Online mechanism design (OMD) addresses the problem of sequential decision making in a stochastic environment with multiple self-interested agents. The goal in OMD is to make value-maximizing decisions despite this self-interest. In previous work we presented a Markov decision process (MDP)-based approach to OMD in large-scale problem domains. In practice the underlying MDP needed to solve OMD is too large and hence the mechanism must consider approximations. This raises the possibility that agents may be able to exploit the approximation for selfish gain. We adopt sparse-sampling-based MDP algorithms to implement efficient policies, and retain truth-revelation as an approximate BayesianNash equilibrium. Our approach is empirically illustrated in the context of the dynamic allocation of WiFi connectivity to users in a coffeehouse.