In a multi-follower Bayesian Stackelberg game, a leader plays a mixed strategy over $L$ actions to which $n\ge 1$ followers, each having one of $K$ possible private types, best respond. The leader's optimal strategy depends on the distribution of the followers' private types. We study an online learning problem for Bayesian Stackelberg game, where a leader interacts for $T$ rounds with $n$ followers with types sampled from an unknown distribution every round. The leader's goal is to minimize regret, defined as the difference between the cumulative utility of the optimal strategy and that of the actually chosen strategies. We design learning algorithms for the leader under different settings. Under type feedback, where the leader observes the followers' types after each round, we design algorithms that achieve $\mathcal O\big(\sqrt{\min\{L\log(nKA T), ~ nK \} \cdot T} \big)$ regret for independent type distributions and $\mathcal O\big(\sqrt{\min\{L\log(nKA T), ~ K^n \} \cdot T} \big)$ regret for general type distributions. Interestingly, these bounds do not grow with $n$ at a polynomial rate. Under action feedback, where the leader only observes the followers' actions, we design algorithms with $\mathcal O( \min\{\sqrt{ n^L K^L A^{2L} L T \log T}, ~ K^n\sqrt{ T } \log T \} )$ regret. We also provide a lower bound of $\Omega(\sqrt{\min\{L, ~ nK\}T})$, almost matching the type-feedback upper bounds.

Bidders in combinatorial auctions face significant challenges when describing their preferences to an auctioneer. Classical work on preference elicitation focuses on query-based techniques inspired from proper learning—often via proxies interfacing between bidders and an auction mechanism—to incrementally learn bidder preferences as needed to compute efficient allocations. Although such elicitation mechanisms enjoy theoretical query efficiency, the amount of communication required may still be too cognitively taxing in practice. We propose a family of efficient LLM-based proxy designs for eliciting preferences from bidders using natural language. Our proposed mechanism combines LLM pipelines and DNF-proper-learning techniques to quickly approximate preferences when communication is limited. To validate our approach, we create a testing sandbox for elicitation mechanisms that communicate in natural language. In our experiments, our most promising LLM proxy design reaches approximately efficient outcomes with five times fewer queries than classical proper learning based elicitation mechanisms.

The role of a market maker is to simultaneously offer to buy and sell quantities of goods, often a financial asset such as a share, at specified prices. An automated market maker (AMM) is a mechanism that offers to trade according to some predetermined schedule; the best choice of this schedule depends on the market maker's goals. The literature on the design of AMMs has mainly focused on prediction markets with the goal of information elicitation. More recent work motivated by DeFi has focused instead on the goal of profit maximization, but considering only a single type of good (traded with a numeraire), including under adverse selection (Milionis et al. 2022). Optimal market making in the presence of multiple goods, including the possibility of complex bundling behavior, is not well understood. In this paper, we show that finding an optimal market maker is dual to an optimal transport problem, with specific geometric constraints on the transport plan in the dual. We show that optimal mechanisms for multiple goods and under adverse selection can take advantage of bundling, both improved prices for bundled purchases and sales as well as sometimes accepting payment "in kind." We present conjectures of optimal mechanisms in additional settings which show further complex behavior. From a methodological perspective, we make essential use of the tools of differentiable economics to generate conjectures of optimal mechanisms, and give a proof-of-concept for the use of such tools in guiding theoretical investigations.

Online market platforms play an increasingly powerful role in the economy. An empirical phenomenon is that platforms, such as Amazon, Apple, and DoorDash, also enter their own marketplaces, imitating successful products developed by third-party sellers. We formulate a Stackelberg model, where the platform acts as the leader by committing to an entry policy: when will it enter and compete on a product? We study this model through a theoretical and computational framework. We begin with a single seller, and consider different kinds of policies for entry. We characterize the seller's optimal explore-exploit strategy via a Gittins-index policy, and give an algorithm to compute the platform's optimal entry policy. We then consider multiple sellers, to account for competition and information spillover. Here, the Gittins-index characterization fails, and we employ deep reinforcement learning to examine seller equilibrium behavior. Our findings highlight the incentives that drive platform entry and seller innovation, consistent with empirical evidence from markets such as Amazon and Google Play, with implications for regulatory efforts to preserve innovation and market diversity.

Pooled testing is a common strategy for public health disease screening under limited testing resources, allowing multiple biological samples to be tested together with the resources of a single test, at the cost of reduced individual resolution. While dynamic and adaptive strategies have been extensively studied in the classical pooled testing literature, where the goal is to minimize the number of tests required for full diagnosis of a given population, much of the existing work on welfare-maximizing pooled testing adopts static formulations in which all tests are assigned in advance. In this paper, we study dynamic welfare-maximizing pooled testing strategies in which a limited number of tests are performed sequentially to maximize social welfare, defined as the aggregate utility of individuals who are confirmed to be healthy. We formally define the dynamic problem and study algorithmic approaches for sequential test assignment. Because exact dynamic optimization is computationally infeasible beyond small instances, we evaluate a range of strategies (including exact optimization baselines, greedy heuristics, mixed-integer programming relaxations, and learning-based policies) and empirically characterize their performance and tradeoffs using synthetic experiments. Our results show that dynamic testing can yield substantial welfare improvements over static baselines in low-budget regimes. We find that much of the benefit of dynamic testing is captured by simple greedy policies, which substantially outperform static approaches while remaining computationally efficient. Learning-based methods are included as flexible baselines, but in our experiments they do not reliably improve upon these heuristics. Overall, this work provides a principled computational perspective on dynamic pooled testing and clarifies when dynamic assignment meaningfully improves welfare in public health screening.

We develop a game-theoretic framework for predicting and steering the behavior of populations of large language models (LLMs) through Nash equilibrium (NE) analysis. To avoid the intractability of equilibrium computation in open-ended text spaces, we model each agent's action as a mixture over human subpopulations. Agents choose actively and strategically which groups to align with, yielding an interpretable and behaviorally substantive policy class. We derive closed-form NE characterizations, adopting standard concave-utility assumptions to enable analytical system-level predictions and give explicit, actionable guidance for shifting alignment targets toward socially desirable outcomes. The method functions as an active alignment layer on top of existing alignment pipelines such as RLHF. In a social-media setting, we show that a population of LLMs, especially reasoning-based models, may exhibit political exclusion, pathologies where some subpopulations are ignored by all LLM agents, which can be avoided by our method, illustrating the promise of applying the method to regulate multi-agent LLM dynamics across domains.

Characterizing revenue-optimal auctions for multi-item, multi-bidder settings remains a fundamental open problem, with no known closed-form solution existing beyond restrictive binary-type instances. This has motivated interest in computational approaches to optimal auction design. In this paper, we introduce the first computational framework that directly tackles the dual problem for multi-item, multi-bidder auctions and dominant-strategy incentive compatibility (DSIC), generating certified revenue upper bounds. Our approach parametrizes Lagrange multipliers with a structurally guaranteed strict flow-conservation property using neural networks, enabling efficient optimization over feasible dual solutions via gradient descent. To bridge the gap between discrete computational methods and theoretical guarantees for continuous types, we develop a novel lifting technique that maps dual certificates from coarse discretizations to fine refinements. We prove that lifting gives valid revenue upper bounds for multi-item, multi-bidder auctions with continuous uniform valuations. Furthermore, we give a generalized lifting construction for arbitrary continuous distributions and demonstrate that these lifted duals converge to the revenue of the original continuous problem in the discrete limit. We validate this computational framework for the dual auction design problem by recovering known analytical mechanisms for canonical instances. For multi-item multi-bidder problems, our framework establishes a small gap between the optimal revenue and best-known DSIC mechanisms, providing computational certificates of near-optimality.

The rapid development of advanced AI agents and the imminent deployment of many instances of these agents will give rise to multi-agent systems of unprecedented complexity. These systems pose novel and under-explored risks. In this report, we provide a structured taxonomy of these risks by identifying three key failure modes (miscoordination, conflict, and collusion) based on agents' incentives, as well as seven key risk factors (information asymmetries, network effects, selection pressures, destabilising dynamics, commitment problems, emergent agency, and multi-agent security) that can underpin them. We highlight several important instances of each risk, as well as promising directions to help mitigate them. By anchoring our analysis in a range of real-world examples and experimental evidence, we illustrate the distinct challenges posed by multi-agent systems and their implications for the safety, governance, and ethics of advanced AI.

Differentiable economics—the use of deep learning for auction design—has driven progress in multi-item auction design with additive and unit-demand valuations. However, there has been little progress for combinatorial auctions (CAs), even in the simplest and yet important single bidder case, due to exponential growth of the bundle space with the number of items. We address this challenge by introducing a deep network architecture for a menu-based CA, which supports the first dominant-strategy incentive compatible (DSIC), revenue-optimizing single-bidder CA. Our idea is to generate a bundle distribution through an ordinary differential equation (ODE) applied to a tractable initial distribution. The BundleFlow method learns suitable ODE-based transforms, one for each menu element, to optimize expected revenue. BundleFlow achieves up to 2.23$\times$ higher revenue than baselines on standard CA testbeds and scales up to 500 items. Compared with other menu-learning baselines, BundleFlow also reduces training iterations by 3.6-9.5$\times$ and cuts training time by about 80% in settings with 50 and 100 items.

Multiagent systems grapple with partial observability (PO), and the decentralized POMDP (Dec-POMDP) model highlights the fundamental nature of this challenge. Whereas recent approaches to addressing PO have appealed to deep learning models, providing a rigorous understanding of how these models and their approximation errors affect agents' handling of PO and their interactions remain a challenge. In addressing this challenge, we investigate reconstructing global states from local action-observation histories in Dec-POMDPs using diffusion models. We first find that diffusion models conditioned on local history represent possible states as stable fixed points. In collectively observable (CO) Dec-POMDPs, individual diffusion models conditioned on agents' local histories share a unique fixed point corresponding to the global state, while in non-CO settings, shared fixed points yield a distribution of possible states given joint history. We further find that, with deep learning approximation errors, fixed points can deviate from true states and the deviation is negatively correlated to the Jacobian rank. Inspired by this low-rank property, we bound a deviation by constructing a surrogate linear regression model that approximates the local behavior of a diffusion model. With this bound, we propose a composite di.usion process iterating over agents with theoretical convergence guarantees to the true state.

Effective human-AI coordination requires artificial agents capable of exhibiting and responding to human-like behaviors while adapting to changing contexts. Imitation learning has emerged as one of the prominent approaches to build such agents by training them to mimic human-demonstrated behaviors. However, current methods struggle to capture the inherent diversity and non-Markovian nature of human behavior and lack the ability to steer behavior at inference time. Drawing inspiration from the theory of human cognitive processes, where inner speech guides action selection before execution, we propose MIMIC (Modeling Inner Motivations for Imitation and Control), a framework that uses language as an internal representation of behavioral intent. MIMIC employs the novel use of vision-language models as linguistic scaffolding to train a conditional variational autoencoder capable of generating inner speech from observations. A diffusion-based behavior cloning policy then selects actions conditioned on current observations and the generated inner speech. MIMIC enables fine-grained steering of behavior at inference time by conditioning the agent on behavior-specific speech. Experiments across robotic manipulation tasks and human-AI collaboration games demonstrate that MIMIC significantly enhances both behavior diversity and fidelity to human demonstrations while enabling nuanced behavioral steering without training on additional demonstrations.

We model a delivery platform facilitating transactions among three sides: buyers, stores, and couriers. In addition to buyers paying store-specific purchase prices and couriers receiving store–buyer-specific delivery compensation from the platform, each buyer has the option to directly tip for delivery from a specific store. An equilibrium consists of prices, compensations, tips, and transactions that clear the market, such that buyers receive deliveries from preferred stores considering the prices and tips they pay, and couriers deliver preferred orders considering the compensations and tips they receive. We illustrate the role of tips in pricing: Without tips, an equilibrium is only guaranteed to exist when there are at least as many couriers as buyers or stores. In contrast, with tips an equilibrium always exists. From an efficiency perspective, the optimal with-tip equilibrium welfare is always weakly larger than the optimal without-tip equilibrium welfare. However, we show that even with tips, efficient equilibria may not exist, and calculating the optimal equilibrium welfare is NP-hard. To address these challenges, we identify natural conditions on market structure that ensure the existence of efficient with-tip equilibria and allow these efficient equilibria to be computed in polynomial time.

This paper investigates the behavior of simulated AI agents (large language models, or LLMs) in auctions, introducing a novel synthetic data-generating process to help facilitate the study and design of auctions. We find that LLMs – when endowed with chain of thought reasoning capacity – agree with the experimental literature in auctions across a variety of classic auction formats. In particular, we find that LLM bidders produce results consistent with risk-averse human bidders; that they perform closer to theoretical predictions in obviously strategy-proof auctions; and, that they succumb to the winner's curse in common value settings. On prompting, we find that LLMs are not very sensitive to naive changes in prompts (e.g., language, currency) but can improve dramatically towards theoretical predictions with the right mental model (i.e., the language of Nash deviations). We run 1,000+ auctions for less than $400 with GPT-4 models (three orders of magnitude cheaper than modern auction experiments) and develop a framework flexible enough to run auction experiments with any LLM model and a wide range of auction design specifications, facilitating further experimental study by decreasing costs and serving as a proof-of-concept for the use of LLM proxies.

Economics studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these agents. Game theory had a substantial impact on economic modeling because it allows us to model the outcome of such economic interaction while taking the incentives of individual agents into account. Mechanism design does the same when designing the rules of economic institutions. Unfortunately, these economic models have turned out to be computationally hard to solve. For example, finding equilibrium in some incomplete-information models of markets with continuous valuation and action spaces are hard in PP, and designing a revenue-maximizing multi-item auction is #P-hard. This computational complexity poses a fundamental barrier in modeling economic systems but is worst-case and considers non-generic instances. Differentiable economics describes a new approach to solving these central problems in the economic sciences. It uses learning algorithms to find or approximate solutions to equilibrium computation or economic design problems. In particular, neural networks and learning algorithms such as Stochastic Gradient Descent have been shown to be very effective. Machine learning has led to breakthroughs in many sciences, and it also holds the potential to fundamentally alter how we analyze and design economic systems.

We study the potential of large language models (LLMs) as proxies for humans to simplify preference elicitation (PE) in combinatorial assignment. While traditional PE methods rely on iterative queries to capture preferences, LLMs offer a one-shot alternative with reduced human effort. We propose a framework for LLM proxies that can work in tandem with SOTA ML-powered preference elicitation schemes. Our framework handles the novel challenges introduced by LLMs, such as response variability and increased computational costs. We experimentally evaluate the efficiency of LLM proxies against human queries in the well-studied course allocation domain, and we investigate the model capabilities required for success. We find that our approach improves allocative efficiency by up to 20 different LLMs and to differences in quality and accuracy of reporting.

In this note, we survey automated mechanism design (AMD): the use of computational techniques to solve mechanism design problems. We describe three distinct but overlapping threads of research: an optimization-based paradigm that formulates mechanism design as linear programming, a line of work on sample complexity and learning theory, and the recent trend of differentiable economics, which has produced state-of-the-art results on a range of problems by borrowing tools and techniques from modern deep learning.

Economics studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these agents. Game theory had a substantial impact on economic modeling because it allows us to model the outcome of such economic interaction while taking the incentives of individual agents into account. Mechanism design does the same when designing the rules of economic institutions. Unfortunately, these economic models have turned out to be computationally hard to solve. For example, finding equilibrium in some incomplete-information models of markets with continuous valuation and action spaces are hard in PP, and designing a revenue-maximizing multi-item auction is # P -hard. This computational complexity poses a fundamental barrier in modeling economic systems but is worst-case and considers non-generic instances. Differentiable economics describes a new approach to solving these central problems in the economic sciences. It uses learning algorithms to find or approximate solutions to equilibrium computation or economic design problems. In particular, neural networks and learning algorithms such as Stochastic Gradient Descent have been shown to be very effective. Machine learning has led to breakthroughs in many sciences, and it also holds the potential to fundamentally alter how we analyze and design economic systems.

Revenue-optimal auction design is a challenging problem with significant theoretical and practical implications. Sequential auction mechanisms, known for their simplicity and strong strategyproofness guarantees, are often limited by theoretical results that are largely existential, except for certain restrictive settings. Although traditional reinforcement learning methods such as Proximal Policy Optimization (PPO) and Soft Actor-Critic (SAC) are applicable in this domain, they struggle with computational demands and convergence issues when dealing with large and continuous action spaces. In light of this and recognizing that we can model transitions differentiable for our settings, we propose using a new reinforcement learning framework tailored for sequential combinatorial auctions that leverages first-order gradients. Our extensive evaluations show that our approach achieves significant improvement in revenue over both analytical baselines and standard reinforcement learning algorithms. Furthermore, we scale our approach to scenarios involving up to 50 agents and 50 items, demonstrating its applicability in complex, real-world auction settings. As such, this work advances the computational tools available for auction design and contributes to bridging the gap between theoretical results and practical implementations in sequential auction design.

Large language models have astounded the world with fascinating new capabilities. However, they currently lack the ability to teach themselves new skills, relying instead on large amounts of human-generated training data. We introduce SECToR (Self-Education via Chain-of-Thought Reasoning), a proof-of-concept demonstration that language models can teach themselves new skills using chain-of-thought reasoning. During the self-learning loop, SECToR asks models to solve addition problems using chain-of-thought reasoning before training the next version of the model to solve those same problems directly without using such reasoning. This process often results in an improved model which can, when again augmented with chain-of-thought reasoning, solve even harder problems than the original model, allowing the self-learning loop to continue. Language models trained via SECToR autonomously learn to add up to the longest-length-digit numbers without access to any ground truth examples beyond an initial supervised fine-tuning phase consisting only of numbers with 6 or fewer digits. Our central hypothesis is that chain-of-thought reasoning can act as a policy improvement operator, similarly to how Monte-Carlo Tree Search is used in AlphaZero (Silver et al., 2017). We hope that this research can lead to new directions in which language models can learn to teach themselves without the need for human demonstrations.