Incentives and co-evolution: Steering linear dynamical systems with noncooperative agents

TL;DR

This work addresses the co-evolution of market dynamics and networked user dynamics by modeling user states with a linear system $x^+ = A x + B u$ and providers as agents in a generalized Nash game whose outcome feeds back as the input. It develops dissipativity-based stability guarantees via IQCs, and proposes a light-touch controller $\kappa(x,\boldsymbol{y})=K\boldsymbol{y}$ to steer the system toward a co-evolutionary equilibrium $\big((I-A)^{-1}BK\boldsymbol{y}^*,\boldsymbol{y}^*\big)$ with exponential convergence rate $\rho<1$. A scalar regulation variant with $K_i=\omega I_m$ yields a tractable optimization (via LMIs/BMIs) and a practical bisection algorithm to trade off regulation intensity against reactivity. The case study on digital regulation of influencer advertising demonstrates substantial computational gains from a dimension-reduction approach, while achieving similar convergence properties, underscoring the method’s relevance for large-scale socio-technical systems.

Abstract

Modern socio-technical systems typically consist of many interconnected users and competing service providers, where notions like market equilibrium are tightly connected to the ``evolution'' of the network of users. In this paper, we model the users' dynamics as a linear dynamical system, and the service providers as agents taking part to a generalized Nash game, whose outcome coincides with the input of the users' dynamics. We thus characterize the notion of co-evolution of the market and the network dynamics and derive dissipativity-based conditions leading to a pertinent notion of equilibrium. We then focus on the control design and adopt the light-touch policy to incentivize or penalize the service providers as little as possible, while steering the networked system to a desirable outcome. We also provide a dimensionality-reduction procedure, which offers network-size independent conditions. Finally, we illustrate our novel notions and algorithms on a simulation setup stemming from digital market regulations for influencers, a topic of growing interest.

Figures (4)

• Figure 1: Multi-level interactions for the application considered in this work (see also § \ref{['sec:case_study']}): $N$ firms paying $m$ influencers, who in turn influence the dynamics of a population of $n_F$ followers.
• Figure 2: Networked system consisting of a population of agents involved in a , whose outcome is affected by the evolution of a discrete-time system.
• Figure 3: Followers' dynamics $x_{F,k}$ in \ref{['eq:LTI_network']} and companies' collective decision vector $\boldsymbol{y}_k$ in \ref{['eq:single_prob_example_schematic']} co-evolution.
• Figure 4: Linear convergence to an equilibrium of the co-evolution dynamics driven by Algorithm \ref{['alg:two_layer']}.

Theorems & Definitions (12)

• Definition 2.1
• Lemma 2.4
• Remark 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4
• Remark 3.5
• Proposition 4.1
• Proposition 5.1
• Remark 5.2