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.

我们研究了一个关于二元决策聚合问题，其中既有真实可信的专家，又有对抗性的专家；我们的目标是设计一个稳健的聚合器来预测真实世界的状态，同时最小化与基准决策之间的预期损失差距。我们证明了在一些条件下，截断均值是最优的聚合器选择，并且很多情况下最优聚合器属于分段线性函数类；我们的结果表明，遗憾值与专家总数无关，只取决于对抗性专家的比例。