## 1 Introduction
### Related Works  
Several studies have addressed the limitations of the LEACH protocol, as highlighted in [13; 14; 15; 16; 17; 18; 19; 20; 21; 22] and have been summarised in Table 1. One of the challenges with LEACH is the uneven distribution of nodes among different CHs, resulting in varying network performance. To mitigate this issue, MaximuM-LEACH [16] has been introduced, aiming to achieve a more balanced load by equalizing the number of nodes within each cluster. This strategy is designed to prolong the overall network lifespan. In [17], a modified protocol named LEACH-Centralized Modified (LEACH-CM) is introduced to improve upon the existing centralized LEACH (LEACH-C). LEACH-CM primarily involves evaluating the distance between a selected CH and a member node, as well as the distance between the member node and the BS for data transmission. Addressing inter-cluster communication issues in LEACH, ref. [19] proposes an enhanced energy-saving strategy using inter-cluster ring routing. This approach abstracts all sensor nodes into a ring system model during network topology formation and implements a ring transmission strategy during data transmission. Additionally, ref. [20] introduces improved computation algorithms for LEACH clustering and enhanced information gathering by CHs, enhancing the rate of combined information flow. Moreover, ref. [14] proposes an enhanced version of LEACH, referred to as enhanced LEACH (E-LEACH), to extend network lifetime and improve energy utilization. Scalability concerns of traditional LEACH are addressed in [21] through the introduction of V-LEACH [15]. Furthermore, ref. [13] presents Energy-LEACH and Multihop-LEACH variations to enhance scalability, while ref. [22] focuses on implementing multihop flat-based and cluster-based protocols to improve scalability. Lastly, ref. [18] introduces the Energy-Efficient Scalable Routing Algorithm (ESRA) as an adaptation of LEACH. In the subject of handling scalability and high node densities in WSNs, one interesting solution is Mean Field Game (MFG), a mathematical framework employed to model decision-making within extensive populations of interacting agents [23; 24]. This concept is particularly important in the field of economics [25], network security [26; 27; 28], and traffic networks [29; 30] and has also been applied to wireless networks [31; 32; 33; 34; 35]. \begin{table}
\begin{tabular}{p{71.1 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Algorithm** & **References** & **Year** & **Key Features** \\ \hline LEACH & [36; 37] & 2000 & Uneven distribution of nodes among CHs, rotating CHs, localized data aggregation and limited network lifespan. \\ \hline Energy-LEACH & [13] & 2009 & Considers the residual energy of nodes as the primary metric for CH selection. \\ \hline Multihop-LEACH & [13] & 2009 & Employs multi-hop communication and selects optimal paths to reduce energy consumption and prolong the network lifetime. \\ \hline E-LEACH & [14] & 2013 & Introduces enhancements in cluster formation and CH selection processes, extension of network lifetime and improved energy utilization. \\ \hline V-LEACH & [15] & 2013 & Improves the lifetime of the network by introducing a vice-CH to mitigate the impact of CH failures. \\ \hline MaximuM-LEACH & [16] & 2018 & Balanced load distribution among clusters and prolonged network lifespan. \\ \hline LEACH-C & [17] & 2021 & Centralized approach, considers distance for CH selection and data transmission and considers active node count for CH determination. \\ \hline ESSRA & [18] & 2022 & Sink node makes decisions about CH selection to reduce energy consumption and it considers residual energy levels and distance from CHs. \\ \hline MFG-LEACH & This work & 2024 & Incorporates the Mean Field Games framework into the data transmission phase of LEACH to select the optimal energy for transmitting. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of LEACH protocol variants.