In this work, using a game-theoretic approach, cost-sensitive mechanisms that lead to reliable Internet-based computing are designed. In particular, we consider Internet-based master-worker computations, where a master processor assigns, across the Internet, a computational task to a set of potentially untrusted worker processors and collects their responses. Several game-theoretic models that capture the nature of the problem are analyzed and mechanisms that, for each given set of cost and system parameters, achieve high reliability are designed. Additionally, two specific realistic system scenarios are studied. These scenarios are a system of volunteering computing like SETI, and a company that buys computing cycles from Internet computers and sells them to its customers in the form of a task-computation service. Notably, under certain conditions, non redundant allocation yields the best trade-off between cost and reliability.