A Multi-Leader Multi-Follower Stackelberg Game for Dynamic Spectrum Sharing in a Blockchain Enabled Cell-Free Massive MIMO Scenario
Abstract
The stringent quality-of-service (QoS) requirements that are being considered for the sixth generation (6G) of mobile networks will necessarily imply the synergistic application of different mechanisms ranging from physical layer processing to radio resource management aspects. In particular, this work blends three complementary techniques that are bound to play a central role in the transition towards 6G. Firstly, cell-free massive MIMO (CF-mMIMO) is often heralded as a foundational physical layer technique that is able to address the demanding spectral efficiencies (SEs) planned for 6G. Secondly, dynamic spectrum sharing (DSS) has shown its potential to optimize the exploitation of the always-scarce radio spectrum. Finally, blockchain technology is used to make these bandwidth transactions fair and accountable. This work presents a general architecture where multiple sellers and buyers of spectrum can dynamically trade spectral resources, leveraging CF-mMIMO-based wireless networks and guided by Stackelberg game theory. Notably, all parameters, operations, and outcomes of this trade protocol are securely recorded on a blockchain using smarts contracts (SCs). Exhaustive numerical results are provided that showcase the effects different network parameters have on the network performance while also revealing the cost associated with blockchain processing when relying on several popular blockchain platforms.
Full text
Received 14 April 2025; revised 20 May 2025; accepted 4 June 2025. Date of publication 9 June 2025; date of current version 7 July 2025. Digital Object Identifier 10.1109/OJCOMS.2025.3578102 A Multi-Leader Multi-Follower Stackelberg Game for Dynamic Spectrum Sharing in a Blockchain Enabled Cell-Free Massive MIMO Scenario GUILLEM FEMENIAS 1,2 (Senior Member, IEEE), M. FRANCISCA HINAREJOS 1,2, FELIP RIERA-PALOU 1,2 (Senior Member, IEEE), JOSEP-LLUÍS FERRER-GOMILA 1,2, AND AMADOR JAUME-BARCELÓ1 1Security and Electronic Commerce Group, University of the Balearic Islands, 07122 Palma, Spain 2Artificial Intelligence Research Institute, University of the Balearic Islands, 07122 Palma, Spain CORRESPONDING AUTHOR: G. FEMENIAS (e-mail: [email protected]) This work was supported in part by MCIN/AEI/10.13039/501100011033, Spain; in part by the European Union (EU) “NextGenerationEU”/PRTR through the Project GERMINAL under Grant TED2021-131624B-I00; in part by MICIU/AEI/10.13039/501100011033, Spain; in part by ERDF, EU through the Project SOFIA-WIND under Grant PID2023-147305OB-C33; and in part by the ERDF “A way of Making Europe” through the Project BLOBSEC under Grant PID2021-122394OB-I00. ABSTRACT The stringent quality-of-service (QoS) requirements that are being considered for the sixth generation (6G) of mobile networks will necessarily imply the synergistic application of different mechanisms ranging from physical layer processing to radio resource management aspects. In particular, this work blends three complementary techniques that are bound to play a central role in the transition towards 6G. Firstly, cell-free massive MIMO (CF-mMIMO) is often heralded as a foundational physical layer technique that is able to address the demanding spectral efficiencies (SEs) planned for 6G. Secondly, dynamic spectrum sharing (DSS) has shown its potential to optimize the exploitation of the always-scarce radio spectrum. Finally, blockchain technology is used to make these bandwidth transactions fair and accountable. This work presents a general architecture where multiple sellers and buyers of spectrum can dynamically trade spectral resources, leveraging CF-mMIMO-based wireless networks and guided by Stackelberg game theory. Notably, all parameters, operations, and outcomes of this trade protocol are securely recorded on a blockchain using smarts contracts (SCs). Exhaustive numerical results are provided that showcase the effects different network parameters have on the network performance while also revealing the cost associated with blockchain processing when relying on several popular blockchain platforms. INDEX TERMS Cell-free massive MIMO, dynamic spectrum sharing, Stackelberg game, blockchain, decentralized autonomous organization. I. INTRODUCTION A. CONTEXT TARGETING the forthcoming wave of new applications and services such as extended reality, autonomous driving, ultrahigh definition video transmission or sub-cm positioning, to name a few, researchers at academic, industrial and standardization entities are currently investigating the foundations of the forthcoming sixth generation (6G) of mobile networks [1],[2]. While many unknowns remain, some specific technologies are bound to have an impact in 6G. Among others, new forms of massive multiple-input multiple-output (mMIMO), novel approaches to spectrum exploitation, and a heavy use of blockchain technology have all been predicted to play a key role in the 6G ecosystem [1], [2],[3],[4]. Despite the incorporation of new frequency bands, the phenomenal surge in traffic demands presents significant challenges in terms of spectral efficiency (SE), which cannot be addressed solely from a physical layer perspective through the use of new mMIMO techniques, but rather require the integration of advanced resource management strategies that judiciously utilize the available spectrum [3]. One such c 2025 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME 6, 2025 5359
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING approach is dynamic spectrum management (DSM), cited in [1] as one of the enabling technologies towards 6G. When relying on DSM, operators are allowed to share and commercialize any portion of assigned spectrum that is temporally underutilized so that this can be exploited by other spectrum-starving operators. Critically, all these spectrum management-related transactions among operators must be accountable, decentralized, and fair. All these requirements can be fulfilled by relying on blockchain technology, yet another of the pillars that will surely play a role in 6G [4]. This work proposes a complete framework to implement a DSM strategy in a blockchain-enabled mMIMO 6G context whose operative is guaranteed to be accountable, fair and privacy-preserving. Over the following paragraphs, each of the constituting technologies are reviewed. Cell-free networks: Over the last decade, mMIMO, rooted in the seminal work of Marzetta [5], has become a cornerstone of fifth generation (5G) networks [6].By deploying a massive number of antennas at the base station (BS), it achieves SEs unattainable in earlier generations. More recently, cell-free massive MIMO (CF-mMIMO) has emerged as a key enabler for future 6G networks [2].A CF-mMIMO network comprises many access points (APs) randomly distributed across the coverage area and connected to a central processing unit (CPU) that handles most baseband processing [7]. Unlike traditional mMIMO, which concentrates antennas at the BS, CF-mMIMO disperses them throughout the service area, placing infrastructure closer to users. This architecture improves macroscopic diversity and, via cooperative processing, mitigates cell-edge effects typical of cellular networks. CF-mMIMO retains all key mMIMO features [7],[8]: channel hardening reduces fading impact, and favorable propagation enables simple linear precoding to achieve near-optimal performance [9]. It is now regarded as more spectrally efficient than co-located mMIMO [8]. Recent research has addressed precoder and combiner design, channel estimation, and power control (see [8],[10] for reviews). Another notable advantage of cell-free networks is their superior energy efficiency compared to co-located systems, thanks to flexible infrastructure activation and minimal cooling needs at the APs [11],[12],[13]. Ongoing studies are also investigating practical pathways for transitioning from cellular to CF-mMIMO architectures [12]. Multi-operator spectrum management: When multiple CF-mMIMO networks coexist in a shared coverage area, each operated by a distinct network operator (NO), spectrum management becomes essential for efficient radio resource allocation, especially across diverse frequency bands envisioned in 6G ecosystems [14]. Traditionally, networks serve users using spectrum statically allocated by regulators, ensuring non-overlapping bandwidths. However, such fixed allocation may lead to inefficiencies, with some networks having surplus spectrum and others experiencing shortages. Dynamic spectrum management allows operators or regulators to allocate bandwidth more flexibly based on user demands. This approach aligns with cognitive radio principles, where secondary operators utilize underused spectrum from primary operators [15],[16]. Beyond spectrum efficiency, the emergence of fourth generation (4G)/5G introduced models like mobile virtual network operators (MVNOs) to optimize resource usage [17],[18]. An MVNO delivers telecom services without owning infrastructure or spectrum, instead purchasing capacity from mobile network operators (MNOs) or spectrum providers (SPs), where SPs may represent central authorities or other M(V)NOs engaged in bandwidth trading. This work explores scenarios in which MVNOs own infrastructure but lack spectrum, requiring them to buy it. Implementing dynamic spectrum sharing (DSS) in such contexts improves utilization by dynamically allocating bandwidth to virtual operators willing to purchase it [19], outperforming static assignments [20]. Allocation occurs periodically or in response to significant network changes [21], often via optimization methods. Game theory is a proven tool to manage competition among NOs/SPs, optimizing spectrum use based on utility functions [22],[23],[24]. Depending on the assumptions, different game structures apply. A key distinction is whether players cooperate and share information (cooperative games), or pursue conflicting goals with limited knowledge of others’ actions (non-cooperative games). Non-cooperative strategies suit environments like ours, where distributed solutions are needed without centralized control and each player maximizes its own benefit. Within non-cooperative games, various designs exist. For instance, prospect theory-based approaches incorporate loss aversion and have been applied successfully in spectrum management [25]. When players lack full knowledge of others’ actions, Bayesian games introduce probabilistic models to capture uncertainty [26]. In this work, we focus on Stackelberg games, a widely used framework for dynamic spectrum allocation in cognitive radio and multi-operator contexts [27],[28]. Stackelberg games feature leaders and followers; leaders know the followers’ responses and optimize accordingly, forming a subclass of perfect information games. In our setup, SPs act as leaders, they own the spectrum and set the price. NOs act as followers, deciding how much bandwidth to purchase at that price. This asymmetry models the authority SPs have over NOs as resource owners. For a comprehensive overview of game theory and Stackelberg games in wireless optimization, see [26]. Blockchain technology and smart contracts (SCs): Managing spectrum resources across multiple SPs and NOs requires infrastructure and protocols to track agreements. Blockchain has emerged as a robust solution for such registration tasks [29]. It is a decentralized, tamperresistant ledger where transaction blocks are sequentially added to form an immutable chain. Each transaction is authenticated by its initiator, preventing later denial [30]. Blockchain networks are typically public or private, with 5360 VOLUME 6, 2025
further classifications into permissioned, permissionless, and consortium types [31]. Public blockchains allow open, permissionless participation, anyone can read, write, and join the consensus. Their high decentralization and node diversity enhance security. For example, Ethereum’s 4,000+ nodes [32] highlight the robustness of public blockchains. This distributed structure increases resistance to Byzantine attacks.1In contrast, private blockchains, managed centrally and limited to approved nodes, improve privacy, efficiency, and scalability but are less resilient to Byzantine failures. Within both public and private settings, permissioned networks restrict access to specific roles, offering controlled decentralization. Permissionless networks are fully open. Consortium blockchains, a type of permissioned network, involve multiple organizations, providing partial decentralization. In summary, public and permissionless blockchains stress openness and decentralization, while private, permissioned, and consortium types focus on control, efficiency, and security. Public blockchains use transaction fees; private and consortium models share infrastructure and costs. Originally designed for cryptocurrency, blockchain now supports broader use cases. SCs address these needs and are implemented on various platforms. They are code snippets stored on blockchain nodes, triggered by authorized users with input parameters. Ethereum introduced SCs via Solidity, running on the Turing-complete Ethereum virtual machine (EVM) [34]. Other public blockchains like Polygon and Binance Smart Chain (BSC) also support the EVM. SCs often need off-chain data but cannot independently access external sources, run periodic tasks, or securely handle sensitive operations like randomness. These limits are addressed via oracles, external entities that link blockchains to outside data [35]. Oracles may be centralized (one trusted source, single point of failure) or decentralized (multiple validators ensuring accuracy and resilience) [36]. Decentralized Autonomous Organization: A decentralized autonomous organization (DAO) is an innovative organizational model that operates without central authority, enabling collective governance by its members [37],[38]. Unlike traditional hierarchies, decisions in DAOs are made through democratic voting, typically facilitated by blockchain. SCs automate and enforce these decisions transparently, eliminating intermediaries, lowering costs, and improving efficiency [39],[40]. Key advantages of DAOs include transparency and community-driven decision-making. Without centralized leadership, members can propose and vote on initiatives, promoting shared ownership and alignment around common goals. This structure ensures decisions reflect collective, not individual, interests. Additionally, all actions, proposals, and 1A Byzantine attack occurs when some nodes in a decentralized network act maliciously, undermining consensus and potentially causing issues like double-spending [33]. votes are immutably recorded on the blockchain, enabling verification and fostering trust while minimizing corruption. Such transparency supports fair governance and facilitates integration with other decentralized applications in the blockchain ecosystem [41]. B. RELATED WORK Several blockchain-based DSS frameworks have recently been proposed, many incorporating Stackelberg game theory. However, most prior works combining Stackelberg games and blockchain address settings different from ours. Specifically, [42],[43],[44] focus on unmanned aerial vehicles; [45],[46] on 5G slicing; and [47],[48] on Internet of Things and Internet of Vehicles applications, respectively. The scenario in [49] is somewhat related but relies on a consortium blockchain, which introduces inherent security risks. Some works [46],[49] explicitly note that a few regulators manage blockchain operations, and most adopt reputation-based miner selection—offering weaker security than public blockchains. In [49], a SC executes the DSS strategy, triggered by a primary node within the network. However, this centralization creates a potential single point of failure, undermining system resilience and reliability. Furthermore, the proposed DSS model requires each organization or operator to contribute infrastructure resources, increasing administrative overhead. Notably, none of these proposals evaluate the operational or maintenance costs of consortium blockchains. Few works address scenarios involving multiple SPs. An exception is [50], where multiple secondary users compete to acquire spectrum from multiple primary users using evolutionary game theory. This was extended in [51] to allow secondary users to purchase bandwidth from secondaryspecific spectrum brokers. However, these studies primarily focus on competitive dynamics, neglecting network-specific characteristics. Another exception is the work in [52], which treats the case of multiple SPs but just considering a single buyer. We refer the interested reader to [53] for a recent survey on blockchain-based dynamic spectrum sharing technologies. In earlier work by the same authors [54], a Stackelberg game theory-based framework [26] was proposed to optimize spectrum utilization for multiple CF-mMIMObased MVNOs with a single SP. However, the single-SP setup oversimplifies blockchain intervention at the cost of reducing the flexibility and generality achievable in multi-SP configurations. C. CONTRIBUTIONS Building on our previous effort in [54], this paper deals with scenarios in which multiple leaders (spectrum traders) and multiple followers (cell-free based MVNOs) are allowed to engage in bandwidth exchange transactions. In particular, the main contributions of this paper are: •A multiple-seller multiple-buyer non-cooperative Stackelberg game model is proposed where spectrum VOLUME 6, 2025 5361
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING sellers seek to maximize the profits of bandwidth trade while NOs (spectrum buyers) aim at maximizing an SErelated utility function. Solving the Stackelberg game involves the SPs (i.e., leaders) setting prices for their available bandwidths by solving a convex optimization problem while subsequently letting the NOs (i.e., followers) determine the amount of bandwidth to be purchased at the set prices by again resorting to convex optimization. Importantly, the game model is build on top of a physical-layer abstraction characterizing a CF-mMIMO-based wireless architecture. •The execution and management of the proposed Stackelberg game model is implemented through various SCs deployed on the blockchain. Moreover, they can be adapted to the terms and conditions negotiated and agreed upon by the sellers and buyers. These SCs control the functions to be executed, who can execute them, and when can they be executed within the Stackelberg game. In addition, an oracle is used to automate the execution of the game, eliminating the need to assign specific tasks to individual participants, streamlining the process and reducing administrative overhead. This approach ensures that spectrum allocation is automated, transparent and traceable, as all actions are recorded on the blockchain, making them verifiable and auditable. •Extensive numerical results are presented to demonstrate the merits of the proposed mathematical framework and the existing trade-offs when implementing different blockchain-based solutions. More in detail, the impact several key parameters (e.g., number of NOs and/or SPs) have on the performance of the system when considering two specific market scenarios (non-differentiated oligopoly (NDO) and differentiated duopoly (DD)) is first assessed. Next, the implementation costs and performance metrics related to the game’s management and execution are analyzed across four different blockchain networks. D. PAPER ORGANIZATION The rest of the paper is organized as follows. Section II presents the system model with a detailed description of all the actors involved in this DSS ecosystem. Section III introduces a cell-free network abstraction in the form of achievable rates that serves, along with the utility functions, to formulate the DSS Stackelberg game described in Section IV. The methodology to solve the Stackelberg game is presented in Section V. Section VI describes the Blockchain-related operations to record all the operative of the spectrum trade. The assessment of the Stackelberg game results in relation to the CF-mMIMO parameters are presented in Section VII while performance results regarding the use of blockchain for different commercial platforms are provided in Section VIII. The main outcomes of this work are recapped in Section IX. II. SYSTEM MODEL In distributed spectrum management, multiple SPs and MNOs, often with competing interests and no pre-existing trust relationships, must coordinate resource allocation and trading. In this context, blockchain offers a tamper-resistant ledger and programmable logic through smart contracts, enabling fair interaction and rule-based access control without the need for a trusted intermediary. This architecture not only improves transparency and auditability but also facilitates the automation and enforcement of agreements, delivering operational and strategic benefits while reducing reliance on centralized entities. Therefore, the proposed system model relies on SCs deployed via blockchain to provide efficient spectrum allocation management. These contracts, as key components of spectrum allocation, require prior deployment to grant NOs and SPs access to their functionalities. Deploying SCs for the proposed spectrum management approach can involve various actors and strategies. SPs, as primary stakeholders, can handle contract deployment directly. Alternatively, delegating this responsibility to specialized entities can streamline the process, ensuring both expertise and accountability. A DAO offers a decentralized governance model where stakeholders collectively oversee contract deployment and management, enabling transparent decision-making and automating certain operations [39],[40]. In this model, a DAO [55],[56] is considered the best choice for ensuring fair decision-making among all parties in managing spectrum sharing. It is assumed that the DAO oversees 6G network-related processes, including deploying SCs for spectrum-sharing agreements service level agreements (SLAs) between SPs and NOs. The DAO can operate either on or off the blockchain, though its design and implementation fall outside the scope of this work. For simplicity, it is assumed that SPs and NOs have agreed to establish and operate a DAO to collectively manage network functionalities. As conceptually shown in Fig. 1, a DSS scenario is considered where SSPs, indexed by s∈S={1,...,S}, sell chunks of bandwidth to Cvirtual CF-mMIMO NOs, indexed by c∈C={1,...,C}. The bandwidth owned by the sth SP is denoted by Bs.Thecth CF-mMIMO network is composed of Mcgeographically scattered APs, with each AP provisioned with an Nc-antenna array. These APs cooperate to jointly serve Kcsingle-antenna mobile stations (MSs) using the same time-frequency resources. All APs in a given CF-mMIMO network are linked to a CPU (or multiple CPUs) through ideal fronthaul links. The performance provided by the cth CF-mMIMO network will be generically represented by an achievable SE ηc(measured in bit/s/Hz). The chunks of bandwidth to be allocated to each of the virtual NOs and the prices they have to pay to each SP are determined through a competitive game aiming at optimizing predetermined utility-related functions. These utility functions are related, on the one hand, to the revenue that each SP will obtain from the sale of the bandwidth, 5362 VOLUME 6, 2025
FIGURE 1. System model depicting the main entities and variables used in this work. Numbering labels represent the steps taken by the proposed scheme that will be detailed in Section VI. and on the other hand, to the benefits that each of the operators can derive from the service they provide to the associated MSs when exploiting the newly acquired bandwidth. The management of spectrum allocation agreements is facilitated by a SC known as the SLA deployed in the blockchain. This SLA registers the SPs and NOs eligible to participate in spectrum allocation processes (offers bandwidth or requests for spectrum), and manages the parameters that govern this process. Additionally, the SLA deploys an MLMF SC used by both SPs and NOs. The MLMF ensures proper bandwidth allocation by executing the DSS algorithm based on the Stackelberg leadership model. Specifically, each SP communicates the available bandwidth to the SC (B1,B2,.., Bs)while the NOs, in turn, communicate their achievable SE (η1,η2,.., ηc). Using these data, and taking into account the utility functions to be optimized, the MLMF implementing the DSS algorithm is executed to determine the optimal chunks of bandwidth to be assigned to the NOs and the prices they have to pay to each SP. This commercial interaction between SP sand CF-mMIMO NO c is characterized by the pair (bcs,pcs), where bcs is the chunk of bandwidth that the NO acquires from the SP, and pcs is the price per unit of bandwidth charged by the SP to the NO (measured in monetary units per Hertz). The SCs should dynamically allocate bandwidth, which can be done at scheduled times or when certain conditions agreed upon by participants are met. However, SCs cannot directly interact with external data and systems. Our system consists of NOs and SPs, so one of them should request the execution of each required function. To prevent a specific entity from handling certain functionalities that must be fulfilled by the SLA, an analysis is performed using an oracle to securely connect off-chain systems to on-chain SCs. III. MODELING THE SE The propagation channels between APs and MSs are defined by smalland large-scale fading parameters. The former are presumed static during the time-frequency coherence block, and the latter (i.e., spatial correlation matrices) remain static over many coherence blocks and they are assumed to be known at the APs and/or the CPU [57], [58],[59],[60]. The channel between MS kand the mth AP in the cth CF-mMIMO network is modeled as hcmk ∼CN(0,Rcmk), where Rcmk ∈CNc×Ncis the spatial correlation matrix, and βcmk =tr(Rcmk)/Nc, captures pathloss, shadowing, and antenna gains. Channels between a MS and different APs are independent, so the collective channel hck =[hT c1k...hT cMck]Tis distributed as hck ∼CN(0,Rck), where Rck =diag(Rc1k,...,RcMck)is block-diagonal. Each CF-mMIMO network relies on time division duplex (TDD), with frames of τfcsamples fitting the coherence block.2Every frame is divided into an uplink (UL) training phase (τpcsamples), an UL payload transmission phase (τuc samples), and a downlink (DL) payload transmission phase (τdcsamples), ensuring τfc=τpc+τuc+τdc. During the UL training phase, τpcorthogonal pilot sequences are allocated deterministically to MSs, allowing some MSs to share the same sequence whenever K>τ pc.ForMSk,letPck denote the set of MSs sharing its pilot sequence. The minimum mean square error (MMSE) channel estimate for hcmk is 2As in co-located cellular massive MIMO networks, TDD relies on the assumption of perfect reciprocity between the uplink and downlink channels so that the uplink acquired channel is also a good estimate of the downlink channel. Models are available to incorporate deviations from perfect reciprocity, collectively known as hardware impairments [61], however, their consideration is beyond the scope of this work. VOLUME 6, 2025 5363
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING given by [8] ˆ hcmk =PpτpcRcmk−1 cmkycmk,(1) where Ppis the pilot-symbol transmit power, and ycmk is the projection of the matrix of signals received at the mth AP on the pilot sequence allocated to MS k, that is, ycmk =Ppτpchcmk +Ppτpc k∈Pk\k hcmk+ncmk,(2) with ncmk ∼CN(0,σ2 uINc), and cmk =EycmkyH cmk=Ppτpc k∈Pk Rcmk+σ2 uINc.(3) The channel estimation error ˜ hcmk =hcmk −ˆ hcmk is independent of the estimate and follows the distribution ˜ hcmk ∼CN(0,Ccmk), where Ccmk =Rcmk −PpτpcRcmk−1 cmkRcmk.(4) To allocate bandwidth and set prices for NOs, this work employs a Stackelberg game based on the SEs offered to MSs. Focusing on UL SEs without loss of generality, the centralized UL operation is assumed, where all baseband processing occurs at the CPU. The achievable UL SE for MS kin the cth CF-mMIMO network (in bit/s/Hz) is [8, Section V.1] ηck =τuc τfc Elog2(1+SINRck),(5) where the pre-log factor τuc/τfcreflects the fraction of the coherence block used for UL transmission. The instantaneous effective signal-to-interference-plus-noise ratio (SINR) is given by (6), as shown at the bottom of the next page, with the expectation taken over the aggregate channel estimates ˆ hck =[ˆ hT c1k...ˆ hT cMck]T. Various combining filters, such as maximal ratio combining (MRC), MMSE, or zero-forcing (ZF), can be used, each leading to different SINRs and SEs. It should be stressed that all the signal processing steps described in this section, most notably, channel estimation, combiner design and power allocation, can all be implemented in a scalable manner. Scalability in a cell-free context refers to the fact that the complexity of all operations remain computationally bounded even when the number of users goes to infinity [8]. IV. FORMULATION OF THE STACKELBERG GAME Let us denote by Bthe C×Smatrix containing the chunks of bandwidth acquired by the CF-mMIMO NOs from the SPs, and by Pthe C×Smatrix containing the prices per unit of bandwidth paid by the NOs to the SPs. Let us also denote by b(CF) c=[bc1...bcS]Tthe vector containing the chunks of bandwidth acquired by the CF-mMIMO NO c from the SSPs, and by b(SP) s=[b1s...bCs]Tthe vector containing the chunks of bandwidth sold by the sth SP to the CNOs. Analogously, let us denote by p(CF) c=[pc1...pcS]T the vector containing the prices per unit of bandwidth paid by the CF-mMIMO NO cto the SSPs, and by p(SP) s= [p1s...pCs]Tthe vector containing the prices per unit of bandwidth charged by the sth SP to the CNOs. The objective of each of the SSPs is to maximize its own revenue obtained from selling the bandwidth to the NOs. Mathematically, the revenue of the sth SP can be formulated as Rsb(SP) s,p(SP) s= C c=1 bcspcs.(7) Note that under the Stackelberg game formulation, the bandwidths b(CF) cacquired by the cth NO are actually a function of the prices p(CF) c. That is, the chunks of bandwidth that a particular CF-mMIMO NO is willing to purchase to the SPs depend on the prices charged by these providers. As the bandwidths available at the SPs are limited, the SPs must determine the prices that maximize the revenue under the aggregate bandwidth constraint. The constrained optimization problem at each of the SPs can then be formulated as PSPs:max pRsb(SP) s,p(SP) s(8a) subject to C c=1 bcs ≤Bs(8b) pcs ≥0∀c∈C.(8c) Inspired by the work of Singh and Vives [62], a suitable utility function for the cth NO can be defined as Ucb(CF) c,p(CF) c=ηc S s=1 αcsbcs − S s=1 pcsbcs −1 2⎛ ⎝ S s=1 βcsb2 cs +2γc s=s bcsbcs⎞ ⎠,(9) where αcs (measured in currency units per bit/s) and βcs (measured in currency units per squared Hz) are positive, and ηc=Kc k=1ηck denotes the aggregate SE (measured in bit/s/Hz) of CF-mMIMO network c. This utility function consists of three parts: the first term (positive) represents the revenue gained from exploiting the purchased spectrum; the second term (negative) corresponds to the payment for spectrum usage; and the third term (also negative) accounts for the cost associated with spectrum substitutability, where −1≤γc≤1is the substitutability coefficient3[62]. Analyzing the structure of the utility 3The substitutability coefficient γcquantifies the degree to which products offered by different suppliers act as substitutes or complements [62].When γc>0, the products are substitutes: purchasing a good from one supplier reduces the consumer’s demand for similar goods from other suppliers. Conversely, when γc<0, the goods are complements, meaning that the consumer derives greater utility from acquiring them jointly, increasing demand for multiple suppliers. In our context, spectrum is modeled as a homogeneous product (i.e., highly substitutable, with γc>0)since the spectrum chunks offered by different SPs are assumed to be equivalent. That is, given a required bandwidth bcs,theNOcis indifferent as to which SP sprovides the additional spectrum, as all are considered functionally interchangeable. 5364 VOLUME 6, 2025
function reveals that, on the one hand, increasing the acquired bandwidth enables a CF-mMIMO NO to improve both spectral efficiency (SE) and profit. On the other hand, acquiring more bandwidth also raises the associated cost. In [63], Niyato et al. observed, within a context similar to the one considered here, that this utility function is concave and effectively captures the saturation of user satisfaction as the achievable SE increases. Furthermore, the first derivative of this quadratic function yields a linear bandwidth demand, which makes the resulting optimization problems analytically tractable. Analyzing the structure of the utility function, it becomes evident that, on the one hand, by increasing the acquired bandwidth, a particular CF-mMIMO NO improves both SE and profit. On the other hand, acquiring more bandwidth increases the cost. Therefore, bandwidth acquisition strategies are needed at the NOs to maximize their own utilities. In fact, the problem that must be solved by the CF-mMIMO NOs (followers’ side) can be formulated as PCFc:max b(CF) c Ucb(CF) c,p(CF) c(10a) subject to bcs ≥0∀s∈S.(10b) Problems PSPs, for all s∈S, and PCFc, for all c∈C, form a non-cooperative Stackelberg game aimed at identifying the Stackelberg equilibrium point (B∗,P∗)where both the SPs and NOs lack incentives to deviate. That is, for any point (B,P)such that B0and P0, the Stackelberg equilibrium point (B∗,P∗)satisfies Rsb(SP) s ∗,p(SP) s ∗≥Rsb(SP) s ∗,p(SP) s∀s∈S,(11a) Ucb(CF) c ∗,p(CF) c ∗≥Ucb(CF) c,p(CF) c ∗∀c∈C.(11b) V. SOLVING THE STACKELBERG GAME A. SUBGAMES OF THE CF-MMIMO NO The first and second derivatives of Uc(b(CF) c,p(CF) c)with respect to bcs are ∂Ucb(CF) c,p(CF) c ∂bcs =ηcαcs −pcs −βcsbcs −γc s=s bcs,(12) and ∂2Ucb(CF) c,p(CF) c ∂2bcs =−βcs,(13) respectively. Since the second derivative is negative, the utility function is concave with respect to bcs and, hence, as the constraint is affine, each of the followers’ games, mathematically modeled in (10), is a convex optimization problem and Karush-Kuhn-Tucker (KKT) conditions apply. Lemma 1: Given a price vector p(CF) c=[pc1...pcS]T,the optimal solution for the chunks of bandwidth that must be acquired by the cth CF-mMIMO NO is b∗ cs =ζcs −cspcs +κcsp(CF) c−s+∀s∈S,(14) with x+=max(0,x),p(CF) c−s=p(CF) c\{pcs}, and ζcs =ηcαcs βcs −γc − γcηcS s=1αcs βcs−γc (βcs −γc)1+γcS s=11 βcs−γc,(15a) cs =1 βcs −γc −γc (βcs −γc)21+γcS s=11 βcs−γc,(15b) κcsp(CF) c−s= γcs=s pcs βcs−γc (βcs −γc)1+γcS s=11 βcs−γc,(15c) where the parameters of the system must be set in order to guarantee that ζcs,cs, and κcs(p(CF) c−s)are positive. Proof: See Appendix. A Thus, it can be observed that if the price is too high, that is, if pcs ≥(ζcs +κcs(p(CF) c−s))/cs,thecth CF-mMIMO NO will not buy any portion of the bandwidth available at the SP sand will be removed from this particular game. B. SUBGAMES OF THE SPs Using the optimal buying strategy of the NOs, mathematically expressed in (14), the optimization problem at the sth SP can be reformulated as PSPR:max p(SP) s C c=1 pcsζcs −cspcs +κcsp(CF) c−s (16) s. t. C c=1ζcs −cspcs +κcsp(CF) c−s≤Bs(16a) 0≤pcs ≤ζcs +κcsp(CF) c−s/cs.(16b) This is a convex optimization problem (note that both the objective function and the constraints are linear functions of pcs)and, again, the KKT conditions apply. SINRck = πckvH ck ˆ hck 2 Kc k=1 k=k πckvH ck ˆ hck 2+vH ckKc k=1Cckvck +σ2 uvck2(6) VOLUME 6, 2025 5365
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING Lemma 2: Given the vectors of prices p(SP) s= [p1s...pCs]T, for all s= s, let us define πs(C)= {cs1,...,csC}as a permutation of the set C={1,...,C}, where the CF-mMIMO NOs are sorted in such a way that ζcs1s+κcs1sp(CF) cs1−s/cs1s≥... ≥ζcsCs+κcsCsp(CF) csC −s/csCs.(17) Given πs(C), let us define τs(C)as the number of elements in the set Cthat match the following condition: ζcscs+κcscspcsc (CF) −s 2cscs ≥c∈πs(C),c<cζc+κcsp(CF) c −s−2Bs c∈πs(C),c<cc .(18) Using this definition, the optimal solution for the prices per unit of bandwidth that must be charged by the sth SP to the CNOs is p∗ cs =Fp(CF) c−s =⎧ ⎪ ⎨ ⎪ ⎩ ζcs+κcsp(CF) c−s 2cs +λs 2∀c∈{cs1,...,cs(τs−1)} ζcs+κcsp(CF) c−s cs ∀c∈{csτs,...,csC}, (19) where λs=τs−1 τ=1ζcsτs+κcsτsp(CF) csτ−s−2Bs τs−1 τ=1csτs ,(20) and ζcs(τs−1)s+κcs(τs−1)sp(CF) cs(τs−1)−s/cs(τs−1)s ≥λs≥ζcsτss+κcsτssp(CF) csτs−s/csτss.(21) Proof: See Appendix. B Lemma 3: An iterative mechanism to update the vector of prices p(CF) cgiven the current prices p(CF) c−sproposed by other SPs in the network, can be obtained from (19) as pcs(t)=Fp(CF) c−s(t−1)∀c∈C,(22) where tis the iteration index. This iterative update converges to a unique fixed point. Note that the convergence of (22) does not depend on the number of users each NO is serving and, indeed, even when K→∞, each CF can be made scalable as stated in Section III, and the solution to (22) is guaranteed to converge. Proof: See Appendix C VI. BLOCKCHAIN-BASED SYSTEM DESIGN AND IMPLEMENTATION Having posed the multi-leader multi-follower (MLMF) problem and proved how it can be solved optimally, it is now mandatory to describe how the information being input into the game and the one resulting from its solution is kept in an auditable manner. In the proposed system model, the SLA manages the deployment of the MLMF SC (referred to as MLMF) on the blockchain, which executes the DSS algorithm. Per SLAdefined rules, both SPs and NOs are notified when to submit their parameters to participate in each game round. Each SP publishes its available bandwidth for a given game execution, uniquely identified by an identifier (id), within the MLMF. NOs request participation by submitting their parameters. Once all data is received, the MLMF notifies the NOs that the game (identified by id) is ready to start and proceeds to solve it. The SLA must also provide SPs and NOs with the address of the deployed MLMF, enabling updates to be disseminated to all participants. To facilitate communication, the MLMF uses events4—a blockchain mechanism for realtime updates and coordination throughout the Stackelberg game. Events are triggered at each DSS stage; for instance, when a SP declares bandwidth or a CF-mMIMO NO submits its parameters. This structured use of events ensures all participants remain informed throughout the process. Interaction between SPs and NOs must meet security standards to ensure trust and reliability in bandwidth allocation. Consequently, all Stackelberg game processes must be transparent and auditable by both parties. This work explores two approaches for resolving the Stackelberg game: 1) the game is solved off-chain (by an oracle or participant), and the SLA publishes results to the MLMF; or 2) the MLMF autonomously solves the game onchain. In both cases, results must be published on the MLMF, communicated to all participants, and remain auditable by any entity, including third parties not directly engaged in the game. It is assumed that neither SPs nor NOs are required to set up, manage, or maintain a blockchain network. A. STACKELBERG GAME SETUP As explained, the proposed solution is divided into two SCs. The first contract, which is called MLMF, specifies the core logic for the provision of updated network parameters from SPs and NOs and the execution of Stackelberg games based on the received parameters. The second contract, which is called SLA, contains the terms and conditions to execute each step of the spectrum allocation protocol defined in the MLMF contract. Specifically, the SLA schedules the execution of the administration or management functions of the MLMF, which are required to be executed when the defined conditions are fulfilled to make the entire system work. Different solutions can facilitate the execution of these functions. While manual or off-chain alternatives may be viable in controlled environments, they present risks when participants have potentially conflicting interests or when strong guarantees of fairness and availability are required. 4An event is a feature in blockchain SCs used to notify external entities or participants of specific changes in the contract state [64]. 5366 VOLUME 6, 2025
FIGURE 2. Sequence diagram of Stackelberg Game Setup. In such cases, decentralized oracle networks offer a more secure and impartial automation layer, reducing reliance on any single party and mitigating the risks of misbehavior or execution failure [65]. Therefore, a decentralized oracle is incorporated into the proposed solution, which leads the SLA to manage the interaction between the oracle and the MLMF. The division of the proposed solution into the MLMF and SLA SCs allows the modification of the terms and conditions in the SLA without redeploying the core logic of the Stackelberg games held in the MLMF. Although in this paper the SLA is designed to handle only one MLMF for simplicity, the SLA could be configured to handle multiple MLMF contracts simultaneously to perform dynamic spectrum allocation. These contracts could differ in terms of setup parameters or use of allocation methods other than Stackelberg games. Even more, each contract could be dedicated to a specific subset of spectrum frequencies or to SPs and NOs with specific technical requirements. AsshowninFig.2, the deployment process of the two SCs begins with the deployment of the SLA by the DAO, which provides the configurable terms and conditions as initial parameters. Then, the SLA deploys the MLMF, which received the network parameters of αcs,βcs and ,aswell as, the blockchain addresses of each participant, both SPs (@SPs)and NOs (@NOs)in order to empower the MLMF to govern participant access, specifying which entities are authorized to engage in the games and execute specific functions of the MLMF. Although an initial list of both the SPs and the NOs will be provided, this list can be updated at any time thereafter. Once the MLMF is deployed, the SLA retrieves the MLMF’s address (@MLMF)as it is needed to execute its functions. Once the SLA and the MLMF are deployed, the DAO retrieves the SLA’s address (@SLA)in order to indicate to the oracle the SC that should be executed. The DAO then registers the oracle’s blockchain address (@oracle)in the SLA to ensure the oracle can perform the authorized functions of the SLA. Following the initialization process of both SLA and MLMF contracts, the execution of the Stackelberg game model can start. B. STACKELBERG GAME EXECUTION The primary objective of this section is to outline the operational logic of the MLMF SC specifically tailored to handle the operations described in the Stackelberg game, as Algorithm 1 Solving the Stackelberg Game Deployment: C,S,andαcs,β cs ∀c∈C,∀s∈S SetupSP: Bs∀s∈S Input: γc,p(CF) c,η c∀c∈C Run Stackelberg game: Evaluate ζcs,cs and κcsp(CF) c−s∀c∈C,∀s∈S modeled in (15a),(15b) and (15c), respectively do p=p Evaluate πs(C)and τs(C)∀s∈S(Lemma 2) satisfying (17) and (18), respectively Evaluate λs∀s∈Smodeled in (20) Evaluate pcs and κcsp(CF) c−s∀c∈C,∀s∈S modeled in (19) and (15c), respectively while C c=1 S s=1p cs −pcs 2≥ p∗=p,b∗ cs (modeled in (14))∀c∈C,∀s∈S Output: Stackelberg point (b∗,p∗) discussed in Section IV and specified in Algorithm 1, with denoting the required accuracy. The multi-leader multi-follower Stackelberg game for our DSS protocol involves the following steps (see Fig. 3): 1) Initialize a game round. As explained in Section II, the SLA manages the conditions under which each game will run (agreed by the DAO). For example, it could be run at scheduled times or when certain conditions are met. This execution is controlled by the oracle, which periodically checks that the conditions in the SLA are met to start a new round of the game. When the conditions are met (executeNeeded,t=tinit), the oracle calls the SLA, which generates a unique identifier (id)associated with this round of the game, and instructs the MLMF to initiate the phase in which SPs provide their available bandwidth (initialize(id)). The MLMF emits an event (initialized)to inform SPs and NOs of the start of a new round. 2) SPs Providing Bandwidth. In this step, each SP publishes the available bandwidth for a specific game round (SetupSP function in Algorithm 1), identified by id, calling the setBandwidth function of the MLMF. The MLMF communicates the bandwidth published by each SP through an event (EventBandwidth),after verifying that the calling SP is authorized (i.e., that its blockchain address is included in the list of authorized addresses to participate in this game). Optionally, in the terms and conditions defined in the SLA, a maximum time limit for the SPs to provide their available bandwidth and participate in the game can be established. 3) Starting Operator Parameters Provision. Similar to Step (1), the oracle checks if the phase for NOs to provide their parameters should be initiated. If the VOLUME 6, 2025 5367
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING FIGURE 10. Gas cost of the MLMF main functions for the four considered scenarios. with S=2. The reason behind the same increment with C=4and S=4is the fact that Band pare matrices of dimensions CxS, so with adding a new NO or SP an additional row or column respectively is stored. The more providers offering bandwidth, the larger the requests made by NOs can be, and consequently, the higher the cost for the NOs (input function Fig. 10b). The cost of the input function increases approximately 22,600 gas units when there is an additional SP, but it doesn’t change when modifying the number of NOs. This is because the input function stores an array of initial prices that has the same length as the number of SP and, hence, when adding an SP an additional value is stored in the blockchain. The cost of the solveStackelberg function (on-chain approach where the game is solved by the MLMF) is presented separately (see Fig. 11) due to the significantly higher gas cost required for its execution compared to other functions. This high gas required is due to the substantial data storage on the blockchain and the numerous and complex operations performed by the SC. When analyzing the effect of the number of SPs, it is evident that this parameter has the most significant impact, as reflected in the initial case. This is primarily due to the increased information storage requirements, as seen in previous cases. Furthermore, when examining the Stackelberg resolution algorithm, it must be executed until reaching an equilibrium point, as explained in Section IV. Therefore, a higher number of iterations results in more operations and, consequently, higher costs. B. COST IN FIAT The cost, expressed in units of gas, is a valuable fixed measure for identifying how changes in different parameters impact the solution and for comparing different solutions. Nevertheless, the economic cost of performing the various operations of the proposed solution also depends on the cryptocurrency price (which can have high volatility) and the gas price (which usually depends on the network conditions) associated with each blockchain network. To evaluate the economic cost of our solution, we chose Polygon, Arbitrum, Fantom, and BNB Chain because they are representative examples of modern blockchains, covering both Layer 1 and Layer 2 technologies. They use different types of consensus methods and offer different advantages, making them ideal for understanding the options available for building scalable and cost-effective blockchain solutions. All these blockchains are EVM-compatible, meaning they support the Ethereum Virtual Machine and allow the deployment of SCs. Polygon and Arbitrum are Layer 2 solutions for Ethereum, designed to make Ethereum faster and cheaper through sidechains and rollups. BNB Chain and Fantom are Layer 1 blockchains, each with unique strengths. 5374 VOLUME 6, 2025
FIGURE 11. GascostofthesolveStackelberg function considering all scenarios. BNB Chain is known for its speed, affordability, and strong connection to the Binance ecosystem, while Fantom uses a DAG-based system to enable fast and low-cost transactions. Even though Ethereum is the most popular public blockchain capable of executing SCs, it has been excluded due to its consistently high fees, as we have observed in other scenarios, and that renders it inviable for the specific problem at hand [32],[54],[67]. As verified in Section VIII-A, the variation in the number of SPs affects the execution of the input function. The variation in both the number of NOs and SPs influences the cost of the solveStackelberg and output functions. However, these variations do not affect the cost of the rest of the functions, which remain constant. Based on this analysis, this section evaluates the cost in fiat currency over time for the previously described cases. The average cost of the functions initialize,setBandwidth, startInput and endInput is shown in Fig. 12 for the scenario with C=4and S=4. In this figure, the impact of cryptocurrency price volatility associated with each blockchain can be observed. There are noticeable peaks, often influenced by the demand for blockchain usage10 or news related to the blockchain’s utility. Table 1provides an overview of the estimated costs (in US dollars) for the scenario with C=4 and S=4(Fig. 12), based on the average, minimum, and maximum gas prices paid over a 12-month period (November 2023 to October 2024). As can be seen, the cost of the initialize,setBandwidth,startInput, and endInput functions is relatively low, with Polygon and Fantom being the two blockchains with the lowest prices. If an operator executes the setBandwidth function, the cost could reach a maximum of $0.172 in Fantom (with an average value of $0.008) and $0.18 in Polygon (with an average value of $0.011). Analyzing Table 1, we can see that the most expensive functions, such as solveStackelberg, can reach prices of $20 in Fantom and $21 in Polygon. It is important to note that in this case, the game is solved on-chain, which adds load and cost. BNB Chain could reach maximum prices of $141, significantly higher than the others. Although these maximums are occasional, they must be taken into account 10The high costs in Polygon from 16/11/2023 to 17/11/2023 were caused by the high demand for mining tokens inspired by Ordinals [68]. when analyzing which blockchain is the most appropriate for each specific scenario. The second most expensive function is output with maximum prices of $1.34 and $1.41 using Fantom and Polygon, respectively. This analysis indicates that Polygon and Fantom offer more favorable average prices and lower standard deviations compared to the other evaluated blockchain platforms. To further examine the functions whose price fluctuates based on the considered scenario, we will present the price graphs using Polygon, as it is a widely adopted Layer 2 blockchain with a higher number of deployed SCs compared to Fantom. 1) IMPACT OF THE NUMBER OF SPs To assess the influence of the number of SPs on the cost in fiat of runnig the functions input,output, and solveStackelberg,wefixC=4and vary the number of SP (from 2 to 8). Table 2a illustrates the cost associated with the three functions. The input and output functions, which are executed by an operator and the SLA, respectively, remain within low price ranges. The input function costs between $0.021 and $0.035, while the output function costs remains between $0.048 and $0.166. On the other hand, running the game on-chain entails much higher costs, ranging from $0.287 (with SP =2)to $7.273 (with SP =8). In addition, it is important to note that the maximum values can reach up to $117 in the case of 8 SPs. These high values are due to factors such as the increased memory requirements and the higher number of iterations needed to converge to an equilibrium point in the game’s resolution process, as analysed in Section IV. 2) IMPACT OF THE NUMBER OF NOs As concluded in Section VIII-A, fixing the number of SPs to S=4, any change in the number of NOs only impacts the execution cost of the output and solveStackelberg functions. As can be observed in Table 2b, the average cost of executing the output, when considering 10 NOs, is approximately $0.20, with observed values ranging from a minimum of $0.038 to a maximum of $3.302. This cost exhibits a notable increase when the game execution occurs on-chain. In such cases, the average cost can reach approximately $3.63, with occasional spikes up to nearly $59. The variation in these values is primarily driven by factors such as network congestion, gas prices, and the complexity of the transactions involved, as previously explained. C. IMPACT OF USING AN ORACLE FOR AUTOMATING THE GAME EXECUTION In this section, we evaluate the suitability of automating and delegating the execution of the SLA operation to invoke the MLMF functions. As explained in Section VI-B,this approach would avoid assigning these functions to one of the game participants. To perform this analysis, we select two well-known decentralized oracles, Chainlink [69] and Gelato [70], which allow execution of SC functions when VOLUME 6, 2025 5375
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING FIGURE 12. Estimated average cost of the initialize,setBandwidth,startInput and endInput functions (measured in USD) over 12-month period (November 2023 to October 2024). TABLE 1. Estimated cost of the proposed solution (in U.S. dollars) for C=4CF-mMIMOnetworksandS=4 SPs and considering the average, minimum and maximum gas prices paid over a 12-month period and their standard deviation (Nov. 2023 to Oct. 2024). predefined conditions are met, as well as execution of offchain operations and publishing the results to an specified SC. As evaluated in Section VIII-B, the execution of on-chain functions comes with an economic cost that is paid by the entity that executes them. Therefore, in the proposed system, the oracle could assume this cost when executing the MLMF functions. The entity that delegates the execution of a function compensates the oracle for this economic cost and provides an additional fee to reward the oracle for participating in the system. In the two evaluated oracles, this additional fee is a percentage of the economic cost of executing the function, usually called the premium [71],[72].The premium can vary between blockchains, for example in Chainlink and Gelato the premium is set to 70% in Polygon, but in Ethereum it is set to 20%. In addition to the premium fee, there is another factor to consider in Chainlink’s billing. Chainlink allows multiple 5376 VOLUME 6, 2025
TABLE 2. Estimated cost of the proposed solution (in U.S. dollars) and considering the average, minimum and maximum gas prices paid over a 12-month period and their standard deviation (Nov. 2023 to Oct. 2024). delegated functions to be executed in a single transaction. This means that Chainlink’s SCs carry out more operations, which leads to extra costs. To cover these additional costs, they are shared among the entities that have delegated their functions to Chainlink. In summary, considering gpis the transaction gas price, gu is the amount of gas used to execute the delegated function, gois the extra amount of gas used by Chainlink11 (cost 0 for Gelato) and pis the premium percentage in the oracle, the transaction cost txCost for executing a function is: txCost =gp(gu+go)(1+p)(25a) Since the economic cost without oracles can be expressed as the transaction gas price (gp)multiplied by the gas used (gu), the overhead (or the extra) cost of executing a single function compared to executing it directly without any oracle (txCost)is given by: txCost =gpgup+gpgo(1+p)(26a) Chainlink and Gelato provide a service to delegate the execution of operations to the nodes of the oracle network instead of encoding them in a SC. In the off-chain version of the proposed solution, this service can be used to compute the Stackelberg game off-chain and publish the results to the MLMF SC. However, this service typically comes with limitations on computational resources. For example, Chainlink limits the maximum data return size to 128 bytes, which restricts the Stackelberg game execution to C=1 and S=2or C=2and S=1. Gelato, on the other hand, does not have this limitation, but has additional restrictions on the number of executions allowed that can be overcome, if required, at an additional cost. The overhead cost of executing the MLMF functions through Chainlink and Gelato (modeled by equation (26a)) has been analyzed considering the average estimated cost of each function in Polygon presented in Table 1. As expected, the use of Chainlink results in higher overhead costs than Gelato. If the resolution of the Stackelberg game resolution is handled off-chain, the total overhead cost is 0.108 for Chainlink and 0.068 for Gelato. If the resolution occurs onchain, the overhead increases to 1.003 for Chainlink and 0.943 for Gelato. 11It is assumed that there are 80,000 units of extra gas, as it is reflected in the documentation [71]. Therefore, the additional cost of automating the SLA functions with an oracle may not appear significant and would help avoid assigning game management responsibilities to specific participants. However, it is important to examine how this cost might impact the long-term execution of the game. 1) BALANCE BETWEEN COST AND LATENCY The financial feasibility of the proposed solution has been evaluated by considering the average cost of executing the MLMF on the blockchain. However, another critical factor influencing the total cost is the latency introduced by blockchain usage. Generally, for a given blockchain, the shorter the execution time required for a function, the higher the fee to be paid [73]. Next, we perform the evaluation, defining a game round as the execution of the main MLMF functions in a scenario with C=4and S=4. We assume that all setBandwidth functions are published in the same blockchain block, as well as all input functions. From now on, we will refer to the initialize,startInput,endInput and output functions as administrative tasks of a game round, as they all need to be executed by the SLA. For the analysis, we use the gas price to reach the minimum delay12 published in PolygonScan [74], BNBScan [75] and FTMScan [76], to get the results for Polygon, BNB Smart Chain and Fantom, respectively.13 The minimum delay found is between 5 and 10 seconds for all these sources. While this represents the minimum reported latency, it should not be interpreted as the lowest achievable delay, as it is influenced by the gas price paid, as previously discussed. Tab. 3summarizes the analysis results. Polygon and Fantom stand out as the blockchains with the lowest transaction costs (as shown in Section VIII-B), especially considering that all analysed blockchains have almost identical publication times. The cost of performing MLMF functions on Fantom is about 50% lower than on Polygon. For example, an operator would pay approximately $0.01 to publish parameters on Fantom, while the cost on Polygon would be approximately $0.018. SPs incur about half the cost of NOs ($0.004 on Fantom and $0.008 on 12We have not found a source to get the estimated transaction delays given the gas price for the blockchains analysed in this work. 13We have not found a source to get the gas price to reach the minimum delay for the Arbitrum network. VOLUME 6, 2025 5377
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING TABLE 3. Trade-off between delay and price for C=4CF-mMIMOnetworksand S=4 SPs; average price for a minimum delay in Polygon, BNB Smart Chain and Fantom. Polygon). The highest administrative costs, particularly when interacting with the Oracle, are around $0.156 on Polygon and $0.079 on Fantom. While these individual costs may seem negligible, recall that the Stackelberg game can be run periodically, leading to significant cumulative costs over time. 2) COST OF RUNNING STACKELBERG ON A REGULAR BASIS In certain scenarios, the Stackelberg game may need to be executed periodically, making it crucial to analyze both the overall management cost of the game and the specific costs incurred by each SP and NO. Table 4summarizes the daily and monthly cost of executing a game round in the two cheapest blockchains (Polygon and Fantom), in a cadence of 10 seconds, 10 minutes or one hour for each participant: the manager (who executes the administration tasks), SPs and NOs. Each SP pays the cost of executing the setBandwidth function to publish its available bandwidth. Meanwhile, each NO pays the cost of executing the input function to publish its network parameters. As observed, the cost of using the blockchain decreases significantly as the frequency of game execution decreases, which is an expected result. For example, when using Polygon and considering the daily cost, one NO’s cost would drop from around $154.14 (when participating in the game every 10 seconds) to just $0.43 when participating once per hour. Similarly, for a SP, the cost would drop from about $66.71 to just $0.19. In the case of Fantom, these costs are reduced by more than half compared to Polygon, as was seen in Section VIII-C.1. For example, an operator would pay about $38.51 versus $66.71 if the game run every 10 seconds. Using an oracle for frequent administrative tasks, such as executing the Stackelberg game every 10 minutes, significantly increases operational costs. In this case, the monthly cost can reach around $673 on Polygon, with about $380 attributed to the oracle overhead. It should be noted that the cost of administration can be distributed among the number of participants in the system. As previously noted, this analysis has assumed C=4and S=4, which would imply eight participants contributing to system maintenance costs. Assuming equal contributions from all participants, the cost per participant of publishing game results with an oracle would be approximately $84 in Polygon and less than $43 in Fantom. D. BLOCKCHAIN PERFORMANCE MEASURED IN TRANSACTIONS PER SECOND In this section, we conduct an analysis of the blockchain performance and scalability, beyond the economic costs and transaction delays associated with blockchain publication. This evaluation considers the number of transactions per second (TPS) that different public blockchains can handle (see Table 5). The maximum theoretical throughput of a blockchain is calculated by dividing the number of transactions that can fit into a block by the duration of the block. The capacity of the block, measured in gas units, determines the maximum number of transactions it can hold. As shown in the Table 5, most blockchains allow up to 30 million gas units per block, except for BNB, which supports up to 140 million gas units. Each blockchain has a predefined block time. For example, Arbitrum has the shortest block time at 250 milliseconds, while Ethereum has the longest at 12 seconds. These factors determine the maximum possible throughput of the network. For example, with a block duration of 250 milliseconds and a block gas limit of 32 million, Arbitrum could theoretically process 1,428 transactions per block, resulting in a maximum theoretical throughput of 6,095 TPS. In contrast, Ethereum currently supports around 119 TPS, although Ethereum 2.0 aims to reach 100,000 TPS in the future. However, these values are theoretical, and actual performance will vary due to fluctuations in block sizes, network congestion, and other operational factors. Looking at the highest recorded TPS, Solana has the highest value with 2,909 TPS, followed by BNB (1,731 TPS) and Arbitrum (1,105 TPS). Real-time throughput over a 30-day period (see Table 5), shows that Solana achieves 679 TPS, Polygon around 40 TPS, and Ethereum only 15 TPS. It is important to highlight that blockchain performance continues to improve over time. For instance, Ethereum aims to reach 100,000 TPS in the future. Meanwhile, Solana has been designed as a high-performance, scalable, and fast blockchain, featuring a block time of just 400 milliseconds, minimal transaction costs, and a theoretical throughput of up to 65,000 TPS. These results indicate that blockchains can support significantly higher loads than those currently utilized, highlighting their potential for future scalability. Remarkably, our proposed system allows the execution of multiple Stackelberg game rounds in parallel, while also enabling the deployment of various MLMF smart contracts to accommodate different system conditions (as explained in Section VI-A). These smart contracts could even be deployed on different blockchains to adapt to changing time and cost requirements over time. This adaptability and versatility of the proposed solution allow it to adjust to different scenarios, ensuring optimal performance and cost efficiency. 5378 VOLUME 6, 2025
TABLE 4. Estimated cost summary for executing a complete Stackelberg game round per day and per month, considering various frequencies and utilizing Polygon and Fantom. TABLE 5. Summary of the analysis of different public blockchains: blockchain parameters and blockchain performance (Data source: Chainspect, blockchain analytics platform https://chainspect.app/dashboard, data accessed on March 28, 2025)). IX. CONCLUSION This work presents a generalized framework for automated, transparent, and fair DSS within 6G networks. Specifically, multiple NOs using the CF-mMIMO topology are considered buyers of bandwidth from multiple SPs. The spectrum trade is modeled as a Stackelberg game, where SPs maximize the price per bandwidth unit, while NOs optimize a utility function balancing SE gains and bandwidth costs. An iterative algorithm is proposed to solve the game, guaranteeing convergence to a unique equilibrium. The game’s organization and resolution require information exchange between SPs and NOs. Blockchain ensures transparency and traceability by recording all actions for auditability. It also enables automated decisions, reducing intermediaries and improving spectrum allocation. The defined SC structure offers adaptability to changes in parameters, participants, or execution rules. Extensive simulations show how network configurations (e.g., number of NOs and SPs) influence game outcomes and pricing. The game-solving protocol is implemented on several blockchain platforms to assess the real costs of the proposed DSS framework. Notably, the blockchain choice significantly affects cost efficiency. Offloading game execution to an oracle was also tested on two commercial platforms, incurring minimal overhead, highlighting the trade-off between reduced computational burden and added costs. On the cell-free side, future work will study the impact of hardware impairments on game outcomes and how mismatched information affects blockchain data requirements. On the blockchain side, the next step involves evaluating the cost and performance of deploying on permissioned blockchains. This includes analyzing network settings such as consensus algorithms, block sizes, and participant roles. The economic costs of running and maintaining nodes will also be assessed, as these factors are critical to security, participant trust, and overall operational expenses. APPENDIX A PROOF OF LEMMA 1 Setting the first derivative of Uc(b(CF) c,p(CF) c)in (12) equal to zero, it is straightforwardly shown that bcs =ηcαcs βcs −γc −pcs βcs −γc −γc βcs −γc S s=1 bcs,(27) VOLUME 6, 2025 5379
FEMENIAS et al.: MULTI-LEADER MULTI-FOLLOWER STACKELBERG GAME FOR DYNAMIC SPECTRUM SHARING for all s∈S. Adding all the chunks of bandwidth in b(CF) c it follows that S s=1 bcs =ηcS s=1αcs βcs−γc−S s=1pcs βcs−γc 1+γcS s=11 βcs−γc .(28) Substituting (28) into (27) and operating leads to bcs =ζcs −cspcs +κcsp(CF) c−s,(29) with ζcs,cs and κcs(p(CF) c−s)defined in (15). Finally, the projection of bcs onto the set of non-negative real numbers provides (14). APPENDIX B PROOF OF LEMMA 2 The Lagrangian of the convex optimization problem PSPR can be expressed as Lp(SP) s,λ s,υs,ξs = C c=1 pcsζcs −cspcs +κcsp(CF) c−s +λsBs− C c=1ζcs −cspcs +κcsp(CF) c−s + C c=1 υcspcs − C c=1 ξcs⎛ ⎝pcs − ζcs +κcsp(CF) c−s cs ⎞ ⎠.(30) where λcs,υcs and ξcs are the Lagrange multipliers. The KKT conditions for this problem are given as follows: Stationarity: ∂Lp(SP) s,λ s,υs,ξs ∂pcs =ζcs +κcsp(CF) c−s −2cspcs +λscs +υcs −ξcs =0,(31a) Complementary slackness: λsBs− C c=1ζcs −cspcs +κcsp(CF) c−s=0,(31b) υcspcs =0∀c∈C,(31c) ξcs⎛ ⎝pcs − ζcs +κcsp(CF) c−s cs ⎞ ⎠=0∀c∈C,(31d) Primal feasibility: C c=1ζcs −cspcs +κcsp(CF) c−s≤Bs,(31e) pcs ≥0∀c∈C,(31f) pcs ≤ζcs +κcsp(CF) c−s/cs ∀c∈C,(31g) Dual feasibility: λs≥0,(31h) υcs ≥0∀c∈C,(31i) ξcs ≥0∀c∈C.(31j) Lemma 4: The prices per unit of bandwidth are always positive, that is, pcs >0for all c∈C. Moreover, the Lagrange multipliers υcs =0for all c∈C. Proof: Let us assume that pcs =0for an arbitrary CFmMIMO network c. In this case, from (31d) it follows that ξcs =0. Thus, using pcs =0and ξcs =0in (31a) yields ζcs +κcsp(CF) c−s+λscs +υcs =0.(32) As stated in Lemma 1, the parameters ζcs,cs, and κcs(p(CF) c−s)are positive. Furthermore, from (31h) and (31i) it is known that λs≥0and υcs ≥0, respectively. Under these conditions, equation (32) is incompatible. The conclusion is then that pcs >0for all c∈Cand from (31c) it follows that υcs =0for all c∈C. Lemma 5: The primal feasibility constraint (31e) is satisfied with equality, that is, C c=1ζcs −cspcs +κcsp(CF) c−s=Bs,(33) and the Lagrange multiplier λs= 0. Proof: Let us assume that λs=0and let us select a particular CF-mMIMO NO cfor which the primal feasibility constraint (31g) is inactive, that is, a CF-mMIMO NO for which pcs <(ζ cs +κcs(p(CF) c−s))/cs.Usingthe complementary slackness condition (31d),itfollowsthat ξcs =0. Hence, using Lemma 4(i.e., υcs =0), and the assumptions that λs=0and ξcs =0in the stationarity condition (31a) entails that the only possible solution would be pcs = ζcs +κcsp(CF) c−s 2cs ,(34) which is incompatible with the fact that, in general, pcs could potentially take any value in the range 0<pcs < (ζcs +κcs(p(CF) c−s))/cs. The conclusion is then that λs= 0 and from the complementary slackness constraint (31b) it holds that the primal feasibility constraint (31e) is active. Let us define the sets CAsand CIsas the sets of NOs for which the dual feasibility constraint (31j) is either active or inactive, respectively. That is, CAs={c:ξcs >0},(35a) CIs={c:ξcs =0}.(35b) Using these definitions and applying Lemma 4in the stationarity condition (31a),itfollowsthat pcs =⎧ ⎪ ⎨ ⎪ ⎩ ζcs+κcsp(CF) c−s cs ∀c∈CAs ζcs+κcsp(CF) c−s 2cs +λs 2∀c∈CIs. (36) Now, using these prices jointly with the equality condition deduced in Lemma 5it can be established that λs=c∈CIsζcs +κcsp(CF) c−s−2Bs c∈CIscs .(37) 5380 VOLUME 6, 2025
Using again Lemma 4in the stationarity condition (31a) leads to the conclusion that ξcs =λscs −ζcs −κcsp(CF) c−s∀c∈CAs,(38) which must be positive and, therefore, it can be inferred that ζcs +κcsp(CF) c−s cs <λ s∀c∈CAs,(39) and results presented in Lemma 2can be straightforwardly deduced. APPENDIX C PROOF OF LEMMA 3 Definition 1: A function F(p)is said to be standard and satisfies the two-sided scalability property if the following conditions hold for all p0: 1) Positivity: F(p)>0 2) Monotonicity: For all pand p,ifpp, then F(p)> F(p). 3) Scalability: For all μ>1,μF(p)>F(μp). 4) Two-sided scalability: For all μ>1,if 1 μppμp, then 1 μF(p)<F(p)<μF(p). Lemma 6: The function Fp(CF) c−s =⎧ ⎪ ⎨ ⎪ ⎩ ζcs+κcsp(CF) c−s 2cs +λs 2∀c∈{cs1,...,cs(τs−1)} ζcs+κcsp(CF) c−s cs ∀c∈{csτs,...,csC}, (40) is a standard function and satisfies the two-sided scalability property. Proof: Positivity: As stated in Lemma 1, the parameters of the system must be set in order to guarantee that ζcs,cs, and κcs(p(CF) c−s)are all positive. Moreover, from Lemma 5and the dual feasibility constraint (31h), we know that λs>0. Then, it follows that F(p(CF) c−s)>0. Monotonicity: For all p(CF) c−sp(CF) c−s, it holds that κcs(p(CF) c−s)>κ cs(p(CF) c−s)and, consequently, F(p(CF) c−s)> F(p(CF) c−s). Scalability: For all μ>1, we have that κcs(μp(CF) c−s)= μκcs(p(CF) c−s)and hence, as ζcs >0, and cs >0it follows that μF(p(CF) c−s)>F(μp(CF) c−s). Two-sided scalability: For all μ>1,wehave that whenever 1 μp(CF) c−sp(CF) c−sμp(CF) c−sthen 1 μκcs(p(CF) c−s)≤κcs(p(CF) c−s)≤μκcs(κcs(p(CF) c−s)) and consequently, 1 μF(p(CF) c−s)≤F(p(CF) c−s)≤μF(κcs(p(CF) c−s)). According to [77],[78],[79], an iterative update of a standard function satisfying the two-sided scalability property converges to the corresponding fixed point and consequently, this completes the proof. REFERENCES [1] W. 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Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1347, Sep. 1995. [78] C. W. Sung and K.-K. Leung, “A generalized framework for distributed power control in wireless networks,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2625–2635, Jul. 2005. [79] T. D. Tran and L. B. Le, “Resource allocation for multi-tenant network slicing: A multi-leader multi-follower Stackelberg game approach,” IEEE Trans. Veh. Technol., vol. 69, no. 8, pp. 8886–8899, Aug. 2020. GUILLEM FEMENIAS (Senior Member, IEEE) received the Telecommunication Engineer and Ph.D. degrees in electrical engineering from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1987 and 1991, respectively. From 1987 to 1994, he worked as a Researcher with UPC, where he became an Associate Professor in 1992. In 1995, he joined the Department of Mathematics and Informatics, University of the Balearic Islands (UIB), Spain, where he became a Full Professor in 2010. He is the Founder of the Mobile Communications Group, UIB. He has been the Project Manager of numerous projects funded by the Spanish and Balearic Islands Governments. In the past, he was also involved with several European projects. His current research interests and activities span the fields of digital communications theory and wireless communication systems, with particular emphasis on radio resource management strategies applied to 5G and 6G wireless networks. On these topics, he has published more than 200 journal and conference papers, as well as some book chapters. Dr. Femenias was the recipient of the Best Paper Awards at the 2007 IFIP International Conference on Personal Wireless Communications and at the 2009 IEEE Vehicular Technology Conference - Spring. He has served for various IEEE conferences as a technical program committee member, as the Publications Chair for the IEEE 69th Vehicular Technology Conference (VTC-Spring 2009) and as a Local Organizing Committee member of the IEEE Statistical Signal Processing in 2016. M. FRANCISCA HINAREJOS received the M.S. degree in telecommunication engineering in 2003, and the Ph.D. degree in computer science from the Technical University of Catalonia in 2010. She is currently an Assistant Professor with the Department of Computer Science, University of the Balearic Islands. She has authored several papers published in national and international conferences and international journals. Her research interests include network security, electronic commerce, security in constrained environments, cybersecurity, and blockchain, and she leads national projects in these areas. FELIP RIERA-PALOU (Senior Member, IEEE) received the B.S./M.S. degree in computer engineering from the University of the Balearic Islands (UIB), Mallorca, Spain, in 1997, the first M.Sc. and Ph.D. degrees in communication engineering from the University of Bradford, U.K., in 1998 and 2002, respectively, and the second M.Sc. degree in statistics from the University of Sheffield, U.K., in 2006. From May 2002 to March 2005, he was with Philips Research Laboratories, Eindhoven, The Netherlands, first as a Marie Curie Postdoctoral Fellow (European Union) and later as a Technical Staff Member. While with Philips, he worked on research programs related to wideband speech/audio compression and speech enhancement for mobile telephony. From April 2005 to December 2009, he was a Research Associate (Ramon y Cajal program, Spanish Ministry of Science) with the Mobile Communications Group, Department of Mathematics and Informatics, UIB. Since January 2010, he has been an Associate Research Professor (I3 Program, Spanish Ministry of Education) with UIB. His current research interests are in the general areas of signal processing and wireless communications. JOSEP-LLUÍS FERRER-GOMILA received the M.S. degree in telecommunications engineering from the Technical University of Catalonia in 1991, and the Ph.D. degree in computer science from the University of the Balearic Islands in 1998, where he is currently an Associate Professor with the Computer Science Department. He has authored several articles published in national and international conferences and international journals. His research interests include network security and electronic commerce, and he leads national projects in this area. AMADOR JAUME-BARCELÓ received the M.S. degree in telecommunication engineering from the University of the Balearic Islands in 2021, where he is currently a Research Assistant with the Computer Science Department. His research interests include cybersecurity and blockchain. VOLUME 6, 2025 5383