Recent work has constructed economic mechanisms that are both truthful and differentially private. In these mechanisms, privacy is treated separately from the truthfulness; it is not incorporated in players’ utility functions (and doing so has been shown to lead to non-truthfulness in some cases). In this work, we propose a new, general way of modelling privacy in players’ utility functions. Specifically, we only assume that if an outcome *o* has the property that any report of player *i* would have led to *o* with approximately the same probability, then *o* has small privacy cost to player *i*. We give three mechanisms that are truthful with respect to our modelling of privacy: for an election between two candidates, for a discrete version of the facility location problem, and for a general social choice problem with discrete utilities (via a VCG-like mechanism). As the number *n* of players increases, the social welfare achieved by our mechanisms approaches optimal (as a fraction of *n*).