Optimal policy design for decision problems under social influence

TL;DR

This work extends the Friedkin-Johnsen opinion model to include personalized policy actions and stochastic disturbances, enabling explicit control of opinion dynamics toward a target choice. By deriving the ergodic properties and the mean-dynamics fixed point, it enables reformulation of the infinite-horizon objective in terms of the mean opinion and develops practical MPC-based policy designs, including conservative and alternative losses, plus estimation-based variants for partial observability. The main contributions are (i) a stochastic, personalized-influence model with external controls and noise, (ii) finite-horizon, tractable policy design strategies with provable contraction behavior, and (iii) a numerical case study illustrating performance–cost trade-offs and robustness to disturbances. The framework provides a principled approach to nudging behavior in social networks under uncertainty, with potential applications in green mobility adoption and other public-interest decisions, while highlighting ethical considerations and directions for future theoretical guarantees and real-world validation.

Abstract

This paper focuses on describing the impact of policy actions on individuals' opinions in the presence of social and external influences toward proposing preliminary nudging strategies to achieve a cost-effectiveness trade-off. To this end, we extend the classical Friedkin and Johnsen model of opinion dynamics to incorporate random factors, such as variability in individual predispositions due to uncontrolled events (e.g., modeling the impact of the weather on daily mobility choices), and describe the impact of personalized policies. Furthermore, we formulate an optimal control problem aimed at fostering the social acceptance of particular actions/choices within the network. Through our analysis and numerical simulations, we illustrate the features of the proposed model in the absence of nudging and the effectiveness of the proposed (optimal) nudging strategies.

Figures (6)

• Figure 1: The clustered social network with $20$ agents considered in our examples.
• Figure 2: Opinion dynamics in the considered network of $20$ individuals.
• Figure 3: Disturbance-free case: states and inputs for the best-performing policies, i.e., (WC) and (E-TV) for scenario 2 and $\lambda=0.25$ over time. The green curves are associated with agents with positive bias ($u_{\mathrm{o},v}=0.8$), while the red ones are associated with agents having $u_{\mathrm{o},v}=0.2$.
• Figure 4: Disturbance-free case: states and inputs for the best-performing policies, i.e., (WC) and (TV) for scenario 2 and $\lambda=0.75$ over time. The green curves are associated with agents with positive bias ($u_{\mathrm{o},v}=0.8$), while the red ones are associated with agents having $u_{\mathrm{o},v}=0.2$.
• Figure 5: Noisy case: states and inputs for (E-WC) and (E-TV) for scenario 2 and $\lambda=0.25$ over time. The green curves are associated with agents with positive bias ($u_{\mathrm{o},v}=0.8$), while the red ones are associated with agents having $u_{\mathrm{o},v}=0.2$.

Theorems & Definitions (9)

• Definition 1: Ergodic process
• Remark 1: Heterogeneous updates
• Proposition 1: Expected opinion dynamics
• Conjecture 1: Ergodicity of latent and manifest opinions
• Proposition 2
• proof
• Remark 2: Input feasibility set
• Remark 3: Nudging decisions & ethical issues
• Remark 4: Optimality over an infinite horizon