We consider a decision aggregation problem with two experts who each make a binary recommendation after observing a private signal about an unknown binary world state. An agent, who does not know the joint information structure between signals and states, sees the experts' recommendations and aims to match the action with the true state. Under the scenario, we study whether supplemented additionally with second-order information (each expert's forecast on the other's recommendation) could enable a better aggregation. We adopt a minimax regret framework to evaluate the aggregator's performance, by comparing it to an omniscient benchmark that knows the joint information structure. With general information structures, we show that second-order information provides no benefit. No aggregator can improve over a trivial aggregator, which always follows the first expert's recommendation. However, positive results emerge when we assume experts' signals are conditionally independent given the world state. When the aggregator is deterministic, we present a robust aggregator that leverages second-order information, which can significantly outperform counterparts without it. Second, when two experts are homogeneous, by adding a non-degenerate assumption on the signals, we demonstrate that random aggregators using second-order information can surpass optimal ones without it. In the remaining settings, the second-order information is not beneficial. We also extend the above results to the setting when the aggregator's utility function is more general.

翻译：我们考虑一个包含两位专家的决策聚合问题，每位专家在观察到关于未知二元世界状态的私人信号后作出二元推荐。一位不了解信号与状态之间联合信息结构的决策者，在观测到专家的推荐后，旨在使自身行动与真实状态相匹配。在此场景下，我们研究额外补充二阶信息（每位专家对另一专家推荐的预测）是否能实现更优的聚合。我们采用最小化遗憾框架来评估聚合器的性能，将其与一个知晓联合信息结构的全知基准进行比较。在一般信息结构下，我们证明二阶信息无法带来收益：没有任何聚合器能优于始终遵循第一位专家推荐的平凡聚合器。然而，当假设专家信号在给定世界状态下条件独立时，积极结果随之显现。当聚合器为确定性时，我们提出了一种利用二阶信息的稳健聚合器，其性能显著优于未使用二阶信息的对应方案。其次，当两位专家同质时，通过对信号添加非退化假设，我们证明使用二阶信息的随机聚合器能够超越未使用该信息的最优聚合器。在其余设定中，二阶信息无法带来收益。我们还将上述结论推广至聚合器效用函数更为一般的情形。