  Prediction markets are often used as mechanisms to aggregate information
about a future event, for example, whether a candidate will win an election.
The event is typically assumed to be exogenous. In reality, participants may
influence the outcome, and therefore (1) running the prediction market could
change the incentives of participants in the process that creates the outcome
(for example, agents may want to change their vote in an election), and (2)
simple results such as the myopic incentive compatibility of proper scoring
rules no longer hold in the prediction market itself. We introduce a model of
games of this kind, where agents first trade in a prediction market and then
take an action that influences the market outcome. Our two-stage two-player
model, despite its simplicity, captures two aspects of real-world prediction
markets: (1) agents may directly influence the outcome, (2) some of the agents
instrumental in deciding the outcome may not take part in the prediction
market. We show that this game has two different types of perfect Bayesian
equilibria, which we term LPP and HPP, depending on the values of the belief
parameters: in the LPP domain, equilibrium prices reveal expected market
outcomes conditional on the participants' private information, whereas HPP
equilibria are collusive -- participants effectively coordinate in an
uninformative and untruthful way.
