Fairness-aware design of nudging policies under stochasticity and prejudices

Abstract: We present an injustice-aware innovation-diffusion model extending the Generalized Linear Threshold framework by assigning agents activation thresholds drawn from a Beta distribution to capture the stochastic nature of adoption shaped by inequalities. Because incentive policies themselves can inadvertently amplify these inequalities, building on this model, we design a fair Model Predictive Control (MPC) scheme that incorporates equality and equity objectives for allocating incentives. Simulations using real mobility-habit data show that injustice reduces overall adoption, while equality smooths incentive distribution and equity reduces disparities in the final outcomes. Thus, incorporating fairness ensures effective diffusion without exacerbating existing social inequalities.

• The paper introduces a stochastic generalized linear threshold model with Beta-distributed adoption thresholds and credibility-weighted social influence, linking reluctance and prejudice to technology diffusion.
• The paper combines model predictive control with separate equity and equality objectives, showing that equitable policies target disadvantaged non-adopters while equality produces a more even distribution of incentives.
• The paper finds that receptivity deficits reduce adoption more sharply than credibility deficits under fair constraints, while combined deficits are hardest to overcome and expose a clear adoption–fairness trade-off.

Overview and contribution

This paper addresses the design of nudging policies for accelerating the diffusion of green technologies—specifically electric vehicles (EVs)—when both stochastic adoption behavior and social prejudices shape the diffusion process. The authors extend their prior deterministic, epistemically-aware Linear Threshold Model (LTM) (Villa et al., 28 Mar 2025) to a stochastic Generalized Linear Threshold Model (GLTM) framework (Kagan et al., 2024), and embed it in a Model Predictive Control (MPC) scheme that jointly optimizes adoption performance and two distinct notions of fairness: equality of resource distribution and equity of outcomes. The framework is evaluated on a real dataset of mobility habits from an EU-wide survey [fiorello2015eu], restricted to $N = 112$ respondents in the metropolitan area of Milan.

The central modeling claim is that credibility deficits—a form of testimonial epistemic injustice in Fricker's sense—directly alter who influences innovation diffusion, and therefore cannot be ignored when designing incentive policies. The paper's empirical finding is twofold: (i) credibility and receptivity deficits reduce achievable adoption rates, with the combined deficit (CRD) being hardest to counteract; and (ii) fairness objectives reshape resource allocation in interpretable ways—equality smooths incentives across agents while equity targets disadvantaged agents—at a measurable cost in overall adoption.

A stochastic, epistemic-informed cascade model

The model operates on a strongly connected influence graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ over $N$ agents. Each agent carries a binary, irreversible adoption state $x_v(t)\in\{0,1\}$ and a reluctance $\rho_v(t)\in[0,1]$ initialized as $\rho_v(0)=1-u^o_v$, where $u^o_v$ is an inherent bias toward the technology. External nudging policies act on reluctance via the linear dynamics

$$\rho_v(t+1) = \rho_v(t) + b_v u_v(t), \qquad b_v \in [-1,0],$$

so interventions lower barriers rather than flip adoption states directly—an interpretation aligned with real instruments such as subsidies, awareness campaigns, and infrastructure provision.

Adoption follows an irreversible threshold cascade: agent $v$ adopts when the credibility-weighted fraction of adopting neighbors exceeds a random activation threshold,

$$x_v(t+1)=1 \iff x_v(t)=1 \;\lor\; \theta_v(N_v^\star(t)) \geq \phi_v(\rho_v(t)),$$

where $\mathcal{G}=(\mathcal{V},\mathcal{E})$0 and the threshold is drawn independently from a Beta distribution whose parameters depend on reluctance:

$\mathcal{G}=(\mathcal{V},\mathcal{E})$1

A key property of this parameterization is that $\mathcal{G}=(\mathcal{V},\mathcal{E})$2: the expected threshold coincides exactly with reluctance, making the stochastic formulation consistent—in expectation—with the deterministic cascade of [villa2024can]. Moreover, the Beta distribution's variance shrinks as its mean approaches 0 or 1, so highly reluctant agents become deterministically conservative, which the authors argue is behaviorally appropriate. High $\mathcal{G}=(\mathcal{V},\mathcal{E})$3 corresponds to Rogers' "conservative" adopter categories; low $\mathcal{G}=(\mathcal{V},\mathcal{E})$4 to early adopters.

The epistemic dimension enters through the distinction between reliability $\mathcal{G}=(\mathcal{V},\mathcal{E})$5 (actual competence) and credibility $\mathcal{G}=(\mathcal{V},\mathcal{E})$6 (perceived competence). The gap $\mathcal{G}=(\mathcal{V},\mathcal{E})$7 formalizes the credibility deficit induced by prejudice against marginalized groups; because $\mathcal{G}=(\mathcal{V},\mathcal{E})$8 weights neighbors by $\mathcal{G}=(\mathcal{V},\mathcal{E})$9 rather than by a credibility-weighted average of neighbor states (as in [villa2025epistemic]), the stochastic GLTM preserves rather than masks the downweighting of discriminated agents. This is a substantive methodological improvement over the deterministic predecessor: prejudice retains its full impact on the contagion process.

The irreversibility assumption is defended on economic grounds—durable-technology investments such as EV purchases are effectively irreversible over short-to-medium horizons (~10 years payback), justifying a finite intervention horizon $N$0 (11 steps, ~6 months at 18 days per step).

Fair MPC policy design

Policy design is deliberately confined to the deterministic reluctance dynamics rather than the stochastic cascade, for two stated reasons: reluctance is the only quantity directly actuated by the policy, and this yields a fully deterministic optimization problem in which stochasticity affects only initial conditions. Resources are not spent on adopters ($N$1 for all $N$2 if $N$3), so the decision-variable dimension shrinks as adoption proceeds.

At each receding-horizon step with prediction horizon $N$4 and per-step budget $N$5, the scheme solves

$N$6

with $N$7 a standard quadratic stage cost plus terminal cost $N$8, where $N$9 solves the discrete Riccati equation for the reluctance dynamics and $x_v(t)\in\{0,1\}$0 guarantees closed-loop stability per standard MPC theory [rawlings2020model]. The fairness cost decomposes as

$x_v(t)\in\{0,1\}$1

where equity penalizes deviation of each non-adopter's reluctance from the population average (steering all agents toward equal proximity to adoption), and equality penalizes dispersion in the allocated incentives themselves. The paper motivates including both: equity-only designs may make advantaged agents feel marginalized (undermining political acceptability), while identical resources can still generate disparate benefits due to socio-economic heterogeneity.

Two caveats are conceded explicitly. First, the budget constraint $x_v(t)\in\{0,1\}$2 holds at every time step without depletion—the same budget recurs indefinitely—which the authors acknowledge as unrealistic and plan to relax. Second, the design assumes policymakers observe each non-adopter's true reluctance; estimation from survey data would introduce errors not yet systematically handled.

Simulation results

The case study uses the Milan mobility network with initial reluctances estimated from survey responses on EV opinions. Reliabilities are drawn from education-dependent uniform distributions ($x_v(t)\in\{0,1\}$3 high EL, $x_v(t)\in\{0,1\}$4 medium, $x_v(t)\in\{0,1\}$5 low), and credibilities are computed by halving $x_v(t)\in\{0,1\}$6 once per membership in prejudiced groups (youth, gender, income)—e.g., an agent with $x_v(t)\in\{0,1\}$7 in three discriminated groups receives $x_v(t)\in\{0,1\}$8. Results aggregate over $x_v(t)\in\{0,1\}$9 Monte Carlo runs across four deficit scenarios: no deficit (ND), credibility deficit (CD), receptivity deficit (RD, with $\rho_v(t)\in[0,1]$0 drawn uniformly from $\rho_v(t)\in[0,1]$1), and combined CRD; each under one-sided ($\rho_v(t)\in[0,1]$2) versus heavily fair-weighted ($\rho_v(t)\in[0,1]$3) designs.

The main findings:

Scenario pair	Observation
ND vs. CD	One-sided and fair policies yield nearly identical adoption trajectories
RD vs. CRD	Unfair (one-sided) design achieves higher adoption, at higher policy effort
CRD overall	Hardest scenario; receptivity deficit hurts more than credibility deficit under fair design

The equivalence in ND/CD is explained structurally: when $\rho_v(t)\in[0,1]$4 is known, the MPC can fully compensate heterogeneous receptivity through larger inputs—but only if fairness constraints do not cap those inputs. Credibility deficits, by contrast, operate through social perception mechanisms that monetary nudges cannot directly mitigate. Under fair weighting, the receptivity deficit dominates the adoption loss by end of horizon, since equality suppresses the compensating large inputs precisely where they matter most. The fairness maps confirm the mechanism: without the equality term, the controller concentrates strong early interventions on disadvantaged agents; adding equality flattens allocation while still favoring them mildly, trading off final adoption rate.

These results support the paper's headline claim—that incorporating fairness ensures effective diffusion without exacerbating existing social inequities—but with an important qualifier visible in the data itself: in deficit-laden populations, fairness weighting measurably lowers peak adoption relative to unconstrained design. The trade-off is acknowledged implicitly ("at the cost of a less widespread adoption") rather than quantified against a welfare criterion.

Limitations and open questions

Several limitations bound the scope of these conclusions. The halving rule for credibilities and the education-based reliability bins are illustrative constructions, not validated measurements; the authors flag them explicitly as examples rather than definitive guidelines. Receptivities $\rho_v(t)\in[0,1]$5 are unknown from survey data and must be simulated. The Monte Carlo sample ($\rho_v(t)\in[0,1]$6) is small for characterizing distributional behavior of a stochastic cascade, and the single 112-agent network limits external validity. The budget-recycling assumption, perfect state knowledge, and the fixed Beta parameterization $\rho_v(t)\in[0,1]$7, $\rho_v(t)\in[0,1]$8 are all simplifying choices whose sensitivity is unexplored. Open questions left by the paper include: how alternative fairness metrics alter the adoption–fairness frontier; whether depletable budgets change the qualitative ranking of deficit scenarios; and how robustness to reluctance-estimation error can be built into the MPC formulation.

Conclusion

The paper makes a concrete contribution at the intersection of opinion dynamics and control: a stochastic GLTM in which activation thresholds are reluctance-driven Beta variables and influence is distorted by credibility deficits, coupled with a receding-horizon controller that treats equality and equity as explicit, separately tunable costs. Its most useful empirical insight is the asymmetry between deficit types—receptivity deficits are compensable by the controller when fairness permits, whereas credibility deficits are not directly mitigable by incentives and propagate through the social perception layer. The framework demonstrates that fairness-constrained nudging remains effective while reshaping allocation toward equitable outcomes, though the adoption penalty under combined deficits and the illustrative nature of the epistemic parameters leave quantitative generalization to future work.