Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Causal Inference from Competing Treatments

Ana-Andreea Stoica, Vivian Y. Nastl, Moritz Hardt

TL;DR

The paper addresses causal inference when multiple treatment administrators compete for attention, proposing a joint game-theoretic framework where rank-based effects attenuate the treatment impact. It introduces a tractable sample-value objective that can be analyzed via Nash equilibria, linking budget allocation to estimation efficiency. The key contributions are (i) a minimax-based estimation-error objective and its tractable surrogate, (ii) a proven pure Nash equilibrium for the sample-value game and an approximation guarantee to the original objective under broad conditions, and (iii) an allocation rule A_{prob} with concentration properties enabling equilibrium analysis and practical budget-spending guidance. This work bridges causal inference and mechanism design, offering principled guidance for conducting adaptive experiments and advertising campaigns under competition, with implications for external validity and decision-making under rank effects.

Abstract

Many applications of RCTs involve the presence of multiple treatment administrators -- from field experiments to online advertising -- that compete for the subjects' attention. In the face of competition, estimating a causal effect becomes difficult, as the position at which a subject sees a treatment influences their response, and thus the treatment effect. In this paper, we build a game-theoretic model of agents who wish to estimate causal effects in the presence of competition, through a bidding system and a utility function that minimizes estimation error. Our main technical result establishes an approximation with a tractable objective that maximizes the sample value obtained through strategically allocating budget on subjects. This allows us to find an equilibrium in our model: we show that the tractable objective has a pure Nash equilibrium, and that any Nash equilibrium is an approximate equilibrium for our general objective that minimizes estimation error under broad conditions. Conceptually, our work successfully combines elements from causal inference and game theory to shed light on the equilibrium behavior of experimentation under competition.

Causal Inference from Competing Treatments

TL;DR

The paper addresses causal inference when multiple treatment administrators compete for attention, proposing a joint game-theoretic framework where rank-based effects attenuate the treatment impact. It introduces a tractable sample-value objective that can be analyzed via Nash equilibria, linking budget allocation to estimation efficiency. The key contributions are (i) a minimax-based estimation-error objective and its tractable surrogate, (ii) a proven pure Nash equilibrium for the sample-value game and an approximation guarantee to the original objective under broad conditions, and (iii) an allocation rule A_{prob} with concentration properties enabling equilibrium analysis and practical budget-spending guidance. This work bridges causal inference and mechanism design, offering principled guidance for conducting adaptive experiments and advertising campaigns under competition, with implications for external validity and decision-making under rank effects.

Abstract

Many applications of RCTs involve the presence of multiple treatment administrators -- from field experiments to online advertising -- that compete for the subjects' attention. In the face of competition, estimating a causal effect becomes difficult, as the position at which a subject sees a treatment influences their response, and thus the treatment effect. In this paper, we build a game-theoretic model of agents who wish to estimate causal effects in the presence of competition, through a bidding system and a utility function that minimizes estimation error. Our main technical result establishes an approximation with a tractable objective that maximizes the sample value obtained through strategically allocating budget on subjects. This allows us to find an equilibrium in our model: we show that the tractable objective has a pure Nash equilibrium, and that any Nash equilibrium is an approximate equilibrium for our general objective that minimizes estimation error under broad conditions. Conceptually, our work successfully combines elements from causal inference and game theory to shed light on the equilibrium behavior of experimentation under competition.
Paper Structure (33 sections, 15 theorems, 122 equations, 3 figures, 1 algorithm)

This paper contains 33 sections, 15 theorems, 122 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Games and objectives
  5. Objective design
  6. Estimation error objective.
  7. Sample value objective.
  8. Main results
  9. Nash equilibrium.
  10. Allocation rule:
  11. A causal inference analysis of the error minimization objective
  12. Data generating process.
  13. Potential outcomes model.
  14. Estimand: ATE but only for the first rank.
  15. An optimality argument
  16. ...and 18 more sections

Key Result

Theorem 2.1

A budget allocation game $\mathcal{G}_{\mathrm{SV}}$ with multiple treatment administrators has a pure Nash equilibrium for all sequences $(\alpha_r)_r$ such that $\alpha_r \in (0,1)$ and $\alpha_1 = 1$. Moreover, for $B^{(a)} \leq n$, an allocation in which all treatment administrators split their

Figures (3)

  • Figure 1: Illustration of the objective set-up and results.
  • Figure 2: Illustration of the data-generating process for an admin. In Part $1$, the admins compete with each other to obtain different ranks for subject slots. After placing the bids, a rank allocation rule is applied. Each admin receives a rank for all slots for which they have bid. For example, the admin depicted in yellow received rank $1$ for subject slots $1$, rank $3$ for subject slots $2$ and $3$, no rank for subject slot $4$ (which happens if an admin does not bid on a slot), and rank $2$ for subject slot $5$. In Part $2$, individuals are sampled at the allocated ranks, with treatment and control assigned in a randomized manner. The collected data for the admin depicted in yellow consists of tuples $(r_i, T_i, Y_i)$ for all individuals, where $r_i$ represents the rank, $T_i$ the binary treatment-control variable, and $Y_i$ the outcome.
  • Figure 3: Example of minimax lower bound with $q=0.5$ and $\sigma^2=4$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 3.3
  • Theorem 3.4: Minimax Lower Bound
  • Proposition 3.5
  • Proposition 3.6
  • Lemma 4.1
  • Conjecture 4.2
  • Proposition 4.3
  • ...and 23 more