**A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers**

Mohamed Baslam, Rachid El-Azouzi, Essaid Sabir, Loubna Echabbi and El-Houssine Bouyakhf

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54380

### **1. Introduction**

156 Game Theory Relaunched

Ontario, Canada.

Washington, D.C., 1996: 100-105.

[16] Redekop J (2000) Increasing marginal cost and the monotonicity of Aumann-Shapley pricing. Working Paper, Department of Economics, University of Waterloo, Waterloo,

[17] Moses F (1989) Effects on Bridges of Alternative Truck Configurations and Weights, NCHRP Contract No. HR 2-16 (b)m, Transportation Research Board, Washington, D.C. [18] Lee D, Garcia-Diaz A (2007) Procedure for Bridge Construction Cost Allocation Based on Game Theory. In Transportation Research Record: Journal of the Transportation Research Board, No., Transportation Research Board of the National Academies,

> Recently, the selfish behavior of customers and Service Providers (SPs) in telecommunications systems has been widely analyzed using game theory with all its powerful solution concepts. It was shown in several works that customer's selfish behavior leads to a network collapse, where a typical prisonerís dilemma situation arises. Despite of the bounty of works and efforts investigated in analyzing market share game, this filed is still an ideal tool to understand interaction among SPs and customers. Indeed, it is common in the literature to assume a single decision action (e.g., cost) through which an equilibrium would be computed. Yet, in order to take into account Quality of Service (QoS), it is necessary to incorporate into the model more than one decision parameter. A simple example is to include both price and some measure of QoS (e.g., delay, throughput, loss probability, etc.). Other multi-criteria models may incorporate, for example, delay and reliability, the latter representing the QoS, price or delay and jitter, etc.

> The competition in terms of prices and QoS among SPs entails the formation of non-cooperative games. We consider multiple SPs (players of the game), where each one seeks to maximize its own revenue, whereby the whole system of SPs would have no incentive to deviate from the Nash equilibrium1 point, i.e., the vector of equilibrium strategies. Yet, such equilibrium point should first mathematically exist. In this chapter, we present a general model for computing a bi-criteria Nash equilibrium for multiple SPs. We shall then analyze the interactions between SPs who won't attract more clients and maximize their respective profits. We address the important problem of Nash Equilibrium characterization with two-component action, when the two components of each provider are the service price and a measure of QoS. Our model is mainly inspired from, [6], where the authors studied a

<sup>1</sup> A Nash equilibrium is a strategy profile where no player has sensitive to deviate unilaterally from its current strategy.

©2013 El-Azouzi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2013 El-Azouzi et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Game Theory 158 Game Theory Relaunched A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers <sup>3</sup>

non-cooperative game for pricing problem considering QoS as an extra decision parameter. The authors build a Markovian model to derive the behavior of customers depending on the strategic actions of the SPs. In contrast to this chapter, we base our study on the concepts of demand for the services of a given SP (defined by linear function that depends on the vectors of prices and QoSs), which is a commonly used function in research related to competitive network and equilibrium models, [9], [5], to calculate the reputation of an SP in the market.

**Organization :**

are drawn in Section 5.

**2.1. Demand model**

where *D*<sup>0</sup>

While *<sup>α</sup><sup>j</sup>*

*<sup>i</sup>* and *<sup>β</sup><sup>j</sup>*

**2.2. Utility model**

*Di*(**p**, **<sup>q</sup>**) = *<sup>D</sup>*<sup>0</sup>

provider *i* to price and QoS of service provider *j*.

*<sup>i</sup>* <sup>−</sup> *<sup>α</sup><sup>i</sup>*

**2. Problem modeling**

The rest of the chapter is organized as follows : in Section 2 we describe the system model and introduce a new demand and utility functions. In Section 3 we formulate the joint price and QoS problem as a non-cooperative game, and investigate existence and uniqueness of a Nash equilibrium solution. Then, we present numerical results obtained from simulations that exploits our joint price and QoS algorithm in Section 4 . Conclusions and future guidelines

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 159

In this chapter, we formulate the interaction among service providers (SPs) as a non-cooperative game. Each SP chooses the Quality of Service to guarantee (it depends on

We consider a system with *N* service providers. Let *pi* and *qi* be, respectively, the tariff/pricing policy and the QoS guaranteed by SP-*i*. Now, each customer seeks to subscribe to the operator which allows him to meet a QoS sufficient to satisfy his/her needs, at suitable price. We consider that behaviors of customer's has been handled by a simple function so called demand functions, see equation (1). This later depends on the price and QoS strategies of all SPs. From a tagged SP's point of view, the question is to set the best pricing strategy and the best QoS (amount of bandwidth to request from the network owner). SPs are supposed to know the effect of their policy on the customer's subscription policy. Whereas from customer's point of

For simplicity, we consider that the demand function *Di* for services of the tagged SP-*i* is linear with respect to the set price *pi* and the promised QoS *qi*, see, [9]. This demand function depends also on prices **<sup>p</sup>**−*<sup>i</sup>* and QoS **<sup>q</sup>**−*<sup>i</sup>* set by the competitors. Namely, the demand function of SP-*i* depends on **p** = [*p*1, .., *pN*] and **q** = [*q*1, .., *qN*]. Eventually, *Di* is decreasing w.r.t. *pi* and increasing w.r.t. *pj*, *j* �= *i*. Whereas it is increasing w.r.t *qi* and decreasing w.r.t. *qj*, *j* �= *i*.

> *αj <sup>i</sup> pj* <sup>−</sup> *<sup>β</sup><sup>j</sup> i qj*

The total revenue of SP-*i* is *Di*(**p**, **q**)*pi*. We assume that we have a single network owner, this latter charges each SP-*i* a cost *ϑ<sup>i</sup>* per unit of requested bandwidth. In order to insure the customers loyalty, the amount of bandwidth *μ<sup>i</sup>* required by SP-*i* should depend on *Di*(.) and

*<sup>i</sup>* is a positive constant used to insure non-negative demands over the feasible region.

*<sup>i</sup>* are positive constants representing respectively the sensitivity of service

, ∀*i* ∈ {1, .., *N*}. (1)

view, the question is to find the SP that has the best price-QoS tradeoff conditions.

Then, the demand functions w.r.t services of SP-*i* can be written as follows:

*<sup>i</sup> pi* <sup>+</sup> *<sup>β</sup><sup>i</sup> i qi* + ∑ *j*,*j*�=*i*

the amount of requested bandwidth) and the corresponding price.

We focus our studies on the non-cooperative games in terms of stable solutions, which are the pure strategy Nash equilibria of the game. We do not consider mixed strategy equilibria, because our environment requires a concrete strategy rather than a randomized strategy, which would be the result of a mixed strategy. Hence, when using the term Nash equilibrium we mean pure strategy exact Nash equilibria unless mentioned otherwise.

We note that the most fundamental assumption in relative works of game theory is rationality. It implies that every player is motivated by increasing his own payoff, i.e. every player is looking to maximize his own utility. John V. Neumann and Morgenstern justified the idea of maximizing the expected payoff in their work in [23]. In this context, all information concerning the game is known to all players, i.e., there is complete information. So, we consider that all players are said to be rational and intelligent. A rational person is one who acts in such a way as to maximize his or her expected payoff or utility as economists would say. An intelligent person is one who can deduce what his or her opponent will do when acting rationally. In fact, humans use a propositional calculus in reasoning, the propositional calculus concerns truth functions of propositions, which are logical truths (statements that are true in virtue of their form). For this reason, the assumption of rational behaviour of players in telecommunications systems is more justified, as the players are usually devices programmed to operate in certain ways. However, there are previous studies that have shown that humans do not always act rationally [10].

#### **Related Works :**

Applying game theory in telecommunications problems is an active research area, in which game-theoretic models have been developed and studied in the last decades, [1, 2, 6, 7, 9, 16, 18]. These models are interested in pricing issues, they proposed non-cooperative game formulations to analyze behaviours of players that selfishly decide their strategies to maximize their respective profits. Other works consider the criteria of price as an implicit parameter, which is determined as a function of the degree of saturation on the network. Typically in these approaches, the price is a shadow price. For more details on those approaches see, [14, 15, 24]. Nonetheless, the price of anarchy has been studied in a large and diverse number of games, e.g., in areas like wireless ad-hoc networks [8, 13], routing and congestion [4, 19], network creation [3], or facility location [22]. In our model, we do not take into account network topology, but rather the effective service proposed by each SP as a single entity. In other words, the price and QoS proposed by an SP will not depend on the source or destination, distance, etc. that underlies the request of each user. After we have proved existence of Nash equilibrium, we propose a joint price and QoS algorithm which allows to learn the equilibrium price and QoS strategies decided by SPs. This is a simple algorithm implementation with lower computational complexity.

#### **Organization :**

2 Game Theory

non-cooperative game for pricing problem considering QoS as an extra decision parameter. The authors build a Markovian model to derive the behavior of customers depending on the strategic actions of the SPs. In contrast to this chapter, we base our study on the concepts of demand for the services of a given SP (defined by linear function that depends on the vectors of prices and QoSs), which is a commonly used function in research related to competitive network and equilibrium models, [9], [5], to calculate the reputation of an SP in the market.

We focus our studies on the non-cooperative games in terms of stable solutions, which are the pure strategy Nash equilibria of the game. We do not consider mixed strategy equilibria, because our environment requires a concrete strategy rather than a randomized strategy, which would be the result of a mixed strategy. Hence, when using the term Nash equilibrium

We note that the most fundamental assumption in relative works of game theory is rationality. It implies that every player is motivated by increasing his own payoff, i.e. every player is looking to maximize his own utility. John V. Neumann and Morgenstern justified the idea of maximizing the expected payoff in their work in [23]. In this context, all information concerning the game is known to all players, i.e., there is complete information. So, we consider that all players are said to be rational and intelligent. A rational person is one who acts in such a way as to maximize his or her expected payoff or utility as economists would say. An intelligent person is one who can deduce what his or her opponent will do when acting rationally. In fact, humans use a propositional calculus in reasoning, the propositional calculus concerns truth functions of propositions, which are logical truths (statements that are true in virtue of their form). For this reason, the assumption of rational behaviour of players in telecommunications systems is more justified, as the players are usually devices programmed to operate in certain ways. However, there are previous studies that have shown that humans

Applying game theory in telecommunications problems is an active research area, in which game-theoretic models have been developed and studied in the last decades, [1, 2, 6, 7, 9, 16, 18]. These models are interested in pricing issues, they proposed non-cooperative game formulations to analyze behaviours of players that selfishly decide their strategies to maximize their respective profits. Other works consider the criteria of price as an implicit parameter, which is determined as a function of the degree of saturation on the network. Typically in these approaches, the price is a shadow price. For more details on those approaches see, [14, 15, 24]. Nonetheless, the price of anarchy has been studied in a large and diverse number of games, e.g., in areas like wireless ad-hoc networks [8, 13], routing and congestion [4, 19], network creation [3], or facility location [22]. In our model, we do not take into account network topology, but rather the effective service proposed by each SP as a single entity. In other words, the price and QoS proposed by an SP will not depend on the source or destination, distance, etc. that underlies the request of each user. After we have proved existence of Nash equilibrium, we propose a joint price and QoS algorithm which allows to learn the equilibrium price and QoS strategies decided by SPs. This is a simple algorithm

we mean pure strategy exact Nash equilibria unless mentioned otherwise.

do not always act rationally [10].

implementation with lower computational complexity.

**Related Works :**

The rest of the chapter is organized as follows : in Section 2 we describe the system model and introduce a new demand and utility functions. In Section 3 we formulate the joint price and QoS problem as a non-cooperative game, and investigate existence and uniqueness of a Nash equilibrium solution. Then, we present numerical results obtained from simulations that exploits our joint price and QoS algorithm in Section 4 . Conclusions and future guidelines are drawn in Section 5.

#### **2. Problem modeling**

In this chapter, we formulate the interaction among service providers (SPs) as a non-cooperative game. Each SP chooses the Quality of Service to guarantee (it depends on the amount of requested bandwidth) and the corresponding price.

We consider a system with *N* service providers. Let *pi* and *qi* be, respectively, the tariff/pricing policy and the QoS guaranteed by SP-*i*. Now, each customer seeks to subscribe to the operator which allows him to meet a QoS sufficient to satisfy his/her needs, at suitable price. We consider that behaviors of customer's has been handled by a simple function so called demand functions, see equation (1). This later depends on the price and QoS strategies of all SPs. From a tagged SP's point of view, the question is to set the best pricing strategy and the best QoS (amount of bandwidth to request from the network owner). SPs are supposed to know the effect of their policy on the customer's subscription policy. Whereas from customer's point of view, the question is to find the SP that has the best price-QoS tradeoff conditions.

#### **2.1. Demand model**

For simplicity, we consider that the demand function *Di* for services of the tagged SP-*i* is linear with respect to the set price *pi* and the promised QoS *qi*, see, [9]. This demand function depends also on prices **<sup>p</sup>**−*<sup>i</sup>* and QoS **<sup>q</sup>**−*<sup>i</sup>* set by the competitors. Namely, the demand function of SP-*i* depends on **p** = [*p*1, .., *pN*] and **q** = [*q*1, .., *qN*]. Eventually, *Di* is decreasing w.r.t. *pi* and increasing w.r.t. *pj*, *j* �= *i*. Whereas it is increasing w.r.t *qi* and decreasing w.r.t. *qj*, *j* �= *i*. Then, the demand functions w.r.t services of SP-*i* can be written as follows:

$$D\_i(\mathbf{p}, \mathbf{q}) = D\_i^0 - \mathfrak{a}\_i^i p\_i + \beta\_i^i q\_i + \sum\_{j, j \neq i} \left[ \mathfrak{a}\_i^j p\_j - \beta\_i^j q\_j \right], \quad \forall i \in \{1, \ldots, N\}. \tag{1}$$

where *D*<sup>0</sup> *<sup>i</sup>* is a positive constant used to insure non-negative demands over the feasible region. While *<sup>α</sup><sup>j</sup> <sup>i</sup>* and *<sup>β</sup><sup>j</sup> <sup>i</sup>* are positive constants representing respectively the sensitivity of service provider *i* to price and QoS of service provider *j*.

#### **2.2. Utility model**

The total revenue of SP-*i* is *Di*(**p**, **q**)*pi*. We assume that we have a single network owner, this latter charges each SP-*i* a cost *ϑ<sup>i</sup>* per unit of requested bandwidth. In order to insure the customers loyalty, the amount of bandwidth *μ<sup>i</sup>* required by SP-*i* should depend on *Di*(.) and on the QoS *qi* it wishes to offer to its customers. Therefore, the net profit of SP-*i* is simply the difference between the total revenue and the fee paid to the network owner:

**3.1. The Nash equilibrium**

with respect to their corresponding strategies, [20].

weighted sum of utility functions, [20].

**Definition 1.** *A QoS vector q*∗ = (*q*∗

[*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *if, for every i* ∈ *N , Ui*(*q*<sup>∗</sup>

which ensures existence of a Nash equilibrium.

formally defined as:

*unique.*

test:

functions.

Considering rationality of service providers, the Nash equilibrium concept is the natural concept solution of the NPQG game. We first will investigate the Nash equilibrium solution for the induced game as defined in the previous section. We will show that a Nash equilibrium solution exists and is unique by using the theory of concave games, [20]. We recall that a non-cooperative game **G** is called concave if all players' utility functions are strictly concave

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 161

According to, [20], a Nash equilibrium exists in a concave game if the joint strategy space is compact and convex, and the utility function that any given player seeks to maximize is concave in its own strategy and continuous at every point in the product strategy space.

is diagonally strictly concave, this implies that the Nash equilibrium point is unique. The notion of diagonal strict concavity means that an individual user has more control over its utility function than the other users have on it, and is proven using the pseudo-gradient of the

**Fixed-Price Game :** Considering some fixed price policy, a Nash equilibrium in QoS is

*<sup>i</sup>* , *q*<sup>∗</sup> −*i*

**Theorem 1.** *A Nash equilibrium in terms of QoS for game G* = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *exists and is*

*Proof.* To prove existence, we note that each SP's strategy space *Qi* is defined by all QoSs in the closed interval bounded by the minimum and maximum QoSs. Thus, the joint strategy space *Q* is a nonempty, convex, and compact subset of the Euclidean space **R***N*. In addition, the utility functions are concave with respect to QoSs as can be seen from the second derivative

In order to prove uniqueness, we follow, [20], and define the weighted sum of user utility

*N* ∑ *i*=1 ) ≥ *Ui*(*q*� *i* , *q*∗ −*i*

<sup>1</sup>, .., *q*<sup>∗</sup>

*∂*<sup>2</sup>*Ui*(**p**, **q**) *∂q*<sup>2</sup> *i*

*ϕ*(**q**, **x**) =

*xiUi*, *xi* > 0 ∀*i*. (4)

*<sup>N</sup>*) *is a Nash equilibrium of the NPQG : G* =

*<sup>i</sup>* ∈ *Qi.*

) *for all q*�

= −2*ϑ<sup>i</sup>* < 0, ∀*i* ∈ *N* , (5)

*xiUi*(*qi*, **<sup>q</sup>**−*i*), (6)

Formally, if the weighted sum of the utility functions with nonnegative weights:

*ϕ* = ∑ *i*=1

$$\mathcal{U}\_i(\mathbf{p}, \mathbf{q}) = D\_i(\mathbf{p}, \mathbf{q}) p\_i - F\_i(q\_i, D\_i)\_\prime \quad \forall i \in \{1, \dots, N\}.$$

where *Fi*(*qi*, *Di*) is the fee paid by SP-*i* (investment of SP-*i*) :

$$F\_{\bar{i}} = \theta\_{\bar{i}} \mu\_{\bar{i}} \left( q\_{\bar{i}\nu} D\_{\bar{i}} \right),$$

where *μ<sup>i</sup>* is the amount of bandwidth required by SP-*i*, such that *ϑ<sup>i</sup>* is a cost per unit of requested bandwidth We assume that the QoS corresponds to the expected delay, also we consider the Kleinrock delay which is a common delay used in Networking Games, so :

$$q\_i = \frac{1}{\sqrt{Delay\_i}} = \sqrt{\mu\_i - D\_i}$$

that mean that:

$$
\mu\_i = q\_i^2 + D\_i
$$

While, the utility function of the SP-*i* is given by the following formula:

$$dL\_i(\mathbf{p}, \mathbf{q}) = D\_i(\mathbf{p}, \mathbf{q}) \left(p\_i - \theta\_i\right) - \theta\_i q\_{i\prime}^2 \quad \forall i \in \{1, \ldots, N\}. \tag{2}$$

#### **3. A non-cooperative game formulation**

For a precise formulation of a non-cooperative game, we have to specify (i) the number of players, (ii) the possible actions available to each player, and any constraints that may be imposed on them, (iii) the objective function of each player which she attempts to optimize. Here we will consider formulation of games where items (i)-(iii) above are relevant.

Let **G** = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] denote the non-cooperative price and QoS game (NPQG), where *N* = {1, .., *N*} is the index set identifying the SPs, *Pi* is the price strategy set of SP-*i*, *Qi* is the QoS strategy set of SP-*i*, and *Ui*(.) is the utility function. Each SP-*i* selects a price *pi* ∈ *Pi* and a QoS measure *qi* <sup>∈</sup> *Qi*. Let the price vector **<sup>p</sup>** = (*p*1, .., *pN*)*<sup>T</sup>* <sup>∈</sup> *<sup>P</sup><sup>N</sup>* <sup>=</sup> *<sup>P</sup>*<sup>1</sup> <sup>×</sup> *<sup>P</sup>*<sup>2</sup> <sup>×</sup> ... <sup>×</sup> *PN*, QoS vector **<sup>q</sup>** = (*q*1, .., *qN*)*<sup>T</sup>* <sup>∈</sup> *<sup>Q</sup><sup>N</sup>* <sup>=</sup> *<sup>Q</sup>*<sup>1</sup> <sup>×</sup> *<sup>Q</sup>*<sup>2</sup> <sup>×</sup> ... <sup>×</sup> *QN* (where T represents the transpose operator). The utility of SP-*i* when it decides the strategy price *pi* to allocate the QoS *qi* is given in equation (2). We assume that the strategy spaces *Pi* and *Qi* of each SP are compact and convex sets with maximum and minimum constraints, For any given user *i* we consider strategy spaces the closed intervals *Pi* = [*pi*, *pi*] and *Qi* = [*qi*, *qi*].

In order to maximize their utilities, each SP-*i* decides a price *pi* and QoS *qi*. Formally, the NPQG problem can be expressed as:

$$\max\_{p\_l \in P\_l, q\_l \in Q\_l} \mathcal{U}\_i(\mathbf{p}, \mathbf{q})\_\prime \,\,\forall i \in \mathcal{N} \,\,. \tag{3}$$

#### **3.1. The Nash equilibrium**

4 Game Theory

on the QoS *qi* it wishes to offer to its customers. Therefore, the net profit of SP-*i* is simply the

*Ui*(**p**, **q**) = *Di*(**p**, **q**)*pi* − *Fi*(*qi*, *Di*), ∀*i* ∈ {1, .., *N*}.

*Fi* = *ϑiμ<sup>i</sup>* (*qi*, *Di*)

where *μ<sup>i</sup>* is the amount of bandwidth required by SP-*i*, such that *ϑ<sup>i</sup>* is a cost per unit of requested bandwidth We assume that the QoS corresponds to the expected delay, also we consider the Kleinrock delay which is a common delay used in Networking Games, so :

<sup>=</sup> *<sup>μ</sup><sup>i</sup>* <sup>−</sup> *Di*

*<sup>i</sup>* , ∀*i* ∈ {1, .., *N*}. (2)

*Ui*(**p**, **q**), ∀*i* ∈ *N* . (3)

difference between the total revenue and the fee paid to the network owner:

*qi* <sup>=</sup> <sup>1</sup>

While, the utility function of the SP-*i* is given by the following formula:

strategy spaces the closed intervals *Pi* = [*pi*, *pi*] and *Qi* = [*qi*, *qi*].

max *pi*∈*Pi*,*qi*∈*Qi*

NPQG problem can be expressed as:

**3. A non-cooperative game formulation**

*Ui*(**p**, **<sup>q</sup>**) = *Di*(**p**, **<sup>q</sup>**)(*pi* <sup>−</sup> *<sup>ϑ</sup>i*) <sup>−</sup> *<sup>ϑ</sup>iq*<sup>2</sup>

*Delayi*

*<sup>μ</sup><sup>i</sup>* = *<sup>q</sup>*<sup>2</sup>

For a precise formulation of a non-cooperative game, we have to specify (i) the number of players, (ii) the possible actions available to each player, and any constraints that may be imposed on them, (iii) the objective function of each player which she attempts to optimize.

Let **G** = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] denote the non-cooperative price and QoS game (NPQG), where *N* = {1, .., *N*} is the index set identifying the SPs, *Pi* is the price strategy set of SP-*i*, *Qi* is the QoS strategy set of SP-*i*, and *Ui*(.) is the utility function. Each SP-*i* selects a price *pi* ∈ *Pi* and a QoS measure *qi* <sup>∈</sup> *Qi*. Let the price vector **<sup>p</sup>** = (*p*1, .., *pN*)*<sup>T</sup>* <sup>∈</sup> *<sup>P</sup><sup>N</sup>* <sup>=</sup> *<sup>P</sup>*<sup>1</sup> <sup>×</sup> *<sup>P</sup>*<sup>2</sup> <sup>×</sup> ... <sup>×</sup> *PN*, QoS vector **<sup>q</sup>** = (*q*1, .., *qN*)*<sup>T</sup>* <sup>∈</sup> *<sup>Q</sup><sup>N</sup>* <sup>=</sup> *<sup>Q</sup>*<sup>1</sup> <sup>×</sup> *<sup>Q</sup>*<sup>2</sup> <sup>×</sup> ... <sup>×</sup> *QN* (where T represents the transpose operator). The utility of SP-*i* when it decides the strategy price *pi* to allocate the QoS *qi* is given in equation (2). We assume that the strategy spaces *Pi* and *Qi* of each SP are compact and convex sets with maximum and minimum constraints, For any given user *i* we consider

In order to maximize their utilities, each SP-*i* decides a price *pi* and QoS *qi*. Formally, the

Here we will consider formulation of games where items (i)-(iii) above are relevant.

*<sup>i</sup>* + *Di*

where *Fi*(*qi*, *Di*) is the fee paid by SP-*i* (investment of SP-*i*) :

that mean that:

Considering rationality of service providers, the Nash equilibrium concept is the natural concept solution of the NPQG game. We first will investigate the Nash equilibrium solution for the induced game as defined in the previous section. We will show that a Nash equilibrium solution exists and is unique by using the theory of concave games, [20]. We recall that a non-cooperative game **G** is called concave if all players' utility functions are strictly concave with respect to their corresponding strategies, [20].

According to, [20], a Nash equilibrium exists in a concave game if the joint strategy space is compact and convex, and the utility function that any given player seeks to maximize is concave in its own strategy and continuous at every point in the product strategy space. Formally, if the weighted sum of the utility functions with nonnegative weights:

$$\mathfrak{gl} = \sum\_{i=1} \mathfrak{x}\_i \mathsf{U}\_{i\prime} \quad \mathfrak{x}\_i > 0 \quad \forall i. \tag{4}$$

is diagonally strictly concave, this implies that the Nash equilibrium point is unique. The notion of diagonal strict concavity means that an individual user has more control over its utility function than the other users have on it, and is proven using the pseudo-gradient of the weighted sum of utility functions, [20].

**Fixed-Price Game :** Considering some fixed price policy, a Nash equilibrium in QoS is formally defined as:

**Definition 1.** *A QoS vector q*∗ = (*q*∗ <sup>1</sup>, .., *q*<sup>∗</sup> *<sup>N</sup>*) *is a Nash equilibrium of the NPQG : G* = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *if, for every i* ∈ *N , Ui*(*q*<sup>∗</sup> *<sup>i</sup>* , *q*<sup>∗</sup> −*i* ) ≥ *Ui*(*q*� *i* , *q*∗ −*i* ) *for all q*� *<sup>i</sup>* ∈ *Qi.*

**Theorem 1.** *A Nash equilibrium in terms of QoS for game G* = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *exists and is unique.*

*Proof.* To prove existence, we note that each SP's strategy space *Qi* is defined by all QoSs in the closed interval bounded by the minimum and maximum QoSs. Thus, the joint strategy space *Q* is a nonempty, convex, and compact subset of the Euclidean space **R***N*. In addition, the utility functions are concave with respect to QoSs as can be seen from the second derivative test:

$$\frac{\partial^2 \mathcal{U}\_l(\mathbf{p}, \mathbf{q})}{\partial q\_i^2} = -2\theta\_l < 0, \ \forall i \in \mathcal{N} \,. \tag{5}$$

which ensures existence of a Nash equilibrium.

In order to prove uniqueness, we follow, [20], and define the weighted sum of user utility functions.

$$\varphi(\mathbf{q}, \mathbf{x}) = \sum\_{i=1}^{N} \mathbf{x}\_i \mathsf{U}\_i (q\_{i\prime} \mathbf{q}\_{-i})\_{\prime} \tag{6}$$

#### 6 Game Theory 162 Game Theory Relaunched A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers <sup>7</sup>

The pseudo-gradient of (6) is given by :

$$\mathbf{g}(\mathbf{q}, \mathbf{x}) = \begin{bmatrix} \mathbf{x}\_1 \nabla \mathcal{U}\_1(q\_1, \mathbf{q}\_{-1}), \dots, \ x\_N \nabla \mathcal{U}\_N(q\_N, \mathbf{q}\_{-N}) \end{bmatrix}^T \tag{7}$$

The Jacobian matrix **J** of the pseudo-gradient (w.r.t. **q**) is written

$$\begin{split} \mathbf{J} &= \begin{pmatrix} \begin{array}{cc} \chi\_{1} \frac{\partial^{2} l\_{1}}{\partial q\_{1}^{2}} & \chi\_{1} \frac{\partial^{2} l\_{1}}{\partial q\_{1} \partial q\_{2}} & \cdots & \chi\_{1} \frac{\partial^{2} l\_{1}}{\partial q\_{1} \partial q\_{N}} \\ \chi\_{2} \frac{\partial^{2} l\_{2}}{\partial q\_{2} \partial q\_{1}} & \chi\_{2} \frac{\partial^{2} l\_{2}}{\partial q\_{2}^{2}} & \cdots & \chi\_{2} \frac{\partial^{2} l\_{2}}{\partial q\_{2} \partial q\_{N}} \end{pmatrix} \\ &\vdots & \vdots & \ddots & \vdots \\ \chi\_{N} \frac{\partial^{2} l\_{N}}{\partial q\_{N} \partial q\_{1}} & \chi\_{N} \frac{\partial^{2} l\_{N}}{\partial q\_{N} \partial q\_{2}} & \cdots & \chi\_{N} \frac{\partial^{2} l\_{N}}{\partial q\_{N}^{2}} \end{pmatrix} \\ &= \begin{pmatrix} -2\chi\_{1}\theta\_{1} & 0 & \cdots & 0 \\ 0 & -2\chi\_{2}\theta\_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & -2\chi\_{N}\theta\_{N} \end{pmatrix}. \end{split}$$

To prove uniqueness we define now the weighted sum of user utility functions

*N* ∑ *i*=1

*<sup>x</sup>*1∇*U*1(*p*1, **<sup>p</sup>**−1), ..., *xN*∇*UN*(*pN*, **<sup>p</sup>**−*N*)

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 163

In order to show that *φ*(**p**, **x**) is diagonally strictly concave in this case we use the following

**Lemma 1.** *If each Ui*(*p*) *is a strictly concave function in pi, each Ui*(*p*) *is convex in p*−*<sup>i</sup> and there is some x* > 0 *such that φ*(*p*, *x*) *is concave in p, then* [*J*(*p*, *x*) + *JT*(*p*, *x*)] *is negative definite, where*

= 0, ∀*i* �= *j*,

+ *N* ∑ *j*�=*i xj* *<sup>∂</sup>*<sup>2</sup>*Uj*(*pi*, **<sup>p</sup>**−*i*) *∂p*<sup>2</sup> *i*

), ∀*i* ∈ *N* . (12)

*xiUi*(*pi*, **<sup>p</sup>**−*i*), (10)

�*<sup>T</sup>* . (11)

*φ*(**p**, **x**) =

From equation (9), we know that *Ui*(**p**) is strictly concave in *pi*. Further

which implies that *Ui*(**p**) is convex in *<sup>p</sup>*−*<sup>i</sup>* as well. Also, we have that

= *xi*

fixed-QoS Nash equilibrium point is then unique and is given by

*<sup>i</sup>* ∈ argmax *pi*∈*Pi*

*p*∗

**3.2. The joint price and QoS game**

the utility given in equation (2) are respectively :

<sup>=</sup> <sup>−</sup>2*xiα<sup>i</sup>*

*∂*2*φ*(**p**, **x**) *∂p*<sup>2</sup> *i*

*∂*<sup>2</sup>*Ui ∂p*<sup>2</sup> *j*

*<sup>∂</sup>*<sup>2</sup>*Ui*(*pi*, **<sup>p</sup>**−*i*) *∂p*<sup>2</sup> *i*

*<sup>i</sup>* < 0, ∀*i*,

then *φ*(**p**, **x**) is concave in *pi* and from *Lemma* 1 we have that [**J**(**p**, **x**) + **J***T*(**p**, **x**)] is negative definite. Thus the weighted sum of utility functions *φ*(**p**, **x**) is diagonally strictly concave. The

> *Ui*(*pi*, **p**<sup>∗</sup> −*i*

As shown in equations (5) and (9), the utility functions *Ui*(**p**, **q**), ∀*i* ∈ *N* , are concave respectively w.r.t. *qi* and *pi*. So, for all, *i* ∈ *N* , the QoS and price conditions which maximizes

> *∂Ui*(**p**,**q**) *<sup>∂</sup>qi* = <sup>0</sup> *∂Ui*(**p**,**q**) *<sup>∂</sup>pi* = <sup>0</sup>

Thus, the computation of Nash Equilibrium can be performed by solving latter system.

⎧ ⎨ ⎩

the pseudo-gradient of this later is given by

lemma proved in, [11].

*J*(*p*, *x*) *is the Jacobian of g*(*p*, *x*)*.*

*g*(**p**, **x**) = �

Thus, **J** is a diagonal matrix with negative diagonal elements. This implies that **J** is negative definite. Henceforth [**J** + **J***T*] is also negative definite, and according to Theorem (6) in, [20], the weighted sum of the utility functions *ϕ*(**q**, **x**) is diagonally strictly concave. Thus the fixed-price Nash equilibrium point

$$q\_i^\* \in \operatorname\*{argmax}\_{q\_l \in Q\_l} \mathcal{U}\_i(q\_{l\nu} \mathbf{q}\_{-i}^\*)\_\prime \,\,\forall i \in \mathcal{N} \,\,\,\tag{8}$$

is unique.

**Fixed-QoS Game :** When fixing the QoS, a Nash equilibrium in terms of price is formally defined as :

**Definition 2.** *A price vector p*∗ = (*p*∗ <sup>1</sup>, .., *p*<sup>∗</sup> *<sup>N</sup>*) *is a Nash equilibrium of the NPQG : G* = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *if, for every i* ∈ *N , Ui*(*p*<sup>∗</sup> *<sup>i</sup>* , *p*<sup>∗</sup> −*i* ) ≥ *Ui*(*p*� *i* , *p*∗ −*i* ) *for all p*� *<sup>i</sup>* ∈ *Pi.*

**Theorem 2.** *A Nash equilibrium in terms of price for the game G* = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *exists and is unique.*

*Proof.* To prove existence, we note that each SP's 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* is a nonempty, 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:

$$\frac{\partial^2 U\_i(\mathbf{p}, \mathbf{q})}{\partial p\_i^2} = -2a\_i^i < 0, \ \forall i \in \mathcal{N} \tag{9}$$

which ensures existence of a Nash equilibrium.

To prove uniqueness we define now the weighted sum of user utility functions

$$\phi(\mathbf{p}, \mathbf{x}) = \sum\_{i=1}^{N} \mathbf{x}\_i \mathsf{U}\_i (p\_{i\prime} \mathbf{p}\_{-i})\_{\prime} \tag{10}$$

the pseudo-gradient of this later is given by

6 Game Theory

*x*1 *∂*2*U*<sup>1</sup> *<sup>∂</sup>q*1*∂q*<sup>2</sup> ··· *x*<sup>1</sup>

. .

*∂*2*U*<sup>2</sup> *∂q*<sup>2</sup> 2

.

−2*x*1*ϑ*<sup>1</sup> 0 ··· 0 0 −2*x*2*ϑ*<sup>2</sup> ··· 0

*∂*<sup>2</sup>*UN <sup>∂</sup>qN <sup>∂</sup>q*<sup>2</sup> ··· *xN*

*<sup>x</sup>*1∇*U*1(*q*1, **<sup>q</sup>**−1), ..., *xN*∇*UN*(*qN*, **<sup>q</sup>**−*N*)

*∂*2*U*<sup>1</sup> *∂q*1*∂qN* ⎞

⎟⎟⎟⎟⎟⎟⎠

*∂*2*U*<sup>2</sup> *∂q*2*∂qN*

> . .

. . *∂*<sup>2</sup>*UN ∂q*<sup>2</sup> *N*

⎞

⎟⎟⎟⎠ .

), ∀*i* ∈ *N* . (8)

*<sup>N</sup>*) *is a Nash equilibrium of the NPQG : G* =

) *for all p*�

*<sup>i</sup>* < 0, ∀*i* ∈ *N* , (9)

*<sup>i</sup>* ∈ *Pi.*

··· *x*<sup>2</sup>

. ... .

. ... .

0 0 ··· −2*xNϑ<sup>N</sup>*

Thus, **J** is a diagonal matrix with negative diagonal elements. This implies that **J** is negative definite. Henceforth [**J** + **J***T*] is also negative definite, and according to Theorem (6) in, [20], the weighted sum of the utility functions *ϕ*(**q**, **x**) is diagonally strictly concave. Thus the

> *Ui*(*qi*, **q**<sup>∗</sup> −*i*

**Fixed-QoS Game :** When fixing the QoS, a Nash equilibrium in terms of price is formally

*<sup>i</sup>* , *p*<sup>∗</sup> −*i*

**Theorem 2.** *A Nash equilibrium in terms of price for the game G* = [*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *exists*

*Proof.* To prove existence, we note that each SP's 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* is a nonempty, 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

<sup>=</sup> <sup>−</sup>2*α<sup>i</sup>*

) ≥ *Ui*(*p*� *i* , *p*∗ −*i*

<sup>1</sup>, .., *p*<sup>∗</sup>

�*<sup>T</sup>* (7)

The pseudo-gradient of (6) is given by :

fixed-price Nash equilibrium point

**Definition 2.** *A price vector p*∗ = (*p*∗

[*N* , {*Pi*, *Qi*}, {*Ui*(.)}] *if, for every i* ∈ *N , Ui*(*p*<sup>∗</sup>

which ensures existence of a Nash equilibrium.

is unique.

defined as :

*and is unique.*

test:

*g*(**q**, **x**) = �

**J** =

The Jacobian matrix **J** of the pseudo-gradient (w.r.t. **q**) is written

⎛

*x*1 *∂*2*U*<sup>1</sup> *∂q*<sup>2</sup> 1

*x*2 *∂*2*U*<sup>2</sup> *<sup>∂</sup>q*2*∂q*<sup>1</sup> *x*<sup>2</sup>

*xN ∂*<sup>2</sup>*UN <sup>∂</sup>qN <sup>∂</sup>q*<sup>1</sup> *xN*

⎛

⎜⎜⎜⎝

. .

> . . . . .

⎜⎜⎜⎜⎜⎜⎝

=

*q*∗

*<sup>i</sup>* ∈ argmax *qi*∈*Qi*

*∂*<sup>2</sup>*Ui*(**p**, **q**) *∂p*<sup>2</sup> *i*

$$\mathbf{g}(\mathbf{p}, \mathbf{x}) = \begin{bmatrix} \mathbf{x}\_1 \nabla \mathcal{U}\_1(p\_{1\prime} \mathbf{p}\_{-1}), \dots, \ x\_N \nabla \mathcal{U}\_N(p\_{N\prime} \mathbf{p}\_{-N}) \end{bmatrix}^T. \tag{11}$$

In order to show that *φ*(**p**, **x**) is diagonally strictly concave in this case we use the following lemma proved in, [11].

**Lemma 1.** *If each Ui*(*p*) *is a strictly concave function in pi, each Ui*(*p*) *is convex in p*−*<sup>i</sup> and there is some x* > 0 *such that φ*(*p*, *x*) *is concave in p, then* [*J*(*p*, *x*) + *JT*(*p*, *x*)] *is negative definite, where J*(*p*, *x*) *is the Jacobian of g*(*p*, *x*)*.*

From equation (9), we know that *Ui*(**p**) is strictly concave in *pi*. Further

$$\frac{\partial^2 \mathcal{U}\_i}{\partial p\_j^2} = 0, \ \forall i \neq j.$$

which implies that *Ui*(**p**) is convex in *<sup>p</sup>*−*<sup>i</sup>* as well. Also, we have that

$$\begin{split} \frac{\partial^2 \phi(\mathbf{p}, \mathbf{x})}{\partial p\_i^2} &= \mathbf{x}\_i \frac{\partial^2 \mathcal{U}\_i(p\_{i\prime} \mathbf{p}\_{-i})}{\partial p\_i^2} + \sum\_{j \neq i}^N \mathbf{x}\_j \frac{\partial^2 \mathcal{U}\_j(p\_{i\prime} \mathbf{p}\_{-i})}{\partial p\_i^2} \\ &= -2\mathbf{x}\_i \boldsymbol{\alpha}\_i^i < 0, \ \forall i \end{split}$$

then *φ*(**p**, **x**) is concave in *pi* and from *Lemma* 1 we have that [**J**(**p**, **x**) + **J***T*(**p**, **x**)] is negative definite. Thus the weighted sum of utility functions *φ*(**p**, **x**) is diagonally strictly concave. The fixed-QoS Nash equilibrium point is then unique and is given by

$$p\_i^\* \in \underset{p\_l \in \mathbb{P}\_l}{\text{argmax}} \,\mathcal{U}\_i(p\_{i\prime} \mathbf{p}\_{-i}^\*), \quad \forall i \in \mathcal{N} \,. \tag{12}$$

#### **3.2. The joint price and QoS game**

As shown in equations (5) and (9), the utility functions *Ui*(**p**, **q**), ∀*i* ∈ *N* , are concave respectively w.r.t. *qi* and *pi*. So, for all, *i* ∈ *N* , the QoS and price conditions which maximizes the utility given in equation (2) are respectively :

$$ \begin{cases} \frac{\partial L(\mathbf{p}, \mathbf{q})}{\partial q\_i} = 0\\ \frac{\partial L(\mathbf{p}, \mathbf{q})}{\partial p\_i} = 0 \end{cases} $$

Thus, the computation of Nash Equilibrium can be performed by solving latter system.

Now, we turn to develop a fully distributed algorithm to learn the two-parameter equilibrium. Designing distributed algorithms that converge quickly to equilibrium is one of the foremost research goals in algorithmic game theory, and convex programs have played a crucial role in the design of algorithms for markets. Assuming that Providers are selfish and choose dynamically each one the best price and QoS that maximize his profiles, the distributed algorithms can be thought of as protocols that players are programmed to follow. The design and analysis of distributed algorithms converging to equilibria in the context of games has also received considerable attention, most commonly convergence of best response dynamics.

where *W*(**p**, **q**) =

*N* ∑ *i*=1

**4. Numerical investigations**

100

**Figure 1.** Price game : Convergence to the Price Nash equilibrium.

120

140

160 180

200

SPs, Best Respense price

220

240 260

280

utilities of all actors at Nash Equilibrium.

parameter values considered in this numerical study. *α*1 <sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>2</sup> <sup>2</sup> *<sup>α</sup>*<sup>1</sup> <sup>2</sup> <sup>=</sup> *<sup>α</sup>*<sup>2</sup> <sup>1</sup> *<sup>β</sup>*<sup>1</sup> <sup>1</sup> <sup>=</sup> *<sup>β</sup>*<sup>2</sup> <sup>2</sup> *<sup>β</sup>*<sup>1</sup> <sup>2</sup> <sup>=</sup> *<sup>β</sup>*<sup>2</sup>

**Table 1.** System parameters used for numerical examples.

*Ui*(**p**, **q**) is a welfare function and *WNE*(**p**, **q**) =

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 165

To clarify and show how to take advantage from our theoretical study, we suggest to study numerically the market share game while considering the best response dynamics and expressions of demand as well as utility functions of SPs. Hence, we consider a system with two SPs seeking to maximize their respective revenues. Table 1 represents the system

0.7 0.3 0.7 0.3 300 250

Figures 1 and 2 present respectively curves of the convergence to Nash Equilibrium Price and to Nash Equilibrium QoS. It is clear that the best response dynamics converges to the unique Nash equilibrium price and QoS. We also remark that the speed of convergence is relatively high (around 9 rounds are enough to converge to the joint price and QoS equilibrium).

1 2 3 4 5 6 7 8 9

Time, t

Next we plot in figures 3 and 4, respectively, the interplay of bandwidth cost (*ϑi*, *i* ∈ {1, 2} on the price and QoS at Nash equilibrium, for both SPs that we consider in this example. On one hand, we note that the equilibrium price for both SPs is increasing with respect to the

*ϑ*<sup>1</sup> = *ϑ*<sup>2</sup> *p*<sup>1</sup> = *p*<sup>2</sup> *p*1, *p*<sup>2</sup> *q*<sup>1</sup> = *q*<sup>2</sup> *q*1, *q*<sup>2</sup> 20 100 1000 0 10

*N* ∑ *i*=1

<sup>1</sup> *<sup>D</sup>*<sup>1</sup>

<sup>0</sup> *<sup>D</sup>*<sup>2</sup> 0

SP−1 : BRp1

SP−2 : BRp2

(t)

(t)

*Ui*(**p**∗, **q**∗) is a sum of

Solutions of equations induces by vanishing the partial derivatives correspond respectively to the best response in terms of QoS *BR<sup>i</sup> <sup>q</sup>*(.), and best response Price *BR<sup>i</sup> <sup>p</sup>*(.), of each SP-*i* as a function of the strategies of its opponents. Since Nash equilibrium point is unique, then a best response-based dynamics would converge to the joint Price-QoS NE. The two-parameters best response dynamics is detailed in Algorithm 1.

#### **Algorithm 1** Best response dynamics

	- **a)** *pt*+<sup>1</sup> *<sup>i</sup>* <sup>=</sup> *BR<sup>i</sup> p*(**p***<sup>t</sup>* , **q***<sup>t</sup>* );
	- **b)** *qt*+<sup>1</sup> *<sup>i</sup>* <sup>=</sup> *BR<sup>i</sup> q*(**p***<sup>t</sup>* , **q***<sup>t</sup>* ).

#### **3.3. Social welfare and price of anarchy**

The concept of social welfare [17] or total surplus [21], is defined as the sum of the utilities of all agents in the systems (i.e. Providers). It is well known in game theory that agent selfishness, such as in a Nash equilibrium, does not lead in general to a socially efficient situation. As a measure of the loss of efficiency due to the divergence of user interests, we use the Price of Anarchy (PoA) [19], this latter is a measure of the loss of efficiency due to actors' selfishness. This loss has been defined in [19] as the worst-case ratio comparing the global efficiency measure (that has to be chosen) at an outcome of the noncooperative game played among actors, to the optimal value of that efficiency measure. A PoA close to 1 indicates that the equilibrium is approximately socially optimal, and thus the consequences of selfish behavior are relatively benign. The term Price of Anarchy was first used by Koutsoupias and Papadimitriou [19] but the idea of measuring inefficiency of equilibrium is older. The concept in its current form was designed to be the analogue of the "approximation ratio" in Approximation Algorithms or the "competitive ratio" in Online Algorithms. As in [12], we measure the loss of efficiency due to actors' selfishness as the quotient between the social welfare obtained at the Nash equilibrium and the maximum value of the social welfare:

$$PoA = \frac{\min\_{\mathbf{p}, \mathbf{q}} \mathcal{W}\_{\text{NE}}(\mathbf{p}, \mathbf{q})}{\max\_{\mathbf{p}, \mathbf{q}} \mathcal{W}(\mathbf{p}, \mathbf{q})} \tag{13}$$

where *W*(**p**, **q**) = *N* ∑ *i*=1 *Ui*(**p**, **q**) is a welfare function and *WNE*(**p**, **q**) = *N* ∑ *i*=1 *Ui*(**p**∗, **q**∗) is a sum of utilities of all actors at Nash Equilibrium.

#### **4. Numerical investigations**

8 Game Theory

Now, we turn to develop a fully distributed algorithm to learn the two-parameter equilibrium. Designing distributed algorithms that converge quickly to equilibrium is one of the foremost research goals in algorithmic game theory, and convex programs have played a crucial role in the design of algorithms for markets. Assuming that Providers are selfish and choose dynamically each one the best price and QoS that maximize his profiles, the distributed algorithms can be thought of as protocols that players are programmed to follow. The design and analysis of distributed algorithms converging to equilibria in the context of games has also received considerable attention, most commonly convergence of best response dynamics. Solutions of equations induces by vanishing the partial derivatives correspond respectively

function of the strategies of its opponents. Since Nash equilibrium point is unique, then a best response-based dynamics would converge to the joint Price-QoS NE. The two-parameters

The concept of social welfare [17] or total surplus [21], is defined as the sum of the utilities of all agents in the systems (i.e. Providers). It is well known in game theory that agent selfishness, such as in a Nash equilibrium, does not lead in general to a socially efficient situation. As a measure of the loss of efficiency due to the divergence of user interests, we use the Price of Anarchy (PoA) [19], this latter is a measure of the loss of efficiency due to actors' selfishness. This loss has been defined in [19] as the worst-case ratio comparing the global efficiency measure (that has to be chosen) at an outcome of the noncooperative game played among actors, to the optimal value of that efficiency measure. A PoA close to 1 indicates that the equilibrium is approximately socially optimal, and thus the consequences of selfish behavior are relatively benign. The term Price of Anarchy was first used by Koutsoupias and Papadimitriou [19] but the idea of measuring inefficiency of equilibrium is older. The concept in its current form was designed to be the analogue of the "approximation ratio" in Approximation Algorithms or the "competitive ratio" in Online Algorithms. As in [12], we measure the loss of efficiency due to actors' selfishness as the quotient between the social welfare obtained at the Nash equilibrium and the maximum value of the social welfare:

*PoA* <sup>=</sup> min**p**,**<sup>q</sup>** *WNE*(**p**, **<sup>q</sup>**)

max**p**,**<sup>q</sup>** *<sup>W</sup>*(**p**, **<sup>q</sup>**) (13)

*<sup>q</sup>*(.), and best response Price *BR<sup>i</sup>*

*<sup>p</sup>*(.), of each SP-*i* as a

to the best response in terms of QoS *BR<sup>i</sup>*

**Algorithm 1** Best response dynamics

*p*(**p***<sup>t</sup>* , **q***<sup>t</sup>* );

*q*(**p***<sup>t</sup>* , **q***<sup>t</sup>* ).

**3.3. Social welfare and price of anarchy**

**a)** *pt*+<sup>1</sup>

**b)** *qt*+<sup>1</sup>

*<sup>i</sup>* <sup>=</sup> *BR<sup>i</sup>*

*<sup>i</sup>* <sup>=</sup> *BR<sup>i</sup>*

best response dynamics is detailed in Algorithm 1.

1: Initialize price and QoS vectors **p** and **q** randomly; 2: For each service provider *i* ∈ *N* at iteration *t* :

To clarify and show how to take advantage from our theoretical study, we suggest to study numerically the market share game while considering the best response dynamics and expressions of demand as well as utility functions of SPs. Hence, we consider a system with two SPs seeking to maximize their respective revenues. Table 1 represents the system parameter values considered in this numerical study.


**Table 1.** System parameters used for numerical examples.

Figures 1 and 2 present respectively curves of the convergence to Nash Equilibrium Price and to Nash Equilibrium QoS. It is clear that the best response dynamics converges to the unique Nash equilibrium price and QoS. We also remark that the speed of convergence is relatively high (around 9 rounds are enough to converge to the joint price and QoS equilibrium).

**Figure 1.** Price game : Convergence to the Price Nash equilibrium.

Next we plot in figures 3 and 4, respectively, the interplay of bandwidth cost (*ϑi*, *i* ∈ {1, 2} on the price and QoS at Nash equilibrium, for both SPs that we consider in this example. On one hand, we note that the equilibrium price for both SPs is increasing with respect to the

**Figure 2.** QoS game : Convergence to the QoS Nash equilibrium.

bandwidth cost. On the other hand, we note that the equilibrium QoS for all SPs is decreasing with the bandwidth cost. When the cost of bandwidth decided by the network owner is cheaper, the SPs invest for more bandwidth, so as to offer better QoS and an attractive price.

10 20 30 40 50 60 70 80 90 100

<sup>=</sup>ϑ<sup>2</sup>

<sup>2</sup> = 1, in the other word, when the sensitivity of an SP

<sup>2</sup> = 0), price of anarchy converges to 1 and so the

<sup>1</sup> <sup>=</sup> *<sup>β</sup>*<sup>2</sup>

<sup>2</sup> = 0.76, *PoA*<sup>∗</sup> =

SP−1 : QoS q1

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 167

SP−2 : QoS q2

at equilibrium

at equilibrium

Cost per unit of requested bandwidth : ϑ<sup>1</sup>

**Influence of** *ϑ<sup>i</sup>* **(cost per unit of requested bandwidth):** Figure 5 shows the PoA variation curve as a function of the providers' bandwidth cost *ϑi*. Without loss of generality, we assume that *ϑ*<sup>1</sup> = *ϑ*2. A special feature is that the Nash equilibrium performs well and the loss of efficiency is only around 8%. This result indicates that the Nash equilibrium of this game is fair and socially efficient. Henceforth, selfish players would not need the help of a third-part regulator (who recommends the players the best strategy profile to achieve their respective best outcomes) to get attracted by the optimum social welfare. However, the network owner can use the value of the bandwidth cost to control the selfishness/aggressiveness of the service

**Influence of** *α* **(Sensitivity of SP-***i* **to his price** *pi***) :** Figure 6 plots the variation curve of price of anarchy with respect to *α* which represents the sensitivity of SP-*i* to his price *pi*. In that figure, we first notice that the price of anarchy increases when *α* increases, the fact that the price of anarchy increases with *α* finds the simple intuition that increasing the sensitivity of SPs to their prices gives more and more freedom to SPs for optimizing the Nash equilibrium.

**Influence of** *β* **(Sensitivity of SP-***i* **to his QoS** *qi***) :** Figure 7 illustrates variations of PoA as a function of, *β*, which is the sensitivity of SPs to their respective own QoS. We first notice that the loss of efficiency is around 8%. Moreover the curve of PoA is concave, this latter mean that

0.925). Surprisingly, the price of anarchy varies slightly (variation of almost 0.001). To explain this behaviour, Figures 8 and 9 depict, respectively, the curves of equilibrium Price and QoS of SP-1 and SP-2. We find that the induced variation of the price is much higher compared to

**Figure 4.** Equilibrium QoSs w.r.t cost per unit of requested bandwidth *ϑi*.

providers, which will improve the whole network performance.

<sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>2</sup>

there are some, *β*<sup>∗</sup> < 1, which optimizes the equilibrium, (*β*<sup>∗</sup> = *β*<sup>1</sup>

<sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>1</sup>

On the other hand, when *α* = *α*<sup>1</sup>

to the price of its competitor is zero (*α*<sup>2</sup>

equilibrium is approximately socially optimal.

Equilibrium QoS

**Figure 3.** Equilibrium Prices w.r.t cost per unit of requested bandwidth *ϑi*.

In the following, we discuss the impact of the system parameters on the system efficiency in terms of Price of anarchy:

**Figure 4.** Equilibrium QoSs w.r.t cost per unit of requested bandwidth *ϑi*.

10 Game Theory

1 2 3 4 5 6 7 8 9

SP−1 : BRq1

SP−2 : BRq2

<sup>=</sup>ϑ<sup>2</sup>

at equilibrium

at equilibrium

SP−1 : price p1

SP−2 :price p2

(t)

(t)

Time t

10 20 30 40 50 60 70 80 90 100

Cost per unit of requested bandwidth : ϑ<sup>1</sup>

In the following, we discuss the impact of the system parameters on the system efficiency in

**Figure 3.** Equilibrium Prices w.r.t cost per unit of requested bandwidth *ϑi*.

bandwidth cost. On the other hand, we note that the equilibrium QoS for all SPs is decreasing with the bandwidth cost. When the cost of bandwidth decided by the network owner is cheaper, the SPs invest for more bandwidth, so as to offer better QoS and an attractive price.

1

240

terms of Price of anarchy:

250

260

270

280

290

Equilibrium price

300

310

320

330

**Figure 2.** QoS game : Convergence to the QoS Nash equilibrium.

1.5

2

2.5

3

SPs, Best Respense QoS

3.5

4

4.5

5

**Influence of** *ϑ<sup>i</sup>* **(cost per unit of requested bandwidth):** Figure 5 shows the PoA variation curve as a function of the providers' bandwidth cost *ϑi*. Without loss of generality, we assume that *ϑ*<sup>1</sup> = *ϑ*2. A special feature is that the Nash equilibrium performs well and the loss of efficiency is only around 8%. This result indicates that the Nash equilibrium of this game is fair and socially efficient. Henceforth, selfish players would not need the help of a third-part regulator (who recommends the players the best strategy profile to achieve their respective best outcomes) to get attracted by the optimum social welfare. However, the network owner can use the value of the bandwidth cost to control the selfishness/aggressiveness of the service providers, which will improve the whole network performance.

**Influence of** *α* **(Sensitivity of SP-***i* **to his price** *pi***) :** Figure 6 plots the variation curve of price of anarchy with respect to *α* which represents the sensitivity of SP-*i* to his price *pi*. In that figure, we first notice that the price of anarchy increases when *α* increases, the fact that the price of anarchy increases with *α* finds the simple intuition that increasing the sensitivity of SPs to their prices gives more and more freedom to SPs for optimizing the Nash equilibrium. On the other hand, when *α* = *α*<sup>1</sup> <sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>2</sup> <sup>2</sup> = 1, in the other word, when the sensitivity of an SP to the price of its competitor is zero (*α*<sup>2</sup> <sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>1</sup> <sup>2</sup> = 0), price of anarchy converges to 1 and so the equilibrium is approximately socially optimal.

**Influence of** *β* **(Sensitivity of SP-***i* **to his QoS** *qi***) :** Figure 7 illustrates variations of PoA as a function of, *β*, which is the sensitivity of SPs to their respective own QoS. We first notice that the loss of efficiency is around 8%. Moreover the curve of PoA is concave, this latter mean that there are some, *β*<sup>∗</sup> < 1, which optimizes the equilibrium, (*β*<sup>∗</sup> = *β*<sup>1</sup> <sup>1</sup> <sup>=</sup> *<sup>β</sup>*<sup>2</sup> <sup>2</sup> = 0.76, *PoA*<sup>∗</sup> = 0.925). Surprisingly, the price of anarchy varies slightly (variation of almost 0.001). To explain this behaviour, Figures 8 and 9 depict, respectively, the curves of equilibrium Price and QoS of SP-1 and SP-2. We find that the induced variation of the price is much higher compared to that of QoS, and subsequently, *β* (Sensitivity of SPs to their QoS) has a smaller impact on the system.

0.5 0.6 0.7 0.8 0.9 1

<sup>1</sup> = *<sup>β</sup>*<sup>2</sup>

at equilibrium

at equilibrium

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

<sup>β</sup> : Sensitivity of SP−i to his QoS qi

<sup>1</sup> = *<sup>β</sup>*<sup>2</sup>

<sup>β</sup> : Sensitivity of SP−i to his QoS qi

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 169

<sup>2</sup> (Sensitivity of SP-*i* to his QoS *qi*)

Price of Anarchy

3

<sup>2</sup> (Sensitivity of SP-*i* to his QoS

4

Sp

5

−1 : QoS q1 at equilibrium

6

7

0.9215

268

270

SP

*qi*)

−1 : price p1 at equilibrium

272

274

**Figure 7.** Price of Anarchy as a function of *β* = *β*<sup>1</sup>

SP−1 : QoS q1

SP−1 : price p1

**Figure 8.** equilibrium Price and QoS of SP-1 as a function of *β* = *β*<sup>1</sup>

0.922

0.9225

0.923

0.9235

Price of Anarchy

0.924

0.9245

0.925

0.9255

**Figure 5.** Price of Anarchy as a function of cost per unit of requested bandwidth *ϑi*.

**Figure 6.** Price of Anarchy as a function of *α* = *α*<sup>1</sup> <sup>1</sup> = *<sup>α</sup>*<sup>2</sup> <sup>2</sup> (Sensitivity of SP-*i* to his price *pi*)

**Figure 7.** Price of Anarchy as a function of *β* = *β*<sup>1</sup> <sup>1</sup> = *<sup>β</sup>*<sup>2</sup> <sup>2</sup> (Sensitivity of SP-*i* to his QoS *qi*)

12 Game Theory

that of QoS, and subsequently, *β* (Sensitivity of SPs to their QoS) has a smaller impact on the

10 20 30 40 50 60 70 80 90 100

= ϑ<sup>2</sup>

Price of Anarchy

Price of Anarchy

Cost per unit of requested bandwidth : ϑ<sup>1</sup>

0.5 0.6 0.7 0.8 0.9 1

<sup>1</sup> = *<sup>α</sup>*<sup>2</sup>

α : Sensitivity of SP−i to his price pi

<sup>2</sup> (Sensitivity of SP-*i* to his price *pi*)

**Figure 5.** Price of Anarchy as a function of cost per unit of requested bandwidth *ϑi*.

system.

0.919

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

**Figure 6.** Price of Anarchy as a function of *α* = *α*<sup>1</sup>

Price of Anarchy

0.92

0.921

0.922

0.923

Price of Anarchy

0.924

0.925

0.926

**Figure 8.** equilibrium Price and QoS of SP-1 as a function of *β* = *β*<sup>1</sup> <sup>1</sup> = *<sup>β</sup>*<sup>2</sup> <sup>2</sup> (Sensitivity of SP-*i* to his QoS *qi*)

Loubna Echabbi

**6. References**

*National Institute of Post and Telecommunication, Madinat Al-Irfane, Rabat, Morocco*

URL: *http://www-sop.inria.fr/mistral/personnel/Eitan.Altman/ntkgame.html*

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design with sel´rsh agents, *Theory of Computing* 4: 77–109.

ad-hoc networks, *Mobile Networks and Applications* 11: 143–159.

concave n-person games, *In Econometrica* pp. 48–251.

URL: *http://www.statslab.cam.ac.uk/frank/elastic.html*

with price and quality-of-service characteristics, *Proc. of ITC, Berlin* . [10] Friedman, J. [1996]. The rational choice controversy, *Yale University Press* .

*"NetGCOOP, Paris, France* .

chains with price and servicecompetition. URL: *http://faculty.fuqua.duke.edu/fernando/bio/*

*Shanghai, China* .

*USA* pp. 9 – 12.

*Telecommunications* 8: 33–37.

[1] Altman, E. & Wynter, L. [2002]. Equilibrium, games, and pricing in transportation and telecommunications network, *submitted to the special issue of "Networks and Spacial Economics", on "Crossovers between Transportation Planning and Telecommunications* .

A Game Theoretic Analysis of Price-QoS Market Share in Presence of Adversarial Service Providers 171

[2] Altman, E. & Wynter, L. [2004]. Equilibrium, games and pricing in transportation and telecommunications networks, *Networks and Spacial Economics, special issue of on:*

[3] Anshelevich, E., Dasgupta, A., Tardos, E. & Wexler, T. [2008]. Near-optimal network

[4] Awerbuch, B., Azar, Y. & Epstein, A. [2005]. The price of routing unsplittable flow, *In*

[5] Baslam, M., Azouzi, R. E., Sabir, E. & Echabbi, L. [2011]. Market share game with adversarial access providers : A neutral and a non-neutral network analysis, *Proc. of*

[6] Baslam, M., Echabbi, L., Azouzi, R. E. & Sabir, E. [2011]. Joint price and qos market share game with adversarial service providers and migrating customers, *Proc. of GameNets,*

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convergence, *IEEE/ACM Transactions on Networking* 7: 961U874. ˝

**Figure 9.** equilibrium Price and QoS of SP-2 as a function of *β* = *β*<sup>1</sup> <sup>1</sup> = *<sup>β</sup>*<sup>2</sup> <sup>2</sup> (Sensitivity of SP-*i* to his QoS *qi*)

#### **5. Conclusion**

In this work, we presented and analyzed a framework to model the complex interactions among SPs as players through a class of two parameter Nash equilibrium models. The model is based on a simple linear demand functions which describe customer behaviour, take into account not only the characteristics of a current SP, (SP-*i*), but also of all other SPs, (SP-*j*, *j i*), the presence of two parameters describing each SP's service price and QoS level. We established uniqueness of a Nash equilibrium point and developed a distributed algorithm to learn it. Then, our proposed algorithm finds very fast the equilibrium price and the equilibrium QoS to be chosen by each provider. Our scheme is different from previous approaches since it involves two varying parameters in a simple implementation and low complexity. Yet, we have obtained some insightful results such as the interplay of bandwidth cost. Results found in this work can be further extended to general network considerations, in particular under non-neutrality perspective or non-linear demand.

### **Author details**

Mohamed Baslam and El-Houssine Bouyakhf *LIMIARF, University of Mohammed V, Faculty of Sciences, Rabat, Morocco*

Rachid El-Azouzi *LIA-CERI, University of Avignon, Avignon, France*

Essaid Sabir *RTSE Laboratory, GREENTIC/ENSEM, Hassan II University, Casablanca, Morocco*

#### Loubna Echabbi

14 Game Theory

at equilibrium

at equilibrium

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

In this work, we presented and analyzed a framework to model the complex interactions among SPs as players through a class of two parameter Nash equilibrium models. The model is based on a simple linear demand functions which describe customer behaviour, take into account not only the characteristics of a current SP, (SP-*i*), but also of all other SPs, (SP-*j*, *j i*), the presence of two parameters describing each SP's service price and QoS level. We established uniqueness of a Nash equilibrium point and developed a distributed algorithm to learn it. Then, our proposed algorithm finds very fast the equilibrium price and the equilibrium QoS to be chosen by each provider. Our scheme is different from previous approaches since it involves two varying parameters in a simple implementation and low complexity. Yet, we have obtained some insightful results such as the interplay of bandwidth cost. Results found in this work can be further extended to general network considerations,

<sup>β</sup> : Sensitivity of SP−i to his QoS qi

<sup>1</sup> = *<sup>β</sup>*<sup>2</sup>

2.5

<sup>2</sup> (Sensitivity of SP-*i* to his QoS

3

Sp

−2 : QoS q2 at equilibrium

3.5

4

4.5

5

210

SP

**5. Conclusion**

**Author details**

Rachid El-Azouzi

Essaid Sabir

*qi*)

−2 : price p2 at equilibrium

211

212

213

214

215

SP−2 : Price p2

SP−2 : QoS q2

**Figure 9.** equilibrium Price and QoS of SP-2 as a function of *β* = *β*<sup>1</sup>

in particular under non-neutrality perspective or non-linear demand.

*LIMIARF, University of Mohammed V, Faculty of Sciences, Rabat, Morocco*

*RTSE Laboratory, GREENTIC/ENSEM, Hassan II University, Casablanca, Morocco*

Mohamed Baslam and El-Houssine Bouyakhf

*LIA-CERI, University of Avignon, Avignon, France*

*National Institute of Post and Telecommunication, Madinat Al-Irfane, Rabat, Morocco*

#### **6. References**


*Simulation of Computer and Telecommunication Systems (MASCOTS 2009), IEEE Computer Society, London, UK* .


© 2013 Mumtaz et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Mumtaz et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Cooperative Game Theory and** 

Senka Hadzic, Shahid Mumtaz and Jonathan Rodriguez

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53930

**1. Introduction** 

**Its Application in Localization Algorithms** 

Game theory is a field of applied mathematics for analyzing complex interactions among entities. It is basically a collection of analytic tools that enables distributed decision process*.*  Game theory (GT) provides insights into any economic, political, or social situation that involves individuals with different preferences. GT is used in economics, political science and biology to model competition and cooperation among entities, and the role of threats/punishments in long term relations. Contemporary social science is based on game theory, economics, and psychology in which mathematical logic is applied. The formation of coalitions or alliances is omnipresent in many applications. For example, in political games, parties, or individuals can form coalitions for improving their voting power. Recently, computer science and engineering have been added to the list of scientific areas applying GT.

While in optimization theory the goal is to optimize a single objective over one decision variable, game theory studies multi-agent decision problems. In social sciences and economics, the focus of game is the design of right incentives/payoffs; in engineering it comes to efficiency – how to design efficient decentralized schemes that take into account incentives. However, there are still similarities when applying game theory to different disciplines. For example, a measurement allocation framework for localization in wireless networks, based on the idea to allocate more measurements to the nodes which contribute more, mimics a capitalist society where the gains are mostly reinvested where more profit is

In general, a game consists of a set of players (decision makers), while each player has its strategy, whereby utility (payoff) for each player measures its level of satisfaction. Each player's objective is to maximize the expected value of its own payoff (Myerson, 1997).

(Srivastava V., et all, 2005) proposed a mapping of network components to game

expected. It also replicates the concept of natural selection in population genetics.

components according to the following table:

[24] Wynter, L. [2001]. Optimizing proportionally fair prices, *INRIA RR. 4311* . URL: *http://www.inria.fr/rrrt/rr-4311.html*
