  Recently, Frazier et al. proposed a natural model for crowdsourced
exploration of different a priori unknown options: a principal is interested in
the long-term welfare of a population of agents who arrive one by one in a
multi-armed bandit setting. However, each agent is myopic, so in order to
incentivize him to explore options with better long-term prospects, the
principal must offer the agent money. Frazier et al. showed that a simple class
of policies called time-expanded are optimal in the worst case, and
characterized their budget-reward tradeoff.
  The previous work assumed that all agents are equally and uniformly
susceptible to financial incentives. In reality, agents may have different
utility for money. We therefore extend the model of Frazier et al. to allow
agents that have heterogeneous and non-linear utilities for money. The
principal is informed of the agent's tradeoff via a signal that could be more
or less informative.
  Our main result is to show that a convex program can be used to derive a
signal-dependent time-expanded policy which achieves the best possible
Lagrangian reward in the worst case. The worst-case guarantee is matched by
so-called "Diamonds in the Rough" instances; the proof that the guarantees
match is based on showing that two different convex programs have the same
optimal solution for these specific instances. These results also extend to the
budgeted case as in Frazier et al. We also show that the optimal policy is
monotone with respect to information, i.e., the approximation ratio of the
optimal policy improves as the signals become more informative.
