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.

翻译：本研究考察了两种类型的二元投注市场：以盈利为主要目标的市场（如体育博彩）和以获取信息为主要目标的市场（如预测市场）。通过阐述信念与价格设定之间的相互作用，我们对两类市场进行了分析，并证明了庄家利润最大化与信息获取这两个目标在根本上是不相容的。一个关键发现是，利润取决于投注者信念（分布）与真实信念之间的偏差，且投注者信念分布的尾部越重，利润就越高。我们的算法贡献在于引入了用于价格设定的在线学习方法。传统上庄家更新价格的频率较低，我们提出了两种算法，在观察到每次投注后指导价格更新，且对投注者信念分布的假设极少。该在线定价算法在最坏局部最优解下实现了 $\mathcal{O}(\sqrt{T})$ 的随机遗憾，或在公平赔率下以高概率对全局最优解实现了 $ \mathcal{O}(\sqrt{T \log T}) $ 的遗憾。更广泛而言，二元投注中盈利与信息寻求之间的内在权衡，可能为理解大规模多智能体行为提供新的视角。