  Prediction markets are long known for prediction accuracy. This study
systematically explores the fundamental properties of prediction markets,
addressing questions about their information aggregation process and the
factors contributing to their remarkable efficacy. We propose a novel
multivariate utility (MU) based mechanism that unifies several existing
automated market-making schemes. Using this mechanism, we establish the
convergence results for markets comprised of risk-averse traders who have
heterogeneous beliefs and repeatedly interact with the market maker. We
demonstrate that the resulting limiting wealth distribution aligns with the
Pareto efficient frontier defined by the utilities of all market participants.
With the help of this result, we establish analytical and numerical results for
the limiting price in different market models. Specifically, we show that the
limiting price converges to the geometric mean of agent beliefs in exponential
utility-based markets. In risk-measure-based markets, we construct a family of
risk measures that satisfy the convergence criteria and prove that the price
can converge to a unique level represented by the weighted power mean of agent
beliefs. In broader markets with Constant Relative Risk Aversion (CRRA)
utilities, we reveal that the limiting price can be characterized by systems of
equations that encapsulate agent beliefs, risk parameters, and wealth. Despite
the potential impact of traders' trading sequences on the limiting price, we
establish a price invariance result for markets with a large trader population.
Using this result, we propose an efficient approximation scheme for the
limiting price.
