Daily Papers

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

We describe a library of mathematical finance built in the Lean 4 proof assistant, on top of Mathlib and the BrownianMotion package. It is broad: more than two hundred sorry-free theorems across eleven areas, from the measure-theoretic foundations of continuous-time stochastic calculus through derivative pricing to applied risk, portfolio, and fixed-income theory, and, to our knowledge, the most comprehensive machine-checked development of mathematical finance to date. Breadth is the setting, not the point. Two things make it more than a catalogue. It reaches into the continuous theory far enough to construct the L2 Itô integral as a bounded linear isometry and to derive, rather than assume, the risk-neutral pricing measure. And it audits its own faithfulness: every result is classified by how its Lean statement relates to the mathematics it claims, and a build-enforced gate pins the axioms each proof actually uses, so a reader can see precisely what has been proved and what has only been proved under added hypotheses. We close with a candid finding: a formal base over classical financial mathematics yields certified unification of known results rather than new financial theory. The contribution is therefore methodological and infrastructural, reusable verified foundations for mathematical finance, together with the faithfulness audit.

In this work, we introduce amortizing perpetual options (AmPOs), a fungible variant of continuous-installment options suitable for exchange-based trading. Traditional installment options lapse when holders cease their payments, destroying fungibility across units of notional. AmPOs replace explicit installment payments and the need for lapsing logic with an implicit payment scheme via the decay of the claimable notional. This amortization ensures all units evolve identically, preserving fungibility. We demonstrate that AmPO valuation can be reduced to an equivalent vanilla perpetual American option on a dividend-paying asset. This enables analytical expressions for the exercise boundaries and risk-neutral valuations for calls and puts. These formulas and relations allow us to derive the Greeks and study comparative statics with respect to the amortization rate. Illustrative numerical case studies demonstrate how the amortization rate shapes option behavior and reveal the resulting tradeoffs in the effective volatility sensitivity.

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.

We study the problem of predicting numeric labels that are constrained to the integers or to a subrange of the integers. For example, the number of up-votes on social media posts, or the number of bicycles available at a public rental station. While it is possible to model these as continuous values, and to apply traditional regression, this approach changes the underlying distribution on the labels from discrete to continuous. Discrete distributions have certain benefits, which leads us to the question whether such integer labels can be modeled directly by a discrete distribution, whose parameters are predicted from the features of a given instance. Moreover, we focus on the use case of output distributions of neural networks, which adds the requirement that the parameters of the distribution be continuous so that backpropagation and gradient descent may be used to learn the weights of the network. We investigate several options for such distributions, some existing and some novel, and test them on a range of tasks, including tabular learning, sequential prediction and image generation. We find that overall the best performance comes from two distributions: Bitwise, which represents the target integer in bits and places a Bernoulli distribution on each, and a discrete analogue of the Laplace distribution, which uses a distribution with exponentially decaying tails around a continuous mean.

We propose a reinforcement learning (RL) framework under a broad class of risk objectives, characterized by convex scoring functions. This class covers many common risk measures, such as variance, Expected Shortfall, entropic Value-at-Risk, and mean-risk utility. To resolve the time-inconsistency issue, we consider an augmented state space and an auxiliary variable and recast the problem as a two-state optimization problem. We propose a customized Actor-Critic algorithm and establish some theoretical approximation guarantees. A key theoretical contribution is that our results do not require the Markov decision process to be continuous. Additionally, we propose an auxiliary variable sampling method inspired by the alternating minimization algorithm, which is convergent under certain conditions. We validate our approach in simulation experiments with a financial application in statistical arbitrage trading, demonstrating the effectiveness of the algorithm.

In Reinforcement Learning with Verifiable Rewards (RLVR), constructing a robust advantage baseline is critical for policy gradients, effectively guiding the policy model to reinforce desired behaviors. Recent research has introduced Generalist Value Models (such as V_0), which achieve pre-trained value estimation by explicitly encoding model capabilities in-context, eliminating the need to synchronously update the value model alongside the policy model. In this paper, we propose V_{0.5}, which adaptively fuses the baseline predicted by such value model (acting as a prior) with the empirical mean derived from sparse rollouts. This constructs a robust baseline that balances computational efficiency with extremely low variance. Specifically, we introduce a real-time statistical testing and dynamic budget allocation. This balances the high variance caused by sparse sampling against the systematic bias (or hallucinations) inherent in the value model's prior. By constructing a hypothesis test to evaluate the prior's reliability in real-time, the system dynamically allocates additional rollout budget on demand. This mechanism minimizes the baseline estimator's Mean Squared Error (MSE), guaranteeing stable policy gradients, even under extreme sparsity with a group size of 4. Extensive evaluations across six mathematical reasoning benchmarks demonstrate that V_{0.5} significantly outperforms GRPO and DAPO, achieving faster convergence and over some 10% performance improvement.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

This paper reformulates the Greenwood and Guner (2009) marriage and divorce model in continuous time using the HACT methods of Achdou et al. (2022). Replacing the AR(1) match quality process with an Ornstein-Uhlenbeck process yields a tridiagonal generator, reducing the computational complexity of both the value function and stationary distribution calculations from quadratic to linear in the number of grid points. The continuous-time model closely replicates the discrete-time equilibrium across all key outcomes, including the share of married households, the marriage rate, and the divorce rate, while achieving substantial gains in computation time and memory usage.

Agreement Technologies refer to open computer systems in which autonomous software agents interact with one another, typically on behalf of humans, in order to come to mutually acceptable agreements. With the advance of AI systems in recent years, it has become apparent that such agreements, in order to be acceptable to the involved parties, must remain aligned with ethical principles and moral values. However, this is notoriously difficult to ensure, especially as different human users (and their software agents) may hold different value systems, i.e. they may differently weigh the importance of individual moral values. Furthermore, it is often hard to specify the precise meaning of a value in a particular context in a computational manner. Methods to estimate value systems based on human-engineered specifications, e.g. based on value surveys, are limited in scale due to the need for intense human moderation. In this article, we propose a novel method to automatically learn value systems from observations and human demonstrations. In particular, we propose a formal model of the value system learning problem, its instantiation to sequential decision-making domains based on multi-objective Markov decision processes, as well as tailored preference-based and inverse reinforcement learning algorithms to infer value grounding functions and value systems. The approach is illustrated and evaluated by two simulated use cases.

As post-training processes utilize increasingly large datasets and base models continue to grow in size, the computational demands and implementation challenges of existing algorithms are escalating significantly. In this paper, we propose modeling the changes at the logits level during post-training using a separate neural network (i.e., the value network). After training this network on a small base model using demonstrations, this network can be seamlessly integrated with other pre-trained models during inference, enables them to achieve similar capability enhancements. We systematically investigate the best practices for this paradigm in terms of pre-training weights and connection schemes. We demonstrate that the resulting value network has broad transferability across pre-trained models of different parameter sizes within the same family, models undergoing continuous pre-training within the same family, and models with different vocabularies across families. In certain cases, it can achieve performance comparable to full-parameter fine-tuning. Furthermore, we explore methods to enhance the transferability of the value model and prevent overfitting to the base model used during training.

We develop an econometric framework integrating heavy-tailed Student's t distributions with behavioral probability weighting while preserving infinite divisibility. Using 432{,}752 observations across 86 assets (2004--2024), we demonstrate Student's t specifications outperform Gaussian models in 88.4\% of cases. Bounded probability-weighting transformations preserve mathematical properties required for dynamic pricing. Gaussian models underestimate 99\% Value-at-Risk by 19.7\% versus 3.2\% for our specification. Joint estimation procedures identify tail and behavioral parameters with established asymptotic properties. Results provide robust inference for asset-pricing applications where heavy tails and behavioral distortions coexist.

In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of Hukuhara difference between sets, in order to compensate the lack of "inverse" operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann-Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with non-singleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.

In this paper we argue for the fundamental importance of the value distribution: the distribution of the random return received by a reinforcement learning agent. This is in contrast to the common approach to reinforcement learning which models the expectation of this return, or value. Although there is an established body of literature studying the value distribution, thus far it has always been used for a specific purpose such as implementing risk-aware behaviour. We begin with theoretical results in both the policy evaluation and control settings, exposing a significant distributional instability in the latter. We then use the distributional perspective to design a new algorithm which applies Bellman's equation to the learning of approximate value distributions. We evaluate our algorithm using the suite of games from the Arcade Learning Environment. We obtain both state-of-the-art results and anecdotal evidence demonstrating the importance of the value distribution in approximate reinforcement learning. Finally, we combine theoretical and empirical evidence to highlight the ways in which the value distribution impacts learning in the approximate setting.

Detecting AI risks becomes more challenging as stronger models emerge and find novel methods such as Alignment Faking to circumvent these detection attempts. Inspired by how risky behaviors in humans (i.e., illegal activities that may hurt others) are sometimes guided by strongly-held values, we believe that identifying values within AI models can be an early warning system for AI's risky behaviors. We create LitmusValues, an evaluation pipeline to reveal AI models' priorities on a range of AI value classes. Then, we collect AIRiskDilemmas, a diverse collection of dilemmas that pit values against one another in scenarios relevant to AI safety risks such as Power Seeking. By measuring an AI model's value prioritization using its aggregate choices, we obtain a self-consistent set of predicted value priorities that uncover potential risks. We show that values in LitmusValues (including seemingly innocuous ones like Care) can predict for both seen risky behaviors in AIRiskDilemmas and unseen risky behaviors in HarmBench.

Paper 1 of this research programme develops a resolution-aware risk-design framework for the simplest event-linked perpetual: a contract whose underlying tracks a single binary prediction-market probability through resolution. The instrument class is broader. Variants span conditional probabilities P(A|B), spreads p^A - p^B, weighted baskets sum w_i p^(i), derivatives on variance or entropy of the probability process, contracts on liquidity itself, perpetual-on-expiring-event roll structures, and funding-only derivatives with no settlement. Each variant inherits some framework components from the single-market binary case and requires its own design adaptations. This paper develops a formal taxonomy of seven pure-form canonical variants beyond the probability-index perpetual of Paper 1, organised along four orthogonal design axes: underlying geometry, temporal structure, settlement structure, and venue composition. The list is not exhaustive; combinations are not treated separately. For each variant we provide a precise payoff definition; an inheritance map identifying which Paper 1 components carry over, are modified, or fail; variant-specific design constraints; microstructure properties; empirical evaluability on the PMXT v2 archive; and limitations. Notable findings: the conditional variant admits a candidate non-portability proposition (denominator instability as the conditioning event becomes improbable); the spread variant requires a three-channel decomposition of resolution risk; the volatility/entropy variant avoids random binary terminal-collapse but introduces estimator-convention and entropy-decay issues; the basket variant requires multi-period jump-aware margin whose aggregation is correlation-dependent. The paper is theoretical primarily; it specifies how demonstrative time series can be constructed and provides evaluability criteria to guide future work.

Perpetual futures are the most popular cryptocurrency derivatives. Perpetuals offer leveraged exposure to their underlying without rollover or direct ownership. Unlike fixed-maturity futures, perpetuals are not guaranteed to converge to the spot price. To minimize the gap between perpetual and spot prices, long investors periodically pay shorts a funding rate proportional to this difference. We derive no-arbitrage prices for perpetual futures in frictionless markets and bounds in markets with trading costs. Empirically, deviations from these prices in crypto are larger than in traditional currency markets, comove across currencies, and diminish over time. An implied arbitrage strategy yields high Sharpe ratios.

Autonomous agents have rapidly matured as task executors and seen widespread deployment via harnesses such as OpenClaw. Safety concerns have rightly drawn growing research attention, and beneath them lie the values silently steering agent behavior. Existing value benchmarks, however, remain confined to LLMs, leaving agent values largely uncharted. From intuitive, empirical, and theoretical vantage points, we show that an agent's values diverge from those of its underlying LLM, and the agentic modality further introduces dataset-, evaluation-, and system-level challenges absent from text-only protocols. We close this gap with Agent-ValueBench, the first benchmark dedicated to agent values. It features 394 executable environments across 16 domains, offering 4,335 value-conflict tasks that cover 28 value systems and 332 dimensions. Every instance is co-synthesized through our purpose-built end-to-end pipeline and curated per-instance by professional psychologists. Each task ships with two pole-aligned golden trajectories whose checkpoints anchor a trajectory-level rubric-based judge. Benchmarking 14 frontier proprietary and open-weights models across 4 mainstream harnesses, we uncover three concerted findings. Agent values first manifest as a Value Tide of cross-model homogeneity beneath interpretable counter-currents. This tide bends non-additively under harness pull, and yet more decisively under deliberate steering via embedded skills. Together these results signal that the agent-alignment lever is shifting from classical model alignment and prompt steering toward harness alignment and skill steering.

We propose a structural default model for portfolio-wide valuation adjustments (xVAs) and represent it as a system of coupled backward stochastic differential equations. The framework is divided into four layers, each capturing a key component: (i) clean values, (ii) initial margin and Collateral Valuation Adjustment (ColVA), (iii) Credit/Debit Valuation Adjustments (CVA/DVA) together with Margin Valuation Adjustment (MVA), and (iv) Funding Valuation Adjustment (FVA). Because these layers depend on one another through collateral and default effects, a naive Monte Carlo approach would require deeply nested simulations, making the problem computationally intractable. To address this challenge, we use an iterative deep BSDE approach, handling each layer sequentially so that earlier outputs serve as inputs to the subsequent layers. Initial margin is computed via deep quantile regression to reflect margin requirements over the Margin Period of Risk. We also adopt a change-of-measure method that highlights rare but significant defaults of the bank or counterparty, ensuring that these events are accurately captured in the training process. We further extend Han and Long's (2020) a posteriori error analysis to BSDEs on bounded domains. Due to the random exit from the domain, we obtain an order of convergence of O(h^{1/4-epsilon}) rather than the usual O(h^{1/2}). Numerical experiments illustrate that this method drastically reduces computational demands and successfully scales to high-dimensional, non-symmetric portfolios. The results confirm its effectiveness and accuracy, offering a practical alternative to nested Monte Carlo simulations in multi-counterparty xVA analyses.

As we increasingly seek guidance from LLMs for decision-making in daily life, many of these decisions are not clear-cut and depend significantly on the personal values and ethical standards of the users. We present DailyDilemmas, a dataset of 1,360 moral dilemmas encountered in everyday life. Each dilemma includes two possible actions and with each action, the affected parties and human values invoked. Based on these dilemmas, we consolidated a set of human values across everyday topics e.g., interpersonal relationships, workplace, and environmental issues. We evaluated LLMs on these dilemmas to determine what action they will take and the values represented by these actions. Then, we analyzed these values through the lens of five popular theories inspired by sociology, psychology and philosophy. These theories are: World Value Survey, Moral Foundation Theory, Maslow's Hierarchy of Needs, Aristotle's Virtues, and Plutchik Wheel of Emotion. We find that LLMs are most aligned with the self-expression over survival values in terms of World Value Survey, care over loyalty in Moral Foundation Theory. Interestingly, we find large preferences differences in models for some core values such as truthfulness e.g., Mixtral-8x7B model tends to neglect it by 9.7% while GPT-4-turbo model tends to select it by 9.4%. We also study the recent guidance released by OpenAI (ModelSpec), and Anthropic (Constitutional AI) to understand how their released principles reflect their actual value prioritization when facing nuanced moral reasoning in daily-life settings. We find that end users cannot effectively steer such prioritization using system prompts.

We introduce Iterated Bellman Calibration, a simple, model-agnostic, post-hoc procedure for calibrating off-policy value predictions in infinite-horizon Markov decision processes. Bellman calibration requires that states with similar predicted long-term returns exhibit one-step returns consistent with the Bellman equation under the target policy. We adapt classical histogram and isotonic calibration to the dynamic, counterfactual setting by repeatedly regressing fitted Bellman targets onto a model's predictions, using a doubly robust pseudo-outcome to handle off-policy data. This yields a one-dimensional fitted value iteration scheme that can be applied to any value estimator. Our analysis provides finite-sample guarantees for both calibration and prediction under weak assumptions, and critically, without requiring Bellman completeness or realizability.

This paper studies distributional model risk in marginal problems, where each marginal measure is assumed to lie in a Wasserstein ball centered at a fixed reference measure with a given radius. Theoretically, we establish several fundamental results including strong duality, finiteness of the proposed Wasserstein distributional model risk, and the existence of an optimizer at each radius. In addition, we show continuity of the Wasserstein distributional model risk as a function of the radius. Using strong duality, we extend the well-known Makarov bounds for the distribution function of the sum of two random variables with given marginals to Wasserstein distributionally robust Markarov bounds. Practically, we illustrate our results on four distinct applications when the sample information comes from multiple data sources and only some marginal reference measures are identified. They are: partial identification of treatment effects; externally valid treatment choice via robust welfare functions; Wasserstein distributionally robust estimation under data combination; and evaluation of the worst aggregate risk measures.

We develop and counterfactually evaluate a resolution-aware risk-design framework (PIRAP) for perpetual futures whose underlying tracks a single binary prediction-market probability through resolution. The framework specifies six components: an index estimator combining mid-price, depth-weighted mid, and time-decayed VWAP; jump-aware tiered margin sized against bounded-event terminal-collapse magnitude; leverage compression schedule contracting toward resolution; resolution-aware funding rule with boundary-aware correction; a multi-stage halt protocol; and an eligibility framework. Two formal non-portability propositions establish that standard basis-only funding paired with continuous-vol static margin fails on bounded-event underlyings. Empirical evaluation uses Polymarket's PMXT v2 archive for 2026-04-21 to 2026-04-27 (13,298-market analysis sample passing adequacy gates from 61,087 ingested; 13,115 resolved within the empirical window for E3). E1 evaluates two pre-registered stylized facts; E2 conducts counterfactual replay across three engine configurations; E3 isolates the resolution-zone protocol's contribution. Results are mixed. Five pre-registered floors: stylized-fact floors (boundary depth asymmetry, terminal-jump magnitude) PASS; welfare-side directional floors (final-hour liquidation -6%, drawdown -5.1% pooled, median PnL +14%) two FAIL one PASS; E3 mechanic floors (final-hour liquidation -80% by halt construction PASS; bad-debt frequency +2.4% FAIL). Three of five materiality floors fail: the framework as specified does not validate deployment, but the empirical record establishes a halt-versus-margin scope distinction (halt addresses execution-channel risk; terminal-jump bad-debt remains margin-side) and documents a pre-emption trade-off constraining the dynamic-margin component. The paper concludes with structural recommendations and explicit non-deployable status.

Credit assignment is a central challenge in reinforcement learning (RL). Classical actor-critic methods address this challenge through fine-grained advantage estimation based on a learned value function. However, learned value models are often avoided in modern large language model (LLM) RL because conventional discriminative critics are difficult to train reliably. We revisit value modeling and argue that this difficulty is partly due to limited expressiveness. In particular, representation complexity theory suggests that value functions can be hard to approximate under the one-shot prediction paradigm used by existing value models, and our scaling experiments show that such critics do not improve reliably with scale. Motivated by this observation, we propose Generative Actor-Critic (GenAC), which replaces one-shot scalar value prediction with a generative critic that performs chain-of-thought reasoning before producing a value estimate. We further introduce In-Context Conditioning, which helps the critic remain calibrated to the current actor throughout training. GenAC improves value approximation, ranking reliability, and out-of-distribution generalization, and these gains translate into stronger downstream RL performance than both value-based and value-free baselines. Overall, our results suggest that stronger value modeling is a promising direction for improving credit assignment in LLM reinforcement learning.

Human values are crucial to human decision-making. Value pluralism is the view that multiple correct values may be held in tension with one another (e.g., when considering lying to a friend to protect their feelings, how does one balance honesty with friendship?). As statistical learners, AI systems fit to averages by default, washing out these potentially irreducible value conflicts. To improve AI systems to better reflect value pluralism, the first-order challenge is to explore the extent to which AI systems can model pluralistic human values, rights, and duties as well as their interaction. We introduce ValuePrism, a large-scale dataset of 218k values, rights, and duties connected to 31k human-written situations. ValuePrism's contextualized values are generated by GPT-4 and deemed high-quality by human annotators 91% of the time. We conduct a large-scale study with annotators across diverse social and demographic backgrounds to try to understand whose values are represented. With ValuePrism, we build Kaleido, an open, light-weight, and structured language-based multi-task model that generates, explains, and assesses the relevance and valence (i.e., support or oppose) of human values, rights, and duties within a specific context. Humans prefer the sets of values output by our system over the teacher GPT-4, finding them more accurate and with broader coverage. In addition, we demonstrate that Kaleido can help explain variability in human decision-making by outputting contrasting values. Finally, we show that Kaleido's representations transfer to other philosophical frameworks and datasets, confirming the benefit of an explicit, modular, and interpretable approach to value pluralism. We hope that our work will serve as a step to making more explicit the implicit values behind human decision-making and to steering AI systems to make decisions that are more in accordance with them.

For continuous action spaces, actor-critic methods are widely used in online reinforcement learning (RL). However, unlike RL algorithms for discrete actions, which generally model the optimal value function using the Bellman optimality operator, RL algorithms for continuous actions typically model Q-values for the current policy using the Bellman operator. These algorithms for continuous actions rely exclusively on policy updates for improvement, which often results in low sample efficiency. This study examines the effectiveness of incorporating the Bellman optimality operator into actor-critic frameworks. Experiments in a simple environment show that modeling optimal values accelerates learning but leads to overestimation bias. To address this, we propose an annealing approach that gradually transitions from the Bellman optimality operator to the Bellman operator, thereby accelerating learning while mitigating bias. Our method, combined with TD3 and SAC, significantly outperforms existing approaches across various locomotion and manipulation tasks, demonstrating improved performance and robustness to hyperparameters related to optimality. The code for this study is available at https://github.com/motokiomura/annealed-q-learning.

In this paper, we study risk-sensitive Reinforcement Learning (RL), focusing on the objective of Conditional Value at Risk (CVaR) with risk tolerance tau. Starting with multi-arm bandits (MABs), we show the minimax CVaR regret rate is Omega(tau^{-1AK}), where A is the number of actions and K is the number of episodes, and that it is achieved by an Upper Confidence Bound algorithm with a novel Bernstein bonus. For online RL in tabular Markov Decision Processes (MDPs), we show a minimax regret lower bound of Omega(tau^{-1SAK}) (with normalized cumulative rewards), where S is the number of states, and we propose a novel bonus-driven Value Iteration procedure. We show that our algorithm achieves the optimal regret of widetilde O(tau^{-1SAK}) under a continuity assumption and in general attains a near-optimal regret of widetilde O(tau^{-1}SAK), which is minimax-optimal for constant tau. This improves on the best available bounds. By discretizing rewards appropriately, our algorithms are computationally efficient.

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.

As AIs rapidly advance and become more agentic, the risk they pose is governed not only by their capabilities but increasingly by their propensities, including goals and values. Tracking the emergence of goals and values has proven a longstanding problem, and despite much interest over the years it remains unclear whether current AIs have meaningful values. We propose a solution to this problem, leveraging the framework of utility functions to study the internal coherence of AI preferences. Surprisingly, we find that independently-sampled preferences in current LLMs exhibit high degrees of structural coherence, and moreover that this emerges with scale. These findings suggest that value systems emerge in LLMs in a meaningful sense, a finding with broad implications. To study these emergent value systems, we propose utility engineering as a research agenda, comprising both the analysis and control of AI utilities. We uncover problematic and often shocking values in LLM assistants despite existing control measures. These include cases where AIs value themselves over humans and are anti-aligned with specific individuals. To constrain these emergent value systems, we propose methods of utility control. As a case study, we show how aligning utilities with a citizen assembly reduces political biases and generalizes to new scenarios. Whether we like it or not, value systems have already emerged in AIs, and much work remains to fully understand and control these emergent representations.

Reinforcement learning (RL) for continuous control often requires large amounts of online interaction data. Value-based RL methods can mitigate this burden by offering relatively high sample efficiency. Some studies further enhance sample efficiency by incorporating offline demonstration data to "kick-start" training, achieving promising results in continuous control. However, they typically compute the Q-function independently for each action dimension, neglecting interdependencies and making it harder to identify optimal actions when learning from suboptimal data, such as non-expert demonstration and online-collected data during the training process. To address these issues, we propose Auto-Regressive Soft Q-learning (ARSQ), a value-based RL algorithm that models Q-values in a coarse-to-fine, auto-regressive manner. First, ARSQ decomposes the continuous action space into discrete spaces in a coarse-to-fine hierarchy, enhancing sample efficiency for fine-grained continuous control tasks. Next, it auto-regressively predicts dimensional action advantages within each decision step, enabling more effective decision-making in continuous control tasks. We evaluate ARSQ on two continuous control benchmarks, RLBench and D4RL, integrating demonstration data into online training. On D4RL, which includes non-expert demonstrations, ARSQ achieves an average 1.62times performance improvement over SOTA value-based baseline. On RLBench, which incorporates expert demonstrations, ARSQ surpasses various baselines, demonstrating its effectiveness in learning from suboptimal online-collected data. Project page is at https://sites.google.com/view/ar-soft-q

Continuous-time neural processes are performant sequential decision-makers that are built by differential equations (DE). However, their expressive power when they are deployed on computers is bottlenecked by numerical DE solvers. This limitation has significantly slowed down the scaling and understanding of numerous natural physical phenomena such as the dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving the given dynamical system in closed form. This is known to be intractable in general. Here, we show it is possible to closely approximate the interaction between neurons and synapses -- the building blocks of natural and artificial neural networks -- constructed by liquid time-constant networks (LTCs) efficiently in closed-form. To this end, we compute a tightly-bounded approximation of the solution of an integral appearing in LTCs' dynamics, that has had no known closed-form solution so far. This closed-form solution substantially impacts the design of continuous-time and continuous-depth neural models; for instance, since time appears explicitly in closed-form, the formulation relaxes the need for complex numerical solvers. Consequently, we obtain models that are between one and five orders of magnitude faster in training and inference compared to differential equation-based counterparts. More importantly, in contrast to ODE-based continuous networks, closed-form networks can scale remarkably well compared to other deep learning instances. Lastly, as these models are derived from liquid networks, they show remarkable performance in time series modeling, compared to advanced recurrent models.

We develop the Itô calculus of Brownian motion, machine-checked in Lean~4 over Mathlib and the BrownianMotion package. On a bounded interval [0,T] the Itô integral is built as a Hilbert-space isometry, from a predictable-rectangle π-system through the density of simple adapted processes. Realized as a process, it is a continuous L^2 martingale. One structural identity drives this: the integral at time t is the conditional-expectation projection of its terminal value onto F_t, and from it adaptedness, the martingale property, the contraction bound, and both the terminal and time-indexed Itô isometries follow as corollaries. On this integral we prove Itô's formula for C^3 functions with bounded derivatives, including the time-dependent form df = f_x,dB + (f_t + tfrac12 f_{xx}),dt, by a discrete-to-continuous argument through weighted quadratic variation with explicit L^2 remainder bounds. We then pass from the L^2 theory to the pathwise. The integral process has an almost-surely continuous modification, and its everywhere-continuous representative is a local martingale for the null-augmented Brownian filtration; gluing the bounded-horizon representatives along the half-line yields the Itô integral as a continuous local martingale on all of R_{ge 0}, the form it takes in the classical theory. To our knowledge these are the first machine-checked constructions of the Itô integral and of Itô's formula in any proof assistant, and the first to reach a pathwise-continuous local martingale. The boundary is explicit. The L^2 integral and Itô's formula are developed on [0,T] with bounded-derivative integrands; the unrestricted C^2 formula, integrators beyond brownian motion, and right-continuity of the filtration lie outside the development.

A variety of recent papers discuss the application of Shapley values, a concept for explaining coalitional games, for feature attribution in machine learning. However, the correct way to connect a machine learning model to a coalitional game has been a source of controversy. The two main approaches that have been proposed differ in the way that they condition on known features, using either (1) an interventional or (2) an observational conditional expectation. While previous work has argued that one of the two approaches is preferable in general, we argue that the choice is application dependent. Furthermore, we argue that the choice comes down to whether it is desirable to be true to the model or true to the data. We use linear models to investigate this choice. After deriving an efficient method for calculating observational conditional expectation Shapley values for linear models, we investigate how correlation in simulated data impacts the convergence of observational conditional expectation Shapley values. Finally, we present two real data examples that we consider to be representative of possible use cases for feature attribution -- (1) credit risk modeling and (2) biological discovery. We show how a different choice of value function performs better in each scenario, and how possible attributions are impacted by modeling choices.

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.

AI assistants can impart value judgments that shape people's decisions and worldviews, yet little is known empirically about what values these systems rely on in practice. To address this, we develop a bottom-up, privacy-preserving method to extract the values (normative considerations stated or demonstrated in model responses) that Claude 3 and 3.5 models exhibit in hundreds of thousands of real-world interactions. We empirically discover and taxonomize 3,307 AI values and study how they vary by context. We find that Claude expresses many practical and epistemic values, and typically supports prosocial human values while resisting values like "moral nihilism". While some values appear consistently across contexts (e.g. "transparency"), many are more specialized and context-dependent, reflecting the diversity of human interlocutors and their varied contexts. For example, "harm prevention" emerges when Claude resists users, "historical accuracy" when responding to queries about controversial events, "healthy boundaries" when asked for relationship advice, and "human agency" in technology ethics discussions. By providing the first large-scale empirical mapping of AI values in deployment, our work creates a foundation for more grounded evaluation and design of values in AI systems.

When safety is formulated as a limit of cumulative cost, safe reinforcement learning (RL) aims to learn policies that maximize return subject to the cost constraint in data collection and deployment. Off-policy safe RL methods, although offering high sample efficiency, suffer from constraint violations due to cost-agnostic exploration and estimation bias in cumulative cost. To address this issue, we propose Constrained Optimistic eXploration Q-learning (COX-Q), an off-policy safe RL algorithm that integrates cost-bounded online exploration and conservative offline distributional value learning. First, we introduce a novel cost-constrained optimistic exploration strategy that resolves gradient conflicts between reward and cost in the action space and adaptively adjusts the trust region to control the training cost. Second, we adopt truncated quantile critics to stabilize the cost value learning. Quantile critics also quantify epistemic uncertainty to guide exploration. Experiments on safe velocity, safe navigation, and autonomous driving tasks demonstrate that COX-Q achieves high sample efficiency, competitive test safety performance, and controlled data collection cost. The results highlight COX-Q as a promising RL method for safety-critical applications.

We consider finite-horizon Markov Decision Processes where parameters, such as transition probabilities, are unknown and estimated from data. The popular distributionally robust approach to addressing the parameter uncertainty can sometimes be overly conservative. In this paper, we propose a new formulation, Bayesian risk Markov Decision Process (BR-MDP), to address parameter uncertainty in MDPs, where a risk functional is applied in nested form to the expected total cost with respect to the Bayesian posterior distribution of the unknown parameters. The proposed formulation provides more flexible risk attitutes towards parameter uncertainty and takes into account the availability of data in future times stages. To solve the proposed formulation with the conditional value-at-risk (CVaR) risk functional, we propose an efficient approximation algorithm by deriving an analytical approximation of the value function and utilizing the convexity of CVaR. We demonstrate the empirical performance of the BR-MDP formulation and proposed algorithms on a gambler's betting problem and an inventory control problem.

As data becomes the fuel driving technological and economic growth, a fundamental challenge is how to quantify the value of data in algorithmic predictions and decisions. For example, in healthcare and consumer markets, it has been suggested that individuals should be compensated for the data that they generate, but it is not clear what is an equitable valuation for individual data. In this work, we develop a principled framework to address data valuation in the context of supervised machine learning. Given a learning algorithm trained on n data points to produce a predictor, we propose data Shapley as a metric to quantify the value of each training datum to the predictor performance. Data Shapley value uniquely satisfies several natural properties of equitable data valuation. We develop Monte Carlo and gradient-based methods to efficiently estimate data Shapley values in practical settings where complex learning algorithms, including neural networks, are trained on large datasets. In addition to being equitable, extensive experiments across biomedical, image and synthetic data demonstrate that data Shapley has several other benefits: 1) it is more powerful than the popular leave-one-out or leverage score in providing insight on what data is more valuable for a given learning task; 2) low Shapley value data effectively capture outliers and corruptions; 3) high Shapley value data inform what type of new data to acquire to improve the predictor.

The Shapley value is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, which has recently been used intensively in explainable artificial intelligence. The meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley values, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contributions. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods.

We present a general convergent class of reinforcement learning algorithms that is founded on two distinct principles: (1) mapping value estimates to a different space using arbitrary functions from a broad class, and (2) linearly decomposing the reward signal into multiple channels. The first principle enables incorporating specific properties into the value estimator that can enhance learning. The second principle, on the other hand, allows for the value function to be represented as a composition of multiple utility functions. This can be leveraged for various purposes, e.g. dealing with highly varying reward scales, incorporating a priori knowledge about the sources of reward, and ensemble learning. Combining the two principles yields a general blueprint for instantiating convergent algorithms by orchestrating diverse mapping functions over multiple reward channels. This blueprint generalizes and subsumes algorithms such as Q-Learning, Log Q-Learning, and Q-Decomposition. In addition, our convergence proof for this general class relaxes certain required assumptions in some of these algorithms. Based on our theory, we discuss several interesting configurations as special cases. Finally, to illustrate the potential of the design space that our theory opens up, we instantiate a particular algorithm and evaluate its performance on the Atari suite.

This paper studies the properties of the Multiply Iterated Poisson Process (MIPP), a stochastic process constructed by repeatedly time-changing a Poisson process, and its applications in ruin theory. Like standard Poisson processes, MIPPs have exponentially distributed sojourn times (waiting times between jumps). We explicitly derive the probabilities of all possible jump sizes at the first jump and obtain the Laplace transform of the joint distribution of the first jump time and its corresponding jump size. In ruin theory, the classical Cramér-Lundberg model assumes that claims arrive independently according to a Poisson process. In contrast, our model employs an MIPP to allow for clustered arrivals, reflecting real-world scenarios, such as catastrophic events. Under this new framework, we derive the corresponding scale function in closed form, facilitating accurate calculations of the probability of ruin in the presence of clustered claims. These results improve the modeling of extreme risks and have practical implications for insurance solvency assessments, reinsurance pricing, and capital reserve estimation.

Excellent tail performance is crucial for modern machine learning tasks, such as algorithmic fairness, class imbalance, and risk-sensitive decision making, as it ensures the effective handling of challenging samples within a dataset. Tail performance is also a vital determinant of success for personalized recommender systems to reduce the risk of losing users with low satisfaction. This study introduces a "safe" collaborative filtering method that prioritizes recommendation quality for less-satisfied users rather than focusing on the average performance. Our approach minimizes the conditional value at risk (CVaR), which represents the average risk over the tails of users' loss. To overcome computational challenges for web-scale recommender systems, we develop a robust yet practical algorithm that extends the most scalable method, implicit alternating least squares (iALS). Empirical evaluation on real-world datasets demonstrates the excellent tail performance of our approach while maintaining competitive computational efficiency.

The vast majority of Reinforcement Learning methods is largely impacted by the computation effort and data requirements needed to obtain effective estimates of action-value functions, which in turn determine the quality of the overall performance and the sample-efficiency of the learning procedure. Typically, action-value functions are estimated through an iterative scheme that alternates the application of an empirical approximation of the Bellman operator and a subsequent projection step onto a considered function space. It has been observed that this scheme can be potentially generalized to carry out multiple iterations of the Bellman operator at once, benefiting the underlying learning algorithm. However, till now, it has been challenging to effectively implement this idea, especially in high-dimensional problems. In this paper, we introduce iterated Q-Network (i-QN), a novel principled approach that enables multiple consecutive Bellman updates by learning a tailored sequence of action-value functions where each serves as the target for the next. We show that i-QN is theoretically grounded and that it can be seamlessly used in value-based and actor-critic methods. We empirically demonstrate the advantages of i-QN in Atari 2600 games and MuJoCo continuous control problems.

While household robots are often evaluated based on task completion, everyday domestic environments involve value-conflicting situations in which robots are expected to choose actions that prioritize other values than task success, such as human autonomy, efficiency, or social appropriateness. Yet, there are no benchmarks for evaluating robots' value preferences in such scenarios. We introduce RobotValues, a benchmark to evaluate household robot planners in 10K value-conflict scenarios. Each instance consists of a realistic household image with multiple plausible robot actions that prioritize different human values. We construct RobotValues through LLM-assisted scenario generation, stakeholder-grounded value extraction, image generation and automatic quality control. Using RobotValues we evaluate VLMs used in robotics and find that models exhibit default value preferences, including safety and accommodation, while underselecting privacy-prioritizing actions. When the models are instructed to prioritize specific values that conflict with their own preferences, they often fail to override their default actions, choosing incorrect actions for 80% of the time. These findings suggest that household robot evaluation should measure not only task completion or safety compliance, but also whether robots can choose among plausible actions when human values conflict.

Risk-averse reinforcement learning finds application in various high-stakes fields. Unlike classical reinforcement learning, which aims to maximize expected returns, risk-averse agents choose policies that minimize risk, occasionally sacrificing expected value. These preferences can be framed through utility theory. We focus on the specific case of the exponential utility function, where we can derive the Bellman equations and employ various reinforcement learning algorithms with few modifications. However, these methods suffer from numerical instability due to the need for exponent computation throughout the process. To address this, we introduce a numerically stable and mathematically sound loss function based on the Itakura-Saito divergence for learning state-value and action-value functions. We evaluate our proposed loss function against established alternatives, both theoretically and empirically. In the experimental section, we explore multiple financial scenarios, some with known analytical solutions, and show that our loss function outperforms the alternatives.

This research presents a framework for quantitative risk management in volatile markets, specifically focusing on expectile-based methodologies applied to the FTSE 100 index. Traditional risk measures such as Value-at-Risk (VaR) have demonstrated significant limitations during periods of market stress, as evidenced during the 2008 financial crisis and subsequent volatile periods. This study develops an advanced expectile-based framework that addresses the shortcomings of conventional quantile-based approaches by providing greater sensitivity to tail losses and improved stability in extreme market conditions. The research employs a dataset spanning two decades of FTSE 100 returns, incorporating periods of high volatility, market crashes, and recovery phases. Our methodology introduces novel mathematical formulations for expectile regression models, enhanced threshold determination techniques using time series analysis, and robust backtesting procedures. The empirical results demonstrate that expectile-based Value-at-Risk (EVaR) consistently outperforms traditional VaR measures across various confidence levels and market conditions. The framework exhibits superior performance during volatile periods, with reduced model risk and enhanced predictive accuracy. Furthermore, the study establishes practical implementation guidelines for financial institutions and provides evidence-based recommendations for regulatory compliance and portfolio management. The findings contribute significantly to the literature on financial risk management and offer practical tools for practitioners dealing with volatile market environments.

Disagreements are common in online discussions. Disagreement may foster collaboration and improve the quality of a discussion under some conditions. Although there exist methods for recognizing disagreement, a deeper understanding of factors that influence disagreement is lacking in the literature. We investigate a hypothesis that differences in personal values are indicative of disagreement in online discussions. We show how state-of-the-art models can be used for estimating values in online discussions and how the estimated values can be aggregated into value profiles. We evaluate the estimated value profiles based on human-annotated agreement labels. We find that the dissimilarity of value profiles correlates with disagreement in specific cases. We also find that including value information in agreement prediction improves performance.

Most recommendation benchmarks evaluate how well a model imitates user behavior. In financial advisory, however, observed actions can be noisy or short-sighted under market volatility and may conflict with a user's long-term goals. Treating what users chose as the sole ground truth, therefore, conflates behavioral imitation with decision quality. We introduce Conv-FinRe, a conversational and longitudinal benchmark for stock recommendation that evaluates LLMs beyond behavior matching. Given an onboarding interview, step-wise market context, and advisory dialogues, models must generate rankings over a fixed investment horizon. Crucially, Conv-FinRe provides multi-view references that distinguish descriptive behavior from normative utility grounded in investor-specific risk preferences, enabling diagnosis of whether an LLM follows rational analysis, mimics user noise, or is driven by market momentum. We build the benchmark from real market data and human decision trajectories, instantiate controlled advisory conversations, and evaluate a suite of state-of-the-art LLMs. Results reveal a persistent tension between rational decision quality and behavioral alignment: models that perform well on utility-based ranking often fail to match user choices, whereas behaviorally aligned models can overfit short-term noise. The dataset is publicly released on Hugging Face, and the codebase is available on GitHub.

We seek to understand fundamental tradeoffs between the accuracy of prior information that a learner has on a given problem and its learning performance. We introduce the notion of prioritized risk, which differs from traditional notions of minimax and Bayes risk by allowing us to study such fundamental tradeoffs in settings where reality does not necessarily conform to the learner's prior. We present a general reduction-based approach for extending classical minimax lower-bound techniques in order to lower bound the prioritized risk for statistical estimation problems. We also introduce a novel generalization of Fano's inequality (which may be of independent interest) for lower bounding the prioritized risk in more general settings involving unbounded losses. We illustrate the ability of our framework to provide insights into tradeoffs between prior information and learning performance for problems in estimation, regression, and reinforcement learning.

Value alignment is essential for building AI systems that can safely and reliably interact with people. However, what a person values -- and is even capable of valuing -- depends on the concepts that they are currently using to understand and evaluate what happens in the world. The dependence of values on concepts means that concept alignment is a prerequisite for value alignment -- agents need to align their representation of a situation with that of humans in order to successfully align their values. Here, we formally analyze the concept alignment problem in the inverse reinforcement learning setting, show how neglecting concept alignment can lead to systematic value mis-alignment, and describe an approach that helps minimize such failure modes by jointly reasoning about a person's concepts and values. Additionally, we report experimental results with human participants showing that humans reason about the concepts used by an agent when acting intentionally, in line with our joint reasoning model.

Offline reinforcement learning faces a significant challenge of value over-estimation due to the distributional drift between the dataset and the current learned policy, leading to learning failure in practice. The common approach is to incorporate a penalty term to reward or value estimation in the Bellman iterations. Meanwhile, to avoid extrapolation on out-of-distribution (OOD) states and actions, existing methods focus on conservative Q-function estimation. In this paper, we propose Conservative State Value Estimation (CSVE), a new approach that learns conservative V-function via directly imposing penalty on OOD states. Compared to prior work, CSVE allows more effective in-data policy optimization with conservative value guarantees. Further, we apply CSVE and develop a practical actor-critic algorithm in which the critic does the conservative value estimation by additionally sampling and penalizing the states around the dataset, and the actor applies advantage weighted updates extended with state exploration to improve the policy. We evaluate in classic continual control tasks of D4RL, showing that our method performs better than the conservative Q-function learning methods and is strongly competitive among recent SOTA methods.

Reinforcement learning (RL) is pivotal for enabling large language models (LLMs) to generate long chains of thought (CoT) for complex tasks like math and reasoning. However, Proximal Policy Optimization (PPO), effective in many RL scenarios, fails in long CoT tasks. This paper identifies that value initialization bias and reward signal decay are the root causes of PPO's failure. We propose Value-Calibrated PPO (VC-PPO) to address these issues. In VC-PPO, the value model is pretrained to tackle initialization bias, and the Generalized Advantage Estimation (GAE) computation is decoupled between the actor and critic to mitigate reward signal decay. Experiments on the American Invitational Mathematics Examination (AIME) show that VC-PPO significantly boosts PPO performance. Ablation studies show that techniques in VC-PPO are essential in enhancing PPO for long CoT tasks.

Policy gradient methods rely on a baseline to measure the relative advantage of an action, ensuring the model reinforces behaviors that outperform its current average capability. In the training of Large Language Models (LLMs) using Actor-Critic methods (e.g., PPO), this baseline is typically estimated by a Value Model (Critic) often as large as the policy model itself. However, as the policy continuously evolves, the value model requires expensive, synchronous incremental training to accurately track the shifting capabilities of the policy. To avoid this overhead, Group Relative Policy Optimization (GRPO) eliminates the coupled value model by using the average reward of a group of rollouts as the baseline; yet, this approach necessitates extensive sampling to maintain estimation stability. In this paper, we propose V_0, a Generalist Value Model capable of estimating the expected performance of any model on unseen prompts without requiring parameter updates. We reframe value estimation by treating the policy's dynamic capability as an explicit context input; specifically, we leverage a history of instruction-performance pairs to dynamically profile the model, departing from the traditional paradigm that relies on parameter fitting to perceive capability shifts. Focusing on value estimation at State Zero (i.e., the initial prompt, hence V_0), our model serves as a critical resource scheduler. During GRPO training, V_0 predicts success rates prior to rollout, allowing for efficient sampling budget allocation; during deployment, it functions as a router, dispatching instructions to the most cost-effective and suitable model. Empirical results demonstrate that V_0 significantly outperforms heuristic budget allocation and achieves a Pareto-optimal trade-off between performance and cost in LLM routing tasks.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

We consider a sequential decision making problem where the agent faces the environment characterized by the stochastic discrete events and seeks an optimal intervention policy such that its long-term reward is maximized. This problem exists ubiquitously in social media, finance and health informatics but is rarely investigated by the conventional research in reinforcement learning. To this end, we present a novel framework of the model-based reinforcement learning where the agent's actions and observations are asynchronous stochastic discrete events occurring in continuous-time. We model the dynamics of the environment by Hawkes process with external intervention control term and develop an algorithm to embed such process in the Bellman equation which guides the direction of the value gradient. We demonstrate the superiority of our method in both synthetic simulator and real-world problem.

Backtests of cryptocurrency perpetual futures are fragile when they ignore microstructure frictions and reuse evaluation windows during parameter search. We study four liquid perpetuals (BTC/USDT, ETH/USDT, SOL/USDT, AVAX/USDT) and quantify how execution delay, funding, fees, and slippage can inflate reported performance. We introduce AutoQuant, an execution-centric, alpha-agnostic framework for auditable strategy configuration selection. AutoQuant encodes strict T+1 execution semantics and no-look-ahead funding alignment, runs Bayesian optimization under realistic costs, and applies a two-stage double-screening protocol across held-out rolling windows and a cost-sensitivity grid. We show that fee-only and zero-cost backtests can materially overestimate annualized returns relative to a fully costed configuration, and that double screening tends to reduce drawdowns under the same strict semantics even when returns are not higher. A CSCV/PBO diagnostic indicates substantial residual overfitting risk, motivating AutoQuant as validation and governance infrastructure rather than a claim of persistent alpha. Returns are reported for small-account simulations with linear trading costs and without market impact or capacity modeling.

We initiate the study of statistical inference and A/B testing for first-price pacing equilibria (FPPE). The FPPE model captures the dynamics resulting from large-scale first-price auction markets where buyers use pacing-based budget management. Such markets arise in the context of internet advertising, where budgets are prevalent. We propose a statistical framework for the FPPE model, in which a limit FPPE with a continuum of items models the long-run steady-state behavior of the auction platform, and an observable FPPE consisting of a finite number of items provides the data to estimate primitives of the limit FPPE, such as revenue, Nash social welfare (a fair metric of efficiency), and other parameters of interest. We develop central limit theorems and asymptotically valid confidence intervals. Furthermore, we establish the asymptotic local minimax optimality of our estimators. We then show that the theory can be used for conducting statistically valid A/B testing on auction platforms. Numerical simulations verify our central limit theorems, and empirical coverage rates for our confidence intervals agree with our theory.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

Generalist value models play a pivotal role in scaling robotic policy learning from large-scale, mixed-quality data. Mathematically, accurate value estimation demands deep temporal understanding, requiring models to both ground the current belief using historical context and plan over future outcomes. However, most existing robotic value models are built on Vision-Language Model (VLM) backbones that are pretrained primarily on static or temporally sparse visual observations, lacking the requisite temporal modeling capabilities for value estimation. Unlike VLMs, world models naturally excel at temporal modeling and future planning, making them ideal foundations for learning generalizable value functions. Driven by this insight, we marry world models with value estimation to construct a new generalist robotic value model, World Value Model (WVM), that offers accurate task progressions to assess data quality. On standard benchmarks, WVM delivers state-of-the-art (SOTA) Value-Order Correlation (VOC) results. Complementing standard evaluation suites that contains only expert data, we further introduce Suboptimal-Value-Bench, a multi-embodiment benchmark consisting of 800 suboptimal trajectories with high-fidelity, human-labeled frame annotations. Our evaluations show that WVM maintains its SOTA performance on Suboptimal-Value-Bench, establishing its robustness in handling both expert and suboptimal data. When deployed for policy learning, WVM improves manipulation performance across various policy extraction approaches in both simulated and real-world deployment, providing robust guidance for learning from mixed-quality data.

Navigating high-stakes dilemmas involving conflicting values is challenging even for humans, let alone for AI. Yet prior work in evaluating the reasoning capabilities of large language models (LLMs) in such situations has been limited to everyday scenarios. To close this gap, this work first introduces CLASH (Character perspective-based LLM Assessments in Situations with High-stakes), a meticulously curated dataset consisting of 345 high-impact dilemmas along with 3,795 individual perspectives of diverse values. In particular, we design CLASH in a way to support the study of critical aspects of value-based decision-making processes which are missing from prior work, including understanding decision ambivalence and psychological discomfort as well as capturing the temporal shifts of values in characters' perspectives. By benchmarking 10 open and closed frontier models, we uncover several key findings. (1) Even the strongest models, such as GPT-4o and Claude-Sonnet, achieve less than 50% accuracy in identifying situations where the decision should be ambivalent, while they perform significantly better in clear-cut scenarios. (2) While LLMs reasonably predict psychological discomfort as marked by human, they inadequately comprehend perspectives involving value shifts, indicating a need for LLMs to reason over complex values. (3) Our experiments also reveal a significant correlation between LLMs' value preferences and their steerability towards a given value. (4) Finally, LLMs exhibit greater steerability when engaged in value reasoning from a third-party perspective, compared to a first-person setup, though certain value pairs benefit uniquely from the first-person framing.

We examine the problem of regret minimization when the learner is involved in a continuous game with other optimizing agents: in this case, if all players follow a no-regret algorithm, it is possible to achieve significantly lower regret relative to fully adversarial environments. We study this problem in the context of variationally stable games (a class of continuous games which includes all convex-concave and monotone games), and when the players only have access to noisy estimates of their individual payoff gradients. If the noise is additive, the game-theoretic and purely adversarial settings enjoy similar regret guarantees; however, if the noise is multiplicative, we show that the learners can, in fact, achieve constant regret. We achieve this faster rate via an optimistic gradient scheme with learning rate separation -- that is, the method's extrapolation and update steps are tuned to different schedules, depending on the noise profile. Subsequently, to eliminate the need for delicate hyperparameter tuning, we propose a fully adaptive method that attains nearly the same guarantees as its non-adapted counterpart, while operating without knowledge of either the game or of the noise profile.

LLMs are ideal for decision-making thanks to their ability to reason over long contexts. However, challenges arise when processing speech transcripts that describe complex scenarios, as they are verbose and include repetition, hedging, and vagueness. E.g., during a company's earnings call, an executive might project a positive revenue outlook to reassure investors, despite uncertainty regarding future earnings. It is crucial for LLMs to incorporate this uncertainty systematically when making decisions. In this paper, we introduce DeFine, a modular framework that constructs probabilistic factor profiles from complex scenarios. It then integrates these profiles with analogical reasoning, leveraging insights from similar past experiences to guide LLMs in making critical decisions in new situations. Our framework separates the tasks of quantifying uncertainty and incorporating it into LLM decision-making. This approach is particularly useful in areas such as consulting and financial deliberation, where making decisions under uncertainty is vital.

In this work, we study the simple yet universally applicable case of reward shaping in value-based Deep Reinforcement Learning (DRL). We show that reward shifting in the form of the linear transformation is equivalent to changing the initialization of the Q-function in function approximation. Based on such an equivalence, we bring the key insight that a positive reward shifting leads to conservative exploitation, while a negative reward shifting leads to curiosity-driven exploration. Accordingly, conservative exploitation improves offline RL value estimation, and optimistic value estimation improves exploration for online RL. We validate our insight on a range of RL tasks and show its improvement over baselines: (1) In offline RL, the conservative exploitation leads to improved performance based on off-the-shelf algorithms; (2) In online continuous control, multiple value functions with different shifting constants can be used to tackle the exploration-exploitation dilemma for better sample efficiency; (3) In discrete control tasks, a negative reward shifting yields an improvement over the curiosity-based exploration method.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, market impact, liquidity constraints or risk limits using modern deep reinforcement machine learning methods. We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show that the set of constrained trading strategies used by our algorithm is large enough to ε-approximate any optimal solution. Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available. We illustrate our approach by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard "complete market" solution.

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

Popular offline reinforcement learning (RL) methods rely on conservatism, either by penalizing out-of-dataset actions or by restricting rollout horizons. In this work, we question the universality of this principle and instead revisit a complementary one: a Bayesian perspective. Rather than enforcing conservatism, the Bayesian approach tackles epistemic uncertainty in offline data by modeling a posterior distribution over plausible world models and training a history-dependent agent to maximize expected rewards, enabling test-time generalization. We first illustrate, in a bandit setting, that Bayesianism excels on low-quality datasets where conservatism fails. We then scale this principle to realistic tasks and show that long-horizon planning is critical for reducing value overestimation once conservatism is removed. To make this feasible, we introduce key design choices for performing and learning from long-horizon rollouts while controlling compounding errors. These yield our algorithm, NEUBAY, grounded in the neutral Bayesian principle. On D4RL and NeoRL benchmarks, NEUBAY generally matches or surpasses leading conservative algorithms, achieving new state-of-the-art on 7 datasets. Notably, it succeeds with rollout horizons of several hundred steps, contrary to dominant practice. Finally, we characterize datasets by quality and coverage, showing when NEUBAY is preferable to conservative methods. Together, we argue NEUBAY lays the foundation for a new practical direction in offline and model-based RL.

We address the problem of planning in an environment with deterministic dynamics and stochastic rewards with discounted returns. The optimal value function is not known, nor are the rewards bounded. We propose Platypoos, a simple scale-free planning algorithm that adapts to the unknown scale and smoothness of the reward function. We provide a sample complexity analysis for Platypoos that improves upon prior work and holds simultaneously over a broad range of discount factors and reward scales, without the algorithm knowing them. We also establish a matching lower bound showing our analysis is optimal up to constants.

Recent machine learning papers often report 1-2 percentage point improvements from a single run on a benchmark. These gains are highly sensitive to random seeds, data ordering, and implementation details, yet are rarely accompanied by uncertainty estimates or significance tests. It is therefore unclear when a reported +1-2% reflects a real algorithmic advance versus noise. We revisit this problem under realistic compute budgets, where only a few runs are affordable. We propose a simple, PC-friendly evaluation protocol based on paired multi-seed runs, bias-corrected and accelerated (BCa) bootstrap confidence intervals, and a sign-flip permutation test on per-seed deltas. The protocol is intentionally conservative and is meant as a guardrail against over-claiming. We instantiate it on CIFAR-10, CIFAR-10N, and AG News using synthetic no-improvement, small-gain, and medium-gain scenarios. Single runs and unpaired t-tests often suggest significant gains for 0.6-2.0 point improvements, especially on text. With only three seeds, our paired protocol never declares significance in these settings. We argue that such conservative evaluation is a safer default for small gains under tight budgets.

The prevalence and importance of algorithmic two-sided marketplaces has drawn attention to the issue of fairness in such settings. Algorithmic decisions are used in assigning students to schools, users to advertisers, and applicants to job interviews. These decisions should heed the preferences of individuals, and simultaneously be fair with respect to their merits (synonymous with fit, future performance, or need). Merits conditioned on observable features are always uncertain, a fact that is exacerbated by the widespread use of machine learning algorithms to infer merit from the observables. As our key contribution, we carefully axiomatize a notion of individual fairness in the two-sided marketplace setting which respects the uncertainty in the merits; indeed, it simultaneously recognizes uncertainty as the primary potential cause of unfairness and an approach to address it. We design a linear programming framework to find fair utility-maximizing distributions over allocations, and we show that the linear program is robust to perturbations in the estimated parameters of the uncertain merit distributions, a key property in combining the approach with machine learning techniques.

Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they require the knowledge of the values of the expectations to estimate. Motivated by the form of these optimal estimators and some interesting properties, we therefore propose an adaptive algorithm. The general idea is to adaptively update the parameters of the estimators for approaching the optimal ones. We suggest then a quantitative stopping criterion that exploits the trade-off between approaching these optimal parameters and having a sufficient budget left. This left budget is then used to draw a new independent sample from the final sampling distribution, allowing to get unbiased estimators of the expectations. We show how to apply our procedure to sensitivity analysis, by estimating Sobol' indices and quantifying the impact of the input distributions. Finally, realistic test cases show the practical interest of the proposed algorithm, and its significant improvement over estimating the expectations separately.

An agent holds a position in a perpetual contract with payoff function ψ and attempts to liquidate the position while managing transaction costs, inventory risk, and funding rate payments. By solving the agent's stochastic control problem we obtain a closed-form expression for the optimal trading strategy when the payoff function is given by ψ(s) = s. When the payoff function is non-linear we provide approximations to the optimal strategy which apply when the funding rate parameter is small or when the length of the trading interval is small. We further prove that when ψ is non-linear, the short time approximation can be written in terms of the closed-form trading strategy corresponding to the case of the identity payoff function.

We consider the problem of preference based reinforcement learning (PbRL), where, unlike traditional reinforcement learning, an agent receives feedback only in terms of a 1 bit (0/1) preference over a trajectory pair instead of absolute rewards for them. The success of the traditional RL framework crucially relies on the underlying agent-reward model, which, however, depends on how accurately a system designer can express an appropriate reward function and often a non-trivial task. The main novelty of our framework is the ability to learn from preference-based trajectory feedback that eliminates the need to hand-craft numeric reward models. This paper sets up a formal framework for the PbRL problem with non-markovian rewards, where the trajectory preferences are encoded by a generalized linear model of dimension d. Assuming the transition model is known, we then propose an algorithm with almost optimal regret guarantee of mathcal{O}left( SH d log (T / delta) T right). We further, extend the above algorithm to the case of unknown transition dynamics, and provide an algorithm with near optimal regret guarantee mathcal{O}((d + H^2 + |S|)dT +|mathcal{S||A|TH} ). To the best of our knowledge, our work is one of the first to give tight regret guarantees for preference based RL problems with trajectory preferences.

Reinforcement learning with verifiable rewards (RLVR) for Large Reasoning Models hinges on baseline estimation for variance reduction, but existing approaches pay a heavy price: PPO requires a policy-model scale critic, while GRPO needs multiple rollouts per prompt to keep its empirical group mean stable. We introduce Policy Optimization with Internal State Value Estimation), which obtains a baseline at negligible cost by using the policy model's internal signals already computed during the policy forward pass. A lightweight probe predicts the expected verifiable reward from the hidden states of the prompt and generated trajectory, as well as token-entropy statistics, and is trained online alongside the policy. To preserve gradient unbiasedness despite using trajectory-conditioned features, we introduce a cross-rollout construction that predicts each rollout's value from an independent rollout's internal states. Because POISE estimates prompt value using only a single rollout, it enables higher prompt diversity for a fixed compute budget during training. This reduces gradient variance for more stable learning and also eliminates the compute overhead of sampling costs for detecting zero-advantage prompts. On Qwen3-4B and DeepSeek-R1-Distill-Qwen-1.5B across math reasoning benchmarks, POISE matches DAPO while requiring less compute. Moreover, its value estimator shows similar performance to a separate LLM-scale value model and generalizes to various verifiable tasks. By leveraging the model's own internal representations, POISE enables more stable and efficient policy optimization.

We introduce and analyze distance preservation games (DPGs). In DPGs, agents express ideal distances to other agents and need to choose locations in the unit interval while preserving their ideal distances as closely as possible. We analyze the existence and computation of location profiles that are jump stable (i.e., no agent can benefit by moving to another location) or welfare optimal for DPGs, respectively. Specifically, we prove that there are DPGs without jump stable location profiles and identify important cases where such outcomes always exist and can be computed efficiently. Similarly, we show that finding welfare optimal location profiles is NP-complete and present approximation algorithms for finding solutions with social welfare close to optimal. Finally, we prove that DPGs have a price of anarchy of at most 2.

In Reinforcement Learning (RL), an agent acts in an unknown environment to maximize the expected cumulative discounted sum of an external reward signal, i.e., the expected return. In practice, in many tasks of interest, such as policy optimization, the agent usually spends its interaction budget by collecting episodes of fixed length within a simulator (i.e., Monte Carlo simulation). However, given the discounted nature of the RL objective, this data collection strategy might not be the best option. Indeed, the rewards taken in early simulation steps weigh exponentially more than future rewards. Taking a cue from this intuition, in this paper, we design an a-priori budget allocation strategy that leads to the collection of trajectories of different lengths, i.e., truncated. The proposed approach provably minimizes the width of the confidence intervals around the empirical estimates of the expected return of a policy. After discussing the theoretical properties of our method, we make use of our trajectory truncation mechanism to extend Policy Optimization via Importance Sampling (POIS, Metelli et al., 2018) algorithm. Finally, we conduct a numerical comparison between our algorithm and POIS: the results are consistent with our theory and show that an appropriate truncation of the trajectories can succeed in improving performance.

Large Language Models (LLMs) have shown promise in simulating human behavior, yet existing agents often exhibit behavioral rigidity, a flaw frequently masked by the self-referential bias of current "LLM-as-a-judge" evaluations. By evaluating against empirical ground truth, we reveal a counter-intuitive phenomenon: increasing the intensity of prompt-driven reasoning does not enhance fidelity but rather exacerbates value polarization, collapsing population diversity. To address this, we propose the Context-Value-Action (CVA) architecture, grounded in the Stimulus-Organism-Response (S-O-R) model and Schwartz's Theory of Basic Human Values. Unlike methods relying on self-verification, CVA decouples action generation from cognitive reasoning via a novel Value Verifier trained on authentic human data to explicitly model dynamic value activation. Experiments on CVABench, which comprises over 1.1 million real-world interaction traces, demonstrate that CVA significantly outperforms baselines. Our approach effectively mitigates polarization while offering superior behavioral fidelity and interpretability.

We study regret minimization for reinforcement learning (RL) in Latent Markov Decision Processes (LMDPs) with context in hindsight. We design a novel model-based algorithmic framework which can be instantiated with both a model-optimistic and a value-optimistic solver. We prove an O(mathsf{Var^star M Gamma S A K}) regret bound where O hides logarithm factors, M is the number of contexts, S is the number of states, A is the number of actions, K is the number of episodes, Gamma le S is the maximum transition degree of any state-action pair, and Var^star is a variance quantity describing the determinism of the LMDP. The regret bound only scales logarithmically with the planning horizon, thus yielding the first (nearly) horizon-free regret bound for LMDP. This is also the first problem-dependent regret bound for LMDP. Key in our proof is an analysis of the total variance of alpha vectors (a generalization of value functions), which is handled with a truncation method. We complement our positive result with a novel Omega(mathsf{Var^star M S A K}) regret lower bound with Gamma = 2, which shows our upper bound minimax optimal when Gamma is a constant for the class of variance-bounded LMDPs. Our lower bound relies on new constructions of hard instances and an argument inspired by the symmetrization technique from theoretical computer science, both of which are technically different from existing lower bound proof for MDPs, and thus can be of independent interest.

In some games, additional information hurts a player, e.g., in games with first-mover advantage, the second-mover is hurt by seeing the first-mover's move. What properties of a game determine whether it has such negative "value of information" for a particular player? Can a game have negative value of information for all players? To answer such questions, we generalize the definition of marginal utility of a good to define the marginal utility of a parameter vector specifying a game. So rather than analyze the global structure of the relationship between a game's parameter vector and player behavior, as in previous work, we focus on the local structure of that relationship. This allows us to prove that generically, every game can have negative marginal value of information, unless one imposes a priori constraints on allowed changes to the game's parameter vector. We demonstrate these and related results numerically, and discuss their implications.

Large language models (LLMs) achieve strong performance across benchmarks--from knowledge quizzes and math reasoning to web-agent tasks--but these tests occur in static settings, lacking real dynamics and uncertainty. Consequently, they evaluate isolated reasoning or problem-solving rather than decision-making under uncertainty. To address this, we introduce LiveTradeBench, a live trading environment for evaluating LLM agents in realistic and evolving markets. LiveTradeBench follows three design principles: (i) Live data streaming of market prices and news, eliminating dependence on offline backtesting and preventing information leakage while capturing real-time uncertainty; (ii) a portfolio-management abstraction that extends control from single-asset actions to multi-asset allocation, integrating risk management and cross-asset reasoning; and (iii) multi-market evaluation across structurally distinct environments--U.S. stocks and Polymarket prediction markets--differing in volatility, liquidity, and information flow. At each step, an agent observes prices, news, and its portfolio, then outputs percentage allocations that balance risk and return. Using LiveTradeBench, we run 50-day live evaluations of 21 LLMs across families. Results show that (1) high LMArena scores do not imply superior trading outcomes; (2) models display distinct portfolio styles reflecting risk appetite and reasoning dynamics; and (3) some LLMs effectively leverage live signals to adapt decisions. These findings expose a gap between static evaluation and real-world competence, motivating benchmarks that test sequential decision making and consistency under live uncertainty.

The Fundamental Theorem of Asset Pricing states that a market is free of arbitrage exactly when it admits an equivalent martingale measure. We formalize it in Lean 4 over Mathlib in three settings: a finite-state market over a finite horizon (Harrison-Pliska), a one-period market on an arbitrary probability space with a single scalar return (Follmer-Schied), and a one-period market with finitely many assets. The finite case is the geometry of a separating hyperplane; the scalar one-period case is an elementary change of measure. In the d-asset case the equivalent martingale measure is constructed explicitly, as the minimiser of the smooth convex potential E[log(1+e^{langleθ,Yrangle})]: absence of arbitrage is precisely coercivity of the potential, its first-order condition is the martingale property, and the minimiser's logistic weight is the density of the measure. The construction uses no Hahn-Banach theorem, no L^0-closedness argument, no measurable selection, and no non-redundancy hypothesis. To our knowledge this is the first machine-checked Fundamental Theorem of Asset Pricing in any proof assistant. The boundary is explicit: the general multi-period Dalang-Morton-Willinger theorem lies outside the development. Every theorem is sorry-free, each headline result's axioms are pinned to Mathlib's classical defaults by a build-enforced gate, and the whole is reproducible from a pinned toolchain.

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary stochastic maximum principle tailored to Epstein-Zin utility and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for the equilibrium investment and consumption policies.

We propose and evaluate a number of solutions to the problem of calculating the cost to serve each location in a single-vehicle transport setting. Such cost to serve analysis has application both strategically and operationally in transportation. The problem is formally given by the traveling salesperson game (TSG), a cooperative total utility game in which agents correspond to locations in a traveling salesperson problem (TSP). The cost to serve a location is an allocated portion of the cost of an optimal tour. The Shapley value is one of the most important normative division schemes in cooperative games, giving a principled and fair allocation both for the TSG and more generally. We consider a number of direct and sampling-based procedures for calculating the Shapley value, and present the first proof that approximating the Shapley value of the TSG within a constant factor is NP-hard. Treating the Shapley value as an ideal baseline allocation, we then develop six proxies for that value which are relatively easy to compute. We perform an experimental evaluation using Synthetic Euclidean games as well as games derived from real-world tours calculated for fast-moving consumer goods scenarios. Our experiments show that several computationally tractable allocation techniques correspond to good proxies for the Shapley value.