# Joint Spreading Factor and Channel Assignment in Multi-Operator LoRaWAN Deployments
## 6 Multi-Operator Game for Channel Selection
### Best Response Algorithm for Computing the NE
Figure 9: Normalized throughput per network as a function of \(S\). \(S\) = 20 km, it remains rather significant. Across all of the networks, our proposal improves the individual network normalized throughput by three times, on average. ## 8 Conclusions

In this paper, we proposed a joint optimal formulation for spreading factor and channel assignment in multi-operator LoRaWAN deployments. The joint problem is decoupled into a set of two games: (a) a spreading factor assignment game and (b) a channel assignment game. We propose semi-distributed and distributed algorithms in order to find the Nash equilibrium for each game. We compare our proposals to the concurrent approaches in LoRaWAN networks and demonstrate the significant gains that they bring in terms of throughput, packet delivery ratios, and energy efficiency. Conceptualization, K.K., H.F. and S.L. ; methodology, K.K., H.F. and S.L. ; software, K.K., H.F. and S.L. ; validation, K.K., H.F., S.L., C.A. and S.M. ; formal analysis, K.K., H.F., S.L., C.A. and S.M. ; investigation, K.K., H.F., S.L., C.A. and S.M. ; resources, K.K., H.F., S.L., C.A. and S.M. ; data curation, K.K., H.F., S.L., C.A. and S.M. ; writing--original draft preparation, K.K. and H.F.; writing--review and editing, K.K., H.F., S.L., C.A. and S.M. ; visualization, K.K., H.F., S.L., C.A. and S.M. ; supervision, K.K., S.L. and C.A. ; project administration, K.K. ; funding acquisition, C.A. All authors have read and agreed to the published version of the manuscript. The APC was funded by Inria, CRI Saclay-Ile-de-France and the LRI, Universite Paris-Saclay. Not Applicable. Data Availability StatementData can be made available upon request. This research was partially supported by Labex DigiCosme (project ANR11-LABEX-0045-DIGICOSME) operated by ANR as part of the program \"Investissement d'Avenir\" Idex Paris-Saclay (ANR-11-IDEX-0003-02). The authors declare no conflict of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l} LoRaWAN & Long Range Wide Area Networks \\ SF & Spreading Factor \\ NE & Nash Equilibrium \\ ADR & Adaptive Data Rate \\ ISM & Industrial Scientific and Medical bands \\ \end{tabular} 

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