Pneumonia causes severe repercussions if not acted upon at the right time. Machine learning has been applied to classify chest X-ray images into pneumonia-positive and pneumonia-negative classes to allow an early diagnosis and support medical experts’ decisions about pneumonia. Nonetheless, the previous attempt focus on binary classification that may not consider the possibility of the presence of uncertain information in chest X-ray images. This method forces the system to have definite and unreliable decision on suspicious cases of pneumonia. The three-way classification can overcome the shortcoming of binary classification by classifying X-ray images into three classes, pneumonia-positive class to categorize instances that represent pneumonia; pneumonia-negative class to categorize instances that represent normal chest X-ray; pneumonia-indecisive class to categorize instances, for which making decision based on the available information is difficult. In this study, chest X-ray images are used to experiment and we employ three-way classification, supported by rough sets, to interpret the doubtful X-ray images that lack complete information. The training set of X-ray images is divided into three distinct regions based on testing and test-treatment thresholds. This threshold pair determines the decision of no treatment, treatment, and delayed treatment. The three regions in three-way classification are termed as positive, negative, and boundary. Definite decisions, such as “treatment” or “no treatment” for pneumonia can be inferred from positive and negative regions, respectively. No decision is made for the boundary region until supporting facts are collected. Conventional rough sets uses a stringent threshold pair that restricts the number of instances being classified for definite decision. Conversely, probabilistic rough sets mitigates this limitation by allowing the use of a modified value for the threshold pair. This theory can classify more instances into positive and negative regions. Hence, we apply probabilistic rough sets to derive three classes, but it is crucial to achieve an appropriate testing and test-treatment threshold pair. In this study, we apply game-theoretic rough sets (GTRS) to determine a suitable threshold pair. The evaluation criteria, accuracy, and coverage of the three-way classification model are played as players in a competitive game. The game is reformulated continuously as the player modifies their strategy. Consequently, a suitable threshold pair can be learned that balances these aspects and achieves an optimal trade-off. By using the achieved threshold pair from GTRS, three classes are defined for the pneumonia classification system. From the first two classes, pneumonia-positive or pneumonia-negative decisions can be made. In the remaining class, we explore X-ray images that lack the crucial information for making inferences. With the third class, the healthcare system would be able to re-examine the indecisive chest X-ray images by conducting tests instead of taking definite decisions based on incomplete data. Based on the experiment, we achieve an accuracy score of 96.25% while covering 64.01% of the test set for certain decisions. The test results are also compared with various other related work and rough sets models. The GTRS based probabilistic rough sets model obtains a greater coverage of data than the conventional rough sets and better accuracy as compared to 0.5-probabilistic rough sets. This thesis is expected to provide an understanding of GTRS from its application perspective.