Optimal promotions of new products on networks

TL;DR

This work develops an exact analytic framework for optimal promotions of new products on networks under the Bass diffusion model, by coupling master equations with Pontryagin's maximum principle to obtain a system of $2^M-1$ boundary-value problems. On structured networks, symmetry reductions yield tractable finite systems, with the infinite complete network limit reducing further to a single boundary-value problem, and the infinite line yielding a four-equation system. Key findings include: the external advertising control $s_p^{opt}(t)$ generally decreases over time while the peer-effect control $s_q^{opt}(t)$ rises from zero to a peak and then declines; horizon length critically affects promotion effectiveness, with short planning horizons producing much larger profit gains and predominantly expansion-driven benefits. The framework is demonstrated across finite and infinite networks, including complete and heterogeneous structures, and shows that uniform promotions are optimal on homogeneous complete networks, while heterogeneity can be incorporated through group-specific controls. Overall, the results illuminate how network structure and planning horizon shape optimal marketing strategies and their economic impact, offering analytic guidance for network-aware promotional campaigns.

Abstract

We present a novel methodology for analyzing the optimal promotion in the Bass model for the spreading of new products on networks. For general networks with $M$ nodes, the optimal promotion is the solution of $2^M-1$ nonlinearly-coupled boundary-value problems. On structured networks, however, the number of equations can be reduced to a manageable size which is amendable to simulations and analysis. This enables us to gain insight into the effect of the network structure on optimal promotions. We find that the optimal advertising strategy decreases with time, whereas the optimal boosting of peer effects increases from zero and then decreases. In low-degree networks, it is optimal to prioritize advertising over boosting peer effects, but this relation is flipped in high-degree networks. When the planning horizon is finite, the optimal promotion continues until the last minute, as opposed to an infinite planning horizon where the optimal promotion decays to zero. Finally, promotions with short planning horizons can yield an order of magnitude higher increase of profits, compared to those with long planning horizons.

Figures (9)

• Figure 1: A) The optimal promotion on a complete homogeneous network with $M=3$ nodes. The dash, dash-dot, and solid lines are $s_p^{\rm opt}(t)$, $s_q^{\rm opt}(t)$, and $s_p^{\rm opt}(t)+s_q^{\rm opt}(t)$, respectively. B) The expected adoption levels $f^{\rm opt}(t)$ in the presence of the optimal promotion (solid) and $f^{0}(t)$ in the absence of a promotion (dashes). Here $p_0=0.01$, $b_p=0.01$, $q_0=0.1$, $b_q=0.1$, $\gamma=1000$, $\theta=0.01$, and $T=\infty$. Here, $\Delta\Pi^{\rm opt}\approx 12\%$.
• Figure 2: A-D) Same as Figure \ref{['fig:pq_opt_M2']}A, for various values of $M$. E) The dependence of $\Delta\Pi^{\rm opt}$ on $M$.
• Figure 3: Same as Figure \ref{['fig:pq_opt_M2']} for an infinite complete network. Here, $\Delta\Pi^{\rm opt}=8.5\%$.
• Figure 4: Same as Figure \ref{['fig:pq_opt']} for T=20. Here, $\Delta\Pi^{\rm opt}=118\%$.
• Figure 5: Same as Figure \ref{['fig:pq_opt']}A for various values of $p_0$. A) $p_0=0.01$. B) $p_0=0.03$.

Theorems & Definitions (45)

• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• Theorem 3: Bass-monotone-convergence-23
• Theorem 4
• proof
• Corollary 2