Prediction markets for future events are increasingly common and they often trade several contracts for the same event. This paper considers the distribution of a normative risk-neutral trader who, given any portfolio of contracts traded on the event, would choose not to reallocate that portfolio of contracts even if transactions costs were zero. Because common parametric distributions can conflict with observed prediction market prices, the distribution is given a nonparametric representation together with a prior distribution favoring smooth and concentrated distributions. Posterior modal distributions are found for popular vote shares of the U.S. presidential candidates in the 100 days leading up to the elections of 1992, 1996, 2000, and 2004, using bid and ask prices on multiple contracts from the Iowa Electronic Markets. On some days, the distributions are multimodal or substantially asymmetric. The derived distributions are more concentrated than the historical distribution of popular vote shares in presidential elections, but do not tend to become more concentrated as time to elections diminishes.