Price Interpretability of Prediction Markets: A Convergence Analysis

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–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 converges to a unique level represented by the weighted power mean of agent beliefs. In broader markets with constant relative risk aversion utilities, we reveal that the limiting price can be characterized by systems of equations that encapsulate agent beliefs, risk parameters, and wealth. Despite the 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. Numerical experiments demonstrate that the accuracy of this approximation scheme outperforms existing approximation methods across various scenarios. Our findings serve to aid market designers in better tailoring and adjusting the market-making mechanism for more effective opinion elicitation.

Funding: This work was supported by the National Natural Science Foundation of China [Grants 71671045, 71971132, 72150002, 72201067, and 72394361], the InnoHK initiative of the Government of the HKSAR, Laboratory for AI-Powered Financial Technologies, the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence [Grant 2023B1212010001], the Shanghai Research Center for Data Science and Decision Technology, and the Key Laboratory of Interdisciplinary Research of Computation and Economics, Ministry of Education, Shanghai University of Finance and Economics.

Supplemental Material: The computer code and data that supports the findings of this study is available within this article’s supplemental material at https://doi.org/10.1287/opre.2022.0417.