Evolution of Information in Prediction Markets

Predictions can be interpreted as probability distributions for future events, updated as information becomes available. We consider political prediction data from PredictIt, a rich data set where millions of dollars are wagered on binary outcomes of elections and other world news. We quantify the gain of information in a time window as the Kullback-Leibler divergence between the market price at the start and end of the window. If world events actually occur according to their predictit prices, then the average of the gain of information should be equal to the average of the decrease in entropy in the same window. As a corollary, the sum of cumulative gains and entropy is expected to be constant as time evolves. These ensemble equalities allow us to make nontrivial tests of the hypothesis that PredictIt market prices are proper probability distributions. They hold within our error bars for the several hundred highest volume events analyzed. In this nontrivial way, trading market prices seem to act as probability distributions. We expect our equivalence can help characterize their temporal evolution.