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Causal Inference on Stopped Random Walks in Online Advertising

Jia Yuan Yu

TL;DR

The paper addresses estimating long-term causal effects of online-advertising treatments under interference, where a treatment alters user trajectories, advertiser budgets, and population size. It develops a Markov-chain framework with stopped random walks to model long-run revenue and introduces a permutation trick to reduce state-space complexity, enabling tractable analysis. A budget-splitting experimental design together with a Markov-chain CLT and a Wald-like equation yields a confidence interval for the long-term treatment effect $\Delta$, including guidance on experiment duration via asymptotic variances and stopping-time considerations. The work advances causal inference in two-sided marketplaces by handling random population sizes, interdependencies, and non-i.i.d. measurements, with practical implications for designing long-running experiments in large online advertising ecosystems.

Abstract

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.

Causal Inference on Stopped Random Walks in Online Advertising

TL;DR

The paper addresses estimating long-term causal effects of online-advertising treatments under interference, where a treatment alters user trajectories, advertiser budgets, and population size. It develops a Markov-chain framework with stopped random walks to model long-run revenue and introduces a permutation trick to reduce state-space complexity, enabling tractable analysis. A budget-splitting experimental design together with a Markov-chain CLT and a Wald-like equation yields a confidence interval for the long-term treatment effect , including guidance on experiment duration via asymptotic variances and stopping-time considerations. The work advances causal inference in two-sided marketplaces by handling random population sizes, interdependencies, and non-i.i.d. measurements, with practical implications for designing long-running experiments in large online advertising ecosystems.

Abstract

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.
Paper Structure (18 sections, 6 theorems, 43 equations, 2 figures)

This paper contains 18 sections, 6 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Markovian dynamics
  3. Random treatment-dependent population
  4. Application to ad auctions
  5. Model
  6. Permutation trick to reduce dimensionality
  7. State-transition dynamics for online advertising
  8. Publisher revenue
  9. Inference
  10. Expected long-term treatment effect
  11. Experimental design: budget-splitting
  12. Main result: confidence interval for $\Delta$
  13. Related Works
  14. Discussions
  15. Proofs
  16. ...and 3 more sections

Key Result

Theorem 3.6

Suppose that the Markov chains $\Phi(v)$, $\Phi^A(v)$, $\Phi(w)$, $\Phi^B(w)$ satisfy Assumptions as:session, as:1, as:2, and as:K. Let $\pi(v)$, $\pi'$, $\pi(w)$, and $\pi"$ denote their respective invariant distributions. Let $\sigma'$ and $\sigma"$ denote the asymptotic variances of $\Phi^A(v)$ a where $z_{\alpha/2} = \Psi^{-1}(1 - \alpha/2)$ and $\Psi$ is the standard Normal distribution funct

Figures (2)

  • Figure 1: Permutation $\sigma$ with $T=6$ that groups page-transitions together by user-session.
  • Figure 2: Auctioneer revenue $r(\cdot,Y)$ for a single auction with fixed bid profile $(1,2)$.

Theorems & Definitions (16)

  • Remark 2.2: Two time scales
  • Definition 3.1: Markov random walk, stopped random walk
  • Remark 3.4: Invariant distribution
  • Theorem 3.6: Confidence interval for $\Delta$
  • Remark 3.7
  • Remark 3.8: Estimating ${\sigma'}^2$ and ${\sigma"}^2$
  • Theorem A.1: CLT jones
  • Theorem A.2: Wald-like equation
  • Lemma A.3: Anscombe for $\Phi(v)$ and $\Phi^A(v)$
  • proof
  • ...and 6 more