Prediction markets are openly traded markets that trade assets which are linked to different potential future outcomes. The purpose of prediction markets is to aggregate information from a diverse set of independent actors. Prediction markets have been shown in numerous studies to have more accurate prediction capabilities than other methods such as polls. The reason for this is that they require the actors trading in the markets (AKA the informants) to put stake in the game and allows them to profit if they are right - while risking loss if they lose.
For a general introduction of prediction markets, it is recommended to read the Wikipedia page on the matter. This article will explore the properties of prediction markets that are most important for a decentralized prediction market, of which Zeitgeist is one example.
Prediction markets can be traced back to writings of Ludwig von Mises and Frederik Hayek, however it is the economist Robin Hansen that is perhaps the best known proponent of them today. Hansen's writings have a very significant theoretic implication on how to implement prediction markets in a blockchain setting. He posits that the primary problem that can be solved is that of "info problems", that is the difficulty of aggregating information among many individuals with all different views of the subject.
In Robin Hansen's paper "Shall We Vote on Values, But Bet on Beliefs?" the author describes the core problem that prediction markets solve as democracy's info problems. That is, there is a hole in which information should be aggregated that leads to inefficient and non-optimal policies to be chosen.
In relation to the info problems described above, Hansen (in the same paper) points out that speculative markets show striking success in their ability to aggregate information. He says "That is, active speculative markets do very well at inducing people to acquire info, share it via trades, and collect that info into consensus prices that persuade wider audiences."