## 1 Introduction  
High-speed rail (HSR), as a prevalent mode of transportation, plays a pivotal role in both economic development and improving people's livelihood. Since 2008, China has recognized high-speed rail as a significant strategic initiative, extensively constructing high-speed railways and promoting nationwide implementation of online ticketing services. However, due to China's unique cultural and developmental patterns, there is typically an ample supply of seated tickets on ordinary weekdays. Conversely, during peak migration periods such as the Spring Festival holidays, seated tickets are in short supply and passengers are often forced to purchase a limited number of standing or unreserved seat tickets. This results in overcrowding within the carages, as passengers with unfixed seats stand in public areas, negatively impacting their travel comfort and satisfaction [1, 2, 3]. To address the issue of train overcrowding and optimize seat utilization, both \"hard\" and \"soft\" approaches are frequently considered [1]. An example of a \"hard\" approach involves expanding the relevant infrastructure to address the supply-demand mismatch. An example of a \"soft\" approach is to influence the behavior of passengers by encouraging them to change their time and mode of train travel [4]. The optimal approach is to integrate both the \"hard\" and \"soft\" strategies, to address overcrowding effectively and maximize the capacity of the railway system [4]. However, the process of expanding and implementing rolling stock and track capacity entails significant time and financial investments [5]. Therefore, this study primarily focuses on employing soft methodologies. Real-time crowding information (RTCI) is an increasingly viable solution within the realm of soft approaches, as it has the potential to influence passenger choice and behavior by providing RTCI for each carriage of an incoming train through visual displays and loudspeakers [6]. The rationale is that data on historical and current passenger flows in contemporary public transportation systems are collected from diverse sources, including Automated Passenger Counts (APC) and Automated Fare Collection (AFC) systems, load-bearing data, and train occupancy information. These datasets can be processed and transmitted to passengers through RTCI, facilitated by load-data analysis and wireless networks [7]. Previous studies have demonstrated that the implementation of RTCI in public transportation systems, such as subways and trains, not only enhances passengers' efficiency in locating available seats, but also significantly contributes to overall satisfaction levels [6; 8; 9; 10]. Although numerous studies have confirmed the positive impact of RTCI on passenger satisfaction, a critical concern with existing RTCI systems lies in the reliability of the provided information [10]. Previous researchers have proposed various techniques for estimating passenger counts on trains, such as weight sensors, closed-circuit television, Wi-Fi, and infrared sensors in the doors. However, all these methods are susceptible to distortions that can introduce errors into RTCI systems [11; 12]. [PERSON] emphasized the risks associated with the accuracy of RTCI which arise from its inherent properties: specifically, the time delay in disseminating crowding information and the influence of passenger reactions induced by information provision on the efficacy of provided RTCI, resulting in disparities between anticipated and actual on-vehicle crowding conditions [10]. [PERSON] hypothesized that participants' pre-existing perceptions regarding the reliability of station information play a crucial role in their decision-making process. Furthermore, if the actual reliability of the information fails to meet these expectations, it is likely to diminish passengers' willingness to modify their behavior [11].