Personalized incentives as feedback design in generalized Nash equilibrium problems

TL;DR

This paper tackles stationary and time-varying generalized Nash equilibrium problems with symmetric interactions where the potential function $\theta(\boldsymbol{x};t)$ is unknown. It proposes a semi-decentralized two-layer scheme: a central coordinator learns the agents' pseudo-gradients from feedback and designs personalized incentives $u_i$ to steer the population toward minimizers of the unknown potential, while agents solve an extended game incorporating these incentives. The authors establish convergence in the stationary case and bounded tracking in the time-varying setting, with performance governed by the learning accuracy and incentive gains; they also validate the approach on a ride-hailing orchestration model, showing improved coordination and congestion reduction through learning-based incentives. The work combines online learning, potential game structure, and regularized equilibrium computation to enable practical coordination in multi-agent systems with privacy and time-varying dynamics.

Abstract

We investigate both stationary and time-varying, nonmonotone generalized Nash equilibrium problems that exhibit symmetric interactions among the agents, which are known to be potential. As may happen in practical cases, however, we envision a scenario in which the formal expression of the underlying potential function is not available, and we design a semi-decentralized Nash equilibrium seeking algorithm. In the proposed two-layer scheme, a coordinator iteratively integrates the (possibly noisy and sporadic) agents' feedback to learn the pseudo-gradients of the agents, and then design personalized incentives for them. On their side, the agents receive those personalized incentives, compute a solution to an extended game, and then return feedback measurements to the coordinator. In the stationary setting, our algorithm returns a Nash equilibrium in case the coordinator is endowed with standard learning policies, while it returns a Nash equilibrium up to a constant, yet adjustable, error in the time-varying case. As a motivating application, we consider the ridehailing service provided by several companies with mobility as a service orchestration, necessary to both handle competition among firms and avoid traffic congestion, which is also adopted to run numerical experiments verifying our results.

Figures (7)

• Figure 1: Personalized incentives as feedback design to steer the population to a point guaranteeing the "minimum" (dis)satisfaction, according to the unknown function $\theta$.
• Figure 2: Residual $\Delta^\star_t$ over the time horizon $T$, normalized by the total number of deployable cars at any $t$, for different values of personalized incentive feedback gains $c(t)$ and $\xi(t)$ in \ref{['eq:pers_feedback']}.
• Figure 3: Optimal strategies pursued by the five companies over a week, coinciding with the number of vehicles deployed at every time interval. The colour line follows the legend in Fig. \ref{['fig:path_length_GP']}.
• Figure 4: Overall service provided by the five firms over a whole week. The colour line follows the legend in Fig. \ref{['fig:path_length_GP']}, while the dotted green line corresponds to the costumer requests obtained from data after a spline interpolation. The dashed red lines coincide with $\underline{x}$ and $\bar{x}$, respectively.
• Figure 5: Profit/cost experienced by the companies over a week. The colour line follows the legend in Fig. \ref{['fig:path_length_GP']}.

Theorems & Definitions (19)

• Definition 1
• Lemma 1
• Proposition 1
• Lemma 2
• Remark 1
• Lemma 3
• Proposition 2
• Remark 2
• Lemma 4
• Theorem 1