In mature markets where total demand is fixed, competitive growth depends not only on attracting customers but also on the timing of their purchases. Yet most quantitative marketing models--whether probabilistic choice, state-dependent demand, or network-based diffusion--capture which option consumers switch to but not when switching occurs, despite substantial heterogeneity in purchase timing driven by frictions, habits, and depletion dynamics.
We address this gap by introducing a heterogeneous-time Markov network that jointly models probabilistic and temporal transitions in purchase behavior. Methodologically, we derive closed-form expressions for market share, sales rates, and inter-purchase times in this heterogeneous-time environment, enabling analytical tractability without distributional assumptions or simulation. Building on these results, we develop a gradient-based optimization framework that respects fixed market demand and identifies minimal, targeted interventions for shifting share. Using IRI panel data, we demonstrate that jointly optimizing switching probabilities and switching times yields significantly larger gains and requires smaller deviations from observed behavior than probability-only approaches.
Our work provides a modern quantitative framework for understanding and optimizing competitive dynamics in mature, zero-sum markets, and highlights the role of temporal heterogeneity as an underexplored driver of firm performance.
