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Innovation diffusion dynamics toward long-term behavioral shifts

Lisa Piccinin, Valentina Breschi, Chiara Ravazzi, Fabrizio Dabbene, Mara Tanelli

TL;DR

This work extends the Friedkin-Johnsen opinion model with saturated-integral dynamics to capture long-term attitudinal shifts under structural nudging policies. It formalizes a multilayer, budget-constrained control framework and develops two policy designs: an Optimized Constant Control Policy (CCP) and a Model Predictive Control (MPC) approach to maximize social adoption while limiting costs. The authors prove asymptotic behavior under static and feedback policies and illustrate the methods via a numerical network study, showing that long-horizon, budget-aware strategies can outperform short-term nudges and prior models in achieving sustained diffusion. The results provide a principled basis for designing centralized fostering policies for greener technologies, with practical implications for policy budgeting and near-term versus long-term trade-offs.

Abstract

Sustainable technologies and services can play a pivotal role in the transition to "greener" habits. Their widespread adoption is thus crucial, and understanding how to foster this phenomenon in a systematic way could have a major impact on our future. With this in mind, in this work we propose an extension of the Friedkin-Johnsen opinion dynamics model toward characterizing the long-term impact of (structural) fostering policies. We then propose alternative nudging strategies that target a trade-off between widespread adoption and investments under budget constraints, showing the impact of our modeling and design choices on inclination shifts over a set of numerical tests.

Innovation diffusion dynamics toward long-term behavioral shifts

TL;DR

This work extends the Friedkin-Johnsen opinion model with saturated-integral dynamics to capture long-term attitudinal shifts under structural nudging policies. It formalizes a multilayer, budget-constrained control framework and develops two policy designs: an Optimized Constant Control Policy (CCP) and a Model Predictive Control (MPC) approach to maximize social adoption while limiting costs. The authors prove asymptotic behavior under static and feedback policies and illustrate the methods via a numerical network study, showing that long-horizon, budget-aware strategies can outperform short-term nudges and prior models in achieving sustained diffusion. The results provide a principled basis for designing centralized fostering policies for greener technologies, with practical implications for policy budgeting and near-term versus long-term trade-offs.

Abstract

Sustainable technologies and services can play a pivotal role in the transition to "greener" habits. Their widespread adoption is thus crucial, and understanding how to foster this phenomenon in a systematic way could have a major impact on our future. With this in mind, in this work we propose an extension of the Friedkin-Johnsen opinion dynamics model toward characterizing the long-term impact of (structural) fostering policies. We then propose alternative nudging strategies that target a trade-off between widespread adoption and investments under budget constraints, showing the impact of our modeling and design choices on inclination shifts over a set of numerical tests.

Paper Structure

This paper contains 15 sections, 5 theorems, 32 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Modeling long-term attitude shifts
  3. Interpreting our model as a multilayer one
  4. A closer look at the model's properties
  5. Expected inclinations with constant policies
  6. Comparison with c26 under static policies
  7. Feedback policies with constraints and budget limitations
  8. Toward optimal nudging policy design
  9. Optimized Constant Control Policy (CCP)
  10. Model Predictive Control (MPC) Fostering Policy
  11. Numerical Example
  12. Policy nudged short-term vs long-term shifts
  13. Analyzing the impact of different budgets
  14. Static vs receding horizon policy
  15. Conclusions

Key Result

Lemma 1

Let Assumption ass:P hold, $x(0) \in [0,1]^{N}$ and $u^{c}(t)=0$ for all $t \in \mathbb{N}_0$. Then, the latent state's expected value satisfies with $\mu_{\infty} \in [0,1]^{N}$.

Figures (4)

  • Figure 1: A multilayer representation of the opinion dynamics model in \ref{['eq:dyn_model']}.
  • Figure 2: The social network considered in our example, featuring $20$ agents and $7$ clusters of agents.
  • Figure 3: Negatively biased scenario: evolution of latent inclinations under different budget constraints.
  • Figure 4: $\Gamma_{\mathrm{sim}}$vs$u_{\mathrm{sim}}^{\Sigma}$: constant vs receding horizon policy.

Theorems & Definitions (13)

  • Remark 1: The validity of \ref{['eq:dyn_model']}
  • Lemma 1: Control-free mean asymptotic opinions
  • proof
  • Remark 2: Lemma \ref{['lemma:steady_free']} and our modeling choices
  • Proposition 1: Asymptotic opinions under static policies
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 3 more