Causal Inference on Stopped Random Walks in Online Advertising

We consider a causal inference problem frequently encountered in online advertising systems, where a publisher (e.g., Instagram, TikTok) interacts repeatedly with human users and advertisers by sporadically displaying to each user an advertisement selected through an auction. Each treatment corresponds to a parameter value of the advertising mechanism (e.g., auction reserve-price), and we want to estimate through experiments the corresponding long-term treatment effect (e.g., annual advertising revenue). In our setting, the treatment affects not only the instantaneous revenue from showing an ad, but also changes each user’s interaction-trajectory, and each advertiser’s bidding policy – as the latter is constrained by a finite budget. In particular, each a treatment may even affect the size of the population, since users interact longer with a tolerable advertising mechanism. We drop the classical i.i.d. assumption and model the experiment measurements (e.g., advertising revenue) as a stopped random walk, and use a budget-splitting experimental design, the Anscombe Theorem, a Wald-like equation, and a Central Limit Theorem to construct confidence intervals for the long-term treatment effect.

💡 Research Summary

The paper tackles a fundamental causal inference problem that arises in modern online advertising platforms: estimating the long‑term impact of a change in the ad‑allocation mechanism (for example, adjusting the auction reserve price) on a publisher’s revenue. Traditional A/B testing assumes independent and identically distributed (i.i.d.) observations, which fails in this setting because the treatment simultaneously influences user behavior, advertiser bidding, and budget consumption, thereby altering the very data‑generating process.

To address these challenges the authors construct a probabilistic framework that models the three‑way interaction among users, advertisers, and the publisher as a coupled Markov chain. Each user session is represented by a sequence of page states (X_n); each advertiser’s remaining budget is captured by a vector (\Omega_n); and the active user at time (n) is denoted by (I_n). The joint process ((X_n,\Omega_n,I_n)_{n\ge1}) evolves according to stationary policies: the publisher sets a reserve price (p_n = p(X_n)), advertisers submit bids (Y_n = Y(X_n,p_n,\Omega_n)), the auction outcome determines the next page state via a transition function (h), and budgets are updated through a slower‑timescale function (g).

Because the treatment changes both the instantaneous revenue per impression and the length of the user‑session trajectory, the population size (i.e., the total number of impressions observed) becomes random and treatment‑dependent. The authors therefore adopt a budget‑splitting experimental design: the total advertising budget is divided into two independent pools, each allocated to either the treatment or control condition. For each condition the cumulative revenue up to a random stopping time (the moment the allocated budget is exhausted) is recorded. This yields two stopped random walks \