We study repeated contract design when a principal observes outcomes but not the actions that generate them. The principal may use any bounded outcome-contingent payment vector, and the agent's best response can make expected profit discontinuous in those payments. For every fixed number m≥2 of outcomes, the minimax regret over T rounds is of order Tm/(m+1), up to logarithmic factors. The upper bound allows arbitrary action spaces and agent heterogeneity, without smoothness or monotone-surplus assumptions. Its key is an effective-dimension reduction that the benchmark can be normalized even when fixed tie-breaking is not shift invariant, after which revealed preference yields a monotone response map in payment-difference coordinates. A learning policy built on a Lipschitz parametrization of this map attains the rate using only observed outcome categories. The lower-bound construction accounts for how incentive losses accumulate across outcome dimensions. It shows that each additional contractible outcome creates a precise and unavoidable increase in the worst-case cost of learning.
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.
Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, who wish to trade a good. We study this problem from the perspective of a profit-maximizing broker within an online learning framework, where the agents' valuations are generated by a smooth adversary. We devise a learning algorithm that guarantees a O~(T) regret bound, which is tight in the time horizon T up to poly-logarithmic factors. This matches the minimax rate for the stochastic i.i.d. case, and is also well separated from the adversarial setting, where sublinear-regret is unattainable. By extending the strong regret guarantees from the i.i.d. case to the smooth adversary, we significantly broaden the scope of settings where such fast rate is achievable, while closing an important gap in the regret landscape of this fundamental economic problem. To overcome the challenges posed by this adversary, we leverage a continuity property of smooth instances and combines this with a hierarchical net-construction of the broker's action space, which is analyzed via algorithmic chaining. We showcase the applicability of these techniques by deriving a similarly tight O~(T) regret bound for a related mechanism design model: the joint ads problem.
Simone Di Gregorio, Paul Dütting, Federico Fusco +1
In this paper, we study regret minimization in repeated games with \emph{adaptive} opponents who can respond based on histories of play. The standard metric of \emph{external regret} in online learning is known to fail to capture such adaptivity. To account for players' counterfactual reasoning, we introduce {\tt Repeated Policy Regret (RP-Regret)}, a game-theoretic metric that measures the difference between the \emph{realized} and the \emph{best-in-hindsight} accumulated utility when all players can \emph{respond} to the history of play. Compared to existing regret notions in this setting, ours is native to repeated game playing, enabling stronger comparators and opponents with fewer constraints, while maintaining the possibility of finding better equilibria when all players minimize it. We first identify necessary conditions for obtaining {\tt RP-Regret} sublinear in time, on the variation of the player's comparator strategies in the regret definition and on the memories of both the comparator and opponents' strategies. We then study additional conditions and provable algorithms to minimize {\tt RP-Regret}, which is by definition \emph{non-convex} in the strategy space. To address this challenge, we propose three algorithms: (i) one based on an optimization oracle, as assumed in some prior work in online non-convex learning; (ii) one that minimizes a convex and \emph{linearized} surrogate of {\tt RP-Regret} at each iteration; (iii) one that directly minimizes {\tt RP-Regret} when opponents change strategies slowly. Furthermore, when all players can run algorithms to minimize the {\tt RP-Regret} (or its linearized variant), certain subgame perfect equilibria of the repeated game can be learned. We also provide experiments showing that minimizing our regret notions can lead to more cooperative solutions with higher utility in games such as Stag-Hunt.