Symmetries, Scaling Laws and Phase Transitions in Consumer Advertising Response
Javier Marin
TL;DR
This paper introduces a physics-inspired framework to model consumer advertising response, leveraging symmetries, scaling laws, and phase transitions to capture nonlinear saturation effects. It proposes a novel equation, $y = C x^{\alpha}(1 - e^{\beta x})^{-{\gamma}}$, with interpretations for baseline effectiveness and three sensitivity indices, designed to reflect symmetry breaking and critical-point dynamics. The model is validated against Michaelis–Menten and Hill benchmarks using three dummy MMM-based datasets, with analyses conducted under standard OLS and restricted regression formats; results highlight the proposed equation’s ability to fit saturation behavior and capture effects near $x = 0$. The work offers practical insights for channel-specific budget allocation and audience structuring, while noting limitations of synthetic data and pointing to future validation on real campaigns with time-varying spend.
Abstract
Understanding how consumers respond to business advertising efforts is essential for optimizing marketing investment. This research introduces a new modeling approach based on the concepts of symmetries and scaling laws in physics to describe consumer response to advertising dynamics. Drawing from mathematical frameworks used in physics and social sciences, we propose a model that accounts for a key aspect: the saturation effect. The model is validated against commonly used models, including the Michaelis-Menten and Hill equations, showing its ability to better capture nonlinearities in advertising effects. We introduce new key parameters like Marketing Sensitivity, Response Sensitivity, and Behavioral Sensitivit, that offer additional insights into the drivers of audience engagement and advertising performance. Our model provides a rigorous yet practical tool for understanding audience behavior, contributing to the improvement of budget allocation strategies.