. pi ≥ ϑi g¯i −

λ 2ρi −1

then



nρi (1 − ρi ) ∂ 2 Πi 

∂ 2 Πi

=− < 0, +



2 ∂ pi ∂ p j λ ∂ pi j=i  if

n( pi −ϑi g¯i ) λ2

(2ρi − 1) +

n λ

 ≤ 0 , i.e. pi ≤ ϑi g¯i −

λ 2ρi −1

then



 ∂ 2 Πi 

∂ 2 Πi

−nρi (1 − ρi ) 2 ( pi − ϑi g¯i ) = (2ρ + − 1) + 3 i

∂ p ∂ p

λ λ ∂ pi2 i j j=i We conclude then if the condition pi > ϑi g¯i − admits a unique Nash Equilibrium.

3λ 2(2ρi −1)

are satisfied, the game G  

4.2 Fixed price game We consider simultaneous setting of both parameters by all S Ps, the joint QoS game arises when the S Pi decides on his both QoS in order to maximize his profit. We suppose that the measure defining the QoS, qi corresponds to some function of the expected delay. We shall use the Kleinrock [14] delay function1 which is a common 1 This function corresponds to a queuing delay in an M/M/1 queue with first-in-first-out discipline or to the more general M/G/1 queue under processor sharing delay.

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delay function used in networking games [24]. So: qi = √

√ 1 = μi − nρi , ∀i ∈ {1, . . . , N } Delayi

(14)

Which means that: μi (n, ρi , qi ) = qi2 + nρi

(15)

From this latter and according to Eq. (5), we have that

gi (qi ) = 1 h i (qi ) = qi2 So, Eq. (6) become:

Πi (p, q) =

  n exp αqiλ− pi  ( pi − ϑi ) − ϑi qi2 .  N αq j − p j 1 + j=1 exp λ

(16)

An important derivative proprety of relation (3) is that: ∂ρi α = ρi (1 − ρi ), ∂qi λ Proof

∂ρi = ∂qi

 ∂

  αqi − pi exp λ  αq − p   j j 1+ Nj=1 exp λ ∂qi 

  αqi − pi ∂ exp λ ∂qi

=

α. exp



αqi − pi λ



1+



N

αq j − p j λ



   αq − p  j j ∂ 1+ Nj=1 exp λ

− ∂qi  2  N αq j − p j 1 + j=1 exp λ

j=1 exp

N





αq j − p j λ



exp

 αqi − pi  λ

  2 αqi − pi α. exp λ

1 + j=1 exp − λ    2  αq j − p j 1 + Nj=1 exp λ  αqi − pi  2   α. exp α exp αqiλ− pi λ  +   =  2   αq j − p j  αq j − p j λ 1 + Nj=1 exp λ 1 + Nj=1 exp λ λ α α 2 α = ρi − ρi = ρi (1 − ρi ) λ λ λ =

λ

 

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Therefor, the profit derivative for SP i is: ∂Πi (qi , q−i ) n.α = ( pi − ϑi ) ρi (1 − ρi ) − 2ϑi qi , ∀i = 1 . . . N ∂qi λ with second derivative   ∂ 2 Πi qi , q−i ∂qi2

=n

 α 2 λ

( pi − ϑi ) ρi (1 − ρi )(1 − 2ρi ) − 2ϑi ,

(17)

(18)

Definition 2 A QoS vector q∗ = (q1∗ , . . . , q N∗ ) is the Nash Equilibrium QoS of the game G = [N , {Pi , Q i }, {Πi (.)}] if: ∗ ∗ ) ≥ Πi (qi , q−i ) ∀(i, qi ) ∈ (N , Qi ), Πi (qi∗ , q−i ∗ = (q ∗ , . . . , q ∗ , q ∗ , . . . , q ∗ ) with q−i 1 i−1 i+1 N

Theorem 3 The game G = [N , { P¯i , Q i }, {Πi (.)}] admits a Nash Equilibrium QoS if and only if ϑi > n2 ( αλ )2 ρi (1 − ρi )(1 − 2ρi ). Proof Equation 18 represent the second derivative of the utility function 16 relative to the price. The condition for this function as strictly concave is: ∂ 2 Πi (qi , q−i ) < 0, ∂qi2  α 2 then n ( pi − ϑi ) ρi (1 − ρi )(1 − 2ρi ) − 2ϑi < 0, λ n  α 2 Finally ϑi > ρi (1 − ρi )(1 − 2ρi ), 2 λ   ¯ Theorem 4 The game G = [N , { Pi , Q i }, {Πi (.)}] admits a unique Nash Equilibrium QoS if and only if ϑi > N 2+2 (1 − 2ρi ). Proof To prove the uniqueness of a Nash Equilibrium, we use the most common method to show uniqueness is the following condition of Rosen [28]. Moulin [23],(see, for example, [21]) which resumed by the following proposition that holds for a concave game [16]: If a concave game satisfies the dominance solvability condition:



∂ 2 Πi 

∂ 2 Πi

+

∂q ∂q < 0. ∂qi2 i j j=i

(19)

we have  α 2 ∂ 2 Πi = −n ( pi − ϑi )ρi (1 − ρi )(1 − 2ρi ), ∂qi ∂q j λ

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so:



∂ 2 Πi

∂q ∂q j=i

i

  



α 2

=

n



λ ( pi − ϑi )ρi (1 − ρi )(1 − 2ρi )

j j=i  α 2 = n(N − 1) ( pi − ϑi )ρi (1 − ρi ) |1 − 2ρi | , λ

(20)

from (18) and (20) we have:



 α 2 ∂ 2 Πi 

∂ 2 Πi

=n + ( pi − ϑi )ρi (1 − ρi ) [1 − 2ρi − 2ϑi



2 ∂qi ∂q j λ ∂qi j=i

+(N − 1) |1 − 2ρi |] , if ρi < 21 , the condition for the expression of the Eq. (19) to be satisfied is: ϑi > N 2 (1 − 2ρi ). But if ρi > 21 , the condition for the expression of the Eq. (19) to be satisfied is: ϑi > N 2+2 (1 − 2ρi ). From these conditions, we can conclude that whatever the value of the ρi ∈ {0, 1}   if ϑi > N 2+2 (1 − 2ρi ) while the QoS equilibrium is unique. 4.3 Joint price and QoS game As shown previously, the profit function of each SP is strictly concave with respect to its own fixed QoS and fixed prices, therefore S Pi s best response for two cases is uniquely defined. Therefore, the best response of price and QoS of SPs are uniquely defined, so, the QoS and price conditions which maximizes the utility given in Eq. 16 are respectively : ⎧ ∂Πi (p, q) ⎪ ⎪ = 0, ⎨ ∂p i

∂Πi (p, q) ⎪ ⎪ ⎩ = 0, ∂qi

Thus, the computation of Nash Equilibrium can be performed by solving latter system. Solutions of equations induces by vanishing the partial derivatives correspond to the best response for joint price and QoS game. Since this system is nonlinear, it is hard to obtain analytical formulas for its solution. For this reason, we use the algorithm 1 presented in the next section to show the convergence towards the Nash Equilibrium point.

5 Learning Nash equilibrium 5.1 Best response algorithm Below, we analyze the competitive price or QoS for N S Ps that maximize their utilities. To do so, we demonstrate the existence and uniqueness of the game equilibrium

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On understanding price-QoS war for competitive market and…

between N S Ps, after we calculate the equilibrium point. To analyze equilibrium of the game, we need to find properties on the utility function. The algorithm 1 is to calculate the best response dynamic. The best response of a player is defined as his optimal strategy which makes him an optimal gain taking into account the strategies of adversaries. The best response dynamic of a player, as the name suggests, is to adapt his strategy with the recent strategies of others and without taking into account the effect of the current strategies of all players on the future game. Algorithm 1 Best Response Dynamic Algorithm 1: Initialization of strategies vectors( prices and/or QoS); 2: For each S Pi i ∈ N at iteration t:   – pit+1 = argmax Πi (pt , qt ) ; p ∈P

i   – qit+1 = argmax Πi (pt , qt ) .

qi ∈Q

5.2 Price of anarchy The concept of the social surplus [19] or total cost [32], is defined as the maximum of the sum of utilities of all agents in the systems (i.e. Providers). It is well known in game theory that selfishness of the agent, as in a Nash equilibrium, typically does not lead to a socially effective situation. As a measure of efficiency loss due to divergence of interests of users, we use the price of anarchy (Po A) [15], the latter is a measure of the loss of efficiency due to the selfishness of the actors. This loss was defined in [15] as the ratio of the worst comparing the measure of the overall efficiency (to be selected) at the end of non-cooperative game played between the actors, to the optimum value of this measure efficiency. A Po A close to 1 indicates that the equilibrium is about socially optimal, and then the consequences of selfish behavior are relatively benign. The term price of anarchy was used by Koutsoupias and Papadimitriou [15]. As in [11], measuring the loss of efficiency due to the selfishness of the actors as the quotient of the social welfare obtained at the Nash equilibrium and the maximum value of social welfare (as in [29]): Po A =

¯ min p,q¯ W N E ( p, q) max p,q¯ W ( p, q) ¯

(21)

N where W ( p, q) ¯ = ¯ is a function of welfare and W N E ( p, q) ¯ = i=1 Πi ( p, q) N ∗ ¯ is a sum of utilities of all actors in the Nash equilibrium. i=1 Πi ( p , q)

6 Numerical analysis of the game To clarify and show how to take advantage from our theoretical study, we suggest in this section to study numerically the market share game while considering the

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n

N

v¯1

v¯2

λ

P1 = P2

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algorithm of best response dynamic (Alg. 1) described in the previous section and expressions of demand as well as utility functions of S Ps. Hence, we consider a system with two homogeneous S Ps seeking to maximize their respective revenues. Until we contraindicate, the parameter values are summarized in Table 1. Figure 1 represent the convergence curves Nash Equilibrium Prices for both SPs. The best dynamic response algorithm used in this framework converges to a unique Nash equilibrium. We also notice that the algorithm has turned approximately 10 iterations, which shows the rapidity of the speed of convergence. Then this simulation of algorithm 1 is able to converge efficiently to the Nash equilibrium price. Figure 2 represent the convergence curves Nash Equilibrium QoS for both SPs. The best dynamic response algorithm used in this framework converges to a unique Nash equilibrium. We also notice that the algorithm has turned approximately 12 iterations, which shows the rapidity of the speed of convergence. Then this simulation of algorithm 1 is able to converge efficiently to the Nash equilibrium QoS. The curves traced in Figs. 3 and 4 shows the convergence to joint price-QoS Nash equilibrium of the game and they show the uniqueness of this equilibrium point, which is difficult to find analytically because of the non-linearity of the equation system given above. The algorithm 1 presented in the previous section has turned 14 iterations to arrive at the equilibrium point.

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Time T Fig. 3 SP1: Convergence to the price and QoS Nash equilibrium

6.1 Impact of users degree of irrationality on the equilibrium price In this simulation, we have undergone a discretization, at a regular step, at the interval of λ. Figure 5 shows the influence of the degree of irrationality on price equilibrium. In the interval where the degree of irrationality is low (λ < 0.35) ie the choice the users is often rational; the S P1 , which starts the games with proportionately high prices, has decreased its price to attract more customer and increased his PayOff. While, the S P2 , which starts the games with motivating prices, began to increase its price to ensure its profit. And with high degrees of irrationality, the S Ps raise their prices without worrying about the opponent’s strategy as clients make irrational decisions.

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6.2 Impact of unit of bandwidth on the equilibrium price In this simulation, we have undergone a discretization, at a regular step, at the interval of ϑi i ∈ {1, 2}. Figure 6 shows the influence of price of unit of bandwidth on price equilibrium. This figure shows that it is clear that when the unit price of bandwidth increases, all S Ps are obliged to increase their price strategy with the goal of keeping at least their current beneficence. 6.3 Impact of users degree of irrationnality on the their gain To validate our model of user gain to the telecommunication networks presented in the Eq. 8, we simulate by varying the degree of irrationality of the users and calculating their gain. Figure 7 shows that when users make decisions about the choice of S P in

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(p ,p )

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λ :Users Degree of Irrationnality Fig. 7 Users Gain w.r.t degree of irrationality

a rational way, their gain in the market increases. While, during a random selection of the S P from users, their gain decreases. In summary, the more customers are confused by their S Ps, the more their gain decreases in this market. 6.4 Impact of users sensitivity on the their gain In this section, we consider a user u that subscribes to the services of the S P1 (ie β1u = 0) and we vary the sensitivity β2u , to the strategy of S P2 , to see its impact on the user’s gain u. We notice from Fig. 8 that when the sensitivity β2u of user u increases, the social gain of u decreases in the telecommunication networks. This result is actually seen in this market, So that when a user subscribes to the services of a S Pi and as soon as the S P j offers services more motivating than the S Pi , The user u feels bad about his choice.

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Gain of Usres at subscribe in SP1

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* 2

G (p ,p ) u

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λ : Degree of Irrationality of Users Fig. 9 Equilibrium QoS w.r.t degree of irrationality of users

6.5 Impact of users degree of irrationality on the equilibrium QoS From Fig. 9, we notice that there is an influence of the degree of irrationality of the customers on the telecommunication networks, especially when we talk about the competition at the QoS level. When customers are not confused, ie they have all the information about what is happening in the market, (λ close to 0) operators will need to improve the QoS (λ < 0.3: the rationality of customers is very strong)to ensure their share in the market. As soon as the information begins to erupt, the customers do not have all the information on the strategies of the operators, the latter block (when 0.3 < λ < 0.5 or 0.7 < λ < 1) or diminish the QoS (when 0.5 < λ < 0.7) especially if they offer prices low enough to keep a very good ratio between supply and investment. So we can conclude that S Ps follow their strategies by observing the behavior of clients to find out if they have any or some information about what is

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PoA

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λ: Degree of Irrationality of Users Fig. 10 Price of anarchy w.r.t degree of irrationality

before them. We also note that the variation of QoS is not strong enough and taking into account the impact of degree of irrationality on the price equilibrium, we can conclude that the strategies of the S Ps surround themselves on which price for what QoS and this ratio (price/QoS) that is the source of the customers confusion. 6.6 Equilibrium efficiency In this part, we will use the concept of the anarchy price presented in Sect. 4 to discuss the effectiveness of the Nash equilibrium. Figure 10 shows the PoA variation curve according to λ, which represents the users degree of irrationnality. In that figure, we first notice that the price of anarchy increases when λ increases. When the degree of irrationality is low, ie users make their choice decisions rationally, the price of anarchy is low. To account for that, the sum of the optimal utilities is greater than the sum of utilities to equilibrium; which shows that the SPs are selfish and each one seeks to maximize its profit. While in the case where the price of anarchy approaches 1, the SPs are not selfish and each one takes in consideration the strategy of his opponent to finally fall into the state of equilibrium.

7 Conclusion In this work we have modeled and studied the impact of customer behavior and their bounded rationality in the telecommunication networks using the mathematical tool of non-cooperative game theory. First, we modeled the competition between S Ps according to two parameters: price and QoS taking into account the probability of choosing the S Ps. On the other hand, we proposed a model of the gain of users in the telecommunication networks. We have demonstrated the existence and uniqueness of the Nash equilibrium Price and QoS, then we applied the Best response Algorithm for

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learning Nash equilibrium. We have shown in the first case that S Ps in the telecommunication networks have an interest in confusing customers, which means that the more customers are irrational, the S Ps earn more. In the second case, the degree of irrationality to a strong influence on prices as the QoS, which shows that the customers are interested in the ratio (price/QoS) what price for which QoS. In the future work, we propose to study this system by taking into account the parallel variation of the two parameters QoS and Price and by considering the variability of the rationality of customers i.e to seek to model the function of rationality. Customer rationality is a function to be modeled depending on a set of parameters such as those related to customer behavior themselves and those related to offers trapped or confused by the S Ps in the telecommunication networks.

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Affiliations Driss Ait Omar1 · M’Hamed Outanoute2 · Mohamed Baslam1 · Mohamed Fakir1 · Belaid Bouikhalene3

B

Driss Ait Omar [email protected] M’Hamed Outanoute [email protected] Mohamed Baslam [email protected]

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D. Ait Omar et al. Mohamed Fakir [email protected] Belaid Bouikhalene [email protected] 1

Information Processing and Decision Support Laboratory, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, Morocco

2

Interdisciplinary Laboratory of Research in Sciences and Technologies, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, Morocco

3

Interdisciplinary Laboratory of Research in Sciences and Technologies, Polydisciplinary Faculty, Sultan Moulay Slimane University, Beni Mellal, Morocco

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