Incentive Design without Hypergradients: A Social-Gradient Method

TL;DR

This paper introduces a hypergradient-free social-gradient flow method for incentive design in multi-agent systems, ensuring convergence to social optima without requiring full knowledge of equilibrium sensitivities.

Contribution

It proposes a novel social-gradient flow approach that guarantees descent and convergence to social optima without hypergradient computation, applicable under information asymmetry.

Findings

The social cost gradient is always a descent direction.

The social-gradient flow converges to the social optimum when equilibria are observable.

Joint strategy-incentive dynamics converge to the social optimum under certain learning rules.

Abstract

Incentive design problems consider a system planner who steers self-interested agents toward a socially optimal Nash equilibrium by issuing incentives in the presence of information asymmetry, that is, uncertainty about the agents' cost functions. A common approach formulates the problem as a Mathematical Program with Equilibrium Constraints (MPEC) and optimizes incentives using hypergradients-the total derivatives of the planner's objective with respect to incentives. However, computing or approximating the hypergradients typically requires full or partial knowledge of equilibrium sensitivities to incentives, which is generally unavailable under information asymmetry. In this paper, we propose a hypergradient-free incentive law, called the social-gradient flow, for incentive design when the planner's social cost depends on the agents' joint actions. We prove that the social cost gradient is always a descent direction for the planner's objective, irrespective of the agent cost landscape. In the idealized setting where equilibrium responses are observable, the social-gradient flow converges to the unique socially optimal incentive. When equilibria are not directly observable, the social-gradient flow emerges as the slow-timescale limit of a two-timescale interaction, in which agents' strategies evolve on a faster timescale. It is established that the joint strategy-incentive dynamics converge to the social optimum for any agent learning rule that asymptotically tracks the equilibrium. Theoretical results are also validated via numerical experiments.