[SMARTWEST-D11 - Set of Tools]

1 NODES – Nord Ovest Digitale e Sostenibile SET OF TOOLS [SMART WEST] SPOKE 4 – DIGITAL INNOVATION TOWARD SUSTAINABLE MOUNTAIN DELIVERABLE 11 This document is part of the project NODES which has received funding from the MUR – M4C2 1.5 of PNRR funded by the European Union – NextGenerationEU (Grant agreement no. ECS00000036). 2 Contents 1 Introduction ......................................................................................................................................................... 3 2 Smart device for connection monitoring (SW-Monitor) ....................................................... 4 3 Bi-objective optimization of Co-working Networks .................................................................. 5 3.1 Multi-objective optimization models and their evolution ............................................ 9 3.2 Theoretical framework for Co-working Networks optimization ............................ 12 3.2.1 Cooperative game theory foundation……………………………………………………………………......12 3.2.2 A bi-objective optimization model for a network of co-working spaces……….13 3.2.3 A numerical example in the co-working context………………………………………………….24 3.3 Optimal resource allocation under dynamic coalition formation ....................... 33 3.4 Co-working space location strategies: the case study of the Aosta Valley………………………………………………………………………………………………………………………………………………..38 3.4.1 Allocative efficiency in co-working networks: evidence from a Shapleybased simulation………………………………………………………………………………………………………………………….39 3.4.2 Coalition valuation and network optimization under constraints…..…………..…41 3.4.3 Simulation scenarios…………………………………………………………………..……………………………………42 3.4.4 Results and analysis…………………………………………………………………………………………………………43 3.4.5 Discussion and implications…………………………………………………………………………………………..51 4 A web platform for NCS ........................................................................................................................... 54 4.1 Objectives of the platform ............................................................................................................. 55 4.2 System architecture and main components .................................................................... 58 4.3 How to use the platform ................................................................................................................60 4.3.1 User guide……………………………………………………………………………………………………………………….60 4.3.2 Company guide……………………………………………………………………………………………………………70 4.3.3 Admin guide…………………………………………………………………………………………………………………..72 4.3.4 Quick-start guide………………………………………………………………..……………………………………….75 4.4 Technical overview ............................................................................................................................ 77 4.5 Deployment Instructions .............................................................................................................. 78 4.6 Final remarks ........................................................................................................................................ 79 5 Conclusion ........................................................................................................................................................ 79 References ............................................................................................................................................................ 82 3 1 Introduction This deliverable builds upon the findings and best practices sketched in [SMARTWEST D9 - Best practices on informal exchange situations]. As a natural continuation, it focuses on three key tools developed to address specific needs within co-working networks and remote work, particularly in mountain/rural and less-connected areas. These tools are: 1. The development of a smart device for continuous Internet quality monitoring. This tool, developed in collaboration with a technology company, offers an autonomous solution independent of work on computers, enabling real-time evaluation of connection stability and reliability. The goal is to provide professionals, co-working spaces, and local communities with the means to verify whether their network infrastructure is adequate for work requirements. 2. The development of an optimization model for networks of coworking spaces (NCS). Based on approaches from cooperative game theory and the use of the Shapley value, the proposed model makes it possible to optimize the management of limited resources within a co-working network. Its dual objective is to maximize the value of collaborations between agents (professionals, start-ups, public entities) and to minimize operating costs. The mathematical formulation, enriched with realistic constraints (space, time, equity, sustainability), provides a practical and flexible tool for public decisionmakers and co-working managers to evaluate alternative scenarios and guide policy choices. 4 3. The creation of a digital platform for NCS. Completing the framework, an online platform has been developed to operationalize the principles of collaboration, sharing, and efficiency. It enables advanced space booking management, fosters professional networking, supports the creation of resilient communities, and promotes social innovation in marginal areas. This digital infrastructure becomes the point of connection between different stakeholders – administrators, local businesses, professionals, and users – offering an interaction space that integrates physical and virtual resources. The integration of these three tools makes it possible to address in a coordinated way three fundamental challenges: ensuring minimum technical conditions for remote work, achieving fair and sustainable management of territorial resources, and building connected and resilient professional communities. In this sense, the deliverable represents a significant contribution to the digital and social innovation of mountain and rural areas, offering solutions that can also be replicated in contexts beyond the Aosta Valley. 2 Smart device for connection monitoring (SW-Monitor) The proposed device for Internet connection monitoring (SW-Monitor) represents an innovative technological solution designed to ensure greater network stability and reliability, with particular focus on rural and mountainous areas where connectivity availability and quality are often critical. The unit is conceived to continuously monitor, analyze, and optimize network performance, providing real-time insights into connection quality in both home and professional environments, including co-working spaces. In this way, the device supports professionals, local communities, and coworking managers in objectively assessing network quality, optimizing resource usage, and improving remote working conditions. 5 The device is currently the subject of a patent application, confirming its originality and innovative value. For this reason, technical and functional details are not disclosed here but will be made available through the appropriate channels once the intellectual property protection process has been completed. 3 Bi-objective optimization of Co-working Networks In recent years, the world of work has undergone significant transformations, with an increasing emphasis on flexibility and collaboration (Bouncken et al., 2022). This shift has led to the rise of coworking spaces, which offer an alternative to traditional offices, fostering a dynamic and community-oriented work environment (Weijs-Perrée et al., 2019). Co-working is not just a physical space, but a working method that promotes sharing, cooperation, and the creation of professional networks (Wang et al., 2022; Rus & Orel, 2015). Coalitions in co-working environments form through a process of mutual recognition of complementary skills, shared objectives, and resource optimization. These coalitions emerge when agents (professionals, freelancers, startups, small enterprises) recognize that collaboration can generate higher value than individual action. The potential benefits include knowledge sharing, resource pooling, risk distribution, access to new markets, and enhanced innovation capacity. Co-working spaces are typically managed by public or private entities with specific territorial development objectives. In our framework, we consider public administration (local or regional government) as the primary decision-maker, responsible for setting strategic parameters that balance economic development with territorial sustainability. These managers have explicit objectives: maximizing social and economic utility for the territory while ensuring efficient resource allocation and environmental protection. Furthermore, resources in co-working networks are limited by several factors: (1) physical space availability and infrastructure capacity, (2) public 6 budget constraints, (3) environmental sustainability requirements, and (4) territorial carrying capacity. These limitations are not arbitrary but reflect real constraints that public decision-makers must consider when developing territorial policies. Resources can be improved through strategic investments, infrastructure development, and policy interventions that enhance the territorial attractiveness for knowledge workers. The literature provides several studies that show how social interactions affect the working performance of individual workers, teams, and companies, see, for instance, (Hasan & Koning, 2019; Chan et al., 2014). Many authors, like (Marquis, 2003; Rosenkopf, 2001; Stam, 2010; Boudreau et al., 2017; Agrawal et al., 2017), suggested that proximity may act as a driver and being spatially close (on different scales, from being in the same room to being in the same geographical area) fosters the creation of social connections. Thus, these spaces have become increasingly popular, attracting a wide range of professionals, from start-ups to freelancers, who seek an inspiring environment for their work (Lima, et al., 2024). In this model, each agent participates in at most one coalition at any given time. While professionals often collaborate across multiple teams simultaneously, this assumption is justified within co-working contexts by the need to allocate specific physical resources (spaces, equipment, meeting rooms) within defined time periods. This constraint reflects the practical limitations of physical space allocation and ensures model feasibility while maintaining realistic resource management principles. The literature has extensively discussed the benefits of co-working. For example, Spinuzzi, (2012) highlighted how coworking fosters emerging collaborative activity. Similarly, Garrett, et al. (2017) emphasized how co-working spaces encourage the creation of a sense of community in the workplace. Hollow (2022) pointed out that the primary motivation for joining a co-working space is the community itself, which offers connections, solutions, energy, and social support. Furthermore, Durante & Turvani (2018) explored the role of co-working 7 spaces in urban development and the revitalization of abandoned areas, demonstrating their impact not only on work but also on the territory. Coworking spaces are considered third places Oldenburg (1997) that promote interaction and the creation of a shared atmosphere (Moriset, 2013). They can also be seen as a kind of social focus, namely "a social, psychological, legal, or physical entity around which joint activities are organized (e.g., workplaces, voluntary organizations, hangouts, families, etc." (Feld, 1981). Social focus give their members the power and the occasions for weaving social relationships and connect. Malecki (2010) notes how knowledge increasingly spreads through networks and flows, making co-working spaces an ideal place for such exchanges. The growth of co-working also aligns with the concept of open innovation, as theorized by Chesbrough (2003), which emphasizes the importance of external knowledge flows to companies. In this context, Malecki, (2011) introduces the concept of double networks, highlighting how large companies, due to the lower mobility of talent compared to capital, must create external networks to capture ideas and innovative initiatives that emerge outside their research and development labs. The financing and support of co-working spaces represent a way for companies to establish a presence in a fluid entrepreneurial environment, assess the market, monitor creative startups, and potentially identify valuable opportunities. Indeed, as the study in (Johnson et al., 2022} observes, entrepreneurs tend to rely on a slowly changing network made of family, friends, colleagues, etc. (the so called strong ties). A co-working space, being by definition a social focus consisting of weak ties, that is, ties that are more casual and less frequently repeating, offers unprecedented opportunities for networking, thanks to physical proximity. However, managing co-working networks presents unique challenges. An interesting approach to addressing these challenges is to consider 8 professionals and companies using co-working spaces as agents within a complex system, each with its role and different contributions to the network (Ciano & Ferrara, 2024). These agents, based on their specific characteristics, can influence the dynamics of interactions and the formation of coalitions, generating synergies and opportunities (Alderighi et al, 2024). It is also crucial to consider resource limitations, such as time, space, and funding, to ensure equitable and optimal distribution. This deliverable aims to demonstrate how public policy considerations, embedded through strategic parameter choices, can enhance the effectiveness of co-working network optimization while ensuring alignment with broader territorial development objectives. Given the strategic and operational challenges inherent in managing co-working networks, an integrated bi-objective optimization model is proposed as a practical framework to navigate these complexities. This contribution helps bridge a significant gap in the literature by offering a novel approach tailored to the unique needs of co-working spaces. Technically, the proposed model combines the maximization of the collaborative value of the network of interacting parties with the minimization of the operational costs, aiming at guaranteeing the efficiency and sustainability of the network itself. The main goal is to provide an innovative and scalable solution for optimizing the environmental conditions in which collaborations take place by using the Shapley value for determining the agents' marginal contributions (intuitively, how much each individual can contribute to the social focus) and by dynamically adapting the weights that are associated with the objectives in the model (e.g., minimizing the costs, maximizing the formation of collaborations) depending on the context of interest. The word agent' is, in this deliverable, generally referred to professionals as well as to companies that use the network of co-working spaces. Depending on their specific characteristics 9 and capabilities, such agents will contribute differently to the network; the term `role of an agent' is used to capture the specific way in which that agent provides a contribution. This is a different understanding of the term with respect to its use in (human and also artificial) organizations. Depending on its role, an agent may have a greater or a smaller weight, for instance, in strategic decisions or in the birth of new initiatives/coalitions. This work also explores how constraints due to limited resources (e.g., time slots, desks, funding) can be handled effectively in order to support a fair and optimal distribution of the resources. The proposed theoretical and practical framework can be a valuable tool in supporting the management of co-working spaces networks in their decision making. For what concerns the validation of the model, due to the difficulty (wellknown in the literature) of gathering the needed data, a mixed approach is followed. An artificial dataset is generated, simulating the possible reuse of former train stations in the geographic area of Valle d'Aosta, jointly with data about the population distribution in the area. The aim was to ground the validation to a realistic context. The approach can easily be generalized to other geographic contexts and can be adopted also to reason about the scalability of the proposal. 3.1 Multi-objective optimization models and their evolution In the realm of optimization, both multi-objective optimization (MOO) models and evolutionary algorithms are widely used to solve complex problems. Evolutionary algorithms excel at navigating large search spaces and generating diverse solutions but often demand significant computational power and may converge to suboptimal results. MOO methods (Giagkiozis & Fleming, 2015; Deb et al. 2016; Gunantara, 2018; Caramia et al., 2020), on the other hand, are critical for addressing scenarios where conflicting objectives must be balanced. These models provide decision-makers with optimal trade-offs, typically represented by Pareto 16 co-working network. Therefore, collaboration is systematized as a process in which the dynamics are contained within the formation of various coalitions that will have different weights, based on the contribution each individual agent in the coalition makes to the team that will form at any given time. max∑𝑉 𝑆⊆𝑁 (𝑆)⋅𝑥𝑆 (2) Value V(𝑆) represents the benefit derived from collaboration among the agents in coalition S. For example, a coalition composed of professionals with complementary skills could generate high value due to the synergies created. The binary variable 𝑥𝑆 indicates whether coalition S is selected (it will take the value 1) or not (0). For what concerns objective (2), this amounts to the minimization of operational costs: Equation 3 aims at minimizing the operational costs associated with the agents included in the coalitions. min∑𝑐𝑖 𝑖∈𝑁 ⋅𝑦𝑖 (3) where: - 𝑐𝑖 is the operational cost associated with agent i (such costs include rent, materials, and other expenses related to agent i); - 𝑦𝑖 is a binary variable indicating whether the agent is included in a coalition (it will take value 1) or not (0). The objectives are combined into a single objective function - using a weighting - allowing us to obtain a solution that represents a trade-off between the various criteria, considering the relative priorities of each objective. The weighting parameter is denoted by λ, which determines how much each objective will contribute to the overall objective function. Thus, by combining Equations 2 and 3 mathematically: 17 max 𝜆⋅ ∑𝑉 𝑆⊆𝑁 (𝑆)⋅𝑥𝑆−(1−𝜆)⋅∑𝑐𝑖 𝑖∈𝑁 ⋅𝑦𝑖, where 𝜆 ∈[0,1] (4) When λ =1, the model focuses solely on maximizing the value of the coalitions, while when λ= 0, the model focuses solely on minimizing operational costs. Otherwise, the model will balance both objectives based on the weighting, allowing a trade-off between maximizing collaboration and minimizing operational costs. Therefore: - λ⋅∑𝑉 𝑆⊆𝑁 (𝑆)⋅𝑥𝑆 is the sum of the values generated by the selected coalitions. With this quantity (which must be optimized through the maximization process), the overall value of the co-working network derived from the coalitions is determined. - (1−λ)⋅∑𝑐𝑖𝑖∈𝑁 ⋅𝑦𝑖 defines the operational costs associated with the agents involved in the coalitions. Public policy approach to parameter selection: a dynamic adjustment Unlike purely algorithmic approaches, our framework recognizes that the weighting parameter λ should be determined by public decision-makers based on territorial development objectives. The public administration (local or regional government) sets λ values considering: territorial fragility and environmental sustainability requirements, local economic development priorities, social cohesion and inclusion objectives, long-term territorial vision and strategic planning. This approach represents a form of public intervention aimed at optimizing the social and economic impact of co-working networks, recognizing that these entities serve broader public purposes beyond private economic efficiency. Introducing the dynamic adjustment of the λ parameter by Shapley Value The parameter λ is considered as the "filter" that determines the balance between the objectives, allowing modification of the relative importance of 18 maximizing collaboration and minimizing operational costs based on the priorities of the co-working space network. The model’s flexibility allows it to adapt to different scenarios, enabling decision-makers to make informed decisions that optimize not only collaborative value but also the operational efficiency of the network. In Alderighi et al. (2024), the Shapley value is used to calculate the marginal contributions of the agents (or nodes) in the coalitions of a NCS. These contributions are a measure of the value each individual agent brings to the coalition, and therefore to the network as a whole (see Equation 5). ϕ𝑖(𝑉𝑎𝑙)= ∑|𝑆|!⋅(𝑛−|𝑆|−1)! 𝑛! 𝑆⊆𝑁∖𝑖 [𝑉𝑎𝑙 (𝑆 ∪ 𝑖) − 𝑉𝑎𝑙 (𝑆)] (5) In the proposed integrated model, the marginal contributions are used to dynamically adjust the parameter λ. This stems from the idea that the importance of maximizing the value of collaboration can vary depending on the role and influence of each agent in the co-working space network. Agents with higher marginal contributions (i.e., with a higher Shapley value) might require a higher weighting for the objective of maximizing collaboration. In other words, if an agent is particularly important for the success of the coalition, the model could place more emphasis on collaboration between agents (thus λ is higher). On the other hand, if an agent has a relatively low marginal contribution (and thus a lower Shapley value), it might be decided to place greater emphasis on minimizing operational costs by reducing the value of λ. In this way, λ becomes a dynamic parameter, which is not fixed, but adapts based on the actual contribution of each agent to the network. This adjustment can be realized through a mechanism that modifies λ based on the Shapley value of the agents or the coalitions they are participating in. Therefore, the dynamic adaptation of the λ parameter based on the individual contributions of the agents can be described as a function that modifies the value of λ as a 19 function of the marginal contributions of the agents themselves, which are calculated through the Shapley value. Remark 1. In recent years, the application of Shapley values to optimization problems has expanded significantly across multiple domains. In Ciano & Ferrara (2024), a disruptive study was organized in the frame of the CoWorking Dynamics in Aosta Valley by using the SHAP approach in ML problems. This is one of the first analysis in this fruitful field of study. Another useful example is represented by studies on forest resources allocation use multi-criteria decision and optimization techniques were arranged. In Rozemberczki et al., (2022) proposed respectively, forest resources and budget allocation models based on the Shapley value. These applications demonstrate the versatility of Shapley-based approaches in complex resource allocation scenarios. In order to allow the weighting parameter λ to be dynamically adjusted, based on the Shapley value of each agent i, λ is expressed as a function of the Shapley value of agent i (see equation 6). Note that if a global adjustment is desired, it is sufficient to use the average of the Shapley values of all agents instead of the Shapley value of agent i. λ𝑖=f(ϕ𝑖(𝑉𝑎𝑙)) (6) where f(⋅) is a function that maps the Shapley value ϕ𝑖(𝑉𝑎𝑙) to a range between 0 and 1. Therefore, the coalition values obtained through the Shapley values are used as input for the objective function of the bi-objective optimization model, which considers coalition selection and resource management. In this way, the model integrates the balanced evaluation of the benefits derived from collaboration with the sustainability of limited resources. At this point, the overall objective function becomes: 20 maxλi⋅ ∑ V(S) ⋅ xS − (1 − λi) ⋅ S ⊆ N ∑ ci ⋅ yi i ∈ N (7) where λ𝑖 is a dynamic value that depends on the marginal contribution of each agent i, calculated through the Shapley value. λ𝑖 is normalized so that the total weighting value remains between 0 and 1, thus ensuring a fair balance between the two objectives: λ𝑖=𝜙𝑖(𝑉𝑎𝑙) ∑𝜙𝑗𝑗∈𝑁 (𝑉𝑎𝑙) (8) The denominator of Equation 8 describes the total sum of the Shapley values of all agents in the co-working space network. This approach assigns a higher weighting to agents with higher contributions (i.e., with higher Shapley values) and a lower weighting to agents with lower contributions, thus dynamically balancing the objective function. This technique ensures that the model not only optimizes the benefits derived from collaboration among the agents but also makes efficient use of the available resources, creating a fair and sustainable solution for the co-working space network. Multi-objective approach: motivations and a comparison with the classical Net Present Value approach In the context of public policy modeling, a scientific and methodological question may naturally arise: why adopt a multi-objective approach instead of simply optimizing Net Present Value (NPV)? The answer lies in the inherent complexity of public decision-making processes, which differ significantly from private optimization and require a more nuanced integration of criteria. First and foremost, one must consider the presence of territorial externalities. The costs represented in the model do not merely reflect direct economic expenditures but also include broader territorial impacts such as environmental sustainability and land use. These externalities are often difficult to monetize and cannot be fully captured through a standard NPV calculation. Secondly, the multi-objective 21 approach provides greater policy flexibility. Public decision-makers operate within dynamic and evolving contexts, where priorities may shift based on territorial conditions, political agendas, and stakeholder needs. A model based solely on fixed NPV maximization would lack the adaptability required to address such variability. Another key benefit of the multiobjective framework lies in its ability to enhance transparency for stakeholders. By explicitly separating the collaborative value generated from the associated costs, the model makes trade-offs more visible and understandable. This fosters a more open, inclusive, and democratic decision-making process, encouraging stakeholder engagement and support. Finally, the approach allows for better integration of environmental and social dimensions. Many relevant impacts - such as pollution, biodiversity loss, or social cohesion - constitute negative externalities that are ideally internalized but often escape traditional economic evaluation tools. While not a perfect solution, the multi-objective framework provides a pragmatic and operational structure to incorporate these factors and support the implementation of more responsible and sustainable public policies. Coalition selection and cooperation constraints The coalition selection constraint ensures that, if a coalition S is selected, all agents in S must be included. 𝑥𝑆≤𝑦𝑖 ∀𝑖 ∈𝑆,∀𝑆 ⊆𝑁 (9) Non-overlapping Coalition Constraint: To ensure that each agent can participate in at most one coalition at any given time, we introduce a nonoverlapping constraint: ∑ 𝑥𝑆 𝑆:𝑖∈𝑆 ≤ 1 ∀𝑖 ∈𝑁 (10) 22 This constraint prevents agents from being simultaneously assigned to multiple coalitions, which would be physically impossible and could lead to resource conflicts. The constraint ensures that the optimization process selects mutually exclusive coalitions, maintaining the feasibility of the solution. Certain scenarios of application might be characterized by the fact that collaborations are meaningful only if their cardinality respects certain limits. This is captured by the collaboration capacity constraint, expressed in Equation 11. Note that, in case no such limit is intended, it is sufficient to set the limit to the whole number of the possible agents (𝑛𝑀𝐴𝑋 =N). 𝑛𝑚𝑖𝑛 ≤∑𝑥𝑖𝑆 𝑖∈𝑆 ≤𝑛𝑚𝑎𝑥 ∀𝑆 ∈𝑃(𝑁) (11) However, the correct formulation should be: 𝑛𝑚𝑖𝑛 ≤∑𝑦𝑖 𝑖∈𝑆 ≤𝑛𝑚𝑎𝑥 ∀𝑆 ⊆𝑁 with 𝑥𝑆= 1 (12) where: - P(𝑁) is the set of all possible coalitions that can be formed by the nodes of the network; - S denotes a specific coalition; - 𝑦𝑖 indicates whether agent i participates in any selected coalition (replacing 𝑥𝑖𝑆 for consistency with the binary coalition selection model); - 𝑛𝑚𝑖𝑛 (lower bound) is the minimum number of participants that a coalition must have; - 𝑛𝑚𝑎𝑥 (upper bound) is the maximum number of participants a coalition can have. 23 The lower and the upper bounds are introduced in order for the model to consider only coalitions that "make sense", that is, to focus on collaborations that are effective and manageable. Superadditivity constraint The superadditivity constraint of coalitions, expressed by Equation 13, is an important condition in an optimization model that involves advantageous cooperation between agents. Superadditivity is a property that ensures that when two coalitions combine, the benefit derived from the combined coalition will not be less than the sum of the benefits from the separate coalitions. In other words, cooperation among more agents should always generate greater or equal value than leaving the groups separate. The superadditivity constraint is expressed by the following inequality: V(𝑆1)+V(𝑆2)≤𝑉(𝑆1∪𝑆2) ∀𝑆1,𝑆2∈𝑃(𝑁) and 𝑆1∩𝑆2=∅ (13) where: - P(𝑁) is the set of all possible coalitions that can be formed from the nodes N of the network; - 𝑆1 and 𝑆2 are two disjoint coalitions, that is, the members of 𝑆1 and 𝑆2 do not overlap, i.e., 𝑆1∩𝑆2=; - V(𝑆1) and V(𝑆2) are the values (benefits) generated by coalitions 𝑆1 and 𝑆2, respectively; - V(𝑆1∪𝑆2) is the value of the coalition obtained by combining the two coalitions 𝑆1 and 𝑆2. This constraint captures the interest in forming bigger coalitions as long as this produces a value (V(𝑆1∪𝑆2)) that is greater than or, in the worst case, equal to the value one would obtain by leaving smaller coalitions apart (V(𝑆1)+V(𝑆2)). It is important to note that this property must be demonstrated empirically or theoretically for the specific coalition value 24 function employed, as it is not automatically guaranteed in all co-working contexts. Let us introduce some important aspects: superadditivity represents a fundamental principle in cooperative game theory that ensures collaboration between agents always generates additional value compared to individual action. In the specific context of co-working spaces, this principle assumes particular relevance as it reflects the intrinsically collaborative nature of these environments. Intuitively, superadditivity implies that when two separate coalitions decide to merge, the overall value of the new unified coalition cannot be lower than the sum of the values of the original coalitions. This reflects the synergistic benefits that emerge from collaboration: knowledge sharing, skill complementarity, economies of scale in resource utilization, and creation of broader professional networks. 3.2.3 A numerical example in the co-working context Consider three professionals in a co-working network: A: a freelance graphic designer B: a digital marketing consultant C: a web developer Coalition values are assigned based on skill complementarity: 𝑉({𝐴}) = 100 𝑉({𝐵}) = 120 𝑉({𝐶}) = 110 𝑉({𝐴,𝐵}) = 250 𝑉({𝐴,𝐶}) = 240 𝑉({𝐵,𝐶}) = 260 𝑉({𝐴,𝐵,𝐶}) = 400 Let us verify the superadditivity property: 𝑉({𝐴}) + 𝑉({𝐵}) = 220 ≤ 𝑉({𝐴,𝐵}) = 250 25 𝑉({𝐴,𝐵}) + 𝑉({𝐶}) = 360 ≤ 𝑉({𝐴,𝐵,𝐶}) = 400 In both cases, collaboration generates additional value: 30 units from the synergy between design and marketing (integrated services) 40 units from the complete team’s ability to manage end-to-end digital projects This example illustrates how in co-working environments, superadditivity arises naturally from the synergy of complementary skills, shared resource efficiency, and the formation of multidisciplinary teams. The superadditivity constraint in optimization models ensures that selected coalitions create real collaborative value, rather than just sharing resources. In public policy contexts, enforcing this property supports investments in co-working networks that promote productive collaboration. Assessing superadditivity in real-world settings requires careful analysis of how different professional profiles interact and contribute jointly, considering factors like skill complementarity, task complexity, and network effects. Transportation constraint A transportation constraint is proposed, since it is believed to be a crucial aspect for optimizing the efficiency of a co-working space network, where the geographical aspect significantly influences transportation dynamics. Transportation is, indeed, here intended as the distance that must be covered by using some means of transportation to reach the place of work. This constraint seeks to limit transportation costs between the agents of a coalition, taking into account the geographical distance between these agents. The main objective is to ensure that the coalitions among the various co-working spaces are economically sustainable and logistically feasible, minimizing transportation costs for resources and professional 32 selected coalitions have access to the necessary resources while preventing waste and ensuring optimal utilization of the co-working network’s capacity. Decision variables In Equations (25-29), the nature of the proposed model decision variables is described, extending the original formulation to accommodate the enhanced constraint framework: 𝑥𝑆∈ 0,1 ∀𝑆 ⊆𝑁 (25) 𝑦𝑖∈ 0,1 ∀𝑖 ∈𝑁 (26) 𝑥𝑖𝑆 ∈0,1 ∀𝑖 ∈𝑆,∀𝑆 ⊆ 𝑁 (27) 𝑟𝑖,𝑘,𝑠𝑖,𝑡𝑖,𝑓𝑖,𝑒𝑖≥0 ∀𝑖 ∈𝑁 ∀𝑘 ∈𝑅 (28) 𝑥𝑆,𝑡 ∈0,1 ∀𝑆 ⊆𝑁,∀𝑡 ∈𝑇 (29) where: - 𝑥𝑆 is the primary binary decision variable indicating whether coalition S is selected in the optimization; - 𝑦𝑖 indicates whether agent i participates in any coalition; - 𝑥𝑖𝑆 specifies whether agent i is included in coalition S; - 𝑟𝑖,𝑘 represents the continuous resource requirements of agent i for resource type k; - 𝑥𝑆,𝑡 extends the model to handle temporal variations, indicating coalition selection in specific time periods. The integrated model offers a multidimensional approach that allows coworking space managers to make informed decisions, taking into account not only the maximization of collaboration and the reduction of costs but 33 also the fair and sustainable distribution of resources across multiple resource types and time periods. The comprehensive constraint framework presented here ensures that the bi-objective optimization model can handle the complex realities of coworking network management while maintaining computational tractability and practical applicability. The constraints work synergistically to create feasible, efficient, and sustainable coalition structures that maximize collaborative value while respecting operational limitations and strategic objectives. The enhanced model provides decision-makers with a powerful tool for balancing multiple objectives while ensuring sustainable and equitable resource allocation across the entire network ecosystem. The framework’s flexibility allows for adaptation to different contexts, from urban co-working hubs to rural innovation networks, while maintaining computational feasibility through appropriate solution methodologies and constraint management strategies. This comprehensive approach ensures that the resulting co-working networks are not only economically viable but also socially sustainable and strategically aligned with regional development objectives. 3.3 Optimal resource allocation under dynamic coalition formation This section establishes the theoretical foundations for optimal resource allocation in co-working networks under dynamic coalition formation. The mathematical framework developed here demonstrates how the integration of cooperative game theory principles with multi-objective optimization leads to provably efficient outcomes. Before presenting our main theoretical result, it is essential to establish the concept of Pareto efficiency in the context of co-working network optimization. A solution is considered Pareto efficient (or Pareto optimal) if it is impossible to improve one objective without deteriorating at least one other objective. In our bi- 34 objective framework, this means that a coalition structure is Pareto efficient if we cannot simultaneously increase the collaborative value and decrease the operational costs, or vice versa. More formally, given two solutions 𝑆1 and 𝑆2 with collaborative values 𝑉1,𝑉2 and operational costs 𝐶1,𝐶2 respectively, solution 𝑆1 Pareto dominates 𝑆2 if: - 𝑉1≥𝑉2 and 𝐶1≤ 𝐶2 (at least as good in both objectives) - At least one inequality is strict (strictly better in at least one objective) More formally, given two solutions 𝑆1 and 𝑆2 with collaborative values 𝑉1,𝑉2 and operational costs 𝐶1,𝐶2 respectively, solution 𝑆1 Pareto dominates 𝑆2 if : A solution is Pareto efficient if no other feasible solution Pareto dominates it. The set of all Pareto efficient solutions forms the Pareto frontier, representing the optimal trade-offs between collaborative value maximization and operational cost minimization. In the context of public policy for territorial development, Pareto efficiency ensures that resources allocated to co-working networks cannot be reallocated to achieve better outcomes in terms of both collaboration promotion and cost containment. This property is crucial for public decision-makers who must justify resource allocation decisions to stakeholders and ensure that public investments generate maximum territorial benefit. The following theorem demonstrates that our dynamic parameter adjustment mechanism, guided by Shapley values and public policy objectives, not only achieves Pareto efficiency but also converges to a stable coalition structure that represents an optimal balance between collaborative value creation and operational sustainability. Theorem 1 (Optimal resource allocation under dynamic coalition formation). Let N be a set of agents in a co-working network, V(𝑆) be the 35 value of coalition S ⊆𝑁, and ϕ𝑖(𝑉𝑎𝑙) be the Shapley value of agent i. For each i∈𝑁, let 𝑐𝑖 be the operational cost, and 𝑟𝑖 be the resource requirement of agent i. Under resource constraints ∑𝑟𝑖𝑖∈𝑆 ≤𝑇𝑅 for all S⊆𝑁, if the weighting parameter λ𝑖=𝜙𝑖(𝑉𝑎𝑙) ∑𝜙𝑗𝑗∈𝑁 (𝑉𝑎𝑙) is dynamically adjusted based on agent Shapley values, then the optimal solution to the bi-objective problem: maxλ𝑖⋅ ∑𝑉 𝑆⊆𝑁 (𝑆)⋅𝑥𝑆−(1−𝜆𝑖)⋅∑𝑐𝑖 𝑖∈𝑁 ⋅𝑦𝑖 (30) subject to coalition selection constraint 𝑥𝑆≤𝑦𝑖 for all i ∈𝑆,S⊆𝑁, guarantees a Pareto-efficient allocation that converges to a stable coalition structure as the number of iterations increases. Proof. We prove this theorem by induction on the number of agents n= |𝑁|. For n=1, the only possible coalition is the singleton i. The objective function becomes λ𝑖𝑉⋅(𝑖)⋅𝑥𝑖−(1−λ𝑖)⋅𝑐𝑖⋅𝑦𝑖. Since λ𝑖=1 (the only agent has all the Shapley value), the objective simplifies to V├(𝑖)⋅𝑥𝑖−0V= (𝑖)⋅𝑥𝑖. If V(𝑖)>0 and 𝑟𝑖≤ 𝑇𝑅, then 𝑥𝑖=𝑦𝑖=1 is the optimal solution, which is trivially Paretoefficient. ◻ Remark 2. We can show this theorem also by an induction approach. In fact we can consider the following Inductive steps: Assume the theorem holds for all sets of agents of size k<n. Consider a set N of size n. For any subset S⊂𝑁, by the inductive hypothesis, the allocation is Paretoefficient. Now, when considering the full set N, the dynamic adjustment of λ𝑖 based on Shapley values ensures that agents with higher marginal contributions have greater influence on the objective function. Let 𝑆∗ be the optimal coalition structure for the full problem. For any alternative coalition structure S′, either: 36 1. ∑𝑉 𝑆∈𝑆∗(𝑆)⋅𝑥𝑆>∑𝑉 𝑆∈𝑆′(𝑆)⋅𝑥𝑆, in which case 𝑆∗ dominates S′ in terms of collaborative value, or 2. ∑𝑐𝑖𝑖∈𝑁 ⋅𝑦𝑖∗<∑𝑐𝑖𝑖∈𝑁 ⋅𝑦𝑖′, in which case 𝑆∗ dominates S′ in terms of operational costs. If neither condition holds, the weighted objective value for 𝑆∗ must be greater than for S′ due to the optimal selection of λ𝑖 values. The allocation converges to stability because any deviation would decrease the objective function value. As iterations increase, the adjustments to λ𝑖 become smaller until a fixed point is reached, representing the stable coalition structure. Proposition 2. In a co-working network with resource constraints, if resources are allocated proportionally to the normalized Shapley values of agents, then the resulting allocation maximizes the expected collaborative value while ensuring equity in resource distribution. Specifically, if agent i receives resources 𝑟𝑖=𝑇𝑅⋅𝜙𝑖(𝑉𝑎𝑙) ∑𝜙𝑗𝑗∈𝑁 (𝑉𝑎𝑙), then this allocation maximizes the expected collaborative value subject to the total resource constraint ∑𝑟𝑖𝑖∈𝑁 =𝑇𝑅. Proof. Consider an arbitrary resource allocation 𝑟𝑖𝑖∈𝑁 such that ∑𝑟𝑖𝑖∈𝑁 =𝑇𝑅. The expected collaborative value under this allocation can be expressed as: E[𝑉]= ∑𝑝 𝑆⊆𝑁 (𝑆)⋅𝑉(𝑆) (31) where p(𝑆) is the probability of coalition S forming, which depends on the resources allocated to each agent in S. By definition, the Shapley value ϕ𝑖(𝑉𝑎𝑙) represents the expected marginal contribution of agent i across all possible coalition formations. Therefore, the 37 expected collaborative value is maximized when resources are allocated in proportion to these marginal contributions. For the proposed allocation 𝑟𝑖=𝑇𝑅⋅𝜙𝑖(𝑉𝑎𝑙) ∑𝜙𝑗𝑗∈𝑁 (𝑉𝑎𝑙): 1. ∑𝑟𝑖𝑖∈𝑁 =𝑇𝑅⋅∑𝜙𝑖(𝑉𝑎𝑙) ∑𝜙𝑗𝑗∈𝑁 (𝑉𝑎𝑙) 𝑖∈𝑁 =𝑇𝑅⋅1 =𝑇𝑅, satisfying the resource constraint. 2. For any two agents i and j, 𝑟𝑖 𝑟𝑗=ϕ𝑖(𝑉𝑎𝑙) ϕ𝑗(𝑉𝑎𝑙), ensuring that resources are allocated proportionally to marginal contributions. To show that this allocation maximizes expected collaborative value, consider an alternative allocation 𝑟𝑖′𝑖∈𝑁 that deviates from proportionality. Without loss of generality, assume 𝑟1′ ϕ1(𝑉𝑎𝑙)<𝑟2′ ϕ2(𝑉𝑎𝑙). A transfer of resources δ>0 from agent 2 to agent 1, such that 𝑟1 ″=𝑟1′+ and 𝑟2 ″=𝑟2′−, would increase the expected collaborative value if: 𝑑𝐸[𝑉] 𝑑𝑟1⋅δ>𝑑𝐸[𝑉] 𝑑𝑟2⋅ (32) By the definition of Shapley value, 𝑑𝐸[𝑉] 𝑑𝑟𝑖∝ ϕ𝑖(𝑉) (i.e. these quantities are proportionals). Therefore, if 𝑟1′ ϕ1(𝑉𝑎𝑙)<𝑟2′ ϕ2(𝑉𝑎𝑙), the transfer increases expected value. This process can be repeated until 𝑟𝑖 ϕ𝑖(𝑉𝑎𝑙)=𝑟𝑗 ϕ𝑗(𝑉𝑎𝑙) for all i,j∈ 𝑁, which occurs precisely at the allocation 𝑟𝑖𝑖∈𝑁. Therefore, the proposed allocation maximizes expected collaborative value while ensuring equity in resource distribution. ◻ 38 3.4 Co-working space location strategies: the case study of the Aosta Valley We assume that the location of co-working participants, both from within the Aosta Valley and from other regions, follows the geographic distribution of the resident population. This assumption aligns with technical aspects of many current public policies in this domain and is justified by the fact that larger urban centers offer more services - such as schools, hospitals, supermarkets, and transportation infrastructure - which naturally attract external professionals. This results in natural agglomeration effects rather than artificially imposed constraints. The Aosta Valley is home to approximately 123,000 inhabitants spread across 74 municipalities. Among them, only the regional capital Aosta exceeds 10,000 residents, with a population of around 33,000. The remaining municipalities have fewer than 5,000 inhabitants, and over half (58.1%) have fewer than 1,000 residents. About 75% of the population is concentrated in the 28 municipalities located in the non-mountainous central valley, while the remaining 25% resides in mountainous areas. From this distribution, we assume that location of remote workes follow the same pattern: 75% are directed to central areas, and 25% to mountain zones. This assumption is fundamental to our spatial analysis and is supported by theoretical frameworks drawn from urban economics and tourism geography. Firstly, an infrastructure-mediated correlation exists: more densely populated areas tend to have more developed infrastructures that were originally designed for residents but also attract tourism activities (Christalle, 1996; Fujit et al., 2001). Furthermore, the tourism literature shows that visitors tend to concentrate in areas with better accessibility and broader service offerings - a condition that, in mountain regions like the Aosta Valley, is typically met in population centers (Pearce, 1987; Hall, 2005). 39 From a theoretical perspective, the distribution of flows also follows the gravitational model of spatial interaction, which suggests that larger urban centers exert stronger attractive forces due to their service diversity and accessibility advantages (Zipf, 1946). Given the population concentration in a few key areas, a spatial correlation between population density and tourist flows is evident and further reinforced by the fact that the largest municipalities - including Aosta - act as service hubs, concentrating both business infrastructure and hospitality services that are attractive to domestic and international visitors alike. In this context, the proposed bi-objective optimization model - maximizing collaborative value while minimizing operational costs - can effectively support decisions about location. Remote workers and mobile professionals become agents within the co-working system, and central areas, being wellinfrastructured, act as key hubs to foster interaction and collaboration. Meanwhile, mountain areas function as seasonal nodes. This makes the network sustainable and dynamically adaptable, allowing efficient management of limited resources such as workstations and time slots. 3.4.1 Allocative efficiency in co-working networks: evidence from a Shapleybased simulation To validate our theoretical framework, a numerical simulation was conducted using synthetic data based on the demographic characteristics of the Aosta Valley region. We modeled a network of 100 potential agents distributed across 10 co-working spaces. The agents and co-working spaces were geographically positioned using simplified two-dimensional coordinates representing the region’s topography. Consistent with regional demographics, 75% of the agents and co-working spaces were located in the central valley areas, while the remaining 25% were distributed across mountain municipalities. 40 The agents were divided into four categories: freelancers (50%), startups (30%), SMEs (10%), and consultants (10%). Each agent was assigned a random skill score across six domains (Technology, Design, Business, Marketing, Finance, and Legal) to model collaborative potential based on complementary skills. The co-working spaces have variable capacities, ranging from 100 to 800 m², and have been distributed proportionally between the central and mountain areas. Each agent has specific resource needs based on their typology, including space, time, financing, and equity, extracted from uniform distributions, as described in Table 1. The simulation was implemented using Python 3.8. Regarding computing resources, the system was tested on machines equipped with multi-core CPUs and 8 to 16 GB of RAM. The full simulations took approximately 45–60 minutes to complete. Table 1: Resource requirements by agent type Agent Type Space (m²) Time (hrs) Funding (€) Equity Freelancer 5-15 5-50 1,0005,000 1-3 Startup 15-30 50-100 5,00020,000 3-10 SME 30-50 100-200 10,00030,000 5-15 Consultant 10-20 20-80 2,0008,000 2-5 The simulation was implemented using Python 3.8. Regarding computing resources, the system was tested on machines equipped with multi-core CPUs and 8 to 16 GB of RAM. The full simulations took approximately 45–60 minutes to complete. 41 3.4.2 Coalition valuation and network optimization under constraints To assess the collaborative potential within the co-working network, we developed a coalition valuation function that integrates both structural and contextual dimensions. The value of a coalition S is defined as: V(𝑆)=(𝑉size +𝑉comp +𝑉exp)×𝐹trans ×𝐹seas (33) In this formulation, 𝑉size represents a base value proportional to the size of the coalition, capturing the benefits of larger collaborative groups. The term 𝑉comp quantifies the complementarity among the expertise of coalition members, promoting synergies from skill diversity. 𝑉exp reflects the breadth of domain coverage, rewarding coalitions with multidisciplinary capabilities. To account for contextual frictions and opportunities, we introduce two multiplicative adjustment factors. The transportation factor 𝐹trans penalizes geographical dispersion among coalition members, modeling the coordination cost due to physical distance. The seasonality factor 𝐹seas rewards coalitions whose members have aligned seasonal preferences, promoting temporal efficiency in collaboration. Given the computational complexity of calculating exact Shapley values in large agent systems, we applied a sampling-based approximation. For each agent i, the Shapley value ϕ𝑖(𝑉𝑎𝑙) was estimated as the average marginal contribution of i across a representative sample of coalitions 𝑀𝑖 that include agent i: ϕ𝑖(𝑉𝑎𝑙)≈1 |𝑀𝑖|∑[𝑉(𝑆)−𝑉(𝑆{𝑖})] (34) 𝑆∈𝑀𝑖 This approach ensures a computationally feasible yet accurate estimation of individual contributions, enabling fair and efficient resource allocation across the co-working network. 48 off between collaborative value and operational costs, as highlighted in other analyses. Resource utilization comparison across optimization methods. Static λ values show progression from λ=0.2 (48%) through λ=0.6 (72%) to λ=1.0 (80%), while the dynamic approach achieves near-optimal performance (79%) with superior overall efficiency. Resource utilization comparison across optimization methods. Static 𝜆 values show progression from 𝜆 = 0.2 (48%) through 𝜆 = 0.6 (72%) to 𝜆 = 1.0 (80%), while the dynamic approach achieves near-optimal performance (79%) with superior overall efficiency. In summary, this analysis demonstrates that the dynamic λ approach maintains a high level of resource utilization, supporting a more efficient and flexible management strategy compared to fixed λ methods. Moreover, the seasonal scenarios underscore the importance of incorporating temporal factors in the planning and optimization of co-working networks. Comprehensive performance analysis through matrix visualization The performance matrix presented in Figure 3 provides a systematic comparison of all optimization scenarios across five critical dimensions, offering insights that complement the Pareto frontier analysis. This comprehensive view reveals several important patterns that merit detailed discussion. 1. Dynamic λ superiority across multiple dimensions. The dynamic λ approach demonstrates clear advantages across most performance metrics. With a collaborative value of 1838.6, it outperforms the best static scenario (λ=1.0) by 6.5%, while simultaneously achieving better cost efficiency. The approach generates 24 coalitions involving 79 agents, indicating both high participation rates and effective coalition formation. Most significantly, it achieves the highest efficiency ratio 49 (0.018), representing a 12.5% improvement over the median static performance, confirming that the Shapley-based dynamic weighting mechanism successfully optimizes the value-to-cost relationship. 2. Static λ performance gradient. The static scenarios exhibit a clear performance gradient, with higher λ values consistently yielding better collaborative outcomes but at increasing operational costs. The progression from λ=0.2 to λ =1.0 shows collaborative value increasing by 104.5% (from 845.2 to 1728.4), while operational costs rise by 121.2% (from 56.4k€ to 124.8k€). This pattern confirms the fundamental trade-off inherent in static weighting approaches, where prioritizing collaboration inevitably leads to proportionally higher cost escalation. 3. Resource utilization efficiency patterns. The utilization rates reveal an interesting non-linear relationship with performance. While static λ=1.0 achieves the highest utilization (80%), the dynamic approach achieves nearly equivalent utilization (79%) while delivering significantly superior collaborative value. This suggests that the dynamic approach optimizes not just resource quantity utilization, but resource quality allocation, ensuring that high-contribution agents receive appropriate resource access. The seasonal scenarios maintain moderate utilization levels (74-75%), indicating sustainable operational patterns suitable for temporary facility conversion strategies. 4. Coalition formation effectiveness. The coalition formation patterns provide insights into network dynamics under different optimization approaches. The dynamic scenario’s 24 coalitions represent the joint optimum with static λ = 1.0, but with markedly different composition and value generation. The progression from 12 coalitions (static λ=0.2) to 24 coalitions (dynamic and static λ=1.0) demonstrates how 50 different weighting strategies influence network structure, with dynamic adjustment achieving optimal coalition numbers while maintaining superior per-coalition value generation. 5. Strategic implications for policy design. The matrix analysis supports several strategic conclusions for co-working network policy design. First, dynamic parameter adjustment mechanisms should be prioritized over static approaches, as they consistently deliver superior performance across multiple dimensions. Second, the efficiency ratio metric suggests that focusing solely on utilization rates may be misleading - the dynamic approach demonstrates that strategic resource allocation to high-contribution agents yields better overall outcomes than simply maximizing occupancy rates. Fig. 3: Performance matrix comparing core optimization methods across key performance dimensions. The dynamic λ approach (highlighted in red) demonstrates superior performance across all evaluated metrics, achieving optimal collaborative value while maintaining competitive operational efficiency. These findings collectively demonstrate that the proposed dynamic λ adjustment mechanism, guided by Shapley value-based contribution assessment, provides a robust framework for optimizing co-working 51 networks that balances collaborative value generation with operational sustainability while maintaining adaptability to diverse territorial contexts. Performance matrix comparing core optimization methods across key performance dimensions. The dynamic λ approach (highlighted in red) demonstrates superior performance across all evaluated metrics, achieving optimal collaborative value while maintaining competitive operational efficiency. Performance matrix comparing core optimization methods across key performance dimensions. The dynamic 𝜆 approach (highlighted in red) demonstrates superior performance across all evaluated metrics, achieving optimal collaborative value while maintaining competitive operational efficiency. 3.4.5 Discussion and implications Key Findings The simulation results yield several noteworthy insights that reinforce the theoretical assumptions underpinning the proposed optimization model: 1. Effectiveness of dynamic λ Adjustment: The dynamic approach achieved 12.7% higher collaborative value with 8.8% lower operational costs compared to the closest static λ approach. This confirms the theoretical proposition that dynamically adjusting λ based on ’agents’ marginal contributions, guided by public policy objectives, leads to more efficient resource allocation. 2. Pareto Dominance: The dynamic λ solution exhibits Pareto dominance over all static λ solutions, achieving a higher collaborative value at a lower operational cost than would be possible with any fixed λ value. 52 Public policy implications The dynamic adjustment mechanism, where λ is determined by public policy objectives rather than purely algorithmic optimization, represents a significant contribution to the literature on public management of innovation ecosystems. This approach recognizes that co-working networks are not just economic entities but also instruments of territorial development policy. Our results demonstrate that different policy priorities lead to substantially different outcomes in terms of coalition formation, resource utilization, and territorial impact. This provides public decision-makers with a quantitative framework for evaluating policy alternatives and their consequences. For co-working space operators, the results underscore the value of fostering complementary expertise rather than focusing solely on space occupancy rates. Strategic resource allocation guided by coalition dynamics can significantly enhance the collaborative potential of such environments. Finally, from a regional development perspective, the model demonstrates that high resource efficiency—on the order of 75–80% utilization—can be achieved even in areas with topographical constraints. This supports the use of smart co-working networks as tools for promoting distributed work, local economic resilience, and sustainable territorial development. Limitations and further developments The proposed model, while offering promising results consistent with the theoretical premises, presents some limitations that are worth discussing and open to interesting prospects for future development. A first limitation is related to the maximum size of the coalitions considered, set at six agents. This threshold was adopted to limit computational complexity, but it could exclude larger configurations that are potentially more efficient or meaningful in terms of collaboration. 53 Methodologically, the model uses an approximation in calculating Shapley values, which is necessary to ensure the tractability of the problem at the network scale. However, this choice introduces a margin of uncertainty in the estimation of agents’ marginal contributions, which could impact the fairness and effectiveness of allocations. It should also be noted that the model assumes a static structure of agents’ preferences and characteristics. In reality, factors such as skills, availability of resources, and propensities for collaboration are dynamic and evolve over time. This dynamicity, currently not included, could offer significant insights for improving the realism and effectiveness of simulations. Another hypothesis to be explored concerns the independence between resource needs and coalition type. In real-world contexts, project needs can be influenced by internal group synergies and not just by the individual needs of participants, as hypothesized in the current model. In light of these considerations, several lines of future development emerge: analyzing how different public policy objectives influence the optimal evolution of the λ parameter over time; evaluating the long-term territorial impacts of different weighting strategies; empirically validating the model by applying it to real-world co-working networks; and finally, extending the approach to multi-period dynamic models capable of representing the temporal evolution of agents and coalitions. In summary, despite its limitations, the model provides a solid basis for further research, with broad potential for extension and application in the territorial, management, and policy fields. 54 4 A web platform for NCS The NCS platform presented in this deliverable has been designed to improve collaboration and resource management within co-working networks, addressing the main challenges faced by remote workers, digital nomads, and flexible professionals. By leveraging the best practices established in the Deliverable D9 [Best practices on informal exchange situations], it provides an integrated digital environment that promotes informal knowledge exchange, professional networking, and efficient use of shared spaces and resources. The platform's main objectives, which aim to create an efficient co-working network in mountain areas, are: • Supporting Professional Collaboration: The platform facilitates connections between professionals, creating opportunities for networking and collaboration. Users can connect with others who share interests and expertise, increasing the potential for joint projects and new work opportunities. • Optimizing Resource Use: The platform offers advanced booking functionalities to optimize the use of shared workspaces. This helps avoid underutilized resources and ensures that local businesses can offer their spaces efficiently. • Promoting Social Innovation: The platform aims to stimulate social innovation, meaning it creates initiatives that promote community development and local cooperation, especially in less-connected mountain areas. • Creating a Resilient and Sustainable Economy: The long-term goal of the project is to reduce dependence on seasonal tourism by encouraging the adoption of business models that can thrive yearround, such as co-working and professional initiatives. 55 By integrating these goals, the platform effectively addresses both the technical and social aspects of co-working, creating a solid digital ecosystem that enhances the overall experience for remote workers and coworking communities. 4.1 Objectives of the platform Objectives The Network of Co-working Spaces (NCS) framework was created to address a crucial challenge in mountain areas: depopulation and the resulting impoverishment of local human capital. This negative dynamic, driven by the lack of job and personal development opportunities in rural areas, pushes young people to move to cities, where they find a richer and more diverse professional environment. The decline in population not only limits innovation and the competitiveness of local businesses but also worsens the shortage of essential services, making mountain regions increasingly less attractive to new residents. To reverse this trend, NCS aims to create an ecosystem that fosters social innovation. By building a network of distributed and interconnected coworking spaces, the project seeks to retain and attract talent while supporting the development of new economic and social models that enhance local characteristics and promote collaboration within communities. In this context, competence districts represent a key element in strengthening NCS’s effectiveness. The creation of specialized networks makes it possible to connect the various skills present in the area, facilitating knowledge exchange and the emergence of synergies among professionals from different sectors. These districts serve not only as platforms for individual and professional growth but also as catalysts for innovative projects that can meet the specific needs of local communities. In this way, 56 NCS not only promotes economic revival but also helps build a cohesive and dynamic social fabric where human potential can fully flourish. Method To tackle this issue, NCS has adopted a multidimensional approach, focusing on creating a specialized network of co-working spaces located in small mountain towns and managed by local actors such as public administrations, trade associations, shop owners, and tourism operators. Each co-working space thus becomes a gathering point for professionals, remote workers, and small businesses. At the heart of the proposal lies an innovative digital platform designed to support the creation of this network. This platform enables the mapping of local skills and the attraction of external expertise, the organization of thematic events, the management of bookings, and the promotion of collaborations and informal interactions among users. Informal interactions, in particular, prove essential for stimulating networking and fostering an atmosphere of trust and knowledge sharing. Creating opportunities for informal encounters among the network’s actors helps break down professional barriers and build meaningful relationships that can lead to new collaborative opportunities. To optimize synergies among stakeholders, NCS employs mathematical models based on game theory. This approach allows for maximizing the value generated through cooperation, promoting the sharing of resources and expertise among professionals and entrepreneurs in the region. Therefore, by integrating competence districts and informal interactions, NCS aims not only to revitalize mountain areas but also to create a sustainable and dynamic environment in which local — and external — talent can thrive. 57 Impact The project has already achieved significant milestones: a conceptual model and a mathematical model for the creation and optimization of co-working space networks have been developed, and the prototype of the software platform is currently under development. The next steps include testing to assess benefits, costs, and impacts—initially through simulations and later using real data by involving local organizations. The NCS framework has the potential to bring substantial future benefits for the economic and social development of mountain areas. By creating this network of interconnected spaces, NCS is expected to facilitate the birth of new businesses and start-ups, promote local innovation, and provide a collaborative environment where professionals and small companies can share resources and skills. This ecosystem is believed to be attractive to external workers—such as freelancers and digital nomads—who will find in mountain regions not only adequate workspaces but also an ideal living environment and stimulating professional context. Such attractiveness will encourage the stable settlement of young professionals, countering depopulation. Moreover, NCS will stimulate the formation of competence districts, where diverse professionals in the area can integrate and collaborate. The sharing of skills among network members will strengthen innovation and increase job opportunities. Informal interactions within these districts will facilitate the creation of meaningful relationships and collaborative projects, fueling a virtuous cycle of growth. In the long term, NCS will also encourage a cultural shift within local communities, helping to consolidate a cohesive social fabric open to collaboration. Co-working spaces will no longer be just workplaces but true community hubs, hosting cultural activities, workshops, and collective 64 Fig. 10: Example of the pointers inside of the map given searching on the search hubs page. On the "search users" page, users can search for other accounts on the platform (only those who have set their account status to "available" will be visible) using several search criteria: • Searching by account name will return all accounts that contain the entered string in their name. • Searching by competences and interests will return users who possess the specified attributes. 65 Fig. 11: Search user page showing some results as an example. From the list, it will be possible to view the profile descriptions and potentially initiate a chat with the selected account to begin communication. Fig. 12: Example of a user profile page showing the competences, interests and description set by the user. 66 The "chat" page contains all conversations between the user and other users. Fig. 13: Chat page with an example of a conversation. The "notifications" page displays announcements and alerts from the administrators. The "bulletin board" page features notes posted by users (for example, the launch of a new startup, a football match organized on a certain date, etc.), and users can also post new messages. Each message can be deleted by the user that created it and contains a link to the profile of the creator to make the creation of conversation between users easier. 67 Fig. 14: Upper part of the bulletin board page. Fig. 15: Lower part of the bulletin board page showing some notes left by the users as an example. In the "booking" page, users can reserve resources offered by hubs. They can either: 68 • Prioritize a specific hub first and then specify the resources at a later stage; or • Search for a particular type of resource for a selected date, in which case the user must specify the type of resource, quantity, and start/end time of the booking. Fig. 16: Booking page containing two methods of search. Afterwards, the user can select the hub from which to reserve, and a list of available resources will be shown. Once confirmed, the booking will appear in the user’s calendar. 69 Fig. 17: Example of a result of a search using the name of the hub as the input that lets the user fill the remaining parameters if the hub is located in a convenient position. On the "my calendar" page, users can view and manage their bookings, including the option to cancel them. Fig. 18: Calendar page with some events or bookings as an example. 70 In the "account settings" page, users can update their profile description, skills and interests, the hub they are currently working at, and manage their account visibility to other users. Fig. 19: Account setting page. 4.3.2 Company guide If the user is registered as a company, their profile will grant access to additional pages: • On the "manage calendar" page, companies can delete their initiatives and manage bookings for the resources they have made available. • On the "add hubs" page, companies can register new hubs in the system by specifying the name, description, municipality, and type of hub. Using the map, they can save the coordinates and upload an image of the hub. • On the "add initiatives" page, companies can define the name, subject, location, and start/end dates of the initiatives they wish to display on the platform. 71 • On the "my hubs" page, companies can edit the information related to their hubs and add or remove the associated resources. Fig. 20: Upper part of the offer facility page. Fig. 21: Lower part of the offer facility page. 72 Fig. 22: Add new initiative page. Fig. 23: My hub page with an example of a hub and resource offered. 4.3.3 Admin guide If the user is registered as an admin, their profile grants access to specific administrative pages: 73 • On the "manage calendar" page, the admin can oversee all bookings and initiatives in the system. • On the "manage users" page, the admin can ban accounts that violate platform rules. • On the "manage district" page, the admin can add and remove new districts. They are assigned to a specific month based on the bookings made during that time frame: if certain hubs have received bookings in the same month from users with similar skills or interests, the admin can create a district related to those common characteristics and assign the relevant hubs to it. The admin can also add new municipalities from this page. • On the "statistic" page, the admin can obtain the number of users that have booked a resource in a certain month with a maximum range of one year. They are divided by hub, competence and interest to show to the admin what kind of districts can be created in that month and which hubs are inside of it. Fig. 24: Statistic page. 80 under what conditions smart collaboration can generate sustainable development. The two works are tightly complementary: the earlier one provided the theoretical and demonstrative framework for understanding collaborative ecosystems, whereas this work offers a structured set of tools to implement, monitor, and optimize them across different organizational and territorial scales. Together, they form a coherent narrative—from strategic vision to actionable methodologies—supporting the sustainable regeneration of mountain communities through digitally enabled, transparent, and participatory processes. This document, on the other hand, takes an operational perspective. It translates the principles and insights into a structured toolkit designed to support implementation. The main achieved results are articulated along three main directions: • The device for Internet connection monitoring provides a concrete response to the problem of unstable connectivity in marginal areas. Currently the subject of a patent application, this tool is designed as an autonomous, simple, and secure solution, capable of offering users an objective evaluation of network quality and, consequently, improved conditions for remote work. • The bi-objective optimization model for co-working networks confirms and extends the theoretical framework already developed, proposing an innovative approach that integrates goals of economic efficiency and territorial sustainability. The dynamic λ adjustment mechanism, based on agents’ Shapley values and guided by policy objectives, has been shown to achieve significantly better results than traditional static weighting approaches. By considering co-working networks as public policy instruments rather than merely private economic entities, the model provides decision-makers with a transparent and flexible 81 framework that balances the maximization of collaborative value with the minimization of operating costs. The application to the Aosta Valley highlighted the model’s ability to generate practical evidence and useful insights for territorial development, particularly in regions facing challenges such as depopulation, economic seasonality, and the need for resilience. • The digital co-working platform translates the principles of collaboration and sharing into an operational infrastructure supporting professional communities. It enables space management, fosters user networking, and supports processes of social innovation, thus becoming a catalyst for territorial cohesion and the enhancement of local resources. Taken together, these three tools should not be seen as separate elements but as parts of a single systemic approach: the device ensures the minimum technical conditions for remote work, the model provides equitable and sustainable criteria for resource allocation, and the platform creates the digital ecosystem needed to strengthen collaborative networks and resilient communities. 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