We do not claim that prediction markets will always be more accurate that other methods. Often other institutions do pretty much the best they can. Rather, compared to other long-lived institutions which make frequently-updated probabilistic forecasts, we claim that well-traded prediction markets will in most situations rarely be much less accurate, and in some situations be much more accurate. Often other institutions are seriously broken, and do much worse than is possible.
While scientifically exact, this statement is too long for people to understand clearly. People need something simpler.