Papers

Reasoning in large language models (LLMs) is often grounded in human text, human demonstrations, and human-generated rationales. For equilibrium reasoning in complex games, however, relying on human data can be suboptimal. In fact, human play is often guided by intuition and heuristics and can deviate substantially from game equilibrium. This discrepancy is amplified in games with mixed-strategy equilibria, where human data is heavily biased toward pure strategies. Consequently, conditioning LLMs on this data yields weak game strategies. To grant LLMs the reasoning capacity in games, in this work, we study how to elicit equilibrium play using solver output. We propose Mixed-Strategy Decision Tree (MDT), which articulates the silent optimality of the equilibrium into sparse strategic rules that both humans and LLMs could understand. Using solver output rather than human annotation allows us to extend the input to arbitrarily new states and continuations. We instantiate this study on No-Limit Texas Hold'em by querying a solver oracle for over 250 million mixed-strategy decisions; MDT together with other techniques \textbf{reduces the $\ell_1$ distance to the equilibrium by $52.6\%$} across $8$ different LLM configurations. A Route-only ablation tests the incremental contribution of the shadow-based contrast, while complete River-endgame and Liar's Dice experiments evaluate strategic fidelity and portability beyond the original NLH communication setting.

Social media recommendation systems create conflict: content producers, content consumers, platform operators, and social pressures struggle to direct the allocation of attention in their favor while also facing competing societal pressures. Producers may feel pressure to optimize for recommendation algorithms and consumers may be exposed to content with negative externalities such as polarization and misinformation. Platforms aim to maximize user engagement, oftentimes resulting in over consumption from consumers. There is mounting pressure from policymakers and broader society to address these issues. Prior methods for constructing social media feeds have focused on recommendation systems. Instead, we propose a, to our knowledge, novel auction based method for feed construction where users bid for the attention of other users. Our mechanism systematically considers producers, consumers, platform operators, and social welfare. We show that our auction is weakly incentive compatible under budget constraints. To balance between producer and consumer welfare, we introduce a tax policy to the auction to increase the cost of content with negative externalities. Simulations over common social network topologies and an empirically observed network show how different feed algorithms prioritize different stakeholders. Our proposed auction based mechanisms produce on average 36.3% higher producer welfare than comparison algorithms on the empirically observed network and 31.4% higher producer welfare than comparison algorithms on the synthetic networks. Our methods also produce more equitable distributions of attention than baseline methods across all evaluated network types. Our proposed mechanism addresses attention allocation at a systematic level, balancing between the needs of different stakeholders.

Infrastructure networks increasingly rely on distributed sensing to detect intrusions before attackers reach valuable assets. Yet sensing devices, communication resources, and edge server capacity are limited, while intelligent attackers can adapt their routes to the deployed defense. Motivated by integrated sensing and communication (ISAC), we study how sensing and processing resources should be allocated under strategic interaction between a defender and an attacker. We formulate their interaction as a graph security game in which the defender deploys sensing actions under resource and false alarm constraints, while the attacker selects routes to valuable targets. We consider simultaneous play and settings in which the attacker observes either a pure defender configuration or a mixed defender strategy. Our analysis characterizes the existence, structure, and computational complexity of the Nash and Stackelberg equilibria, showing how the attacker's observation of the defense affects equilibrium behavior and when optimal strategies become difficult to compute. We develop algorithms that construct effective pure configurations and refine restricted games for mixed Nash and mixed Stackelberg play. On enumerable instances, their solutions have small mean normalized differences from fully enumerated references; the methods also apply when exhaustive strategy enumeration is impractical. We also identify conditions under which Nash and mixed Stackelberg payoffs are ordered or coincide.

Consider a revenue-maximizing seller who can access a binary signal about two bidders` joint values. We explore what kind of information is most valuable to the seller by studying three classes of signals, each capturing a distinct dimension of bidders` values: their overall level (demand), their relative strength (ranking), and their dispersion while preserving bidder anonymity (competitiveness). We characterize the optimal signal and corresponding auction mechanism within each class, and find that competitiveness signals are particularly effective. Under certain regularity conditions, the optimal competitiveness signal yields at least as much revenue as any ranking signal or demand signal. Moreover, for signals that induce a monotone allocation, the optimal competitiveness signal yields at least as much revenue as any other binary signal.

Deciding which of two agents is stronger means playing games until skill outweighs luck, and every game costs money, model inference, or expert time. Since the number of games needed is unknown, fixed-budget evaluations either keep paying after the result is settled or stop before the agents can be told apart, while naive optional stopping with an ordinary confidence interval invalidates the stated level. We make such an evaluation stop as soon as its evidence suffices, with the guarantee intact. The Action-Informed Value Assessment Tool (AIVAT) reduces variance in imperfect-information games through conditional mean-zero corrections, by a median $54\times$ across 15 LLM agent configurations spanning 71,439 paired Heads-Up No-Limit Hold'em (HUNL) hands, but does not say when to stop. We combine AIVAT with continuously monitored Confidence Sequences (CSs) into anytime-valid AIVAT (AV-AIVAT), whose online value model learns only from past games so that no game scores its own correction. At the nominal 95% level and a target precision of $\pm1$ Big Blind, raw outcomes need a median $74\times$ as many hands as AIVAT-corrected outcomes to stop under the Asymptotic CS (AsympCS). Exact finite-sample certification uses the Empirical-Bernstein CS (EB-CS), which needs an independently justified bound on corrected payoffs. We establish such a bound structurally for Leduc hold'em and characterize a width floor set by the CS's bet cap and that bound, which governs how much of a variance gain becomes earlier stopping; the descriptive HUNL EB-CS runs show a median $1.37\times$ stopping-time ratio. AV-AIVAT turns variance reduction into efficient, auditable early stopping while separating asymptotic screening from exact certification, so an evaluation can stop the moment its evidence suffices and hand a third party everything needed to recheck the verdict at that very stopping time.

We give a formal mechanism design model for the continuous participatory governance of a deployed AI agent. The mechanism is built on the principle that governance should control an AI agent through resource allocation so as to make authorization self enforcing via compute budgets. The mechanism seeks to establish the Safe AI paradigm that compute is an effective governance lever. We situate our work as a compliance or commons overlay on a deployer. One governance period is an extensive form game in which verified human stakeholders arrive sequentially and contribute, on a provision or a rejection market, in a governance currency that is deliberately distinct from the agents compute. A funding aggregator turns raw contributions into breadth weighted effective supports - a two threshold gate with hysteresis converts net support into a binary authorization that, through a coupling map bounded by an exogenously certified safety ceiling, releases a metered compute budget - realized in hardware as a signed compute license so that the decision is self-enforcing. We characterize the class of agents the mechanism can govern and isolate manipulation of the governing electorate by the governed agent as the central open problem. We also introduce several challenges addressing manipulation of governing electorate by the governed agents.

We study the existence of fair and efficient allocations of indivisible goods under the balancedness constraint, which requires that any two agents' bundles differ in size by at most one. Our main result establishes the existence of balanced allocations that satisfy envy-freeness up to one good (EF1) and fractional Pareto optimality (fPO) for arbitrary additive valuations. This generalizes a recent result of Kawase et al. (2026), which establishes existence only for personalized bivalued valuations or when there are at most two distinct valuation types. Our proof applies the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a weighted-welfare duality framework and develops a novel price-interlacing lemma to overcome barriers encountered by prior work. We extend this technique to category constraints, also known as partition-matroid constraints. In this setting, we establish the existence of an fPO allocation satisfying a weaker, category-sensitive relaxation of EF1, under which envy can be eliminated by removing at most one good from each category. All proofs in this paper were obtained using GPT-5.6-Sol with guidance from the authors. The authors verified the proofs, expanded the exposition, and simplified the arguments with assistance from GPT-5.6-Sol and Claude Fable 5.

Approval-based committee (ABC) elections select a fixed-size committee of candidates based on voters' approval preferences. We study a setting where elected members may resign and formalize resignation monotonicity: when we re-run the voting rule after resignations, a resignation monotone rule must still elect all remaining previously winning candidates. We show that many prominent ABC rules fail this property, and by connecting our notion to prior work on ABC elections with dynamic candidate sets, we show that the justified representation (JR) axiom is incompatible with resignation monotonicity. It turns out that fractional committee voting does not suffer from this issue: rules based on using maximum flows in the network representation of an election instance can satisfy both representation axioms and resignation monotonicity. We design an integral version of maximum flow, the Maximum Payment Rule (MPR), and show that it satisfies a relaxed form of resignation monotonicity that only requires there to be some way of replacing the resigning candidates in a way that guarantees the PJR+ axiom. MPR is NP-hard to compute in general, but becomes tractable in structured domains. Finally, we study a strategic setting where a losing candidate may introduce new weak candidates to the election to try to become a winner. We show that all resignation monotone rules as well as many sequential rules are immune, but that the PAV rule can be manipulated in this way.

We study double auctions for markets in which goods are valuable in bundles, such as data, model weights, and fine-tuned AI assets. A key friction in such markets is No Assembly: a platform may be unable, for legal or technical reasons, to combine components supplied by different sellers into a single bundle. We formulate a combinatorial buyer's-bid double auction under this constraint. Under explicit stability and price-influence conditions (maintained in general, and for two goods derived from local price-taking and a feedback bound), each bundle submarket inherits the large-market discipline of single-good double auctions: bid shading vanishes, and clearing prices concentrate on competitive levels and track the common value (price discovery). The key incentive step, that bidding on a bundle creates no first-order strategic distortion beyond the single-good logic, is proved for two goods; for larger item sets it remains a maintained condition. Multi-agent reinforcement-learning simulations decompose the welfare loss and indicate that No Assembly, not strategic shading, is the binding finite-market friction, with both losses small in moderately thick markets and declining with complementarity amongst goods.

We study the expected average rank achieved by the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism in i.i.d.\ matching markets. While student-proposing Deferred Acceptance gives students an expected average rank of logarithmic order, we prove that EADA's expected average rank is at most $4\log\log n+O(1)$. Therefore, EADA improves the asymptotic order of students' assignments. At the cost of a weaker bound, $O((\log\log n)^2)$, we extend this conclusion to a much larger class of mechanisms. Namely, every Pareto-efficient mechanism that weakly Pareto-dominates DA breaks DA's logarithmic barrier. These are the first asymptotic guarantees for the expected average rank of EADA and of the broader class of Pareto-efficient improvements of DA. The conclusions extend to many-to-one markets with bounded quotas and random markets with correlated preferences.

Generative models can reproduce an observational distribution while encoding an incorrect causal structure. We study a sequential game in which a structural causal generator proposes observational and interventional distributions, while an adversarial experimentalist selects interventions intended to maximally falsify the generator. The discriminator is therefore not merely a real-versus-synthetic classifier: it is indexed by an intervention and tests whether the generator reproduces the corresponding post-intervention law. We introduce Adversarial Causal Intervention Falsification (ACIF), formulate oracle and implementable versions of the game, and distinguish three objects that are often conflated: observational fit, interventional equivalence over an admissible query class, and point identification of a structural causal model. For finite model and intervention classes, we prove: (i) an exact reduction of the adversarial objective to a worst-intervention integral probability metric; (ii) identification up to interventional equivalence, with point identification under a separating intervention family; (iii) existence of mixed-strategy equilibria; (iv) finite-sample uniform convergence and margin-based model-selection guarantees; and (v) a logarithmic elimination guarantee for a disagreement-driven sequential design under a balanced-separation condition. We also give a complete linear-Gaussian example in which two observationally indistinguishable causal directions are separated by a single well-chosen intervention. The framework clarifies what an adversarial causal discriminator can and cannot certify, and provides a principled bridge between causal generative modeling, active causal discovery, and experimental design.

The El Farol Bar game is a classical model of coordination under uncertainty that traditionally treats the venue as a passive constraint. In this work, we reconceptualize the problem by modeling the bar as a strategic player endowed with AI-driven learning capabilities. We extend the original framework in two principal directions: first, by introducing partial observability, whereby agents observe only subsets of past attendees; and second, by transforming the bar from a passive capacity threshold into an active mechanism designer that adjusts pricing policies to balance revenue, utilization, and sustainability constraints. Agents employ AI-based learning to form beliefs and adapt attendance strategies under incomplete information, while the bar applies policy learning to optimize dynamic pricing. The resulting two-sided learning system frames coordination as a co-evolutionary process between boundedly rational agents and an adaptive institution, offering insights into congestion management, resource allocation, and mechanism design in complex adaptive systems.

I study Full Justified Representation (FJR) in approval-based multiwinner elections under both the Hare and Droop quota conventions. I introduce a descending-budget algorithm in which voters distribute their remaining budgets across their current representation gaps and candidates are purchased whenever the resulting offers cover a common price. With candidate price $\lambda_H=n/k$, the algorithm returns a Hare-FJR committee; with candidate price $\lambda_D=n/(k+1)$, it returns a committee satisfying the more demanding Droop-FJR axiom of Casey and Elkind. The two guarantees share a historical-payment invariant and a terminal row–column accounting argument, while the Droop proof requires a new residual-budget argument when all $k$ paid seats are filled. Both variants are deterministic once the voter and candidate orders are fixed and use $O(kmn)$ rational operations.

In approval-based multiwinner voting, voters submit an approval ballot on which basis a committee of fixed size $k$ has to be selected. Compared to single-winner voting this makes it possible to represent minorities in the selection of the committee as well. Thus, one of the most desirable goals of multiwinner voting is to satisfy proportional representation. In this paper we propose a variant of the Greedy Justified Candidate rule that satisfy the notion of Full Justified Representation (FJR). This variant can be computed in polynomial time answering the open question of whether an FJR committee can always be computed efficiently. Additionally, we answer an open question whether committees returned by MES can always be extended in a way to satisfy FJR negatively.

Our results show that the existence of a short high-utility protocol already suffices for efficient communication. In particular, in a game with $n$ possible observations and $m$ actions: (1) For any achievable target utility $\alpha$, we give an algorithm with $\mathrm{poly}(n, m, 1/\epsilon)$ runtime that designs a protocol achieving utility at least $\alpha-\epsilon$ using only $2^{\mathcal O(CC_\alpha(G))}/\epsilon^2$ bits of communication. Here, $CC_\alpha(G)$ is the minimum number of bits used by any protocol, even a computationally inefficient one, to achieve utility $\alpha$. (2) We prove that this exponential dependence on $CC_\alpha(G)$ is tight up to a constant. That is, unless $\mathrm P=\mathrm{NP}$, no polynomial-time algorithm can in general find optimal protocols using fewer than $2^{CC_\alpha(G) -2}$ bits. We note that our results strictly weaken the assumptions required by prior work in the multi-agent information aggregation literature, filling a gap that had remained elusive even for games with constant $CC_\alpha(G)$. In particular, prior guarantees for agreement-based information aggregation rely on structural assumptions such as informational substitutes or weak learnability. We show that these assumptions already imply $CC_\alpha(G) = O(1)$ and are therefore more restrictive conditions than required by our protocol to succeed. On a technical level, our results involve a novel strengthening of the Frieze-Kannan weak regularity lemma and yield the following powerful polynomial-time transformation tool: for every communication game $G$, it constructs a game $\hat G$ that is a coarsening of the agents' observation spaces into constant-size partitions, such that $G$ and $\hat G$ are indistinguishable with respect to every short communication protocol. This coarsening theorem is the engine behind our algorithm and may be of independent interest.

Power indices, originating in cooperative game theory, quantify each player's influence on the outcome of a given game. Originally designed to distribute profits or costs among players and to analyse the fairness of voting systems, power indices have recently gained prominence as methods for attributing outputs of AI-based systems to inputs, thus facilitating explainability. However, selecting the appropriate power index for a given explanation task is an understudied problem. To address this, we organise power indices along three attribution dimensions: single-player, set-based, and cardinality-based. For each dimension, we review the corresponding power indices, generalise existing ones where applicable, and analyse which formal principles they satisfy. We provide proofs for properties that are missing in the literature and show that moving to the cardinality-based setting removes player-identity information while preserving some index-level distinctions. Using concrete examples, we illustrate how the choice of dimension and index affects the resulting attributions in practice, and offer guidance for practitioners seeking to select a suitable power index for a given application context.

Centralization in maximal extractable value (MEV) markets is a significant concern for blockchain systems, as persistent concentration of economic power can weaken competition, reduce openness, and undermine the decentralization goals of permissionless protocols. While much of the existing analysis has focused on builders, validators, and block-building markets, this paper studies centralization from the perspective of searcher competition. We develop a heterogeneous model in which searchers differ in opportunity coverage and execution efficiency, and we analyze how auction design affects fairness, decentralization, and security among searchers competing for the same MEV opportunity. To evaluate searcher competition, we introduce two metrics: a Shapley-weighted Jain fairness index, which measures whether rewards are proportional to searchers' marginal contributions, and an expected-reward Herfindahl-Hirschman Index (HHI), which measures concentration in long-run searcher rewards. Using these metrics, we first analyze the standard first-price, winner-take-all auction as a benchmark. Our analysis shows that, under searcher heterogeneity, first-price competition can reward rank dominance rather than marginal contribution, leading to concentrated rewards and weaker contribution-adjusted fairness. Motivated by these limitations, we propose an entry-filtered Shapley-capped auction mechanism that distributes searcher rewards more fairly and broadly among admitted high-quality submissions. Designing such a mechanism in a permissionless blockchain environment is challenging: searchers may create Sybil identities by submitting copied or degraded versions of the same execution strategy, and validators may collude with searchers to increase joint payoff. We address these concerns through Bayesian security constraints for copied-code Sybil deviations and validator-searcher coalition deviations.

We give an existential transfer framework for converting continuous fair division theorems into guarantees for indivisible items arranged on a path. This allows continuous envy-freeness and consensus results to translate directly into EF$k$-type guarantees for indivisible allocations. Combining this method with connected cake-cutting theorems, we obtain connected allocations satisfying envy-freeness up to one good and one chore for identical valuations and for arbitrary valuations when the number of agents is a prime power. Combining this method with the equicardinal necklace-splitting theorem of Jojić et al., we show that, for any prime-power number $r$ of bundles and $n$ arbitrary valuation functions, there exists an allocation in which every bundle is the union of at most $n$ intervals, and the bundles satisfy consensus up to $n$ goods and $n$ chores. This result is the first EF$k$-type guarantee for consensus fair division with non-additive valuations beyond the halving case. Envy-freeness constraints can be imposed simultaneously at the cost of one additional interval and one additional item in each guarantee. As a consequence, when the number of agents is a prime power, every instance with monotone valuations admits an EF$2$ allocation whose bundle sizes differ by at most two.

Budget-feasible mechanism design is a classic framework introduced by Singer, but there is still a wide gap between existing upper and lower bounds. In this paper, we significantly advance the state of the art. First, without computational constraints, we show that there exists a universally truthful budget-feasible mechanism with the following approximation ratios: - $3$ for monotone submodular valuations and $e+1$ for nonmonotone submodular valuations, improving over $3.798$ and $9.742$, respectively. - $e+1$ for XOS valuations, improving over $28$. In large markets, our approximation can be improved deterministically to $e$. - $2e+1$ for subadditive valuations, improving over $33$. In large markets, our approximation can be improved deterministically to $2e$. Moreover, for subadditive valuations, we obtain a constant-approximation mechanism that runs in polynomial time using demand queries. This improves over the previous best approximation of $O(\log \log n)$, resolving a long-standing open problem going back to Dobzinski, Papadimitriou, and Singer, who conjectured that a constant approximation requires exponentially many demand queries. We obtain these results through a simple and unifying framework based on non-truthful indirect mechanisms, recently coined compensation design. In particular, through a potential argument, we establish constant price-of-stability bounds for compensation design based on marginal-contribution payment rules, which we then translate into truthful direct mechanisms. For subadditive valuations, the core of the argument is a new smoothing lemma showing that every subadditive function can be approximated within a factor of $2$ by a self-bounding function. This is also of independent interest, readily addressing an open question in multiwinner elections by showing the existence of a $2e$-approximate core even under subadditive valuations.

Agentic AI systems routinely transport conclusions across biological, clinical and financial contexts, and the emerging safeguard is local verification: checking at each step that the entity is representable in the chosen tool, that parameters are compatible, and that outputs cohere with the plan. We prove this class of safeguard is structurally incomplete. Modelling a covering of context space by its nerve and evidence by a real-valued 1-cochain, an agent chaining evidence performs path integration: its conclusion is path-independent if and only if the cochain is exact, and disagreement between valid reasoning paths is exactly the holonomy of a first Cech cohomology class. Hodge decomposition partitions evidence conflict into a gradient part (calibration), a curl part (local inconsistency, visible at triple overlaps) and a harmonic part. Our central result is that no family of simplex-supported consistency checks can distinguish omega from omega+h for harmonic h, which nonetheless generates non-zero disagreement between valid paths; detection requires a statistic on a cycle basis. The resulting procedure, Ksetra, estimates by coboundary projection and gates abstention on the harmonic component, which we give a mechanism: it arises from effect modification combined with overlap-specific population composition, and vanishes to machine precision when effect modification is absent. The degrees of freedom of an evidence network partition into calibration, coherence and transport, yielding an exact F-test for the existence of a global claim; we quantify its distortion under unequal precision and supply the precision-whitened form that restores exactness. Foreign exchange, where the arbitrage-free null makes the cochain exactly a coboundary, serves as a calibration bench: the test is correctly sized, fires on loop arbitrage, and ignores triangular arbitrage.

Swap regret governs the rate at which uncoupled learning dynamics converge to correlated equilibria in multiplayer general-sum games. Under full-information feedback, the best previous guarantee when every player follows the same dynamics grows logarithmically in the horizon $T$. We construct uncoupled dynamics under which every player incurs only $O(nm^2\sqrt{\log m\log T})$ swap regret, where $n$ is the number of players and $m$ bounds the number of actions per player. To our knowledge, this is the first sublogarithmic individual guarantee in this setting, and it implies that the time-averaged product distribution of play is an $O(nm^2\sqrt{\log m\log T}/T)$-approximate correlated equilibrium. The key algorithmic choice is to combine the Blum–Mansour reduction with optimistic follow-the-regularized-leader using a hybrid regularizer that separately weights negative Shannon entropy and the log-barrier: the entropy controls the optimistic prediction error, whereas the log-barrier controls the transition-matrix movement through its Bregman divergence. A new sensitivity theorem for stationary distributions of Markov chains, which involves neither mixing parameters nor the smallest transition probability, transfers this control to the played strategies and yields a simpler analysis without local-norm or self-concordance arguments. The guarantee is preserved by an adversarially robust variant that additionally ensures $O(nm^2\sqrt{\log m\log T}+\sqrt{mT\log m})$ swap regret against arbitrary utility sequences, and by a horizon-free variant that requires no prior knowledge of $T$.

Three-player auction bridge is an updated version of the worldwide-recognized auction bridge card game. Dynamic partners and opponents are the key factors in this game. This dynamic property reduces the bias and increases the authenticity of this game. The article provides insight on statistical learning and insight for a three-player auction bridge game. An exact algorithm for various scenarios is developed along with the corresponding winning strategies to explore different game plannings and the computational intelligence involved. Based on the algorithm, we simulate data and define the probabilistic values to win a certain game.

We study fair allocations of indivisible goods among agents with heterogeneous monotone valuations. As fair we consider the allocations that are envy-free-up-to-any-good (EFX). Finding if EFX alloca- tions always exist, even for agents with additive valuations, is a major open problem in Fair Division. Christodoulou et al. (2023) introduced the (multi-hyper)graph setting, where agents and goods are represented by vertices and edges of a graph, respectively, and only the endpoints of an edge may have non-zero marginal value for it. We show that for hypergraphs with girth at least 4 and agents with general monotone valuations there always exists an EFX allocation and can be constructed in polynomial time. We generalize our approach to also show that multi-hypergraphs with girth (on the simple hypergraph) at least 4 always admit an EFX allocation, as long as there exists a single vertex whose incident edges have multiplicity at most the size of that edge minus 2; our construction in this case needs pseudo-polynomial time.

As mobility service providers (MSPs) and energy providers (EPs) expand electric vehicle ecosystems, models are needed to understand their interactions within a three-sided market. Existing frameworks often overlook the temporal interdependencies between mobility and charging demands. We address this gap by proposing a bilevel problem as an assignment game overseen by a market regulator. The upper level optimizes service pricing to maximize platform profitability. The lower level models a multi-stakeholder equilibrium using a scalable, link-based Perturbed Utility Route Choice (PURC) framework. The evaluation time frame is divided into discrete intervals, capturing the temporal lag between mobility and charging demand via an empirical affine function. We solve the model by chronologically decomposing the lower level into interacting mobility service and recharge subnetworks. Numerical experiments on the expanded Nguyen-Dupuis network reveal several key insights. First, a critical charging capacity threshold exists; operating below it forces a severe reduction in the deployable fleet and creates localized transit deserts. Second, modeling endogenous operating costs reveals a concave profit trajectory, demonstrating that total profit maximizes at a specific fleet size just before market saturation. Third, optimal dynamic pricing operates within a narrow range, where peak pricing acts as a steady revenue driver and off-peak pricing serves as a highly sensitive operational buffer. These findings provide actionable strategies for coordinating fleet sizing and charging infrastructure deployment.

Apportionment converts fractional entitlements into integer seat allocations. Quota requires each state to receive the floor or ceiling of its standard quota, whereas population monotonicity prohibits a state whose population weakly increases from losing a seat to a state whose population weakly decreases. Gölz, Peters, and Procaccia recently removed the order-preservation assumptions used in classical incompatibility results by giving a five-state construction; together with the three-state Webster possibility result, this left four states as the unresolved boundary. We prove that no deterministic four-state apportionment solution satisfies both axioms under their definition, which permits the compared house sizes to differ. The proof is a finite logical gadget. Conditional on one quota choice at a central profile, twelve auxiliary profiles encode three bits and force a frustrated cycle. The argument uses only transfers between states whose populations are unchanged and assumes neither anonymity, neutrality, order preservation, homogeneity, nor other regularity conditions. It also applies to relative population monotonicity. Geometrically, the result is a global compatibility obstruction for quota-constrained lattice rounding, rather than an average-distortion bound.

Multi-level apportionment allocates integer seats through a hierarchy of groups. Schmidt-Kraepelin, Suksompong, and Wijaya proved that, at every fixed house size, lower and upper quota can be satisfied simultaneously; they also constructed house-monotone rules satisfying either quota separately. They left open whether one rule can satisfy lower quota, upper quota, and house monotonicity together, even when quota is required only relative to the root. We give a negative answer. For a full binary comb with $D$ equally entitled leaves, every house-monotone allocation sequence induces a sequence of seat recipients. Quota for the nested comb groups would force every grid-aligned prefix discrepancy to be below one. A midpoint embedding then bounds the full interval discrepancy by this quantity plus $1/2$, contradicting Schmidt's logarithmic lower bound. Conversely, a binary van der Corput seat schedule has comb-prefix error at most $\log D/(3\log 2)+1$. Thus the optimal worst-case error on the comb is $\Theta(\log D)$, and for sufficiently large finite $D$ no house-monotone quota rule exists. The proof isolates a static–dynamic gap: each house size admits a quota-feasible allocation, but the feasible allocations cannot be embedded into one monotone path. In quantization language, the result characterizes the order of the embedded-quantization penalty for progressive one-hot rounding on the comb.

We study prior-independent auction design when bidder values are independently and identically distributed and the seller knows only a scale-invariant shape restriction on their distribution, but neither the distribution nor the scale of values. We show that the maximin problem over a broad class of dominant-strategy incentive-compatible mechanisms reduces without loss to scale-free mechanisms. For any $n\ge 2$ monotone-hazard-rate bidders, the second-price auction without a reserve is maximin optimal over this class, including randomized mechanisms that may allocate to a lower bidder. We derive its exact guarantee for every $n$ and the sharp exponential rate at which its loss relative to the Bayesian optimum vanishes. Many familiar auctions are standard: they allocate only to a highest bidder, although incentive compatibility does not require this. For two regular bidders, we solve the standard problem exactly: its optimal mechanism mixes the second-price auction with a relative-markup auction and achieves a worst-case ratio of approximately $0.524413$. We construct a nonstandard mechanism that sometimes allocates to the lower bidder and achieves approximately $0.524829$, proving that standardness is strictly costly. The contrast is driven by tail restrictions: monotone hazard rate makes lower-rank allocation unhelpful, whereas regularity permits it to improve worst-case revenue.

In noisy social dilemmas, intended actions are stochastically corrupted before execution, so an observed defection may reflect hostile intent or action error. Standard Markov Decision Process (MDP) formulations treat executed actions as states, structurally precluding this distinction and causing systematic over-retaliation. We introduce a Partially Observable MDP (POMDP) formulation encoding opponent intentions as latent states and executed actions as noisy observations, solved within the active inference (AIF) framework with a cost function that decomposes into epistemic and pragmatic components that jointly address inferring current intent and learning how intent evolves. In the Iterated Prisoner's Dilemma with symmetric noise, we derive a critical noise threshold governing cooperation collapse, connecting it to a fixed-point condition on learned priors. Experiments reveal that the value of intention inference is context-dependent: the POMDP provides consistent advantages against conditionally cooperative opponents, but mutual intention inference under sufficient noise produces correlated belief-driven collapse. The advantage is specific to games where intent attribution is decision-relevant.

Monderer and Shapley (1996) showed that a game is a potential game precisely when the players' second-order cross-differences agree pair by pair. This paper asks what can be built from them when the agreement fails. The resulting MS-potential, assembled from their common-interest part, is unique up to separable payoff terms and exists precisely when a higher-order MS-condition holds; on exact potential games it recovers the potential up to the players' main effects. A least-squares construction extends the MS-potential to all finite games and induces the \MS-decomposition: every game splits into a common-interest MS-potential game and a residual absorbing every player's individualistic effects. Everything read off the second differences is invariant under the transformations that leave strategic content untouched, relabelling, non-strategic translation and action duplication, where the CMOP decomposition is not; only the least-squares extension fails, since it averages and centres. The central result is an identity: when all players have equally many actions, an augmentation of the MS-potential coincides, up to the additive constant, with the potential of Candogan et al. (2011). If action counts are unequal they diverge, and no bound on that divergence is established here. Both rest on the same uniform weighting of the players' actions.

Computing certain autobidding equilibria is PPAD complete in the worst case. Yet such instances rarely arise in practice, where advertisers running simple, decentralized learning strategies usually converge quickly. We show there is no contradiction: the hardness requires atomicity and vanishes once the value distribution is nonatomic, as it is in real world markets. To bridge worst case hardness and practical convergence, we introduce diffuse analysis, a beyond worst case framework that studies equilibrium computation when bidder values are drawn from general nonatomic distributions. Under this framework, the autobidding equilibrium becomes a separately monotone generalized Nash equilibrium (GNE). For this GNE, we give the first solver with last iterate linear convergence. Thus, the equilibrium has polynomial diffuse complexity, matching the convergence observed in real-world markets. Concretely, our framework subsumes the budget pacing and the throttling equilibrium as special cases when the payment rule is a convex combination of first and second price.