We investigate the mismatched distortion-rate problem formulated by Lapidoth in 1997. The single-letter characterization of the distortion-rate tradeoff is an open problem. When the communication rate is large enough, the mismatched distortion-rate problem is deeply related to the mechanism design problem formulated by Jackson and Sonnenschein in 2007. Inspired by this solution, we formulate a new converse bound that matches the achievability bound in four special cases: 1) when the rate is large enough, 2) when the source is binary and uniform and the decoder output is binary, 3) when the encoder distortion measure does not depend on the source symbols, 4) when the sum of the two distortion measures is zero. An important feature of the model is the stochastic post-processing of the codewords. By carefully selecting the randomness of the decoding function, the encoder is encouraged to truthfully reveal the source sequence.