We consider a binary decision aggregation problem in the presence of both truthful and adversarial experts. The truthful experts will report their private signals truthfully with proper incentive, while the adversarial experts can report arbitrarily. The decision maker needs to design a robust aggregator to forecast the true state of the world based on the reports of experts. The decision maker does not know the specific information structure, which is a joint distribution of signals, states, and strategies of adversarial experts. We want to find the optimal aggregator minimizing regret under the worst information structure. The regret is defined by the difference in expected loss between the aggregator and a benchmark who makes the optimal decision given the joint distribution and reports of truthful experts. We prove that when the truthful experts are symmetric and adversarial experts are not too numerous, the truncated mean is optimal, which means that we remove some lowest reports and highest reports and take averaging among the left reports. Moreover, for many settings, the optimal aggregators are in the family of piecewise linear functions. The regret is independent of the total number of experts but only depends on the ratio of adversaries. We evaluate our aggregators by numerical experiment in an ensemble learning task. We also obtain some negative results for the aggregation problem with adversarial experts under some more general information structures and experts' report space.

翻译：我们研究了同时存在诚实专家与对抗性专家的二元决策聚合问题。诚实专家在适当激励下会如实报告私有信号，而对抗性专家可以任意报告信息。决策者需要设计鲁棒聚合器，基于专家报告预测世界真实状态。决策者未知具体信息结构——即信号、状态与对抗性专家策略的联合分布。我们旨在寻找在最差信息结构下使遗憾最小化的最优聚合器，其中遗憾定义为聚合器损失与基于联合分布及诚实专家报告作出最优决策的基准损失之差。证明表明：当诚实专家对称且对抗性专家数量有限时，截断均值法为最优策略——即剔除部分最低和最高报告后对剩余报告取平均。此外，在多数场景下最优聚合器属于分段线性函数族，且遗憾值与专家总数无关，仅取决于对抗者比例。我们通过集成学习任务的数值实验验证了所提聚合器的有效性。同时，针对更一般的信息结构与专家报告空间，我们获得了对抗性专家聚合问题的若干消极结论。