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Non-cooperative scheduling games can be used to coordinate residential loads in order to achieve a common goal while accounting for individual consumer's interests, privacy, and autonomy. However, a significant portion of the residential flexibility—Thermostatically Controlled Loads (TCLs) such as water and space heating/cooling appliances—has not been fully addressed under this game theoretic approach: their comfort constraints and integer control were not considered. This paper presents a method for properly including TCLs in this framework and discusses its application in energy communities. Specifically, we propose a general mathematical formulation for considering users' comfort in non-cooperative games. We model the integer nature of the TCLs control with binary variables and show that optimal or close to optimal (less than 1%) solutions are reached. Moreover, different total cost functions can be used depending on the market context and the objective of the demand management program. To illustrate and discuss these aspects in practical applications, we used a case study of an energy community in Spain. The results show that the TC solutions are optimal or only 0.80% worse than optimal; different total cost functions result in different results (load curve smoothing or peak load reduction); consumers' comfort is respected; and the proposed game model cooperates with consumers in order to minimize community's costs. [ABSTRACT FROM AUTHOR]