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Unmanned Aerial Vehicle (UAV)-assisted Multi-access Edge Computing (MEC) systems have emerged recently as a flexible and dynamic computing environment, providing task offloading service to the users. In order for such a paradigm to be viable, the operator of a UAV-mounted MEC server should enjoy some form of profit by offering its computing capabilities to the end users. To deal with this issue in this paper, we apply a usage-based pricing policy for allowing the exploitation of the servers’ computing resources. The proposed pricing mechanism implicitly introduces a more social behavior to the users with respect to competing for the UAV-mounted MEC servers’ computation resources. In order to properly model the users’ risk-aware behavior within the overall data offloading decision-making process the principles of Prospect Theory are adopted, while the exploitation of the available computation resources is considered based on the theory of the Tragedy of the Commons. Initially, the user’s prospect-theoretic utility function is formulated by quantifying the user’s risk seeking and loss aversion behavior, while taking into account the pricing mechanism. Accordingly, the users’ pricing and risk-aware data offloading problem is formulated as a distributed maximization problem of each user’s expected prospect-theoretic utility function and addressed as a non-cooperative game among the users. The existence of a Pure Nash Equilibrium (PNE) for the formulated non-cooperative game is shown based on the theory of submodular games. An iterative and distributed algorithm is introduced which converges to the PNE, following the learning rule of the best response dynamics. The performance evaluation of the proposed approach is achieved via modeling and simulation, and detailed numerical results are presented highlighting its key operation features and benefits.