# Strategic Bandwidth Pricing and Allocation in a 5G/6G Network Slicing Stackelberg Game

Orland-Médy Saizonou, Patrick Maillé, Bruno Tuffin

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Orland-Médy Saizonou, Patrick Maillé, Bruno Tuffin. Strategic Bandwidth Pricing and Allocation in a 5G/6G Network Slicing Stackelberg Game. ISNCC 2025 - 12th International Symposium on Networks, Computers and Communications, Oct 2025, Paris, France. pp.1-6. hal-05268398

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# Strategic Bandwidth Pricing and Allocation in a 5G/6G Network Slicing Stackelberg Game

Orland-Médy Saizonou

Patrick Maillé

Bruno Tuffin

Inria, Univ. Rennes, CNRS, IRISA

IMT Atlantique, IRISA, UMR CNRS 6074

Inria, Univ. Rennes, CNRS, IRISA

Rennes, France

Rennes, France

Rennes, France

orland-medy.saizonou@inria.fr

patrick.maille@imt.fr

bruno.tuffin@inria.fr

**Abstract**—Network slicing, a key feature of next-generation mobile networks, has underexplored economic implications. We model a Stackelberg game where a mobile operator sells a slice to a service provider offering two service tiers, showing that slicing boosts revenues and improves resource allocation.

**Index Terms**—Network slicing, 5G networks, 6G networks, resource allocation, pricing

# I. INTRODUCTION

The rapid growth of demand in wireless communications and global connectivity has led to the development of 5G and now the targeted 6G networks, aiming to offer customized services like Augmented Reality (AR), Virtual Reality (VR), mobile tele-surgery, or cloud gaming. 5G and 6G networks provide Mobile Network Operators (MNOs) with the capability to offer sufficient Quality of Service (QoS) to users for each type of service, thanks to *wireless network virtualization*, that is the process of abstracting and decoupling the physical wireless network infrastructure into multiple virtualized network instances, sharing the same physical wireless resources while operating independently.

Among the new concepts that have been proposed for 5G and the upcoming future networks for network virtualization, probably one of the most important one is network slicing. Slicing allows to divide a single physical network into multiple “slices”. Each slice operates as an independent network, customized for specific needs, including bandwidth, latency, and security [1]. But the implementation of slicing capabilities requires the creation of a supporting business model to ensure an efficient use of the network, limiting slice tenants regarding the size of their allocated slice to avoid a potential unnecessary reservation of resources [2]. Network slicing offers an opportunity for MNOs and Service Providers (SPs) to attract more end users and generate a larger return on investment, offering the most appropriate QoS at the most appropriate price thanks to market segmentation.

In this paper we propose to analyze the best strategies for MNOs, SPs, and End Users (EUs) in a competitive/selfish environment using non-cooperative game theory [3].

The model we introduce considers one MNO offering an Internet access to EUs, one SP offering two service classes with different QoS for a given (additional) service type. It for instance illustrates a practical situation in Germany where the MNO Deutsche Telekom has developed secure 5G network

slicing technology that directly connects to a private cloud, for mobile cloud gaming, and addressing enterprise concerns over high-bandwidth. Viveport, a SP (VR Content Provider) needing QoS buys bandwidth from Deutsche Telekom in order to offer a premium service called Infinity in addition to a basic one called Infinity Vista, hence requiring two specific slices<fnref n="1" />. Since decisions by the various participants occur at three different time scales, we examine our model as a Stackelberg game [4]–[6].

Note that studying economic models of network slicing has already been the topic of several works. Among the notable works, several are based on auction-related mechanisms. In both [7] and [8], the authors propose a pricing scheme for slices based on the Kelly mechanism where SPs, also often called Network Slice Tenants simultaneously submit a bid to the MNO, obtain a portion of the total resource in proportion to the submitted bids and pay depending on how much they bid. The authors of [9] consider the same set of actors as in the previous model but use the Bertrand-based price mechanism [10] in the game between SPs, while the authors of [11] use the Vickrey-Clarke-Groves auctions [12]. In contrast, we introduce EUs as actors in our paper, leading to a Stackelberg-based game with decisions taken at different time scales.

Some other works also consider EUs as actors. [13], [14] and [15] also consider a model with MNO, SPs and EUs in a Stackelberg game where the MNO sells all its capacity to SPs. In contrast, our paper allows EUs to use MNO’s service without using SPs’ services, adding a strategic choice for the MNO.

Some other works, even closer to us, allow the MNO to keep a part of its capacity to directly provide service to customers, and analyze decisions through a Stackelberg game. It is the case of [16] where customers’ behavior is represented by a Markov process solved by dynamic programming. Nonetheless the analysis is designed to investigate the interest of cooperation between the SP and the MNO, not that of slicing (note that in [16], it is not slicing which is modeled but the introduction of a *virtual* operator, which can be seen as an equivalent problem). Our paper also sees the SP activity as *complementary* to MNO’s one, not substitutable, therefore not in competition. Also, one of the main differences with the

<footnote n="1">¹See https://www.viveport.com/joininfinity</footnote>
closest paper by some of the co-authors [4] with a similar model, is that user behavior is analyzed using a discrete choice model while we consider here the Wardrop equilibrium notion, and again the offered services are substitutable.

The remainder of this paper is organized as follows. Section II presents the model with actors involved, their decisions, and their utilities as well as the decisions, leading to a Stackelberg game. Section III solves the user equilibrium using the Wardrop principle, showing its existence and uniqueness in terms of service prices and slice sizes. Section IV analyzes the corresponding and anticipated decisions made by the SP, and by the MNO. Section V analyzes the simplified settings where the SP offers a single service option and where there is even no SP, and compares the outputs. We conclude the document in Section VI with some suggestions for future work.

## II. SYSTEM MODEL

We present here a model for the interactions between an MNO, an SP seeking a capacity slice to possibly offer two services, and EUs choosing between their services.

### A. Mobile network operator

The MNO holds a fixed capacity $C$, which it can divide into two slices: one for Internet with capacity $C_I$, and another, with capacity $C_S = C - C_I$, sold to the SP. The MNO earns revenue by charging the SP a total price $P(C_S)$, with $P(\cdot)$ an increasing function, and by offering Internet access to EUs at price $q_I$.

### B. Service provider

The SP buys a slice of capacity $C_S$ from the MNO to support its service, which is offered in two versions: premium and basic, with respective capacities $C_P$ and $C_B$ such that $C_P + C_B = C_S$, and prices $q_P$ and $q_B$, respectively (as those capacities will be dimensioned in a fixed manner by the SP, they can be considered as belonging to two distinct (sub-)slices).

### C. End-users

We consider a total mass $M > 0$ of infinitesimal EUs, each choosing among four service options: Internet only ($I$), Internet with premium service ($P$), Internet with basic service ($B$), or no service (0). Access to the SP services requires a subscription to Internet. Let $d_X$ denote the mass of users choosing option $X$, with $X \in \{I, P, B, 0\}$.

### D. Utility functions

From the previous settings, we define the utility functions guiding each actor's strategy.

The MNO aims to maximize its total revenue:

$$R = P(C_S) + (d_P + d_B + d_I)q_I \quad (1)$$

where $P(C_S)$ is the income from selling capacity to the SP, and the second term is the revenue from EUs subscribing to Internet access.

The SP's objective function will be:

$$R_S = d_P q_P + d_B q_B - P(C_S) \quad (2)$$

where revenues from premium and basic services are offset by the slice cost.

EUs will select an option maximizing their utility. For Internet only ($I$) we consider a utility of the form:

$$U_I = \ln \left( 1 + \frac{C_I}{d_P + d_B + d_I} \right) - q_I \quad (3)$$

where $\frac{C_I}{d_P+d_B+d_I}$ represents the Internet capacity per user mass unit, and the logarithm in the valuation is a standard assumption representing diminishing returns [17].

For the two SP-based options $X \in \{P, B\}$, we consider utilities of the form

$$U_X = U_I + \ln \left( 1 + \alpha_X \frac{C_X}{d_X} \right) - q_X, \quad (4)$$

with $\alpha_X$ capturing sensitivity to SP service quality. The utility of opting out is:

$$U_0 = 0. \quad (5)$$

EUs choosing the option(s) with the highest utility leads to a user distribution that is derived in Section III.

### E. Order of decisions

We assume that decisions occur at different time scales, with each actor anticipating the responses of those who follow. This setup forms a Stackelberg game [18] where:

* At the highest time scale, the MNO sets the Internet price $q_I$ and the slice price function $P(C_S) = \gamma C_S^2$, aiming to maximize revenue $R$ in (1), with $\gamma > 0$ as its decision variable.

* At a smaller time scale, the SP chooses the slice size $C_S$, its allocation ($C_P, C_B$), and service prices ($q_P, q_B$) to maximize revenue $R_S$ in (2).

* At the smallest scale, EUs choose services to maximize utility, leading to a Wardrop equilibrium [19].

## III. USER EQUILIBRIUM

Infinitesimal EUs choose the option with the highest utility among $U_I, U_P, U_B$, and $U_0$ as defined in Subsection II-D. The resulting user distribution corresponds to the Wardrop equilibrium notion, where only options with maximal utility have non-zero demand [19]. The following proposition established its existence and uniqueness in terms of prices and capacities (explicit expressions omitted for brevity, see in the extended version [20]).

**Proposition 1.** Given non-negative capacities $C_P, C_B, C_I$ and strictly positive prices $q_P, q_B, q_I$, a unique user equilibrium ($d_P, d_B, d_I, d_0$) exists, whose expression can take four different forms depending on the values:

* Domain $\beta_1$: if $M < \frac{\alpha_P C_P (1 + \frac{C_I}{M})}{e^{q_P+q_I} - (1 + \frac{C_I}{M})} + \frac{\alpha_B C_B (1 + \frac{C_I}{M})}{e^{q_B+q_I} - (1 + \frac{C_I}{M})}$, then only Options P and B are chosen.

* Domain $\beta_2$: if $\frac{\alpha_P C_P (1 + \frac{C_I}{M})}{e^{q_P+q_I} - (1 + \frac{C_I}{M})} + \frac{\alpha_B C_B (1 + \frac{C_I}{M})}{e^{q_B+q_I} - (1 + \frac{C_I}{M})} \le M$ and $\frac{C_I}{e^{q_I-1}} \le \frac{\alpha_P C_P}{e^{q_P-1}} + \frac{\alpha_B C_B}{e^{q_B-1}}$, Options P, B, and 0 are chosen.

* Domain $\beta_3$: if on the other hand $\frac{\alpha_P C_P}{e^{q_P-1}} + \frac{\alpha_B C_B}{e^{q_B-1}} < M \le \frac{C_I}{e^{q_I-1}}$, Options P, B, and I are chosen.
• Finally, Domain $\beta_4$: if $\frac{\alpha_P C_P}{e^{q_P - 1}} + \frac{\alpha_B C_B}{e^{q_B - 1}} < \frac{C_I}{e^{q_I - 1}} < M$, all options are chosen.

## IV. SERVICE PROVIDER AND MNO DECISIONS

In the Stackelberg game, the MNO leads and the SP follows. Using backward induction, the SP maximizes its revenue in (2), based on the user equilibrium computed in Proposition 1, while the MNO anticipates its response when optimizing its own revenue in (1).

### A. Decisions of the service provider

For a given value of $\gamma$, the SP optimizes $(q_P, q_B, C_S, C_P)$, with $C_B = C_S - C_P$ and $q_P > q_B$, again using the user equilibrium from Proposition 1. For $\gamma = 0.25$, $q_I = 0.5$, and the model parameters given in Table I, we obtain numerically

$$ (q_P, q_B, C_S, C_P) = (0.46, 0.0, 3.48, 3.48) \quad (6) $$

Figures 1, 2, and 3 illustrate this result by showing $R_S$ as $q_P$, $C_S$, and $C_P$ vary, with shifts in user demand domains. We omit the figure for varying $q_B$, since $q_B$ has no impact on $R_S$. With $C_P = C_S$, the SP allocates all capacity to the premium service. This result is intuitive: given identical utility forms and $\alpha_P > \alpha_B$, the premium option dominates in user choice.

<caption>TABLE I
PARAMETERS USED IN OUR NUMERICAL EXPERIMENTS</caption>

$$ \frac{M}{100} \quad \frac{C}{15} \quad \frac{\alpha_P}{3} \quad \frac{\alpha_B}{2} $$

<figure><img src="image_2.png" /><figcaption>Fig. 1. SP revenue optimization for varying $q_P$ values.</figcaption></figure>

### B. MNO decisions

For our parameter values, we numerically found that the MNO optimizes its revenue in Equation (1) by selecting $\gamma = 0.23$ and $q_I = 0.10$ (see Figures 5 and 4). As shown in Table II, the SP purchases $C_S = 5.32$ but allocates all capacity to premium service ($C_P = C_S, C_B = 0, q_B = 0$), leading to zero demand for the basic option. Most EUs choose premium service ($d_P = 91.98$), with few opting for Internet only ($d_I = 0.06$) or no service ($d_0 = 7.96$). The SP earns $R_S = 8.21$ and the MNO earns $R = 15.71$. Offering both services provided no added benefit in this scenario. We now compare this outcome with optimized single-service strategies.

<figure><img src="image_3.png" /><figcaption>Fig. 2. SP revenue optimization when $C_S$ varies.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. SP revenue optimization when $C_P$ varies, with the optimal value of $C_S$ marked.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. MNO total revenue optimization for varying $\gamma$ values when SP offers two services.</figcaption></figure>

<table><caption>TABLE II
OUTPUT OF THE STACKELBERG GAME WHEN THE SP OFFERS TWO SERVICES</caption><thead><tr><th>γ</th><th>q<sub>I</sub></th><th>C<sub>S</sub></th><th>C<sub>P</sub></th><th>q<sub>P</sub></th><th>q<sub>B</sub></th><th>d<sub>B</sub></th><th>d<sub>P</sub></th><th>d<sub>I</sub></th><th>d<sub>0</sub></th><th>R</th><th>R<sub>S</sub></th></tr></thead><tbody><tr><td>0.23</td><td>0.10</td><td>5.32</td><td>5.32</td><td>0.16</td><td>0.0</td><td>0.0</td><td>91.98</td><td>0.06</td><td>7.96</td><td>15.71</td><td>8.21</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Fig. 5. MNO total revenue optimization for varying $q_I$ values when SP offers two services.</figcaption></figure>

## V. COMPARISON WITH THE SP DELIVERING ONLY ONE SERVICE (PREMIUM OR BASIC), AND WITH THE MNO USING ALL ITS CAPACITY FOR INTERNET SERVICE

To benchmark the main outcome, we compare it with cases where the SP offers only one service (premium or basic) or is absent. For each case, we first derive the closed-form user equilibrium, then perform numerical optimization at the SP (if present) and MNO levels. The decision order follows Section II-E, with adjustments: under single-service offerings, only one capacity $C_X$ and price $q_X$ (for $X \in \{P, B\}$) are optimized. In the no-SP case, the SP stage is removed, and the MNO optimizes only $q_I$, with no pricing function $P(\cdot)$.

### A. Scenario where the SP offers a single service

We consider the SP offering a single service $X \in P, B$ and purchasing capacity $C_X$ from the MNO. Its revenue is

$$R_S^{(X)} = d_X q_X - P(C_S). \qquad (7)$$

The MNO revenue is in this context

$$R^{(X)} = (d_X + d_I)q_I + P(C_S). \qquad (8)$$

EUs choices $X, I, 0$ depend on utilities $U_X, U_I, U_0$, leading to a unique equilibrium defined by prices and capacities (2). We can see explicit expressions in the extended version [20].

**Proposition 2.** Given capacities ($C_X, C_I$) $\ge 0$ and strictly positive prices ($q_X, q_I$), a unique equilibrium ($d_X^*, d_I, d_0$) exists, defined over four domains.

* Domain $\beta_5$: If $M < \frac{\alpha_X C_X (1 + \frac{C_I}{M})}{e^{q_X + q_I} - (1 + \frac{C_I}{M})}$, only option X is chosen.

* Domain $\beta_6$: If $\frac{\alpha_X C_X (1 + \frac{C_I}{M})}{e^{q_X + q_I} - (1 + \frac{C_I}{M})} \le M$ and $\frac{C_I}{e^{q_I - 1}} \le \frac{\alpha_X C_X}{e^{q_X - 1}}$, Options X, and 0 are chosen.

* Domain $\beta_7$: If on the other hand $\frac{\alpha_X C_X}{e^{q_X - 1}} < M \le \frac{C_I}{e^{q_I - 1}}$, options X and I are chosen.

* Domain $\beta_8$: Finally, if $\frac{\alpha_X C_X}{e^{q_X - 1}} < \frac{C_I}{e^{q_I - 1}} < M$, all options are chosen.

1) Basic service and Internet case: We assume here that the SP can only offer the Basic service. The decisions are as follows.

* The SP now maximizes its revenue in (7) by choosing $q_B, C_B$. Relying on the user equilibrium in Proposition 2, the SP determines the optimal basic price and the capacity. As an example, for $\gamma = 0.25$ and $q_I = 0.5$ we numerically obtain ($q_B, C_B$) = (0.28, 2.94). Figures 6 and 7 confirm these values by showing $R_S^{(B)}$ as $q_B$ and $C_B$ vary, with shifts in user demand domains.

<figure><img src="image_7.png" /><figcaption>Fig. 6. SP total revenue optimization for varying $q_B$ values when it offers just basic service.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. SP total revenue optimization for varying $C_B$ values when it offers just basic service.</figcaption></figure>

* We numerically find that the MNO chooses $\gamma(B) = 0.32$ and $q_I = 0.12$ to maximize its revenue in (8), as illustrated in Figures 9 and 8. Table III summarizes the outcome of that interaction.

<table><caption>TABLE III
OUTPUT OF THE STACKELBERG GAME WHEN THE SP OFFERS ONLY BASIC SERVICE</caption><thead><tr><th>γ<sup>(B)</sup></th><th>q<sub>I</sub></th><th>C<sub>B</sub></th><th>q<sub>B</sub></th><th>d<sub>B</sub></th><th>d<sub>I</sub></th><th>d<sub>0</sub></th><th>R<sup>(B)</sup></th><th>R<sub>S</sub><sup>(B)</sup></th></tr></thead><tbody><tr><td>0.32</td><td>0.12</td><td>2.92</td><td>0.06</td><td>94.44</td><td>0.31</td><td>5.25</td><td>14.10</td><td>2.94</td></tr></tbody></table>

2) Premium service and Internet case: When the authorized service is the premium one, the outcome is exactly the same as when both services are authorized, since the forbidden service
<figure><img src="image_9.png" /><figcaption>Fig. 8. MNO total revenue optimization for varying $\gamma^{(B)}$ values when SP offers just basic service.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. MNO total revenue optimization for varying $q_I$ values when SP offers just basic service.</figcaption></figure>

(the basic one) is not even implemented when allowed. That outcome is reminded in Table IV.

<table><caption>TABLE IV
OUTPUT OF THE STACKELBERG GAME WHEN THE SP OFFERS ONLY PREMIUM SERVICE</caption><thead><tr><th>γ<sup>(P)</sup></th><th>q<sub>I</sub></th><th>C<sub>P</sub></th><th>q<sub>P</sub></th><th>d<sub>P</sub></th><th>d<sub>I</sub></th><th>d<sub>0</sub></th><th>R<sup>(P)</sup></th><th>R<sub>S</sub><sup>(P)</sup></th></tr></thead><tbody><tr><td>0.23</td><td>0.10</td><td>5.32</td><td>0.16</td><td>91.98</td><td>0.06</td><td>7.96</td><td>15.71</td><td>8.21</td></tr></tbody></table>

Comparing with Table III, we observe that the premium-only case is more profitable to the SP, which could be expected as users value that service more when capacity-per-user is the same. Note that this situation would also be preferred by the MNO: the SP being able to make more revenue from capacity incentivizes it to purchase more capacity.

## B. Case where the MNO uses all its capacity (no SP)

In this case, the MNO is the only one leader, aiming to maximize its revenue:

$$R = d_I q_I \qquad (9)$$

As all capacity goes to Internet service, $P(\cdot)$ is irrelevant. The MNO sets $q_I$ to maximize (9), based on Proposition 3. The explicit expressions are available in the extended version [20].

**Proposition 3.** Given $C_I \ge 0$ and $q_I \ge 0$, there exists a unique user ($d_I, d_0$), that can take the two following forms depending on parameter values:

* Domain $\beta_9$: If $M \le \frac{C_I}{e^{q_I-1}}$, Only option I is chosen.

* Domain $\beta_{10}$: If $M > \frac{C_I}{e^{q_I-1}}$, options I and 0 are chosen.

<figure><img src="image_11.png" /><figcaption>Fig. 10. MNO total revenue optimization for varying $q_I$ values.</figcaption></figure>

As shown in Figure 10, the MNO's revenue peaks with a domain shift. Table V shows that the MNO captures most demand, earning less than in previous cases. This illustrates the revenue advantage of selling capacity to the SP.

<table><caption>TABLE V
OUTPUT OF THE STACKELBERG GAME WHEN THE MNO IS THE ONLY PROVIDER</caption><thead><tr><th>q<sub>I</sub></th><th>d<sub>I</sub></th><th>d<sub>0</sub></th><th>R</th></tr></thead><tbody><tr><td>0.14</td><td>99.82</td><td>0.18</td><td>13.97</td></tr></tbody></table>

# VI. CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH

This paper analyzes network slicing in a competitive setting involving an MNO, an SP, and end users. We show that slicing enhances both efficiency and profitability: the MNO benefits by selling capacity rather than by using it solely, and the SP maximizes gains by offering a single premium service. Future work could extend this model to include multiple SPs with revenue-sharing or budget constraints, or explore the MNO's optimal capacity allocation across competing SPs.

## ACKNOWLEDGMENT

The authors acknowledge the support from France 2030 program, operated by the Agence Nationale de la Recherche (ANR), under the grant ANR-22-PEFT-0002.

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