A fundamental problem of hypothesis testing with finite inventory in e-commerce

TL;DR

This paper investigates how finite inventory constraints in e-commerce A/B testing can lead to high false positive and negative rates, challenging the assumption of independent groups and providing analytical tools for better experiment design.

Contribution

It introduces a simplified 2D random walk model to analyze false positive rates under finite inventory constraints in hypothesis testing.

Findings

High false-positive rates can occur due to inventory constraints.

Finite inventory can also cause high false-negative rates.

The paper provides a closed-form expression for false-positive probability.

Abstract

In this paper, we draw attention to a problem that is often overlooked or ignored by companies practicing hypothesis testing (A/B testing) in online environments. We show that conducting experiments on limited inventory that is shared between variants in the experiment can lead to high false positive rates since the core assumption of independence between the groups is violated. We provide a detailed analysis of the problem in a simplified setting whose parameters are informed by realistic scenarios. The setting we consider is a $2$-dimensional random walk in a semi-infinite strip. It is rich enough to take a finite inventory into account, but is at the same time simple enough to allow for a closed form of the false-positive probability. We prove that high false-positive rates can occur, and develop tools that are suitable to help design adequate tests in follow-up work. Our results also show that high false-negative rates may occur. The proofs rely on a functional limit theorem for the $2$-dimensional random walk in a semi-infinite strip.