Multi-agent Adaptive Mechanism Design

TL;DR

This paper addresses learning-incentive sequential mechanism design when agents’ beliefs are initially unknown. It introduces DRAM, a distributionally robust adaptive mechanism that learns beliefs online, enforces truthfulness with high probability, and reduces payments as estimates improve, achieving a regret bound of $\tilde{O}(\sqrt{T})$. The authors establish a matching lower bound, show robustness to misspecification and delayed feedback, and extend to general estimators (DRAM+). Through warm-start phases and adaptive epochs, the framework provides a principled approach to learning incentives while controlling costs, with broad applicability to information elicitation and peer-prediction tasks in uncertain environments.

Abstract

We study a sequential mechanism design problem in which a principal seeks to elicit truthful reports from multiple rational agents while starting with no prior knowledge of agents' beliefs. We introduce Distributionally Robust Adaptive Mechanism (DRAM), a general framework combining insights from both mechanism design and online learning to jointly address truthfulness and cost-optimality. Throughout the sequential game, the mechanism estimates agents' beliefs and iteratively updates a distributionally robust linear program with shrinking ambiguity sets to reduce payments while preserving truthfulness. Our mechanism guarantees truthful reporting with high probability while achieving $\tilde{O}(\sqrt{T})$ cumulative regret, and we establish a matching lower bound showing that no truthful adaptive mechanism can asymptotically do better. The framework generalizes to plug-in estimators, supporting structured priors and delayed feedback. To our knowledge, this is the first adaptive mechanism under general settings that maintains truthfulness and achieves optimal regret when incentive constraints are unknown and must be learned.

Figures (3)

• Figure 1: An image labeling example. Nature samples an unlabeled image with an unknown ground truth, which is then independently observed by multiple agents. Each agent's observation (type) is private to herself. The agents then report to the principal and receive rewards in the end. Lying or lazy behavior is possible, since the principal does not know the ground truth or the agents' observations. One objective is to incentivize truthful behavior via reward mechanisms based on only agents' reports.
• Figure 2: Minimum reward gap between truthful reporting and other pure strategies across 1000 runs of a sequential labeling game. Negative gap means the constraints are violated. In this simulation, the minimum gap distribution is well separated from $0$, meaning that truthful reporting dominates other strategies by a considerable margin, and $\texttt{DRAM}$ guarantees truthfulness even with spare robustness.
• Figure 3: Average cumulative regret over time across 1000 runs of a sequential labeling game. The first $\sim10^5$ rounds are warm-start phase, and then comes with doubling epoch lengths. The regret curve is piecewise linear, as the expected round-wise regret within each epoch stays unchanged. The geometric epoch schedule ensures $O(\sqrt{T})$ regret.

Theorems & Definitions (20)

• Remark 1
• Remark 2
• proposition 1: Truthfulness is necessary for maximal quality
• theorem 1: Optimal cost of a two–player peer-prediction mechanism
• Remark 3
• theorem 2: Robustness to distributional misspecification
• theorem 3: Bounds on payments of robust mechanism
• corollary 1: Bounds on achievable robustness
• theorem 4: Cost of robustness
• lemma 1: Fact checking under diagonal dominance