Navigating the Lobbying Landscape: Insights from Opinion Dynamics Models

TL;DR

The study investigates how lobbying strategies influence public opinion within an opinion-dynamics framework by introducing budget-constrained lobbyists who send costly, time-staged signals. Agents hold two probabilistic models, an optimistic and a pessimistic one, and update their beliefs with a Bayesian-like social-learning rule, modulated by under-reaction and confirmation bias, e.g. $p_{i,t}=w_{i,t}\pi_o+(1-w_{i,t})\pi_p$ and $\lambda_{i,t}=\phi_i|1-s_t-w_{i,t-1}|+(1-\phi_i)\lambda_i$. The model reveals two regimes under lobbying: a lobbyist-influence regime where a single lobbyist can dominate, and a peer-effect regime where polarization persists; with symmetric lobbyists, oscillations emerge, while frontloading is advantageous under peer-dominant dynamics and backloading under lobbyist-dominant dynamics. When two opposing lobbyists interact, convergence is delayed or fails, producing sustained oscillations; timing strategies create phase-like boundaries, and larger budgets expand the lobbyist-influence region. The framework thus links cognitive biases, strategic signaling, and network structure to potential real-world lobbying dynamics, offering a path for empirical validation and policy-relevant insights into influence operations.

Abstract

While lobbying has been demonstrated to have an important effect on public opinion and policy making, existing models of opinion formation do not specifically include its effect. In this work we introduce a new model of lobbying-driven opinion influence within opinion dynamics, where lobbyists can implement complex strategies and are characterised by a finite budget. Individuals update their opinions through a learning process resembling Bayes-rule updating but using signals generated by the other agents (a form of social learning), modulated by under-reaction and confirmation bias. We study the model numerically and demonstrate rich dynamics both with and without lobbyists. In the presence of lobbying, we observe two regimes: one in which lobbyists can have full influence on the agent network, and another where the peer-effect generates polarisation. When lobbyists are symmetric, the lobbyist-influence regime is characterised by prolonged opinion oscillations. If lobbyists temporally differentiate their strategies, frontloading is advantageous in the peer-effect regime, whereas backloading is advantageous in the lobbyist-influence regime. These rich dynamics pave the way for studying real lobbying strategies to validate the model in practice.

Figures (8)

• Figure 1: Baseline (no-lobbyist) scenario. The average effective number of clusters and the final average subjective probabilities of agents are represented as a function of strength of directional motivated reason $\phi$ and the degree of under-reaction $\lambda$. In the case where $\lambda=1.0$, $\phi = 0.0$, the number of clusters metric is not applicable, since the agents do not change their opinions over time from their initial conditions, randomly drawn from an uniform distribution in [0,1].
• Figure 2: Opinion evolution. Examples of the evolution of the agent's subjective probabilities in the baseline scenario with and without final consensus of the network.
• Figure 3: Average number of iterations in the baseline (no-lobbyist) scenario. The average number of iterations to reach out the equilibrium in the network is represented as a function of strength of directional motivated reason $\phi$ and the degree of under-reaction $\lambda$. The colorbar is in logarithmic scale.
• Figure 4: One-lobbyist scenario. The effective number of clusters and average subjective probability of the final opinion distribution is represented as a function of the strength of directional motivated reason $\phi$ and the degree of under-reaction $\lambda$ in presence of a lobbyist. The lobbyists supports the pessimistic model, has a budget $B = 10,\!000$ to send its signals and can be active for a time horizon $T=100$.
• Figure 5: Budget sensitivity analysis in one-lobbyist scenario. The average subjective probability of the final opinion distribution is represented as a function of strength of directional motivated reason $\phi$ and the degree of under-reaction $\lambda$ for different values of $B$, the budget of the lobbyist. The lobbyist supports the pessimistic model and can be active for a time horizon $T = 100$.