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Abstract: In this paper we study optimization problems with verifiable one-parameter selfish agents introduced by Auletta et al. [V. Auletta, R. De Prisco, P. Penna, P. Persiano, The power of verification for one-parameter agents, in: Proceedings of the 31st International Colloquium on Automata, Languages and Programming, ICALP, in: LNCS, vol. 3142, 2004, pp. 171–182]. Our goal is to allocate load among the agents, provided that the secret data of each agent is a single positive real number: the cost they incur per unit load. In such a setting the payment is given after the load completion, therefore if a positive load is assigned to an agent, we are able to verify if the agent declared to be faster than she actually is. We design truthful mechanisms when the agents’ type sets are upper-bounded by a finite value. We provide a truthful mechanism that is -approximate if the underlying algorithm is -approximate and weakly-monotone. Moreover, if type sets are also discrete, we provide a truthful mechanism preserving the approximation ratio of its algorithmic part. Our results improve the existing ones which provide truthful mechanisms dealing only with finite type sets and do not preserve the approximation ratio of the underlying algorithm. Finally, we give applications for our payment schemes. Firstly, we give a full characterization of the problem by using our techniques. Even if our payment schemes need upper-bounded type sets, every instance of can be “mapped” into an instance with upper-bounded type sets preserving the approximation ratio. In conclusion, we turn our attention to binary demand games. In particular, we show that the Minimum Radius Spanning Tree admits an exact truthful mechanism with verification achieving time (and space) complexity of the fastest centralized algorithm for it. This contrasts with a recent truthful mechanism for the same problem [G. Proietti, P. Widmayer, A truthful mechanism for the non-utilitarian minimum radius spanning tree problem, in: Proceedings of the 17th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA, ACM Press, 2005, pp. 195–202] which pays a linear factor with respect to the complexity of the fastest centralized algorithm. Such a result is extended to several binary demand games studied in literature. [Copyright &y& Elsevier]