Mean-Field Control of Adherence in Participation-Coupled Vehicle Rebalancing Systems

Abstract

Human driver participation is a critical source of uncertainty in Mobility-on-Demand (MoD) rebalancing. Drivers follow platform recommendations probabilistically, and their willingness to comply evolves with experienced outcomes. This creates a closed-loop feedback in which stronger recommendations increase participation, participation increases congestion, congestion lowers allocation success, and realized allocations update adherence beliefs. We propose a microscopic stochastic model that couples (i) belief-driven participation, (ii) Poisson demand, (iii) uniform matching, and (iv) Beta--Bernoulli belief updates. Under a large-population closure, we derive a deterministic mean-field recursion for the population adherence state under platform actuation. For i.i.d. Poisson demand and constant recommendation intensity, we prove global well-posedness and invariance of the recursion, establish equilibrium existence, provide uniqueness conditions, and show global convergence in the regime where platform recommendations are no weaker than baseline participation. We then define steady-state adherence and throughput, characterize the induced performance frontier, and show that adherence and throughput cannot, in general, be simultaneously maximized under uniform time-invariant actuation. This yields a throughput-maximization problem with an adherence floor. Exploiting the monotone frontier structure, we show the optimal uniform time-invariant policy is the maximal feasible recommendation intensity and provide an efficient bisection-based algorithm.

Figures (6)

• Figure 3: Mean-field trajectory versus microscopic simulation for $K=100$ agents under strong heterogeneity ($\alpha_i(0),\beta_i(0)\sim U(1,50)$, $p_i\sim U(0,1)$). Demand follows $D(t)\sim\mathrm{Poisson}(80)$ with recommendation intensity $u=0.9$. The dashed and dotted curves show the Monte Carlo averages of the pooled and direct population means across $100$ runs, illustrating that the mean-field approximation remains accurate even under large heterogeneity.
• Figure 4: Closed-loop adherence dynamics: microscopic stochastic model (finite $K$) and its mean-field approximation obtained via large-population averaging.
• Figure 5: Illustration of equilibrium structure. Left: the fixed-point map $y=s(x)$ and the diagonal $y=x$ (see \ref{['eq:q_a_s_of_x']}). For $u=0.6$ the curves intersect once, yielding a unique equilibrium, while for $u=0.05$ multiple intersections appear. Right: derivative $s'(a)$ over the interval $[a_{\min},a_{\max}]$. When $\sup s'(a)<1$ the contraction condition of \ref{['thm:eq_unique']} holds (unique equilibrium), whereas when $\sup s'(a)>1$ the condition fails, allowing multiple equilibria.
• Figure 6: Convergence trajectories of the adherence state $\bar{x}(t)$ for different constant controls $u$. Each trajectory converges to a control-dependent equilibrium.
• Figure 7: Log--log decay of the adherence error $|e(t)|$ for different controls $u$. The dashed line shows a reference curve of order $C/t$.

Theorems & Definitions (23)

• Remark B.1: Measurability
• Example B.1: Accuracy of the mean-field approximation
• Lemma C.1: Well-posedness and invariance
• proof
• Theorem C.1: Existence
• proof
• Lemma C.2: Slope of $g$ (Poisson demand)
• proof
• Theorem C.2: Uniqueness
• proof