Game theory in these setting is built on subjective probabilities! The standard solution concept in incomplete-information games is even known as Bayes-Nash equilibrium.

The LMSR is stronger strategically than Nash equilibrium, assuming everyone participates only once. In that case, it's a dominant strategy to be honest, rather than just a best response. If people participate multiple times, the Bayes-Nash equilibrium is harder to characterize. See Gao et al (2013)] for the best current description, which roughly says you shouldn't reveal any information until the very last moment. The paper has an overview of the LMSR for anyone interested.