The Network Economic Game Theory Model of Price and Quality Competition in a Service-Oriented Internetand Quality Competition in a Service-Oriented Internet

In this section, I develop a network economic game theory model for a multi-provider service-oriented network with heterogeneous markets of users. The network structure of the problem, which depicts the direction of the content flows, is given in Figure 3.3. See Figure 3.4 for a graphic depiction of the financial payments

in this general model. I assume m content providers, a typical one denoted by CP_i; {i = 1, . . . , m}, n network providers, denoted by N P_j; {j = 1, . . . , n}, and o markets of users, denoted by u_k; {k = 1, . . . , o}. These providers compete under the Nash concept of noncooperative behavior to set their prices and quality levels so as to maximize their utilities, which are in the form of profits.



Figure 3.3. The Network Structure of the Multi-Provider Model’s Content Flows



Figure 3.4. Graphic of the Multi-Provider Model with a Focus on Payments

To receive a unit of content service from CP_i with quality q_ci, which is transmitted by N P_j with quality q_sj, a user pays p_ci and p_sj to the CP_i and N P_j, respectively.

The content providers also pay the network providers for transferring their content to the users. Each network provider N P_j has a fixed transmission fee p_tj that he charges the CPs per unit of content. I group the p_tj, p_sj, q_sj, p_ci, and q_ci for i = 1, . . . , m;

j = 1, . . . , n, into vectors p_t, p_s, q_s, p_c, and q_c, respectively.

The users are heterogeneous in their demands and signal their preferences through a demand function d_ijkfor the content produced by content provider i and transmitted by N P_j to demand market k:

dijk = dijk(pc, qc, ps, qs), ∀i, j, k. (3.32)

In this game theory model, the demand d_ijk does not only depend on the price and quality of CP_i and N P_j, but also on the prices and quality levels of the other content and network providers as a result of competition among the providers. Moreover, unlike the specialized, illustrative model in Section 3.1, the demand functions above need not be linear, as in (3.1), and in the work of Altman, Legout, and Xu (2011) and El Azouzi, Altman, and Wynter (2003).

Herein, if p_sj and p_ci (q_sj, and q_ci) decrease (increase), d_ijk naturally goes up, but it decreases if the price (quality) of the other providers decreases (increases). I now describe the behavior of the content providers.

Each content provider CP_i produces distinct (but substitutable) content of specific quality q_ci, and sells at a unit price of p_ci. The total supply of CP_i, SCP_i, is given by:

SCPi =

n

X

j=1 o

X

k=1

dijk, i = 1, . . . , m. (3.33)

Each CP_i has a production cost, CC_i, which is a function of his supply and his quality of service:

CC_i = CC_i(SCP_i, q_ci), i = 1, . . . , m. (3.34) I assume that the production cost functions are convex, continuous, and continuously differentiable functions.

Also, I assume that the content providers are profit-maximizers, where the profit or utility of CP_i; i = 1, . . . , m, which is the difference between his total revenue and his total cost, is given by the expression:

U_CPi = U_CPi(p_c, q_c, p_s, q_s) =

n

X

j=1

(p_ci− p_tj)

o

X

k=1

d_ijk− CC_i. (3.35)

Let K_i¹ denote the feasible set corresponding to CP_i, where K¹_i ≡ {(p_ci, q_ci) | p_ci ≥ 0 and q_ci ≥ 0}.

I now describe the behavior of the network providers.

A network provider N P_j; j = 1, . . . , n, is distinguishable by means of his quality q_sj, the fee p_tj that he charges each content provider to transfer one unit of content to the users, and the fee p_sj that he charges users to transfer them one unit of content. By charging p_tj, I have a two-sided market. Here, as in Section 3.1, the p_tjs are assumed to be an exogenous parameter in this multi-provider model. I assume that all content providers are connected to all network providers and, subsequently, to all users. The total amount of content of services transported by N Pj, T N Pj, is given by:

T N P_j =

m

X

i=1 o

X

k=1

d_ijk, j = 1, . . . , n. (3.36) N P_j incurs the cost, CS_j, of maintaining his network based on the offered quality and the total traffic passing through his bandwidth:

CS_j = CS_j(T N P_j, q_sj), j = 1, . . . , n. (3.37) Similar cost functions were used in Altman, Legout, and Xu (2011), where it was noted that the (transport) network provider has to cover the costs of operating the backbone, the last mile, upgrades, etc. I also assume that these cost functions are convex, continuous, and continuously differentiable functions. The utility of N Pj; j = 1, . . . , n is defined as the difference between his income and his cost, that is:

UN Pj = UN Pj(pc, qc, ps, qs) = (psj+ ptj)T N Pj− CSj. (3.38) Let K²_j denote the feasible set corresponding to N P_j, where K²_j ≡ {(p_sj, q_sj) | p_sj ≥ 0 and q_sj ≥ 0}.

I now consider the Nash equilibrium that captures the providers’ behavior.

Definition 3.2: Nash Equilibrium in Price and Quality A price and quality level pattern (p^∗_c, q^∗_c, p^∗_s, q_s^∗) ∈ K³ ≡Qm

i=1K¹_i ×Qn

j=1K²_j, is said to constitute a Nash equilibrium if for each content provider CP_i; i = 1, . . . , m:

UCPi(p^∗_ci, ˆp^∗_ci, q_c^∗_i, ˆq^∗_ci, p^∗_s, q_s^∗) ≥ UCPi(pci, ˆp^∗_ci, qci, ˆq^∗_ci, p^∗_s, q_s^∗), ∀(pci, qci) ∈ K¹_i, (3.39)

where

pˆ^∗_ci ≡ (p^∗_c1, . . . , p^∗_ci−1, p^∗_ci+1, . . . , p^∗_cm) and ˆq_c^∗_i ≡ (q_c^∗₁, . . . , q_c^∗_i−1, q_c^∗_i+1, . . . , q_c^∗_m), (3.40)

and if for each network provider N P_j; j = 1, . . . , n:

U_{N Pj}(p^∗_c, q_c^∗, p^∗_s

j, ˆp^∗_s

j, q^∗_s

j, ˆq_s^∗

j) ≥ U_{N Pj}(p_sj,p^∗_c, qˆ^∗_c, p^∗_s

j, q_sj, ˆq_s^∗

j), ∀(p_sj, q_sj) ∈ K²_j, (3.41)

where

pˆ^∗_sj ≡ (p^∗_s1, . . . , p^∗_sj−1, p^∗_sj+1, . . . , p^∗_sn) and ˆq_s^∗_j ≡ (q_s^∗₁, . . . , q_s^∗_j−1, q^∗_sj+1, . . . , q_s^∗_n). (3.42)

According to (3.39) and (3.41), a Nash equilibrium is established if no provider can unilaterally improve upon his profits by selecting an alternative vector of quality levels and prices.

Theorem 3.4: Variational Inequality Formulations of Nash Equilibrium for the Service-Oriented Internet

Assume that the provider utility functions are concave, continuous, and continuously differentiable. Then (p^∗_c, q_c^∗, p^∗_s, q_s^∗) ∈ K³ is a Nash equilibrium according to Definition 3.2 if and only if it satisfies the variational inequality:

−

+

Variational inequality (3.44) can be put into standard form (cf. (2.1)): determine X^∗ ∈ K³ such that:

hF (X^∗), X − X^∗i ≥ 0, ∀X ∈ K, (3.45) where F (X) is a continuous function such that F (X) : X 7→ K ⊂ R^N, and K is a closed and convex set. The term h·, ·i denotes the inner product in N -dimensional Euclidean space. I define X ≡ (p_c, q_c, p_s, q_s), and F (X) ≡ (F_pc, F_qc, F_ps, F_qs). The

3.3. Explicit Formulae for the Euler Method Applied to Vari-ational Inequality (3.45) with F (X) Defined by (3.46) – (3.49)

For computational purposes, the Euler method (cf. Section 2.4), which is a discrete-time algorithm serving as an approximation to the continuous-discrete-time trajectories, is utilized.

The elegance of this procedure for the computation of solutions to this network economic model of the service-oriented Internet can be seen in the following explicit formulae. Indeed, variational inequality (3.44) yields the following closed form expressions, at each iteration, for the price and quality levels of each content and network provider i = 1, . . . , m; j = 1, . . . , n:

q_s^{τ +1}_j = max



0, q_s^τ_j + a_τ(

m

X

i=1 o

X

k=1

∂d_ijk

∂q_sj × (p^τ_sj + p_tj) − ∂CS_j(T N P_j, q_s^τ_j)

∂q_sj )



. (3.53)

Notice that all the functions to the left of the equal signs in (3.50)-(3.53) are evaluated at their respective variables computed at the τ -th iteration.

I now provide the convergence result. The proof is direct from Theorem 5.8 in Nagurney and Zhang (1996).

Theorem 3.5: Convergence

In this service-oriented Internet model, assume that F (X) = −∇U (p_c, q_c, p_s, q_s) is strongly monotone. Also, assume that F is uniformly Lipschitz continuous. Then, there exists a unique equilibrium price and quality pattern (p^∗_c, q_c^∗, p^∗_s, q_s^∗) ∈ K³ and any sequence generated by the Euler method as given by (3.50)-(3.53), where {a_τ} satisfies P∞

τ =0a_τ = ∞, a_τ > 0, a_τ → 0, as τ → ∞ converges to (p^∗_c, q_c^∗, p^∗_s, q_s^∗).