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Toulis, Panagiotis

Planned experiments are the gold standard in reliably comparing the causal effect of switching from a baseline policy to a new policy. % One critical shortcoming of classical experimental methods, however, is that they typically do not take into account the dynamic nature of response to policy changes. For instance, in an experiment where we seek to understand the effects of a new ad pricing policy on auction revenue, agents may adapt their bidding in response to the experimental pricing changes. Thus, causal effects of the new pricing policy after such adaptation period, the {\em long-term causal effects}, are not captured by the classical methodology even though they clearly are more indicative of the value of the new policy. % Here, we formalize a framework to define and estimate long-term causal effects of policy changes in multiagent economies. Central to our approach is behavioral game theory, which we leverage to formulate the ignorability assumptions that are necessary for causal inference. Under such assumptions we estimate long-term causal effects through a latent space approach, where a behavioral model of how agents act conditional on their latent behaviors is combined with a temporal model of how behaviors evolve over time.

The ideal estimation method needs to fulfill three requirements: (i) efficient computation, (ii) statistical efficiency, and (iii) numerical stability. The classical stochastic approximation of (Robbins, 1951) is an iterative estimation method, where the current iterate (parameter estimate) is updated according to some discrepancy between what is observed and what is expected assuming the current iterate has the true parameter value. Classical stochastic approximation undoubtedly meets the computation requirement, which explains its widespread popularity, for example, in modern applications of machine learning with large data sets, but cannot effectively combine it with efficiency and stability. Surprisingly, the stability issue can be improved substantially, if the aforementioned discrepancy is computed not using the current iterate, but using the conditional expectation of the next iterate given the current one. The computational overhead of the resulting implicit update is minimal for many statistical models, whereas statistical efficiency can be achieved through simple averaging of the iterates, as in classical stochastic approximation (Ruppert, 1988). Thus, implicit stochastic approximation is fast and principled, fulfills requirements (i-iii) for a number of popular statistical models including generalized linear models, M-estimation, and proportional hazards, and it is poised to become the workhorse of estimation with large data sets in statistical practice.

The effect of an intervention in an economic mechanism, for example an increase in the reserve price of an auction, is causal if the observed effect is better than the counterfactual, i.e., the effect that would be observed under no intervention. As mechanisms are populated by dynamical systems of interacting agents, their response to an intervention fluctuates until the system reaches a new equilibrium. Effects measured in the new equilibrium, the long-term causal effects, are more representative of the value of interventions. However, the statistical estimation of long-term causal effects is difficult because it has to rely, for practical reasons, on data observed before the new equilibrium is reached. Furthermore, agent actions do not only depend on the mechanism that the agents are situated in but also on the behavior of others, which complicates the causal evaluation. In this paper, we formalize this problem of estimating long-term causal effects under the potential outcomes framework of causal inference [17, 21]. We develop an estimation method that relies on a data augmentation strategy, where agents are assumed to adopt, at each timepoint, a behavior that is latent. This allows us to leverage existing work in behavioral game theory and time-series analysis of compositional data. Our method identifies the long-term causal effects under a set of assumptions that we formulate explicitly. We illustrate our method on a dataset from a real-world behavioral experiment, and discuss open problems to stimulate future research.