Motivated by applications such as cloud platforms allocating GPUs to users or governments deploying mobile health units across competing regions, we study the constrained dynamic allocation of a reusable resource to a group of strategic agents. Our objective is to simultaneously (i) maximize social welfare, (ii) satisfy multidimensional long-term cost constraints, and (iii) incentivize truthful reporting. We begin by numerically evaluating primal-dual methods widely used in constrained online optimization and find them to be highly fragile in strategic settings - agents can easily manipulate their reports to distort future dual updates for future gain. To address this vulnerability, we develop an incentive-aware framework that makes primal-dual methods robust to strategic behavior. Our primal-side design combines epoch-based lazy updates - discouraging agents from distorting dual updates - with dual-adjust pricing and randomized exploration techniques that extract approximately truthful signals for learning. On the dual side, we design a novel online learning subroutine to resolve a circular dependency between actions and predictions; this makes our mechanism achieve eO( T)social welfare regret (where T is the number of allocation rounds), satisfies all cost constraints, and ensures incentive alignment. This eO( T) performance matches that of non-strategic allocation approaches while additionally exhibiting robustness to strategic agents.

However,such assumption does not always hold inreality. Itisoften the case that arm pulls are performed by multiple different agents whose individual goals are not aligned with the system, and the principal can only observeagents' actions. One typical example is the individual buyers (agents) and the online shopping platform (the principal).

We study Bayesian automated mechanism design in unstructured dynamic environments, where a principal repeatedly interacts with an agent, and takes actions based on the strategic agent's report of the current state of the world. Both the principal and the agent can have arbitrary and potentially different valuations for the actions taken, possibly also depending on the actual state of the world. Moreover, at any time, the state of the world may evolve arbitrarily depending on the action taken by the principal. The goal is to compute an optimal mechanism which maximizes the principal's utility in the face of the self-interested strategic agent.We give an efficient algorithm for computing optimal mechanisms, with or without payments, under different individual-rationality constraints, when the time horizon is constant. Our algorithm is based on a sophisticated linear program formulation, which can be customized in various ways to accommodate richer constraints.

Algorithmic decision making is increasingly prevalent, but often vulnerable to strategic manipulation by agents seeking a favorable outcome. Prior research has shown that classifier abstention (allowing a classifier to decline making a decision due to insufficient confidence) can significantly increase classifier accuracy. This paper studies abstention within a strategic classification context, exploring how its introduction impacts strategic agents' responses and how principals should optimally leverage it. We model this interaction as a Stackelberg game where a principal, acting as the classifier, first announces its decision policy, and then strategic agents, acting as followers, manipulate their features to receive a desired outcome. Here, we focus on binary classifiers where agents manipulate observable features rather than their true features, and show that optimal abstention ensures that the principal's utility (or loss) is no worse than in a non-abstention setting, even in the presence of strategic agents. We also show that beyond improving accuracy, abstention can also serve as a deterrent to manipulation, making it costlier for agents, especially those less qualified, to manipulate to achieve a positive outcome when manipulation costs are significant enough to affect agent behavior. These results highlight abstention as a valuable tool for reducing the negative effects of strategic behavior in algorithmic decision making systems.

Large Language Models (LLMs) are increasingly applied to domains that require reasoning about other agents' behavior, such as negotiation, policy design, and market simulation, yet existing research has mostly evaluated their adherence to equilibrium play or their exhibited depth of reasoning. Whether they display genuine strategic thinking, understood as the coherent formation of beliefs about other agents, evaluation of possible actions, and choice based on those beliefs, remains unexplored. We develop a framework to identify this ability by disentangling beliefs, evaluation, and choice in static, complete-information games, and apply it across a series of non-cooperative environments. By jointly analyzing models' revealed choices and reasoning traces, and introducing a new context-free game to rule out imitation from memorization, we show that current frontier models exhibit belief-coherent best-response behavior at targeted reasoning depths. When unconstrained, they self-limit their depth of reasoning and form differentiated conjectures about human and synthetic opponents, revealing an emergent form of meta-reasoning. Under increasing complexity, explicit recursion gives way to internally generated heuristic rules of choice that are stable, model-specific, and distinct from known human biases. These findings indicate that belief coherence, meta-reasoning, and novel heuristic formation can emerge jointly from language modeling objectives, providing a structured basis for the study of strategic cognition in artificial agents.

In this work, we argue that the order of play in strategic classification is fundamentally determined by the relative frequencies at which the decision-maker and the agents adapt to each other's actions.