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 .
Incentive design studies how a central authority can influence strategic agents through payments, subsidies, or taxes, so that individual objectives align with collective welfare. This paper introduces a No-Regret Adaptive Incentive Design (RAID) framework for nonlinear games with continuous action spaces and private agent costs. In this framework, the authority (planner) designs incentives that regulate the Nash equilibrium toward a socially optimal action profile, while simultaneously learning agents' unknown preferences from repeated strategic responses. We formulate the RAID problem and construct a least-squares estimator whose strong consistency requires only diminishing excitation. Leveraging this weak excitation requirement, we propose a switching incentive policy that alternates between probing (exploration) and estimate-based (exploitation) incentives. The resulting policy achieves an O(t−0.5) parameter estimation rate and accumulates O(t0.5logt) squared social-cost regret, almost surely. We further extend the framework to an endogenous-noise response model, where standard least-squares estimation is biased due to an error-in-variables correlation between the noise and agent responses. We utilize a repeated-sampling estimator and corresponding switching policy that retain the same almost-sure convergence and regret rates. Numerical experiments validate the effectiveness and predicted convergence rates of the method.
Georgios Vasileiou, Lantian Zhang, Silun Zhang
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, 10044, Sweden.
Incentive design constitutes a foundational paradigm for influencing the behavior of strategic agents, wherein a system planner (principal) publicly commits to an incentive mechanism designed to align individual objectives with collective social welfare. This paper introduces the Regret-Minimizing Adaptive Incentive Design (RAID) problem, which aims to synthesize incentive laws under information asymmetry and achieve asymptotically minimal regret compared to an oracle with full information. To this end, we develop the RAID algorithm, which employs a switching policy alternating between probing (exploration) and estimate-based incentivization (exploitation). The associated type estimator relies only on a weaker excitation condition required for strong consistency in least squares estimation, substantially relaxing the persistence-of-excitation assumptions previously used in adaptive incentive design. In addition, we establish the strong consistency of the proposed type estimator and prove that the incentive obtained asymptotically minimizes the planner's average regret almost surely. Numerical experiments illustrate the convergence rate of the proposed methodology.
Georgios Vasileiou, Lantian Zhang, Silun Zhang
Dept. of Mathematics at the KTH Royal Institute of Technology, Stockholm, 10044, Sweden.
We study the consequences of information asymmetries and misaligned incentives in settings with multiple independent agents. We model an interaction between a Sender, who holds vital private information but cannot act, and a Receiver, who must make decisions but is dependent on the Sender's information. We find that the Sender learns an optimal communication strategy that the Receiver reliably acts on. Importantly, this strategy is highly sensitive to the degree of conflict in the agents' rewards and the amount of environmental information the Receiver can already observe. We introduce a mechanism allowing the agents to form linear contracts, where a price is established for the information. We demonstrate that the Sender learns to use these payment structures to improve its rewards, though this comes at a cost of "fairness" between agents as the Sender is able to extract much of the Receiver's surplus. This raises questions about fairness, contract design, and learning in the context of multi-agent systems.