Robust Decision Aggregation with Second-order Information
Yuqi Pan Zhaohua Chen Yuqing Kong
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.