二进制评分规则，激励精度
Binary Scoring Rules that Incentivize Precision
摘要：
所有适当的评分规则都激励专家预测\emph{accurately}（报告他们的真实估计），但并非所有适当的评分规则都同样激励\emph{precision}。而不是把专家的信念作为外生的，我们考虑一个模型，在这个模型中，一个理性的专家可以通过反复支付固定成本来内生地完善他们的信念，并通过适当的评分规则来激励他们这样做。具体来说，我们的专家旨在预测明天掷出的带有偏见的硬币将正面朝上的概率，并且可以以每次投掷$c$的成本在今天掷硬币任意次数。我们的第一个主要结果定义了一个适当的评分规则的\emph{incentivization index}，并证明了该指数的预期误差的专家的估计（其中翻转的数量今天选择自适应预测的预期收益最大化）。我们的第二个主要结果发现了独特的评分规则，优化激励指数在所有适当的评分规则。我们还考虑扩展到最小化的$\ell^{th}$时刻的错误，并再次提供了一个激励指数和最佳的适当的评分规则。在某些情况下，所得的评分规则是可微的，但不是无限可微的。在这些情况下，我们进一步证明，最佳的多项式评分规则可以一致近似。最后，我们通过我们的措施比较常见的评分规则，并包括模拟确认我们的措施的相关性，即使在域之外，它可证明适用。
英文摘要：
All proper scoring rules incentivize an expert to predict \emph{accurately} (report their true estimate), but not all proper scoring rules equally incentivize \emph{precision}. Rather than treating the expert's belief as exogenously given, we consider a model where a rational expert can endogenously refine their belief by repeatedly paying a fixed cost, and is incentivized to do so by a proper scoring rule. Specifically, our expert aims to predict the probability that a biased coin flipped tomorrow will land heads, and can flip the coin any number of times today at a cost of $c$ per flip. Our first main result defines an \emph{incentivization index} for proper scoring rules, and proves that this index measures the expected error of the expert's estimate (where the number of flips today is chosen adaptively to maximize the predictor's expected payoff). Our second main result finds the unique scoring rule which optimizes the incentivization index over all proper scoring rules. We also consider extensions to minimizing the $\ell^{th}$ moment of error, and again provide an incentivization index and optimal proper scoring rule. In some cases, the resulting scoring rule is differentiable, but not infinitely differentiable. In these cases, we further prove that the optimum can be uniformly approximated by polynomial scoring rules. Finally, we compare common scoring rules via our measure, and include simulations confirming the relevance of our measure even in domains outside where it provably applies.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
