  We examine two types of binary betting markets, whose primary goal is for
profit (such as sports gambling) or to gain information (such as prediction
markets). We articulate the interplay between belief and price-setting to
analyse both types of markets, and show that the goals of maximising bookmaker
profit and eliciting information are fundamentally incompatible. A key insight
is that profit hinges on the deviation between (the distribution of) bettor and
true beliefs, and that heavier tails in bettor belief distribution imply higher
profit. Our algorithmic contribution is to introduce online learning methods
for price-setting. Traditionally bookmakers update their prices rather
infrequently, we present two algorithms that guide price updates upon seeing
each bet, assuming very little of bettor belief distributions. The online
pricing algorithm achieves stochastic regret of $\mathcal{O}(\sqrt{T})$ against
the worst local maximum, or $ \mathcal{O}(\sqrt{T \log T}) $ with high
probability against the global maximum under fair odds. More broadly, the
inherent trade-off between profit and information-seeking in binary betting may
inspire new understandings of large-scale multi-agent behaviour.
