He, Xue-Zhong and Treich, Nicolas (2011) Necessary and Sufficient Conditions for Prediction Market Accuracy.

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Abstract

We consider a binary-event prediction market in which traders have
heterogeneous prior beliefs. We derive conditions so that the prediction
market is accurate in the sense that the equilibrium price equals
the mean of traders’ beliefs. We show that the prediction market is
accurate i) for all distributions of beliefs if and only if the utility function
is logarithmic, and ii) for all strictly concave utility functions if
and only if the distribution of beliefs is symmetric about one half. We
also exhibit a necessary and sufficient condition for the equilibrium
price to be always above or below the mean beliefs for all symmetric
beliefs distributions.