The Price of Incentivizing Exploration: A Characterization via Thompson Sampling and Sample Complexity

We consider incentivized exploration: a version of multiarmed bandits where the choice of arms is controlled by self-interested agents and the algorithm can only issue recommendations. The algorithm controls the flow of information, and the information asymmetry can incentivize the agents to explore. Prior work achieves optimal regret rates up to multiplicative factors that become arbitrarily large depending on the Bayesian priors and scale exponentially in the number of arms. A more basic problem of sampling each arm once runs into similar factors. We focus on the price of incentives: the loss in performance, broadly construed, incurred for the sake of incentive compatibility. We prove that Thompson sampling, a standard bandit algorithm, is incentive compatible if initialized with sufficiently many data points. The performance loss because of incentives is, therefore, limited to the initial rounds when these data points are collected. The problem is largely reduced to that of sample complexity. How many rounds are needed? We address this question, providing matching upper and lower bounds and instantiating them in various corollaries. Typically, the optimal sample complexity is polynomial in the number of arms and exponential in the “strength of beliefs.”

Funding: This work was supported by the National Science Foundation [Grant 1650114 (Graduate Research Fellowship)] and a William R. and Sara Hart Kimball Stanford Graduate Fellowship.