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We consider incentivized exploration: a version of multi-armed bandits wherethe choice of arms is controlled by self-interested agents, and the algorithmcan only issue recommendations. The algorithm controls the flow of information,and the information asymmetry can incentivize the agents to explore. Prior workachieves optimal regret rates up to multiplicative factors that becomearbitrarily large depending on the Bayesian priors, and scale exponentially inthe number of arms. A more basic problem of sampling each arm once runs intosimilar factors.  We focus on the price of incentives: the loss in performance, broadlyconstrued, incurred for the sake of incentive-compatibility. We prove thatThompson Sampling, a standard bandit algorithm, is incentive-compatible ifinitialized with sufficiently many data points. The performance loss due toincentives is therefore limited to the initial rounds when these data pointsare collected. The problem is largely reduced to that of sample complexity: howmany rounds are needed? We address this question, providing matching upper andlower bounds and instantiating them in various corollaries. Typically, theoptimal sample complexity is polynomial in the number of arms and exponentialin the "strength of beliefs".

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