Modelling the coevolution of opinion dynamics and decision making in social dilemmas

Abstract: This paper proposes a mathematical model for the coevolution of actions and opinions for a population facing a social dilemma. In particular, we assume each person participates in a Public Goods Game (PGG), with their action being to cooperate or defect, and holds an opinion about which action they prefer. We propose a payoff function that combines the PGG with the Friedkin--Johnsen model from opinion dynamics to form a coevolutionary game. According to a discrete-time process, players asynchronously update their actions and opinions, aiming to maximise their individual payoff for the coevolutionary game using myopic best-response. We study the equilibria and provide conditions for the existence of the all-defection and all-cooperation consensus equilibria. We also establish conditions for global convergence to the all-defection equilibrium.

• The paper presents a coevolutionary model that integrates discrete cooperation/defection decisions with continuous opinion dynamics using a modified Friedkin–Johnsen framework.
• The paper derives explicit conditions under which consensus (all-cooperation or all-defection) emerges, highlighting the balance between material incentives and social conformity.
• The paper demonstrates global stability results for consensus equilibria, offering insights into the design of interventions in collective action scenarios.

Modelling the Coevolution of Opinion Dynamics and Decision Making in Social Dilemmas

Introduction

This paper introduces a rigorous coevolutionary game-theoretic model that synthesizes binary action selection—cooperate or defect—in repeated Public Goods Games (PGG) with continuous opinion evolution on a social influence network. The central premise is to account for the empirical observation that social dilemma behaviors arise from the dynamic interplay of individual beliefs and collective social influence, with explicit mathematical distinction between actions and underlying attitudes or preferences. The model departs from classical approaches by integrating the Friedkin–Johnsen (FJ) opinion dynamics mechanism and considering an agent's intrinsic drive for consistency between opinions and actions, thus capturing phenomena such as motivated reasoning and cognitive consonance within the decision process itself.

Mathematical Framework

Agents are embedded in a weighted, possibly undirected, social network. At each discrete time step, each agent holds a binary action ($x_i \in \{0,1\}$, for defection/cooperation) and a continuous opinion ($y_i \in [0,1]$ supports cooperation). The PGG payoff is parameterized by a public good multiplier $r$ and population size $n$; payoffs are linear in the number of cooperators, corresponding to the standard PGG with uniform contribution. Opinion dynamics follow the FJ model, featuring neighbor influence modulated by a stochastic weight matrix and optional stubborn attachment to a prior prejudice $u_i$, here set to zero in the main results for analytic tractability.

A novel payoff function for each agent combines three components:

1. PGG incentive: Immediate material payoff from the public-good interaction ($\pi^a$).
2. Social conformity: Utility from holding opinions congruent with neighbors (FJ mechanism, $\pi^o$).
3. Self-consistency: Penalty for divergence between action and opinion, reflecting cognitive dissonance.

Critically, the agent uses a myopic best-response process to simultaneously select both their action and update their opinion to maximize the total convex-combination payoff.

Analytical Characterization of Equilibria

Nash equilibria of the repeated coevolutionary game correspond to system equilibria, where neither actions nor opinions change for any agent under best-response. The analysis isolates consensus equilibria—states of unanimous action and/or opinion—and provides explicit conditions for their existence and stability.

Existence of Consensus Equilibria

• All-Defection Consensus: Always exists. Under mild parameter regimes (specifically, when relative weighting of public-good disincentives outweighs the joint force of conformity and self-consistency), this state is globally attractive. The precise condition is:

$$\frac{\beta_i \lambda_i}{\beta_i + \lambda_i} \leq 2\alpha_i \left(1 - \frac{r}{n}\right) \quad \forall i,$$

where $\alpha_i$, $\beta_i$, $y_i \in [0,1]$0 are agent-level weights for PGG, social, and self-consistency payoffs, respectively.

• All-Cooperation Consensus: Exists if and only if

$y_i \in [0,1]$1

implying collective action is sustained in equilibrium if social and cognitive factors jointly compensate for material incentives to defect.

Notably, the analysis proves that any action consensus at equilibrium implies the same consensus in opinions, due to the structure of the best-response update and the spectral properties of the underlying influence matrix.

Uniqueness and Global Stability

If the all-defection consensus condition strictly holds for all agents, it not only exists but is the unique equilibrium, and the system globally converges to it from any initial condition, given a symmetric, irreducible (i.e., connected), and row-stochastic influence network.

The convergence proof leverages a Lyapunov potential argument: after action convergence, the remaining opinion dynamics become a contraction in the FJ style, with the unique minimizer of the global potential corresponding to uniform defection-supporting opinions.

Mixed Equilibria

While the main results focus on consensus regions, the paper notes that for intermediate parameter regimes, the structure allows for equilibria with mixtures of cooperators and defectors. These cases are left open for further analysis regarding their existence, stability, and possible selection dynamics.

Implications

Theoretical Consequences

The explicit coupling of discrete actions (materially relevant in game-theoretic settings) and continuous opinions (richly structured by social influence) advances the understanding of social decision making in dilemmas beyond previous continuous-opinion/discrete-action models. In particular, the explicit self-consistency term formalizes the cognitive friction between belief and action, producing equilibria dependent on the joint effect of incentive structures and social conformity mechanisms.

The analytic results reveal parameter regimes where social mechanisms alone are insufficient to support collective action if material incentives are too sharply misaligned—providing a formal underpinning for why free-riding predominates in certain societies. Conversely, sufficiently strong social pressure and cognitive consonance drive can compensate for incentive misalignment and stabilize cooperation, offering insight into how social movements may persist or fail.

Practical Relevance

The model forms a basis for designing interventions targeting not only incentive structure (e.g., adjusting $y_i \in [0,1]$2 in institutional design) but also the topology and weighting of influence networks and the salience of norm/attitude consistency. Applications include forecasting or steering collective behavior in open-participation movements, protest mobilization, climate-change collective action, or organizational cooperation.

Directions for Future Work

Several extensions are suggested:

• Explicit characterization of regions of attraction for cooperation-dominant equilibria and mixed states.
• Exploration of heterogeneity in prejudice ($y_i \in [0,1]$3) and its effect on equilibria structure and convergence.
• Relaxation to more general PGG payoff forms and nonlinearities.
• Stochastic best-response and experimentation beyond myopic optimization.
• Empirical validation via data from observed protest or collective-action episodes.

Conclusion

This paper rigorously defines and analyzes a coevolutionary game-theoretic model of opinion-action coupling in social dilemmas, delineating explicit conditions for consensus equilibria and their stability. The findings formalize the interplay of incentive, social, and cognitive factors in shaping collective behavior, contributing to both theoretical understanding and the design of mechanisms or interventions in multi-agent social systems. Future work is anticipated to expand on nonconsensus equilibria and incorporate richer psychological and network-based heterogeneity.