In evaluating prediction markets (and other crowd-prediction mechanisms),
investigators have repeatedly observed a so-called "wisdom of crowds" effect,
which roughly says that the average of participants performs much better than
the average participant. The market price---an average or at least aggregate of
traders' beliefs---offers a better estimate than most any individual trader's
opinion. In this paper, we ask a stronger question: how does the market price
compare to the best trader's belief, not just the average trader. We measure
the market's worst-case log regret, a notion common in machine learning theory.
To arrive at a meaningful answer, we need to assume something about how traders
behave. We suppose that every trader optimizes according to the Kelly criteria,
a strategy that provably maximizes the compound growth of wealth over an
(infinite) sequence of market interactions. We show several consequences.
First, the market prediction is a wealth-weighted average of the individual
participants' beliefs. Second, the market learns at the optimal rate, the
market price reacts exactly as if updating according to Bayes' Law, and the
market prediction has low worst-case log regret to the best individual
participant. We simulate a sequence of markets where an underlying true
probability exists, showing that the market converges to the true objective
frequency as if updating a Beta distribution, as the theory predicts. If agents
adopt a fractional Kelly criteria, a common practical variant, we show that
agents behave like full-Kelly agents with beliefs weighted between their own
and the market's, and that the market price converges to a time-discounted
frequency. Our analysis provides a new justification for fractional Kelly
betting, a strategy widely used in practice for ad-hoc reasons. Finally, we
propose a method for an agent to learn her own optimal Kelly fraction.