What You Jointly Know Determines How You Act: Strategic Interactions in Prediction Markets

Abstract

The primary goal of a prediction market is to elicit and aggregate information about some future event of interest. How well this goal is achieved depends on the behavior of self-interested market participants, which are crucially influenced by not only their private information but also their knowledge of others' private information, in other words, the information structure of market participants. In this paper, we model a prediction market using the now-classic logarithmic market scoring rule (LMSR) market maker as an extensive-form Bayesian game and aim to understand and characterize the game-theoretic equilibria of the market for different information structures. Prior work has shown that when participants' information is independent conditioned on the realized outcome of the event, the only type of equilibria in this setting has every participant race to honestly reveal their private information as soon as possible, which is the most desirable outcome for the market's goal of information aggregation. This paper considers the remaining two classes of information structures: participants' information being unconditionally independent (the I game) and participants' information being both conditionally and unconditionally dependent (the D game). We characterize the unique family of equilibria for the I game with finite number of participants and finite stages. At any equilibrium in this family, if player i's last stage of participation in the market is after player j's, player i only reveals his information after player j's last stage of participation. This suggests that players race to delay revealing their information, which is probably the least desirable outcome for the market's goal. We consider a special case of the D game and cast insights on possible equilibria if one exists.