<p>The collection and integration of health data through electronic health records (EHRs) has improved the overall quality of healthcare. By centralizing all healthcare institutions’ operations in a single record and making it available to authorized individuals, blockchain applications offer a safe, rapid, and secure way to store EHRs. Healthcare providers require clarity on the specific blockchain technology necessary for managing medical records. In this regard, it is essential to understand this technology in terms of cost, speed, security, block size, and its capacity for handling extensive volumes of EHRs. To cope with this limitation, this study divides blockchain technology into two classes: private blockchain and public blockchain, where each blockchain is characterized in terms of its operation and installation costs, processing speed, block size capacity, and level of security. To provide a rational and robust blockchain environment, doctors and patients involve themselves in a novel coalition game to decide which blockchain technology is beneficial for handling their EHRs. Shapley value analysis introduces two novel concepts: fair blockchain and secure blockchain, providing a rational and reliable decision-making process for doctors and patients to determine their policy toward blockchain application when dealing with their EHRs. The study suggests that under a fair and secure blockchain platform, medical doctors and patients prefer private blockchain to public blockchain when dealing with large-scale EHR data. Furthermore, they exhibit the highest Shapley value within the coalition, indicating a strong motivation to utilize blockchain technology in their EHR interactions. Numerical results are performed based on two sample data sets where theoretical findings are justified.</p>