Optimal Policy Design for Repeated Decision-Making under Social Influence

TL;DR

This work extends the Friedkin–Johnsen opinion dynamics model by incorporating random uncontrollable perturbations and external budgeted nudges, while enforcing a time-scale separation between rapid individual decisions and slower social imitation. It establishes ergodicity and convergence to a steady-state mean $x^*=(I-\Lambda \bar P)^{-1}(I-\Lambda)\eta^0$, and shows that $E[y(t)]=x^*$ in the long run, enabling model-based policy design. Two policy classes are developed: (i) constant nudges (MFCCP and MBCCP) that optimize asymptotic acceptance via a quadratic program, and (ii) a Model Predictive Control framework that handles transients, with both oracle and estimated-mean variants. Numerical results on a 100-node modular network demonstrate that adaptive MPC approaches substantially improve acceptance of a target action under budget constraints, with performance depending on the time-scale separation parameter $\alpha$ and network topology.

Abstract

In this paper, we present a novel model to characterize individual tendencies in repeated decision-making scenarios, with the goal of designing model-based control strategies that promote virtuous choices amidst social and external influences. Our approach builds on the classical Friedkin and Johnsen model of social influence, extending it to include random factors (e.g., inherent variability in individual needs) and controllable external inputs. We explicitly account for the temporal separation between two processes that shape opinion dynamics: individual decision-making and social imitation. While individual decisions occur at regular, frequent intervals, the influence of social imitation unfolds over longer periods. The inclusion of random factors naturally leads to dynamics that do not converge in the classical sense. However, under specific conditions, we prove that opinions exhibit ergodic behavior. Building on this result, we propose a constrained asymptotic optimal control problem designed to foster, on average, social acceptance of a target action within a network. To address the transient dynamics of opinions, we reformulate this problem within a Model Predictive Control (MPC) framework. Simulations highlight the significance of accounting for these transient effects in steering individuals toward virtuous choices while managing policy costs.

Figures (4)

• Figure 1: The social network with $100$ nodes considered in the numerical example.
• Figure 2: Free evolution of the system: evolution of hidden and manifested options and their time averages over time.
• Figure 3: Control cost vs social cost for different policies for modular graphs with inter-cluster connectivity $\gamma=0.9$ for $r\in[10^-12,5]$ and different values of $\alpha$. Dotted lines indicate sample means, and shaded areas represent empirical standard deviations across 20 Monte Carlo simulations. The blue curve is used as a reference and represents the average social cost when no control is enacted.
• Figure 4: Box and scatter plot showing the relative shift of social cost for $50$ realizations of random modular graphs with $100$ nodes for each value of the inter-cluster connectivity parameter $\gamma$ for different values of $\alpha$.

Theorems & Definitions (25)

• Definition 1: Ergodicity of a random process
• Remark 1: Dynamics and external factors
• Proposition 1: Expected hidden inclinations' dynamics
• proof
• Corollary 1: Expected acceptance variables
• proof
• Theorem 1: Ergodicity of hidden inclinations
• proof
• Theorem 2: Ergodicity of acceptance variables
• proof