Papers

We present Cogentic, a multi-agent harness for automated proof discovery on open research problems. While frontier language models can generate strong mathematical ideas in a single shot, single-shot generation is often insufficient for open problems that require exploring multiple competing conjectures, overcoming subtle technical obstructions, and retaining intermediate progress over a long horizon. Cogentic addresses these challenges through an iterative prove-verify loop in which an orchestrator allocates a population of independent provers across distinct proof directions, subjects their output to adversarial verification by several specialized components, and promotes confirmed intermediate results into a persistent verified ledger that later rounds build on. The harness is designed to be able to solve research-level math and theoretical computer science problems. Using either Gemini 3.1 Pro or an early version of Gemini 4 Argon as the base model, Cogentic produced novel results on open problems across online learning, auction theory, and mechanism design. Each result was independently verified by domain experts and is developed in full in companion papers. We list these results, and new ones as they are verified, at https://sites.google.com/view/cogentic .

Autobidding has become a dominant paradigm in online advertising by enabling advertisers to set high-level goals while algorithms handle real-time bid optimization. A prominent example is the Return-on-Spend (RoS) value maximizer, which maximizes total value subject to an aggregate value-per-spend constraint. While most work on mechanism design for autobidders focuses on one-sided markets, many real-world platforms involve strategic behavior on both sides. We initiate the study of two-sided markets with autobidders. We first establish a stark negative result: for the challenging objective of liquid gains from trade (LGFT) in the prior-free setting, broad classes of utility-truthful, budget-balanced mechanisms have unbounded Price of Anarchy (PoA) once value-maximizing agents are present, even in double-auction environments. We then give two positive results. In the prior-free repeated double-auction setting, McAfee's Trade Reduction mechanism has unbounded PoA for LGFT but achieves constant PoA for liquid welfare under RoS bidding, showing that off-the-shelf mechanisms retain meaningful welfare guarantees. With distributional information regarding the private costs and values, we design a two-sided mechanism that is incentive compatible for each agent under its corresponding objective, individually rational, ex-ante weakly budget balanced, and achieves first-best liquid welfare, equivalently optimal LGFT, whenever at least one side of the market consists of RoS value maximizers. This result applies to matching markets with general downward-closed feasibility constraints. It contrasts sharply with the Myerson–Satterthwaite impossibility theorem~\citep{MS83}, which rules out first-best efficiency with incentive compatibility, individual rationality, and budget balance even in bilateral trade when both the buyer and the seller are quasi-linear utility maximizers.

Electric ride-pooling must jointly manage passenger service, routes, batteries, and finite chargers; a locally useful relocation can reduce later service. We introduce a discrete-event ride-pooling simulator with ordered passenger stops, pickup and ride-time constraints, battery reserves, charger travel, and finite first-come, first-served (FCFS) charging queues. On top of a common dispatcher, we add a learned repair layer that moves uncollected passengers between vehicles, uses a supervised score to estimate later effects, and uses a two-stage proximal-policy-optimization (PPO) policy to focus computation on promising moves. Every executed move must satisfy route, battery, and time constraints and provide an immediate route-cost improvement. For larger instances, we limit the number of candidate moves and apply several non-conflicting repairs at once. On 12 held-out Manhattan episodes with 100 requests, 30 EVs, and one plug per station, supervised repair reduces operational cost by 6.57% versus no repair (12/12 wins) and serves 74.92 rather than 72.75 requests. Two-stage PPO stays within 1.43% of supervised repair with 95.4% fewer exact trials, while our experiments scale to 8,000 requests and 2,400 EVs with cost reductions across all three larger development settings. Queue-aware charging reduces waiting by 41.0%; under a declared 1.5-kW queue-idle load, operational electricity per served request falls 1.73%.

The economic theory of LLM pricing treats tokens as a homogeneous commodity considering aggregate token count as the main features buyers and sellers consider. We model inference as a service market where buyers have three-dimensional private information - willingness-to-pay, task volume, and time preference - and utility depends on latency slack alongside token quantities. Our main result is a separation theorem: discrete hardware tiers induce endogenous self-selection on time preferences, reducing three-dimensional screening to standard one-dimensional screening within each tier. We derive the cost structure from GPU inference physics - compute-bound prefill and bandwidth-bound decode - and characterize optimal tiered mechanisms via virtual-value techniques. Optimal per-task prices are volume-independent, providing theoretical grounding for flat per-token API pricing. We verify the mechanism empirically by calibrating to 8-GPU clusters of H100 and B200 hardware. The separation theorem holds in 83% of 105 tested configurations overall, rising to 96% at economically relevant WTP scales. A seller adopting two-tier pricing under the optimal mechanism captures 26-66% higher profit than the best single-tier alternative, with gains driven by efficient cross-tier allocation in regimes where hardware costs are a significant fraction of per-request value.

When inference demand exceeds available compute capacity, model providers must decide which requests should be served first. Users have different tolerances for delay from an LLM API, but current priority pricing schemes compress these differences into coarse fixed-price service tiers. We design an inference auction that allows users to bid for faster service. Our auction allocates priority in an economically efficient way without sacrificing latency, and we develop fast algorithms for implementing prices that incentivize truthful bidding. We also design an autobidding agent for our inference auction, where users specify an inference budget and the autobidder dynamically adjusts its bids over time to maximize user utility subject to the budget constraint. Experiments validate the practicality of our auction: it increases system welfare while maintaining the cache utilization and latency advantages of SGLang, a state-of-the-art inference serving framework.

Prophet Inequalities for online auctions, as many classical auction settings, traditionally consider agents who strive to maximize quasi-linear utility, i.e. their obtained value minus the price paid. This simplified view of the agents' objectives does not capture real-world settings where agents are almost always constrained by budgets, and can have different objectives altogether (e.g., very prominently, value maximizers in autobidding). We generalize classic prophet inequality auction settings to accommodate unknown, varying types of agent objectives and budgets, modeling them as stochastic input similar to the traditional valuation distributions. In this new model, we analyze (anonymous) posted-price mechanisms, one main category of which are the balanced prices central to many auction results in the prophet inequality model. We demonstrate both the resilience and limits of balanced price mechanisms for generalized prophet inequalities with stochastic agent objectives, budgets and valuations: our generalization of the balanced prices framework incorporates budget constraints and mixed objectives. We show constant competitive ratios for some the most central settings, spanning all the way up to agents with subadditive valuation functions. On the way, we establish trade-offs between competitive ratio and agents' price-sensitivity, and give a general reduction from the budgeted setting to budget-free value maximizers.

We study the efficiency of auctions with budget-constrained bidders and derive tight price of anarchy (PoA) bounds for coarse correlated equilibria (CCE). To this end, we develop a general framework that reduces proving PoA guarantees to finding feasible solutions to the dual of a linear program. The LP is formulated over a small number of per-bidder equilibrium statistics, incorporates constraints implied by the auction setting and equilibrium behavior, and captures a novel smoothness notion tailored to budget-constrained environments; we provide a reduction from classical smoothness to this new notion. Our approach enables simple and tight PoA analyses across a broad range of auction formats. Using our framework, we obtain tight liquid welfare guarantees for simultaneous first-price, second-price, and all-pay auctions, simultaneous auctions with restricted uniform bidding interfaces, discriminatory and uniform price multi-unit auctions, and generalized first-price position auctions. Beyond budget-constrained settings, we derive new lower bounds for the budget-free uniform price auction and generalized second-price auction; in particular, our lower bound for the uniform price auction matches the upper bound of $3.146$ by de Keijzer et al. (2013).

Human decision-making reflects not only material incentives, but also concern for others' gains and the ability to understand others. However, the extent to which sensitivity to fairness interacts with aspects of social cognition and affects sequential social interactions requires further understanding. Here, we address this question through a behavioural experiment on sequential exchanges of resources, represented by the Incremental Centipede Game (ICG). The design comprises a "zero-end" treatment (n=164), where the ICG has a deadline yielding no payoff, and a "fair-end" treatment (n=162), where the last interaction leads to a fair outcome. The ICG, which is repeated twice, is complemented by two individual tasks: the Social Value Orientation task (SVO), to elicit social preferences, and the Cognitive Reflection Test (CRT), to assess participants' deliberation. Our results show that the fair outcome, though uncertain, prompts participants to move beyond self-interest. Indeed, an evolutionary model shows that the fair split shifts play towards a mutually beneficial strategy when players are sufficiently prosocial and use probabilistic strategies. A second game repetition reinforces this pattern, with adaptation towards cooperation. Although CRT scores do not predict behaviour in the ICG, participants who deviate from game-theoretic rationality exhibit stronger prosociality in the SVO. Notably, in the fair-end treatment, some non-rational players are individualists, suggesting that the fair split may function as a nudge. In conclusion, our findings suggest that certain personal traits, such as social preferences, allow individuals to overcome self-interest to better align with their peers and achieve better collective outcomes in mixed-motive games.

We study the bridge location problem with optional preferences, where two separated regions each contain one prelocated facility. Each agent has a private location and a private preference specifying a nonempty subset of the two facilities in which she is interested. Her individual cost is measured by one of three natural variants: the maximum, the sum, or the minimum of her distances to the facilities in which she is interested. The social planner must design deterministic strategyproof mechanisms that elicit truthful reports, choose the location of a connecting bridge, and approximately minimize either the social cost or the maximum cost. Our main results are as follows. For the social cost objective, we design optimal mechanisms for the max-variant and sum-variant costs, and provide a 3-approximation mechanism for the min-variant cost, and establish a lower bound of 2 for the min-variant. For the maximum cost objective, we give 5/3-approximation mechanisms for the max-variant and sum-variant costs, and a 3-approximation mechanism for the min-variant cost; we also prove a common lower bound of 5/3 that applies to all three cost variants under the maximum cost objective.

Recent advances in large language models have made automatic game generation increasingly feasible, yet reliably improving generated games beyond a playable version remains challenging. Naive iterative refinement can easily overfit a small set of test cases, producing fragile games with unresolved bugs, missing behaviors, and poor generalization to broader player interactions. We introduce RSIGame, an autonomous agentic game development framework with recursive self-improvement. RSIGame organizes development into complementary local and global loops. Concretely, a local explore-diagnose-improve loop broadly explores the executable game, diagnoses and prioritizes discovered issues, and performs evidence-grounded revision, where an evolving checklist continually accumulates new testing and improvement guidance. A global loop tracks overall quality, preserves the best checkpoint, and detects saturation or regression over long-horizon development. Beyond test-time improvement, RSIGame further internalizes successful development experience into the generator through training. Across 140 GameCraft-Bench tasks, two game engines, and five generators, RSIGame consistently improves game quality under matched development budgets. Notably, experience internalization enables Qwen3.8-27B to reach 61.38 on Godot and 58.53 on Phaser, exceeding GPT-5.5 one-shot scores while reducing Qwen's generation tokens by 11 times.

Optimistic matrix mirror-prox (OMMP) computes $\epsilon$-approximate Nash equilibria in quantum zero-sum games with an $O(1/\varepsilon)$ average-iterate guarantee [arXiv:2311.10859]. We investigate whether this dependence on accuracy is tight and whether geometric last-iterate convergence can be guaranteed. We study these questions through explicit games with one qubit per player. First, we prove an $\Omega(1/\varepsilon)$ lower bound for the uniform-average output that includes the maximally mixed initial state, independently of the regularizer and step size. Second, we construct a fixed game on which optimistic gradient descent-ascent (OGDA), initialized at the maximally mixed state, has last-iterate Frobenius distance to equilibrium $\Theta(1/t)$ and duality gap $\Theta(1/t^3)$ for every sufficiently small fixed step size. A separate fixed game exhibits arbitrarily long delays in reducing the initial error by a constant factor across a family of initial states. Finally, we give a fixed game with a unique, strictly complementary equilibrium on which optimistic matrix multiplicative weights updates (OMMWU) converge only polynomially from the maximally mixed state for every fixed positive step size. The last-iterate Frobenius distance and quantum relative entropy from the equilibrium to the iterates decay as $\Theta(1/t)$, while the duality gap decays as $\Theta(1/t^2)$.

We study noncooperative multi-vehicle games among heterogeneous autonomous vehicles, where each vehicle adopts a decentralized closed-loop policy based on its own state, and optimizes an objective that depends on other vehicles through potentially asymmetric interaction weights. We develop an $\alpha$-potential game framework that reduces the computation of an approximate Nash equilibrium (NE) to the minimization of a single auxiliary $\alpha$-potential function. We explicitly construct this $\alpha$-potential, establish the existence of its minimizers, and characterize the equilibrium approximation error $\alpha$ in terms of interaction asymmetry. We further introduce vehicle-specific scaling to reduce the effective interaction asymmetry, thereby tightening the equilibrium approximation and, in important cases, recovering an exact NE despite asymmetric interactions. We also derive social-efficiency guarantees for the potential-selected policies, revealing how the interaction structure shapes worst-case efficiency. Numerical experiments demonstrate the flexibility of the framework in capturing heterogeneous vehicle interactions, collision and obstacle avoidance, lane changing and overtaking under different traffic configurations, and priority-based intersection crossing.

Auto-bidding is now widely adopted in online advertising platforms, allowing advertisers to specify high-level campaign objectives–such as maximizing total value subject to a return-on-spend (ROS) constraint–rather than manual per-query bids. A central question in algorithmic mechanism design is characterizing the worst-case efficiency loss, or Price of Anarchy (PoA), across auction formats in this prior-free setting. While randomized auctions are known to strictly improve efficiency over deterministic mechanisms for two bidders, two fundamental questions have remained open: (1) what is the optimal PoA for two bidders, and (2) can any mechanism beat the barrier of 2 for general $n \ge 3$ bidders? We resolve both questions using the family of $r$-proportional first-price auctions ($\text{pFPA}_r$), in which each bidder wins with probability proportional to their bid raised to an exponent $r > 0$ and pays their bid upon winning. First, for two bidders, we prove that the standard proportional first-price auction ($r = 1$) achieves a tight $\text{PoA} \le 1.5$, complemented by a matching lower bound showing that no anonymous, monotone mechanism can do better. Second, for general $n \ge 2$ bidders, setting $r = 2n$ achieves $\text{PoA} \le 2 - \frac{1}{4n+1} = 2 - \Omega(1/n)$ across all undominated bid profiles, breaking the deterministic barrier of 2 for every finite $n$ and asymptotically matching the known $2 - \Theta(1/n)$ lower bound.

We design auctions for the sale of a single item with unlimited supply given a single prediction of the revenue-maximizing uniform price. This departs from prior work on auctions with predictions which typically assumes predictions of every bidder's value. Our main result is a characterization of the Pareto frontier for consistency and robustness attainable by any universally truthful auction. We show that a mechanism that randomizes between posting the predicted price and conducting an optimal prior-free fallback auction is Pareto-optimal. We then extend our results to achieve graceful degradation of revenue as a function of the prediction accuracy. Finally, we study how such a prediction can be obtained from historical market data through the lens of learning theory. Together, our results give an end-to-end account of how a price prediction can be learned and used robustly to maximize auction revenue.

Simulation-augmented generation (SAGE) is a recent technical proposal in which models simulate individuals' viewpoints at inference time in order to provide more representative answers to contentious user queries. A core challenge for SAGE is making inference-time simulation efficient without sacrificing representation quality. We introduce the first formalization of this problem, based upon an axiom from proportional clustering known as metric proportional justified representation+ (mPJR+) which is the strongest proportionality axiom known to always be satisfiable by centroid-based clustering. We prove that to proportionally represent the viewpoints of a population of $n_H$ humans on a given prompt, we need only create simulations of $n \ll n_H$ individuals, and at inference time, need only dynamically route to $k \ll n$ of those simulations based upon the prompt. This twofold reduction still yields approximate proportional representation guarantees for the entire population. Empirically, across two domains-political questions and personal advice-our proposed routing algorithm achieves higher mPJR+ satisfaction rates than $k$-means-based or random selection baselines.

IREG-PRM+ normalizes the cumulative regret vector by its own norm and attains optimal regret without knowledge of the payoff scale. Run unmodified on zero-sum matrix games, it converges linearly in the last iterate, and no analysis explains why. The obstacle is that the algorithm has no fixed step size to analyze: the step size is a state variable, the inverse of a regret norm that the trajectory itself moves. Every proved linear rate for regret-matching dynamics comes from restarting or modifying the update. We identify the mechanism as norm saturation: the regret norm rises to a finite limit and freezes the step size. We prove that it always does, with an explicit bound, and that saturation forces the last-iterate Nash gap to vanish on every matrix game; pointwise convergence follows whenever the equilibrium is unique. Near a unique strictly complementary equilibrium the active support freezes in one step, and the one-round Jacobian on that support has a closed form. The last-iterate then converges linearly at a closed-form rate, provided one scale-invariant quantity stays below one: the saturated step size times the largest singular value of the value-centered payoff submatrix on the support. On the $216$-instance testbed, the $184$ instances with a resolvable limit all satisfy it. The same analysis gives a ratio certificate: observable norm-increment ratios bound the unobservable Nash-gap ratio up to a constant that enters once and does not accumulate with the iteration count. Its slope-two law holds on $96.1\%$ of the instances where the slope is measurable, and the same increment monitors progress in extensive-form games, where best-response passes can be scheduled sparsely. The code is available at https://github.com/lbn187/NormCert.

We give an explicit solution to the five expert prediction with expert advice partial differential equation (PDE) in the finite-time horizon setting. The solution formula establishes that the adversary's rank strategy $(1,0,1,0,0)$ is globally optimal, and the COMB strategy $(1,0,1,0,1)$ is optimal exactly on the set where $x_1=x_2$ and $x_3=x_4$. The formula is derived from the solution of the geometric-stopping problem given in our companion paper through the transform principle of Bayraktar, Ekren and Zhang, which links the two problems by a Laplace transform. Inverting the transform term by term expresses the solution through a series of Gaussian and complementary error function kernels. The optimality of $(1,0,1,0,0)$ is reduced to the signs of $41$ one-variable Gaussian series, which are certified with computer assistance by Poisson summation, first-mode domination and interval arithmetic on $1616$ rational cells. The proofs of our main theorems, certificates included, are also formalized in the Lean proof assistant.

We study fair and strategy-proof allocation of multiple divisible resources with Leontief utilities, motivated by cloud computing. The canonical mechanism, Dominant Resource Fairness (DRF), satisfies sharing incentive (SI), envy-freeness (EF), strategy-proofness (SP), and Pareto optimality (PO), but can be highly inefficient in terms of utilitarian social welfare. Under the classical approximation benchmark, no mechanism satisfying even one of SI, EF, and SP can improve on the trivial worst-case guarantee. We therefore adopt the recently introduced fair-ratio benchmark, which compares a mechanism only with the welfare-maximizing allocation that itself satisfies SI and EF. For two resources, we first introduce Adaptive-Speed Fairness (ASF), a unified parametric framework that captures previous mechanisms as special or boundary cases. Every ASF mechanism satisfies SI, EF, and PO, and we derive a general sufficient condition that guarantees SP. Optimizing within this framework yields a strategy-proof mechanism with asymptotic fair-ratio $2/(2\sqrt{2}-1)\approx 1.094$, substantially improving the previous best guarantee $3-\sqrt{3}\approx1.268$. We complement this upper bound with a lower bound of $1.0789$ for all ASF mechanisms. To overcome this limitation, we introduce Corrected Resource Balancing (CRB), which uses the full resource-load structure together with a one-agent incentive correction. CRB satisfies SI, EF, SP, and PO and achieves fair-ratio at most $1+1/n$. Together with a general $1+\Omega(1/n)$ lower bound, this establishes the optimal asymptotic order $1+\Theta(1/n)$ for two resources. Finally, for every $m\ge3$, any deterministic mechanism satisfying SI and SP has fair-ratio exactly $m$. The same lower bound holds for randomized mechanisms satisfying ex-post SI and truthfulness in expectation.

We study how well simple mechanisms approximate gains from trade (GFT) in two-sided matching markets with independent buyer values and seller costs, where feasible outcomes form an arbitrary downward-closed family of matchings. This model includes bilateral trade and double auctions as special cases. We focus on the Generalized Random-Offerer (GRO) mechanism, which is an equal mixture of the Generalized Sellers-Offering Mechanism (GSOM) and the Generalized Buyers-Offering Mechanism (GBOM). We determine GRO's exact worst-case approximation ratio with respect to first-best GFT, showing that it is $e$. This improves the previous $3.15$ approximation guarantee (Babaioff et al. STOC 2026) for the same mechanism. In bilateral trade, the result implies a $1/e$ guarantee for the random-offerer mechanism, improving the previous $1/\pi$ bound (Jo 2026). Our analysis uses a two-dimensional inequality in quantile space to compare first-best GFT with optimal one-sided auction profits. For each potential trade, we identify the region of buyer values and seller costs for which the first-best allocation selects that trade, and then randomly shrink this region in quantile space, yielding posted-price rules whose expected profits can be evaluated exactly. We establish tightness of the $e$ approximation by constructing markets with regular type distributions, pairwise disjoint trading edges, and a single knapsack constraint. On these instances, GRO's expected GFT approaches a $1/e$ fraction of first-best GFT, therefore ruling out any better guarantee even under these restrictions.

How much social welfare must a market lose because values are private? We determine the sharp ratio of second-best to first-best welfare under independent nonnegative types, Bayesian incentive compatibility, interim individual rationality, and no expected budget deficit. The ratio is approximately $0.8882516903$ for arbitrary priors and $0.9113893681$ when buyers have monotone hazard rates and sellers are unrestricted. The arbitrary-prior guarantee holds for every downward-closed family of feasible matchings; the MHR guarantee holds when every matching of a compatibility graph is feasible. Both constants are \revise{attained}{sharp} already in bilateral trade. The proof keeps the sellers' initial endowment inside the budget Lagrangian. For arbitrary priors, a common threshold decomposition reduces the problem to three-parameter power-law distributions. For monotone hazards, common transformations of buyer and seller scores reduce it to shifted capped exponential buyers; a typewise allocation certificate and an explicit worst-case seller satisfy the same boundary equation. The bilateral-to-matching framework, with a welfare endowment charge and an MHR-preserving packing argument, transfers these affine inequalities without loss. We give exact variational characterizations of both constants and reproducible interval certificates for their numerical evaluation. Unrestricted signed transfers also permit pointwise strong budget balance.

In this paper, we study the matroid base packing problem from the perspective of cooperative game theory and define the associated matroid base packing game. This model extends the network strength game of Baiou and Barahona (2020) from graphic matroids to general matroids. Building on the exchange structure of matroid, we develop a graph theoretic framework for analyzing the core and the nucleolus in this game. This framework yields a compact characterization of core nonemptiness and enables polynomial time algorithms for testing core nonemptiness, finding a core allocation, and computing the nucleolus. In particular, the approach provides a broader and more structured framework for the nucleolus computation problem arising in network strength games studied by Baiou and Barahona (2020), covering both the nonempty core and empty core cases.

Most successes of superhuman game-playing algorithms are in games with discrete actions, yet in auctions, robotics, sports, or trading, actions are nearly continuous. Prior techniques either rely on expert-designed discretizations or are sample inefficient. We present a scalable policy-gradient algorithm for large sequential games with continuous or mixed discrete and continuous actions. It combines magnetic mirror descent with a mixture of Gaussians reparametrization, trained via self-play. We show that it approximates equilibrium in games where gradient descent fails. In sequential games, it outperforms neural fictitious self-play and matches or outperforms the final strategies of policy space response oracles with 3.5–5.5$\times$ fewer samples. In heads-up no-limit Texas hold'em, it performs on par with Slumbot.

A coalition that deviates once can profit many times when what it sells keeps working. In a threshold-encrypted mempool, a leading defense against maximal extractable value (MEV), a quorum of the decryption committee that sells its decryption capability to a front-runner exposes every later block that the capability still decrypts. We ask how large a penalty, such as slashable stake, deters this kind of collusion. In our repeated game, a single leak by any coalition in a monotone family of authorized coalitions (for example, any $k$ of the $n$ committee members) unlocks a set of future rounds, costs a one-time penalty, and ends the coalition's participation. We show that every dynamic deviation reduces to choosing a leak time, so deterrence holds if and only if each coalition's penalty covers the largest discounted value that a single leak reaches. Without discounting, over $T$ rounds of unit value full reuse needs a penalty of $T$ while binding each leak to its own round needs $1$, so no penalty that is constant in the horizon deters unbounded reuse; a reuse window of $w$ rounds costs at most $w$ times the largest per-round value. The cheapest profile of per-party stakes that deters every coalition solves a covering linear program. For blockchain design, per-epoch keys cut the required stake from the value of a key's lifetime to the value of one epoch; we calibrate the gap on Ethereum front-running data and place Ferveo and Shutter in the model. The analysis extends to sealed-bid auctions, multi-authority voting, and federated learning under a shared key.

We consider a data seller who designs pricing mechanisms over multiple datasets to maximize revenue from budget-constrained buyers. The seller offers multiple datasets and assigns each a pricing function that maps the quantity purchased to a total payment. The goal is to design these pricing functions to maximize revenue, anticipating that buyers—who trade off accuracy gains against cost—choose bundles optimally subject to their budget constraints. Prior work [Chaudhury et al., 2026] studies such optimal pricing under the restriction that each dataset is assigned a linear price, and shows that computing optimal linear prices is computationally intractable. In contrast, we allow each dataset to be priced via a general function and show that this additional flexibility can not only increase the revenue but also restore tractability, yielding a surprising simultaneous improvement in economic performance and computational efficiency. Even when pricing functions are only required to be monotone and lower-continuous, optimal pricing admits a highly structured and simple form: each pricing function is piecewise linear and convex (PLC), and the optimal solution can be computed in polynomial time. Moreover, the total number of kinks across all pricing functions is bounded by the number of buyers. Consequently, when datasets significantly outnumber buyers, most pricing functions are effectively linear. We further empirically study the structure of optimal pricing by analyzing the number of kinks and the revenue gap between optimal nonlinear pricing and optimal linear pricing on simulations generated from a real dataset.

Discrete fair division is the problem of dividing a discrete set of goods among agents in a fair manner. In this setting, one of the most sought-after notions of fairness is envy-freeness up to any good (EFX). In 2023, Christodoulou, Fiat, Koutsoupias, and Sgouritsa introduced the idea of a graphical valuation, where the fair division problem is represented by a simple graph where vertices are the agents and the edges are the goods, and each vertex only values incident edges. They showed that an EFX allocation always exists, while determining the existence of an EFX orientation is NP-hard. They posed a open question of determining which graphs always admit an EFX orientation regardless of valuation. These graphs, called strongly EFX orientable graphs, were first studied by Zeng and Mehta in 2025, who demonstrated that all such graphs have chromatic number at most 3, and bipartite graphs always admit an EFX orientation regardless of valuation. In this manuscript, we finish resolving this question by giving a polynomial time characterization of strongly EFX orientable graphs. In particular, we show that a connected graph $G$ is strongly EFX orientable if and only if either of the following is true: (1) $G$ is bipartite, or (2) the block decomposition of $G$ contains exactly one nonbipartite block $B$, and there exists a vertex $v \in B$ such that the degree of $v$ within $B$ is 2 and $G-v$ is bipartite. This proof was discovered by AI, with human intervention to break the problem into the appropriate subproblems.

Modern Security Operations Centers struggle with delayed manual incident response, enabling adversaries to advance through the Cyber Kill Chain during early stage reconnaissance. While classical game theoretic defense models optimize strategic resource allocation, they rely on static utility matrices that fail to adapt to dynamic telemetry. This paper presents an integrated, metrics driven decision engine that bridges quantitative risk parameterization and continuous automated response time. Common Vulnerability Scoring Systems exploitability parameters are mapped to attacker success probabilities and evaluate defender log distributions via Factor Analysis of Information Risk Monte Carlo simulations. Real time SIEM logs streams are modeled as Poisson process arrival rates, dynamically updating defender posterior threat belief through sequential Bayesian filtering. A closed form threshold is derived by framing the interaction as a dynamic Bayesian Stackelberg game, where the expected unmitigated risk exceeds proactive containment cost. Parameterized against empirical data from the 2023 MGM Resorts and Caesars Entertainment cyber incident, simulation results demonstrate that the engine suppresses transient background noise while triggering automated SOAR network isolation within seconds of adversarial probing. Multi parameter sensitivity analysis confirms that the decision boundary dynamically adjusts to live perimeter vulnerability, offering a control theoretic foundation for sub minute automated threat containment.

The tragedy of the commons poses a multi-agent safety problem: reward-seeking agents can deplete a shared resource, and cooperation among its users does not itself specify how much must be preserved. We make preservation an explicit requirement by formulating a regenerative commons as a constrained Markov game or a constrained multi-agent MDP with a designer-specified depletion budget. We develop a nonstationary Lagrangian framework that constructs a policy sequence from solutions of unconstrained games or cooperative control problems. Extending earlier time-average constructions, we introduce average-epoch solution concepts for reset episodes with discounted rewards and terminal costs. We prove a reward-independent feasibility certificate, cooperative feasibility and approximate optimality against feasible policy mixtures, and an extension to unbiased sampled costs. For self-interested agents, a constrained Nash certificate quantifies the price-dispersion term introduced by deviations that redistribute budget across epochs. Under the stated assumptions on solver accuracy and multiplier updates, these results give constrained policy-sequence guarantees using solutions of unconstrained problems. Experiments with constrained IPPO and MAPPO in a Gordon-Schaefer fishery examine how depletion budgets shape stock retention, harvest rewards, and price adaptation.

Arctic product-mix auctions allow budget-constrained bidders to express preferences over multiple substitute goods. We give the first strongly polynomial algorithm to find the competitive equilibria when the auctioneer has separable convex piecewise-linear costs. It combines a strongly polynomial reduction to the costless Arctic auction with an existing strongly polynomial algorithm for costless Arctic auctions. We also give a polynomial-time Turing reduction from costless Arctic auctions to linear Fisher markets, and a strongly polynomial many-to-one reduction to linear Arrow-Debreu exchange markets. If a good is sold in positive quantity in two competitive equilibria, its price is the same in both.

In this work we initiate the study of competitive algorithms with arrival-time incentive compatibility for random-order online bipartite matching in settings where the users care only about receiving service (matched vs. unmatched) and not which offline resource serves them, while the platform's objective is to maximize total matching reward. This captures applications such as ride-sharing where the users primarily care about being matched to a ride while the platform internalizes the cost of dispatching a distant driver; dispatching homogeneous service requests to heterogeneous servers (cloud/edge routing); and assigning customer requests to a pool of providers with different flexibility (e.g., English-only vs. bilingual agents). Our main question is: Is constant-competitive matching possible for incentive-compatible, random-order edge-weighted matching on complete bipartite graphs? Motivated by the LP-based treatment of incentive compatibility in the classical secretary problem by Buchbinder et al., we impose a constraint that the ex ante probability of selection is equalized across all arrival positions. We answer our main question in the affirmative and propose the first constant-competitive algorithm for incentive compatible edge-weighted random-order online matching on complete bipartite graphs. The competitive ratio of our algorithm is parameterized by the imbalance factor $k := n/m$ – where $n$ and $m$ are the numbers of online and offline nodes, respectively, and $k$ is a positive integer. In particular, we obtain a competitive guarantee of the form $c_k - O(1/\sqrt{m})$ where $c_1 \approx 0.162$ and $c_k \to 0.02308\ldots$ as $k \to \infty$. We also present algorithms with strictly improved competitive ratio of $\approx 0.07 + O(1/m)$ for the binary-weighted case ($0$-$1$ rewards).

Influence maximization is a central algorithmic challenge in network analysis, aiming to identify a set of $k$ seed nodes in a graph with $n$ nodes and $m$ edges that maximizes the expected cascade of information under standard diffusion models. The seminal work of Borgs, Brautbar, Chayes, and Lucier (SODA'14) yielded a fundamental breakthrough\footnote{The conference version of their paper originally claimed a runtime of $\tilde O_{\epsilon}(n+m)$, but this was subsequently corrected to a runtime of $\tilde O_{\epsilon}((n+m)k)$ in an updated version of the paper that is available online. We validate the necessity of this additional factor $k$ in Section~\ref{sec:lowerbound} by demonstrating that if their algorithm is restricted to a runtime budget of $\tilde{O}_{\epsilon}(n+m)$, the approximation ratio deteriorates to $O(k^{-1/4})$.} for this problem by achieving an $\tilde O_{\epsilon}((n+m)k)$ time algorithm for approximating the solution within a factor of $1-1/e-\epsilon$. In the years since, numerous efforts have attempted to improve the runtime of this algorithm; however, these works have been successful in only shaving logarithmic factors or improving the dependence on $\epsilon$, leaving the existence of an almost linear-time algorithm as an open question. In this work, we resolve this long-standing open question. We present a novel algorithm that approximates the influence maximization problem within a factor of $1-1/e-\epsilon$ in time $\tilde{O}_{\epsilon}(n+m)$, effectively removing the multiplicative dependence on $k$ from the time complexity.