Bankruptcy Problem in Network Sharing: Fundamentals, Applications and Challenges

Network sharing has already been adopted by mobile network operators as a reliable and effective countermeasure to the constantly increasing network cost. The introduction of 5G mobile communications is expected to revolutionize the telecommunications world. However, it also brings new challenges to network sharing, as new technologies (e.g., network virtualization, cloud architectures, and so on) and types of resources (e.g., computation and storage) come to the fore. Despite their inherent differences, the common characteristic of the emerging resources is that they are limited and, in many cases, insufficient to cover the rising traffic demands. In this article, we highlight the equivalence between the distribution of a limited number of resources and the bankruptcy problem, where the demands of different agents over a given commodity exceed its total quantity. In particular, i) we review the fundamentals and existing solutions of the bankruptcy problem; ii) we provide potential applications of this problem in mobile network sharing; and iii) we list challenges and open issues for the application of the bankruptcy game in the mobile communications domain. The main goal of our work is to identify new research lines that will foster 5G network sharing.

lished. In this context, this challenging situation of sharing limited resources among various interested parties can be formulated as a bankruptcy problem, which is a distribution problem that involves the allocation of a given amount of a single commodity among a group of agents, when this amount is insuffi cient to satisfy all their demands [4].
Taking into account the similarities of limited resource sharing with the bankruptcy problem and the fact that bankruptcy theory has not yet been fully exploited in mobile network sharing scenarios, the main goal of this article is to identify new research lines in order to pave the way for the application of the bankruptcy problem in network sharing. To that end, our contribution is mainly threefold. First, we present the fundamentals of the bankruptcy problem, along with a list of possible solutions and the key ideas behind these solutions. Then, we focus on the telecommunications domain and we provide potential applications in existing and future mobile networks, where resource sharing can be modeled and solved as a bankruptcy problem. Finally, we list some intriguing open research lines and challenges for the application of the bankruptcy theory in wireless networks.
bAnkruPtcy Problem: bAckground The origins of the bankruptcy problem go several centuries back, following the fundamental human need for fair division of commodities. 1 In principle, each bankruptcy problem is characterized by an entity E ∈  that has to be divided among N agents, whose individual claims C i , i ∈ N, add up to an amount higher than the total entity, that is, Apparently, as it is not possible for all claims to be satisfied, different approaches may be followed for the division. In this section, we present a toy example along with some of the most popular solutions and the outcome allocations (A i ) they yield, summarized also in Table 1. bAnkruPtcy Problem: toy exAmPle Without loss of generality, we assume that a total entity of E = 45 units has to be allocated to three claimants, whose claims are C 1 = 10, C 2 = 20 and C 3 = 30, respectively. As the sum of all claims is greater than the total entity, that is,  3 i=1 C i = 60 > 45, it is not possible to fulfi ll all claims.

eQuAl shArIng
Equal sharing is the simplest form of sharing a commodity among a set of interested agents. Equal sharing completely neglects the individual claims of the involved parties and divides the commodity into equal shares that are allocated to each agent, respectively. It is worth noting that, following this approach, it is possible that one agent receives an amount higher than the one they claimed. In our example, the entity would be divided into three equal parts, that is, A 1 = A 2 = A 3 = 15. Obviously, the fi rst claimant would receive a quantity higher than the requested, while the requirements of the other two claimants would not be satisfi ed.

wAter fIllIng
The water fi lling solution is another simple method that is mainly used to avoid that a claimant is awarded with a portion higher than the requested. In this case, we can think of the claims as tanks of diff erent height (i.e., proportional to each claim) that are gradually filled with part of the entity. Once the lower tank is fi lled, we continue only with the remaining tanks, maintaining always the same "water" level in the tanks to be filled. However, one important disadvantage of this approach is that the losses are not equally shared, as usually the smaller claim is completely satisfi ed and the losses aff ect mainly the high claims. Figure 1 demonstrates the water fi lling allocation in four diff erent cases. In our toy example, following this approach, all claimants would receive 10 units (until the demand of the fi rst claimant is fulfi lled) and then the remaining 15 units would be equally split to the other two claimants.

loss shArIng
To overcome the unbalanced loss issue, one possible solution would be to observe the problem from a different point of view and focus on the fair distribution of losses instead of sharing the entity. More specifi cally, as the sum of the claims is always higher than the entity, we may estimate their difference and then split this difference equally to the N agents. You may also notice that in case that there are big discrepancies between the claims, this approach may induce negative values. For instance, if E = 45, C 1 = 5 and C 2 = 60, the loss sharing method would allocate A 1 = -5 and A 2 = 50. In this case, agent 1 should subsidize agent 2 with five units. However, in such extreme cases, the accepted solution is to allocate the whole amount to the higher claim. Focusing in our particular example, the total loss of 15 units would be allocated equally to the three players and, as a result, each of them would lose five units, that is, A 1 = 5, A 2 = 15 and A 3 = 25.

ProPortIonAl shArIng
This is another approach that explicitly takes into account the claims, as the sharing takes place proportionally to the claims. Proportional division is a quite appealing approach and it is also adopted in several real life use cases (e.g., in case of natural disasters, insurance companies pay off the losses with a fixed amount per dollar). It can be also easily proven that the proportional sharing of the losses provides the same solution, so we avoid dissatisfaction due to unbalanced losses. In our example, the second and the third claimant would receive an amount two and three times higher, respectively, than the first player.

contested gArment rule
This method is one of the oldest sophisticated ones and named after the problem of sharing a garment. The main logic of this approach lies in the fact that the division takes place on the disputed part (i.e., the part that both agents claim), while the higher claimant receives the undisputed part. For instance, considering the case of E = 125, C 1 = 100 and C 2 = 200, both claimants would receive 50 units (as the common claimed part is 100), while the remaining 25 units would be allocated to the second claimant. Although beyond the scope of this article, the solution in the n-player problem 2 is not straightforward and it tries to: provide half of the claimed amount to each claimant, and equalize the losses among claimants. In our example, the solution would be A 1 = 5, A 2 = 15 and A 3 = 25.

shAPley VAlue
As it has turned out that the bankruptcy problem can be formulated as a cooperative game, the Shapley value can provide another alternative sophisticated solution. The Shapley value was introduced in 1953 by Lloyd Shapley [6] and assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. For the computation of the Shapley value, the order of arrival of the different requests is taken into account, 3 while to remove any unfairness, all possible combinations of arrival are considered. For instance, in our example, the allocation according to the Shapley value would be A 1 = 6 2/3, A 2 = 14 1/6 and A 3 = 24 1/6, as demonstrated in detail in Fig. 2.

PotentIAl APPlIcAtIons In 5g mobIle communIcAtIons
In this section, we will try to shed some light on the potential applications of the bankruptcy problem in the telecommunications domain. Apparently, we are moving through a digital era, where the user demands and requirements are constantly increasing, rendering the network resources scarce. In the following, we will present a list of possible research areas (some of them are also illustrated in Fig. 3), where the sharing of the resources can be formulated as a bankruptcy problem.

network resource mAnAgement
The management of radio resources constitutes one of the most traditional problems in wireless communications and there are already some works that have treated it as a bankruptcy problem [7]. With the introduction of 5G-driven bandwidth hungry applications (e.g., AR, 4K video streaming, and so on), the radio resources become even more valuable, as it is not trivial to allocate the bandwidth in a fair and efficient manner. However, the fact that the new applications come with clear requirements and key performance indicators (KPIs) promotes the formulation of the problem as a bankruptcy case, where the application requirements correspond to the claims.
In upcoming 5G networks, it is expected that the exploitation of high frequency bands and mmWave technology will provide a solution to this problem, as spectrum will be abundant and no longer a constrained resource. However, in such scenarios, other network parts can constitute a bottleneck, for example, the transport network or the fronthaul link in C-RAN topologies. Hence, bankruptcy game theory could still be a valuable tool for overcoming these bottlenecks by efficiently sharing the limited network resources.

InfrAstructure shArIng
Infrastructure sharing is a quite broad concept that has emerged during the last decade and, in its basic form, involves the sharing of passive (e.g., cooling, warehouses) and active (e.g., antennas, base stations) network equipment. The entry of C1=100 C2=200 C3=300 C1=100 C2=200 C3=300 C1=100 C2=200 C3=300 C1=100 C2=200 C3=300 new stakeholders, such as mobile virtual network operators, in the telecommunications landscape has already motivated the research community to study the efficient sharing of the underlaid network infrastructure, using advanced analytical tools from game and auction theory [2]. With 5G evolution, the networks become more dense (hence the number of small base stations increases); however, the number of interested parties that are willing to have access and control over the deployed infrastructure is also growing. In addition, the concept of a trusted third party that provides the network infrastructure (also known as infrastructure provider) also gains ground, escalating further the problem of a fair and profitable allocation of the 5G small cells to the network operators. To that end, bankruptcy theory can be considered as an alternative solution to this problem, being complementary to traditional game theoretic approaches.

energy shArIng
Energy commodities have always been precious in real life, as well as in various industrial domains. In particular, the information and communication technologies (ICT) sector is considered as one of the most power demanding industries, responsible for ∼2 percent of global greenhouse gas emissions. To confront this issue, mobile operators are increasingly adopting the utilization of renewable (or clean) energy sources, such as wind, water or solar. Although clean energy sources contribute significantly to the reduction of CO2 emissions, their power supply is intermittent and cannot be accurately predicted, thus being unreliable for mobile networks. The use of batteries could partially provide a solution to this issue; however, battery capacity is also limited and efficient sharing mechanisms should be put in place.
Bankruptcy theory would be an appropriate framework to model this problem in different use cases. For instance, in the context of network sharing, the mobile operators that share a base station could claim different amounts of energy according to their traffic demands [8]. Another possible application would be in single-operator scenarios, where many base stations are powered by a pool of batteries. In this case, the various applications could have different claims according to their criticality and bankruptcy modeling could solve the problem.

loAd sheddIng
Load shedding is a common technique that was primarily introduced in information systems (mainly in web services) to avoid overloading the system, and thus making it unavailable for all users. The main idea behind shedding part of the load is that some requests may be ignored so the rest of the request can be efficiently served. In the same context, with the smart grid revolution, load shedding is required when there is an imbalance between electricity demand and electricity supply.
Apparently, efficient schemes for handling situations of this kind should be properly devised, as load shedding can be very inconvenient and irritating to the users that are rejected by the respective system (either requests by operators/end users in cloud data centers or energy requests in smart grid scenarios). To that end, bankruptcy theory can provide new insights and solutions to the specific problem by allocating the resources in a fair manner, while maximizing the number of users that can be served.
cloud And edge comPutIng Cloud computing infrastructures are expected to have a leading role in upcoming 5G networks thanks to two important enablers. First, network virtualization gradually transforms the existing rigid mobile networks to flexible software platforms, suitable to accommodate sophisticated virtual network functions (VNFs) that need computational resources for their execution. In addition, the adoption of novel computing paradigms, such as fog computing or MEC, brings the computational resources at the edge of the network.
The main idea behind the aforementioned new paradigms is that by running applications and performing processing tasks closer to the end user, network congestion is reduced (e.g., less signaling in the core network), while the performance of existing and new applications can be considerably improved (e.g., we are able meet the strict delay requirements of ultra low latency traffic). Nonetheless, despite the potential advantages of edge computing, there are also important cost and space limitations that do not allow their vast implementation in future networks. Hence, the limited edge resources become valuable, while the claims over these resources is expected to increase, as the user density and requirements are also increasing.
Apparently, the allocation of the insufficient edge resources to a number of interested parties can be inherently formulated as a bankruptcy problem, where the claimants could be either the different mobile operators that request computing resources by a third party that acts as an infrastructure provider, or even the end users that compete for these resources in single-operator scenarios [9].

content cAchIng
Content caching in mobile networks has been introduced as an effective countermeasure for the reduction of the increased backhaul data rates due to the constantly growing data traffic [10]. Caching can be applied in various scenarios and topologies, such as C-RAN, heterogeneous networks, and device-to-device communications, among others. However, regardless of the use case, caching is used to bring popular and viral contents close to the end user so as to improve the user quality of experience with lower delays, and the network utilization for the operator by avoiding the repeated transmission of the same content over the network links. Similar to the cloud computing case, the deployment of cache servers in all network elements is not feasible, as it can be either costly or impossible due to space constraints. On the other hand, the number of over-the-top (OTT) content providers is still increasing, while they are heavily interested in obtaining access to these resources in order to enhance their quality of service. This conflicting situation, that is, a set of OTT providers that compete over a limited set of resources to cache their contents, can be modeled as a bankruptcy problem. In a similar context, multiple virtual MNOs could have the role of the claimants, asking for caching resources from the infrastructure provider, who has also to take into account the popularity of the different contents for estimating its potential gains. In all cases, bankruptcy theory could be employed to yield efficient solutions and useful insights [11].

reVenue shArIng
As the bankruptcy framework was initially applied to economic problems, its application to financial scenarios is quite straightforward. More specifically, 5G is expected to further boost the already high revenues of mobile networks, while the coexistence of multiple claimants is very possible to generate conflicting situations where the total claimed amount will be higher than the actual profit. It is also worth noting that the issue of revenue sharing can appear in different parts of the network with different entities as players.
For instance, in the various infrastructure sharing cases, where the operators share different parts of the network, different claims may arise according to their contributions (e.g., one operator may provide the infrastructure, another the spectrum, and so on) and their subjective perspectives. Another example comes from the invasion of OTT content providers in mobile networks and the new relationship dynamics with mobile operators that are formed. In this case, the mobile operators could claim part of the OTTs revenues, as the OTTs operate freely over mobile networks thanks to network neutrality. In the same context, as the mobile operators assert that content providers exploit their deployed networks, they could even ask for compensation in order to share the network cost. In all these cases, bankruptcy theory could facilitate the solution of the problem.

unmAnned AerIAl VehIcles (uAVs)
The flexible deployment along with the strong lineof-sight links of UAVs are expected to make them an important component of the future 5G and beyond wireless networks toward the provision of ubiquitous anytime connectivity [12]. Besides the straightforward application of the bankruptcy problem for spectrum sharing among the UAVs and the fixed network network infrastructure, the introduction of aerial networks implies additional challenges that need to be addressed.
In particular, the energy supply of UAVs has been identified as one major issue in aerial networks, since apart from the possibility of communication interruption, the risk of physical damages (as UAVs may hover over crowded areas) is also looming. To deal with this problem, the idea of strategically deploying charging stations in urban and rural areas has been recently proposed [13] in the literature. However, as UAVs gain ground in mobile communications and the number of UAV operators is expected to increase, the limited places in the charging stations should be claimed and allocated efficiently.

oPen reseArch lInes And chAllenges
This section presents some challenges and possible extensions for the application of the bankruptcy framework in the telecommunications domain.

comPutAtIonAl comPlexIty
As we have seen in the previous sections, the bankruptcy problem can have several applications in the mobile communications domain, while there are also many possible solutions for each one of these applications. Therefore, the selection of the appropriate approach to each use case is something that should be carefully decided, and to that end, computational complexity is a parameter that should be definitely taken into account. More specifically, there are some very simple possible solutions of low complexity (e.g., equal share), while the complexity of some other solutions grows exponentially and can be even NP-hard in some cases (e.g., the Shapley value).
In upcoming 5G networks, the time frame for making a decision in each problem will also determine the range of the acceptable solutions. In offline methods, such as revenue distribution or energy sharing, where the decision can be made in a longer timescale, the application of more sophisticated approaches could yield fairer and more efficient results. On the other hand, in cases where decisions have to be made "on-the-fly," for example, spectrum allocation or MEC assignment, solutions of lower computational complexity that are able to be executed quickly should be adopted. Of course, the selection of the solution is not always straightforward, as there can be cases where the time frame is dynamic, for example, in charging station sharing for UAVs, the available time to make a decision depends on the location of the UAVs and the distance they have to cover. In such cases, the study of the trade-off between the time execution and the efficiency of the results could reveal quite interesting insights.
5g end-to-end (e2e) network slIcIng 5G is characterized by the entry of various vertical industries with strict but distinct requirements. Applications such as AR and 4K video require very high data rates and bandwidth; autonomous driving or smart-grid connections need extremely low latency; while smart home deployments are based on the simultaneous connections of a plethora of sensors in the network. The aforementioned applications drive respectively the three main traffic types in 5G, that is, enhanced Mobile Broadband (eMBB), Ultra Reliable and Low Latency Communication (URLLC) and massive Machine Type Communication (mMTC).
To fulfill the different requirements of these applications, the concept of e2e network slicing has been introduced, which defi nes the formation of virtual networks through resource reservation across the different network domains. For an efficient slice creation, it is possible that a series of different bankruptcy problems need to be jointly solved, as diff erent types of resources are required. For instance, to experience an uninterrupted interactive AR service in a CRAN network, the reservation of fronthaul, spectrum and cache resources would be essential.

gAme theory
The bankruptcy problem is inherently connected with game theory, as it concerns strategic interaction between rational decision-makers. This is also the main reason for modeling this problem as a cooperative game, where the Shapley value can provide a fair solution. However, due to the nature of the specifi c problem and the confl icting interests among the claimants, non-cooperative game theoretic approaches could be also considered [14].
Non-cooperative game theory focuses on cases where there is no collaboration among the players, while the main individual goal of each player is to achieve a stable solution, where their received utility will not depend on the actions of the other players. This solution point is called Nash equilibrium and it has been proven that, in this point, none of the players can improve their payoff by unilaterally changing strategy. In particular, in bankruptcy problems, the utility functions of the diff erent agents could correspond to their allocated amount, while the claims can be the outcome of the game, that is, the strategies that will constitute the Nash equilibrium. Although, in general, cooperative game theory can achieve even optimal results, non-cooperative solutions also have some nice attributes (e.g., stability, user satisfaction, and so on) that make them appealing in scenarios where cooperation is hard to be achieved.

mAchIne leArnIng
Machine learning [15] is another emerging technology that could be exploited to provide novel alternative solutions in bankruptcy problems. Machine learning provides systems with the capability to automatically learn and improve from experience without being explicitly programmed. In particular, as the main goal of the bankruptcy problem solutions is to off er fair solutions that satisfy the requirements of the various claimants, machine learning could enable the system to learn through satisfaction measures of previous distributions so as to improve the allocation and achieve near-optimal solutions.
Supervised learning, where the system learns a function that maps an input to an output based on an existing training set (i.e., sets of past inputs-outputs), is a class of machine learning whose application would be quite straightforward in sharing scenarios (in this case, a given allocation would be the input, while the user satisfaction or fairness indices could be potential outputs). Moreover, reinforcement learning is another area of machine learning that could be considered mainly for new applications of bankruptcy problems, where existing training sets do not yet exist. In such scenarios, the distributions of the commodities could be made so as to maximize some cumulative reward, such as the user satisfaction.
conclusIon This article has shed some light on the potential applications of the bankruptcy problem in mobile communications. Taking into account the similarities of the bankruptcy problem with the sharing of limited resources in emerging 5G wireless networks, we tried to identify new research lines that will foster 5G network sharing. To that end, fi rst we introduced the intuition behind the bankruptcy problem along with a set of proposed solutions. Then, we focused on the telecommunications domain, quoting a list of possible applications that includes a broad range of emerging techniques and concepts, from energy and spectrum sharing to UAVs and MEC use cases. Finally, we highlighted the challenges and open issues for the smooth application of the bankruptcy problem in wireless networks, including also possible advancements based on game theoretic and machine learning tools.