In incentive design problems, a system planner seeks to steer 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 and optimizes incentives using hypergradients-the total derivatives of the planner's objective with respect to incentives. However, computing the hypergradient 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 social cost depends on the agents' joint action. We prove that the social cost gradient is 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. We establish that the joint strategy-incentive dynamics converge to the social optimum for any agent learning rule that asymptotically tracks the equilibrium. Finally, we validate theoretical results via numerical experiments.
Figures & tables
Fig. 1: Φ(x∗(p(t))) and ∥p(t)−p†∥2 , under ( 5 ) with initial conditions in \interiorPc∗ . Trajectories correspond to flows from the maximal and minimal sublevel sets across 100 initial conditions.
Fig. 2: Median (solid) and min–max envelope (shaded) of ∥xk−x∗(pk)∥2 (top) and ∥pk−p†∥2 (bottom) under ( 8 ), with fNE (left) and fBR (right), over 100 initial conditions sampled from \interiorX×Pc , with c=0.8c∗ .
Fig. 3: Iterates (xk,pk)k (orange) generated by ( 8 ) from x0=(0,−0.5)⊤ and p0=(−3,−3)⊤ . Blue curve indicates the social-gradient flow ( 5 ) from p(0)=p0 and t∈[0,20] . Contours are level sets of Φ(x) (left) and Φ∗(p) (right). Solid lines are the boundaries ∂X and ∂Pc , with c=0.9c∗ .
Fig. 4: ∥xk−x∗(pk)∥2 and ∥pk−p†∥2 under iteration ( 8 ), with initial condition x0=(0,−0.5)⊤ and p0=(−3,−3)⊤ . Lines depict variations of the timescale separation degree βk/ak=O(k−γ) for different γ>0 .