Algorithmic decision-making in high-stakes domains often involves assigning decisions to agents with incentives to strategically modify their input to the algorithm. In addition to dealing with incentives, in many domains of interest (e.g. lending and hiring) the decision-maker only observes feedback regarding their policy for rounds in which they assign a positive decision to the agent; this type of feedback is often referred to as apple tasting (or one-sided) feedback. We formalize this setting as an online learning problem with apple-tasting feedback where a principal makes decisions about a sequence of $T$ agents, each of which is represented by a context that may be strategically modified. Our goal is to achieve sublinear strategic regret, which compares the performance of the principal to that of the best fixed policy in hindsight, if the agents were truthful when revealing their contexts. Our main result is a learning algorithm which incurs $\tilde{\mathcal{O}}(\sqrt{T})$ strategic regret when the sequence of agents is chosen stochastically. We also give an algorithm capable of handling adversarially-chosen agents, albeit at the cost of $\tilde{\mathcal{O}}(T^{(d+1)/(d+2)})$ strategic regret (where $d$ is the dimension of the context). Our algorithms can be easily adapted to the setting where the principal receives bandit feedback -- this setting generalizes both the linear contextual bandit problem (by considering agents with incentives) and the strategic classification problem (by allowing for partial feedback).

Strategic Apple Tasting

The contextual bandit problem is a theoretically justiﬁed framework with wide applications in various ﬁelds. While the previous study on this problem usually requires independence between noise and contexts, our work considers a more sensible setting where the noise becomes a latent confounder that aﬀects both contexts and rewards. Such a confounded setting is more realistic and could expand to a broader range of applications. However, the unresolved confounder will cause a bias in reward function estimation and thus lead to a large regret. To deal with the challenges brought by the confounder, we apply the dual instrumental variable regression, which can correctly identify the true reward function. We prove the convergence rate of this method is near-optimal in two types of widely used reproducing kernel Hilbert spaces. Therefore, we can design computationally eﬃcient and regret-optimal algorithms based on the theoretical guarantees for confounded bandit problems. The numerical results illustrate the eﬃcacy of our proposed algorithms in the confounded bandit setting.

Dual Instrumental Method for Confounded Kernelized Bandits

Machine Learning algorithms often prompt individuals to strategically modify their observable attributes to receive more favorable predictions. As a result, the distribution the predictive model is trained on may differ from the one it operates on in deployment. While such distribution shifts, in general, hinder accurate predictions, our work identifies a unique opportunity associated with shifts due to strategic responses: We show that we can use strategic responses effectively to recover causal relationships between the observable features and outcomes we wish to predict. More specifically, we study a game-theoretic model in which a principal deploys a sequence of models to predict an outcome of interest (e.g., college GPA) for a sequence of strategic agents (e.g., college applicants). In response, strategic agents invest efforts and modify their features for better predictions. In such settings, unobserved confounding variables can influence both an agent's observable features (e.g., high school records) and outcomes. Therefore, standard regression methods generally produce biased estimators. In order to address this issue, our work establishes a novel connection between strategic responses to machine learning models and instrumental variable (IV) regression, by observing that the sequence of deployed models can be viewed as an instrument that affects agents' observable features but does not directly influence their outcomes. Therefore, two-stage least squares (2SLS) regression can recover the causal relationships between observable features and outcomes. Beyond causal recovery, we can build on our 2SLS method to address two additional relevant optimization objectives: agent outcome maximization and predictive risk minimization. Finally, our numerical simulations on semi-synthetic data show that our methods significantly outperform OLS regression in causal relationship estimation.

Strategic Instrumental Variable Regression: Recovering Causal Relationships From Strategic Responses.

We study a multi-armed bandit learning setting where a social planner incentivizes a set of heterogeneous agents to efficiently explore the set of available arms. At each round, an agent arrives with their unobserved private type that determines both their prior preferences across the actions as well as their action-independent confounding shift in the rewards. The planner provides the agent with an arm recommendation that may alter their belief and incentivize them to explore potentially sub-optimal arms. Under this setting, we provide a novel recommendation mechanism that views the planner’s recommendations as a form of instrumental variables (IV) that only affect agents’ arm selection but not the observed rewards. We construct such IVs by carefully mapping the history–the interactions between the planner and the previous agents–to a random arm recommendation. Despite the unobserved confounding shift in the rewards, the resulting IV regression provides reliable estimates on the mean rewards of the actions and enables the social learning process to minimize regret over the long term.

Incentivizing Bandit Exploration: Recommendations as Instruments

Randomized experiments can be susceptible to selection bias due to potential non-compliance by the participants. While much of the existing work has studied compliance as a static behavior, we propose a game-theoretic model to study compliance as dynamic behavior that may change over time. In rounds, a social planner interacts with a sequence of heterogeneous agents who arrive with their unobserved private type that determines both their prior preferences across the actions (e.g., control and treatment) and their baseline rewards without taking any treatment. The planner provides each agent with a randomized recommendation that may alter their beliefs and their action selection. We develop a novel recommendation mechanism that views the planner's recommendation as a form of instrumental variable (IV) that only affects an agents' action selection, but not the observed rewards. We construct such IVs by carefully mapping the history -- the interactions between the planner and the previous agents -- to a random recommendation. Even though the initial agents may be completely non-compliant, our mechanism can incentivize compliance over time, thereby enabling the estimation of the treatment effect of each treatment, and minimizing the cumulative regret of the planner whose goal is to identify the optimal treatment.

Incentivizing Compliance with Algorithmic Instruments

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Incentivizing Bandit Exploration: Recommendations as Instruments

Incentivizing Compliance with Algorithmic Instruments