Incentivizing Exploration with Heterogeneous Value of Money

TL;DR

This paper extends a model for incentivized exploration in multi-armed bandits to agents with heterogeneous and non-linear utilities for money, proposing a convex programming approach for optimal signaling policies.

Contribution

It introduces a convex program-based method for designing optimal, signal-dependent policies under heterogeneous agent utilities, improving worst-case guarantees.

Findings

Convex program derives optimal policies for heterogeneous utilities.

Worst-case guarantees are tight, matching 'Diamonds in the Rough' instances.

More informative signals lead to better approximation ratios.

Abstract

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.