Mean Field Drift of Intentions

• Mean Field Drift of Intentions is a framework that models the stochastic evolution of agent intentions via mean-field games, kinetic equations, and random perturbations.
• It integrates both discrete and continuum approaches to capture how endogenous social interactions and exogenous common noise drive consensus and coordination breakdown in multi-agent systems.
• Analytical and simulation studies reveal that noise levels critically determine the emergence of stable intention fields and the rate of consensus formation.

Mean Field Drift of Intentions describes the stochastic perturbation and macroscopic evolution of intention distributions among interacting agents, arising from both endogenous mechanisms (social influence, local interactions) and exogenous shocks (common noise). The concept integrates mean-field games, kinetic equations, and stochastic processes, elucidating how group-level consensus, diversity, and coordination costs emerge in systems where agents adjust their intentions or states algorithmically, subject to both strategic goals and random disturbances.

1. Frameworks and Model Primitives

The formalization of Mean Field Drift of Intentions occurs in discrete-time and continuum models. In the discrete-time mean-field-type games (MFTGs), the population is partitioned into $m$ teams, each represented by an agent whose state $X_t^i$ lies in the finite set $S = \{0, 1, ..., G-1\}$. The agent’s action $a_t^i$ is chosen from $A = S$ (Lambrecht et al., 30 Dec 2025). Dynamics are modulated by:

• Global common noise: $$\xi_t^0 = (u_t, Z_t) \in [0, 1] \times \mathbb{R}_+^K$$, where $u_t \sim \mathrm{Unif}[0, 1]$ and $Z_t$ is a $K$-dimensional vector of iid $\mathrm{Exp}(1)$ random variables, with $K = |S|^m$. All noises are temporally independent.

Alternatively, for continuum models, agents indexed by spatial position $x \in \mathbb{R}^n$ and intention $w \in \mathbb{R}$ interact via various kernels, leading to time-dependent intention fields (1311.0810, Degond et al., 2016).

2. Dynamics of Mean Field Drift

The central mechanism involves randomized evolution of intention laws, integrating mean-field measures and stochastic perturbations. At each discrete-time step:

• The joint law $\mu_t \in \mathcal{P}(S^m)$ encodes the population mean-field state.
• The marginal state-action law $\operatorname{pr}(\mu_t)$ is perturbed by $Z_t$ via:

$$[Z_t \cdot \operatorname{pr}(\mu_t)](x) = \frac{Z_t(x) \operatorname{pr}(\mu_t)(x)}{\sum_{y \in S^m} Z_t(y) \operatorname{pr}(\mu_t)(y)}$$

• The next state $X_{t+1}$ is sampled using the Blackwell–Dubins function:

$$X_{t+1} = \rho_{S^m}([Z_t \cdot \operatorname{pr}(\mu_t)], u_t)$$

The perturbed distribution $[Z_t \cdot \operatorname{pr}(\mu_t)]$ defines the Mean Field Drift of Intentions, capturing the random displacement of the population’s nominal coordination.

In hydrodynamic and kinetic models, the evolution is captured by stochastic PDEs of the Edwards–Wilkinson type:

$$\partial_t I(x, t) = D \nabla^2 I(x, t) + \eta(x, t)$$

where $I(x, t)$ is the coarse-grained intention field, $D$ is a spatial diffusion coefficient, and $\eta(x, t)$ is white noise.

3. Cost Structure and Equilibrium Existence

Each agent aims to minimize the total discounted cost:

$$J^i = \mathbb{E}\left[ \sum_{t=0}^{\infty} \gamma^t f^i(X_t^i, a_t^i, \mu_t) \right], \quad \gamma \in (0,1)$$

The stage-wise cost function is:

$$f^i(x^i, a^i, \mu) = |x^i - x_*^i| + \sum_{j \neq i} w_j^i \sum_{y \in S} |x^i - y| \mu_j(y)$$

with $w_j^i \in \{-1, 0, 1\}$ reflecting attraction, indifference, or repulsion to other teams' states (Lambrecht et al., 30 Dec 2025).

Under the following conditions:

• (A) Finite, compact state/action spaces,
• (B) Map $F$ continuous almost everywhere,
• (C) Cost bounded and Lipschitz in $\mu$,
• (D) Discount factor $\gamma \in (0,1)$,
• (E) Transition kernel $P$ absolutely continuous w.r.t. a reference measure,

a stationary closed-loop Nash equilibrium exists in the infinite-horizon mean-field-type game, established via a lifting to Markov games on $\mathcal{P}(S^m)$ and application of Dufour–Prieto–Rumeau fixed-point theory (Lambrecht et al., 30 Dec 2025).

4. Macroscopic Evolution and Diffusion of Intentions

For agent-based and kinetic models, the microscopic update rule is:

$$\varphi_{\alpha,i} \to \varphi_{\alpha,i} + \gamma (\varphi_{\beta,j} - \varphi_{\alpha,i}) + \eta$$

$\eta$ is a zero-mean noise of variance $\Sigma^2$ (1311.0810, Degond et al., 2016). Aggregating over space and time yields:

Boltzmann Equation:

$$\partial_t P_\alpha(\varphi) = -W_\alpha(\varphi) P_\alpha(\varphi) + \sum_\beta f(|R_\alpha - R_\beta|) \iint d\psi d\psi' P_\alpha(\psi) P_\beta(\psi') G(|\psi - \psi'|) \int d\eta Q(\eta) \delta[\varphi - \psi - \gamma(\psi' - \psi) - \eta]$$

When local equilibrium exists (condensation: $\Sigma^2 < \Sigma_c^2 = \gamma(1-\gamma)\zeta^2$), the intention field obeys:

$$\partial_t I(x, t) = D \nabla^2 I(x, t) + \eta(x, t)$$

Spatial correlations in equilibrium display logarithmic decay:

$$C(r) \simeq \frac{\Delta}{2\pi D}[\ln L - \ln r], \quad r \gg \ell$$

where $\Delta$ is the coarse-grained noise amplitude, $L$ the system size, and $\ell$ the cutoff (1311.0810).

5. Mean-Field Fokker–Planck Formalism and Consensus

In the grazing-collision regime $\gamma \ll 1$, consensus emerges as the outcome of drift and diffusion under symmetric or non-symmetric binary interactions (Degond et al., 2016):

Fokker–Planck Equation:

$$\partial_t f(x, w, t) = -\partial_w \left\{ A[f](x, w, t) f(x, w, t) \right\} + \partial_w^2 \left\{ D[f](x, w, t) f(x, w, t) \right\}$$

with coefficients:

• $$A[f] = \gamma H(P*f) \int (v-w) P(w, v) f(x, v, t) dv$$
• $D[f] = \frac{\gamma \sigma^2}{2} H(P*f) (P*f)$

Symmetric interactions ($H \equiv 1$) yield a conservative macroscopic equation for mean intention $m(x, t)$:

$$\partial_t(\rho m) + \nabla_x \cdot (C_s \rho^2 \nabla_x m) = 0$$

Non-symmetric (Motsch–Tadmor style, $H(g)=g$) yield:

$$\partial_t(\rho m) + \nabla_x \cdot (C_a \rho \nabla_x m) = 0$$

The speed of consensus depends on density: non-symmetric interaction accelerates consensus in low-density regimes while symmetric interaction dominates in dense populations (Degond et al., 2016).

6. Coordination, Breakdown, and Interpretation

Mean Field Drift of Intentions quantifies the impact of common noise and interaction structure on coordination:

• When agents concentrate the intention law $\operatorname{pr}(\mu_t)$ on a single state $x$, the random perturbation $[Z_t \cdot \operatorname{pr}(\mu_t)]$ collapses to $\delta_x$, nullifying the drift. Conversely, uncoordinated populations experience pronounced stochastic drift, interpretable as a cost of coordination failure under shocks (Lambrecht et al., 30 Dec 2025).
• In opinion models, the existence of a local condensed equilibrium ($\Sigma^2 < \Sigma_c^2$) is necessary for a meaningful intention field to persist. Exceeding the critical noise threshold induces breakdown, abolishing long-range spatial correlations and field structure (1311.0810).
• The logarithmic decay of spatial correlations in the stationary intention field reproduces empirical phenomena in voting and opinion patterns (1311.0810). Analytical findings are corroborated by extensive agent-based simulations.

7. Comparative Overview and Physical Interpretation

The table below summarizes the essential mechanistic distinctions found in the cited models:

Model Context	Drift Mechanism	Equilibrium/Field Existence
Discrete MFTG (Lambrecht et al., 30 Dec 2025)	Common noise perturbation ($Z_t$)	Stationary Nash exists under (A–E)
Agent-based (Deffuant-Weisbuch) (1311.0810)	Social convergence + idiosyncratic noise	Gaussian field exists if $\Sigma^2 < \Sigma_c^2$
Continuum Fokker–Planck (Degond et al., 2016)	Symmetric/Non-symmetric collision kernel	Consensus speed and conservation law

Physical relevance: Drift of intentions encapsulates the tension between individual targeting/strategy and population-level coordination, under risk of random perturbation and incomplete consensus. The formal derivations elucidate how intention fields propagate, stabilize, or fragment depending on the strength of social cohesion, the nature of stochastic inputs, and interaction topology. The theoretical regime boundaries mark transitions between ordered consensus states and uninhibited diversity.