Papers

Learning algorithms are often used to make decisions in repeated multi-agent environments. When another player understands how a learner adapts from past experience, that player can plan strategically across rounds to influence the learner's future behavior. Recent work shows that optimizing against Replicator Dynamics, the continuous-time analogue of Multiplicative Weights Update, is tractable in zero-sum games but can be hard in unrestricted general-sum games. We study the first structured class beyond zero sum: bimatrix games satisfying $\text{rank}(A+B)=1$, for which Nash equilibria can be computed in polynomial time. Our main result shows that this equilibrium tractability does not extend to planning against learning dynamics. Unless $\mathsf{P}=\mathsf{NP}$, approximating the optimizer's optimal continuous-time reward within a fixed additive constant is NP-hard even when $\text{rank}(A+B)=1$, the learner starts from the uniform state, and the optimizer is restricted to constant strategies. The hardness persists for bounded payoff matrices and polynomially bounded horizons. We complement this result with structural characterizations of several tractable special cases. Thus rank-one games already separate efficient equilibrium computation from strategic planning against a learning opponent.

We give a two-player zero-sum repeated game between a learner and nature whose value identity generates Bayesian updating and an exact accounting of exponential-weights regret at once, and supplies the comparator-class variational form that a wide class of concentration phenomena share. The terminal payoff is the most a comparator can gain at fixed relative entropy from the prior, and the one-step constraint is an information budget on nature's move under the learner's mixed action. With the learner's move otherwise unrestricted, Gibbs/Bayes weights emerge as its unique Bellman equalizer – the mixed action that makes the per-round loss independent of which direction nature moves – with log-partition functions playing the role of value functions. The regret decomposes exactly into three parts: a per-round information loss reflecting the variation in observed outcomes, an additive retempering drift that accounts exactly for any change of measurement scale between rounds, and the information the comparator carries relative to the prior. The variance and bounded-range proxies that drive standard regret bounds are looser relaxations of this decomposition, which holds generally and governs them all. Both players' strategies are read off from the decomposition term by term, and repeated play yields an information-theoretic ledger of self-play in place of the usual quadratic-variation surrogate. The same comparator-class geometry accounts for the classical large-deviation bounds, and methods across bandits, posterior sampling, aggregation, and boosting are specializations of the one regret decomposition.

We propose a randomized social choice rule called Mixed Integrated Veto (MIV) with metric distortion of $5/2$, improving the previous best upper bound of $2.75271$. MIV is the equal mixture of Maximal Lotteries and Integrated Veto, a new rule built on the Simultaneous Veto process of Kizilkaya and Kempe. Rather than returning the candidate surviving longest, Integrated Veto assigns each candidate probability proportional to its average score over the whole process.

We study a dynamic coalition-formation process in the tradition of Konishi and Ray (2003): players repeatedly form and dissolve binding agreements, evaluate states by discounted long-term expected payoffs, and hold self-confirming beliefs about the process. States and payoff sharing follow Heitzig and Kornek (2018): a state is a hierarchy of nested agreements, and the members of a new agreement share the surplus it generates, measured against the state without that agreement. All payoff assumptions are structural. We prove that every grand state ever reached is absorbing, and that every absorbing state is grand, for every discount factor. A grand state is actually reached, almost surely, for small discount factors, for three players, and, at every discount factor, whenever distributional stakes are smaller than each player s share of the efficiency gain. Otherwise the process can fail only by cycling for ever among non-grand states. We give exact necessary conditions on such a cycle, decidable for a fixed candidate cycle by linear programming, and exhibit, under an earlier and weaker notion of profitability, a four-player payoff structure whose only closed class is a cycle of two pairs forming and dissolving alternately. Under the present definition, and under either termination rule, no equilibrium traverses a cycle on a fixed schedule: somebody always reaches a state they would rather not leave, and the axioms give them the floor. Whether arrival can fail by cycling at random is open. The axioms are not merely postulated: we exhibit a bargaining game proposal, amendment by substitutes voted on by their own signatories, final unanimity, and an arbitrarily small delay on failure whose equilibria satisfy them as the delay vanishes, and which settles each period in its first round.

Regret matching+ (RM+) is parameter free, scale invariant, and central to large game solving, but its only general individual-regret guarantee grows as $\sqrt{T}$. A recent ICLR result used this envelope to prove that RM+ reaches an $\epsilon$-stationary point of a smooth objective over a product of simplices in $O(\epsilon^{-4})$ iterations, or $O(\epsilon^{-8})$ from the standard zero initialization. We give an exact one-step conservation law for RM+. It states that forward utility gain pays for both squared state motion and growth of the regret-state norm. Norm growth is at most $\sqrt{m-1}$ times forward gain for $m$ actions, and the coefficient is sharp. This yields four results for unmodified RM+. Its regret on any utility path is controlled by centered temporal variation. Its regret is uniformly bounded under alternating play in every finite exact potential game, resolving an open question and making squared activation gaps summable. Both certified lazy and ordinary cyclic play attain an $\epsilon^{-2}$ exponent. On any smooth, possibly nonconcave simplex objective, RM+ finds an $\epsilon$-KKT point in $O(\epsilon^{-2})$ iterations. Most broadly, for a smooth objective over an arbitrary product of simplices, cyclic block RM+ attains the same $O(\epsilon^{-2})$ exponent from arbitrary initialization, with an explicit trajectory-dependent constant. The proof controls the finite objective loss caused by low-state blocks and then self-bounds every block state and the total squared path length. Complete proofs cover zero states, sharpness, common-profile stationarity, and robust gain dominance. Oracle-normalized diagnostics compare RM+ with predictive and smooth extra-gradient variants on graphical potential games and dense nonconvex objectives.

We study the trade-off between firm-side fairness and coalition stability in many-to-one matching markets with transferable payments. For a fixed matching $X$, we characterize the largest supportable core factor by a bottleneck financing problem: $\alpha(X)=1/\Phi(X)$, where $\Phi(X)=\min_{z\ge0}\max_i R_i(X,z)$. This yields a polynomial-time linear program and local sensitivity formulas for one-worker reallocations. We then develop a maximum-edge round algorithm and a broader class of mutual-top safe choices. Every safe execution is EF1 and, with $t=\delta(A)$ denoting the minimum positive-edge quality, guarantees $\alpha(X)\ge\max\{t,1/[m-(m-1)t]\}$ and $SW(X)/OPT\geq t+(1-t)/m$. These bounds give finite-firm lower and upper bounds for the EF1–core minimax frontier, with exact results for two firms and for three firms when $\delta\le1/2$; as the number of firms grows, the tight scale-free stability rate is $\delta$. We also extend the financing formulation to stronger $EFX^+$ fairness and capacity-constrained markets.

We give an exact computer-assisted counterexample to a direct common-degree quartic extension of the affine active-network shape theorem for demand-dependent Price of Anarchy. The instance is a directed network with five vertices, six edges, and three origin–destination paths, all with positive rational costs $c_e(x)=a_e+b_ex^4$. Exact rational interval certificates show that all three paths carry positive Wardrop flow throughout the demand interval $[17,24]$, while \[ \operatorname{PoA}(21)>\operatorname{PoA}(17),\qquad \operatorname{PoA}(21)>\operatorname{PoA}(24). \] Continuity therefore forces an interior local maximum despite a constant equilibrium active network. Krawczyk inclusions isolate the Wardrop and social-optimum KKT solutions, elementary boundary inequalities certify constant support, and interval social costs certify both strict comparisons. At differentiability points, we also derive a serial-edge identity explaining how a common quartic edge can alter the derivative of PoA without changing either route split. The proof and certificate use exact rational arithmetic for every interval and sign decision.

Reinforcement learning (RL) theory fundamentally depends on probability theory through the Markov chain. There is a deep connection between probability theory and potential theory. This paper reviews that connection and explores the potential-theoretic viewpoint for core reinforcement learning representations and algorithms under a fixed-policy assumption. This viewpoint may offer a path for improved sample efficiency and formal constraints that can be applied to RL. When the fixed-policy assumption is relaxed, the linear potential theory framework can be naturally extended to the nonlinear case.

This article surveys emerging directions in the mathematics of democracy. It uses three case studies — voting theory, participatory budgeting, and deliberative democracy — to highlight how contemporary challenges motivate rigorous mathematical research that incorporates real-world data, institutional constraints, and implementation feasibility. Within each case, we highlight active and promising research frontiers, evidence of real-world impact, practical applications, and opportunities for getting involved.

Peer prediction seeks to incentivize agents to truthfully report an observed signal by rewarding joint sets of reports without observing a ground truth. Following the generalization of information-theoretic mutual information introduced in Kong and Schoenebeck (2019), we call a function of a joint distribution over signals a mutual information when it is non-negative and disincentivizes garbling reports for all information structures. An unbiased estimator for a mutual information takes some number of samples from the distribution and returns rewards for both agents, such that the expected reward is equal to the mutual information. We seek to characterize the set of mutual informations with unbiased estimators for a given number of samples. We show that for three or fewer sampled report pairs, the only mutual information with an unbiased estimator is trivially zero, and for four or five samples with a binary report space, the Determinant Mutual Information (DMI) of Kong (2024) is the unique mutual information (up to a scalar multiple). We further show that DMI ceases to be unique at six samples. We provide an improved estimator of DMI for any given number of samples and characterize its convergence rate. We also examine mutual information estimators that accept a randomized number of samples. First, we show that mutual information estimators on an ex-ante bounded number of samples (termed "stop-short estimators") can achieve a lower variance than an equivalent fixed-sample estimator (for DMI). Second, we introduce the class of scoring-rule-based mutual informations and identify in this family a mutual information that can be estimated with under three samples in expectation.

Liquid democracy permits voters to vote directly or delegate their votes to others. Existing algorithmic analyses assign each voter a single scalar parameter, interpreted as an independent probability of voting for the ground truth. This representation is inadequate when delegation is fixed before public information changes different voters' reliability in different ways. In this work, we study a minimal common-signal model of this phenomenon. Delegation occurs before a public binary signal is realised, while voting occurs afterwards. Conditional on the signal, sink votes are independent, and each voter has a signal-specific competence; marginally, correctness events are correlated through the common signal. We show that delegation based on average competence is not a safe scalarisation: it can violate do-no-harm, and a beneficial delegation rule may send votes to voters with lower average competence. We present and analyse three novel delegation mechanisms for this setting. The first is a conservative intersection mechanism that delegates only to neighbours whose competence exceeds the delegator's by a prescribed margin in every signal state; the conservative intersection mechanism inherits all the guarantees of delegation in the scalar competence setting. The next mechanism is a confounded-set mechanism that enable the delegation to neighbours who are favoured in one state and worse by at most a prescribed tolerance in the other. For this mechanism, we prove expected-margin bounds, and identify the weight-dispersion and mechanism-concentration conditions needed to obtain majority-correctness guarantees. Finally, on bounded-in-degree graphs, a multi-round certified-path mechanism propagates nonnegative two-dimensional path certificates; it is acyclic, yields statewise terminal competence improvement, and gives uniform bounds on path length and terminal voting weight.

Motivated by modern marketplaces, where the platform or the seller routinely gathers detailed user profiles, we study a novel learning theoretic model that simultaneously involves information and mechanism design. Specifically, we consider the economic setting recently introduced by Bergemann et al. (2022), where in addition to the menu of quality-price pairs, the seller offers information on the value of the match between product quality and buyer's taste via a signaling scheme. We relax the assumption that the seller knows the buyers' belief about the distribution of tastes and study the sample requirements of designing a revenue maximizing scheme. We consider both the batch setting where we have access to data from a set of i.i.d. buyers and an online demand query model where we observe the buyers' behaviors to seller's schemes. Despite the apparent non-convexity of the problem, we also give the first FPTAS to compute a scheme that maximizes the revenue within an arbitrarily small additive loss, which was left open by Bergemann et al. (2022). Overall, this brings a new learning perspective in asymmetric economic settings where buyers and sellers know different types of information.

Budget-constrained advertisers commonly rely on two control mechanisms: pacing scales bids, whereas throttling randomizes participation. We prove that, in second-price auctions, these two different mechanisms share the same sharp approximation-hardness threshold. For pacing, computing a $\gamma$-approximate equilibrium is $\mathsf{PPAD}$-hard for every constant $\gamma\in[0,1)$. For throttling, computing a $\delta$-approximate equilibrium is $\mathsf{PPAD}$-hard for every constant $\delta\in(0,1)$. At parameter $1$, the complementarity requirement becomes vacuous and the all-zero solution is feasible. That is, approximation does not eliminate the fixed-point barrier at any nontrivial parameter value.

We find no evidence of critical scaling in the Schelling segregation model, in either the Moore neighborhood or its dense-spectrum extension to Chebyshev radii up to $r_0 = 6$ ($k = 168$ neighbors). On periodic grids up to $L = 320$ with 50 trials per point (> 12,500 runs), every finite-size scaling diagnostic in the Moore baseline fails: the per-$L$ $T_c$ does not drift, Var$(S) \sim L^{-2.02 \pm 0.09}$ matches trivial averaging, $\gamma/\nu \approx 0$, and the scaling collapse never reaches a finite optimum. The 8-site Moore neighborhood restricts satisfaction to ratios $j/k$ with $k \leq 8$, giving $S(T)$ a staircase structure with 23 rational thresholds; discreteness alone does not forbid criticality (cf. the Ising model), but the scaling evidence rules it out empirically. A branching-ratio calculation predicts subcritical cascades of mean size $1/(1-R)$ and is validated by perturbation experiments to within 15%; the multiscalar dissimilarity length stays finite across the transition. The dense-spectrum extension strengthens the negative verdict: across $r_0 \in {3,4,5,6}$ on $L \in {40,80,160}$ the Binder cumulant has no $L$-curve crossing and the per-$L$ $T_c$ drift is monotonic and unsaturated; at $r_0 = 4$, extending to $L = 320$ gives $\alpha = -2.70$, below the critical boundary $\alpha = -2$, dissolving an apparent $\alpha = +0.81$ signal visible only on $L \in {40,80}$. The mechanism is the absence of long-range correlation in equilibrium plus deterministic high-$k$ dynamics, not the staircase structure. With a Beta-distributed heterogeneous tolerance, the intolerant tail drives segregation even at moderate population-average tolerance. The staircase theorem and cascade mechanism together account for the Schelling transition without invoking critical phenomena.

We study the strategyproof placement of \(k\) facilities on the real line for \(n\) agents who privately report their locations, without monetary transfers. For two facilities, the Proportional Mechanism of Lu, Sun, Wang, and Zhu (2010) is strategyproof in expectation and achieves a constant-factor approximation to the optimal social cost. Whether such a guarantee is possible for three facilities in the standard model, where each agent is served by her nearest open facility, has remained open. We resolve this question affirmatively by introducing the Random-Anchor Volume mechanism. The mechanism first opens a facility at the report of a uniformly random agent, called the anchor, and then jointly selects two additional reports, assigning each pair probability proportional to the product of the two consecutive gaps formed by the pair and the anchor. We prove that the mechanism is strategyproof in expectation and has expected social cost at most \(8 OPT_3\), where \(OPT_k\) denotes the minimum social cost achievable using at most \(k\) facilities. The mechanism naturally extends to every \(k\geq 2\) by selecting \(k-1\) additional reports with probability proportional to the product of the consecutive gaps among them and the anchor. Under truthful reporting, this generalization has expected social cost at most \(4(k-1)OPT_k\). Its incentive guarantee, however, has a sharp boundary: the mechanism is strategyproof in expectation for \(k\in\{1,2,3\}\), but is manipulable for every \(k\geq 4\).

Nearly a decade ago, Calude et al. showed that parity games can be solved in quasi-polynomial time. This result is now understood in terms of universal trees. By reduction to parity games, the quasi-polymonial result can benefit all omega-regular games. However, beyond such reductions, and with the exception of Rabin games, our understanding of the role of universal trees in direct solutions is still quite limited. In this work, we refute the common view that universal trees are relevant only for games that admit memoryless winning strategies. We contribute a full understanding of how universal trees interact with Zielonka trees for the solution of Streett and Emerson-Lei games. As a consequence, we show that winning regions and strategies in Streett games with $n$ vertices, $m$ edges, and $k$ pairs can be computed in time $O(mk\log(k)k!|U(n,k)|)$, where $U(n,k)$ is a universal tree for $n$ leaves and depth $k$. This improves upon the best previously known complexity result for Streett games, which relied on reduction to parity games and their quasi-polynomial solution. Furthermore, we show that winning regions and strategies for Emerson-Lei games with $n$ vertices, $m$ edges, and $c$ colors can be computed in time $O(mc\log(c)c!|U(n,c/2)|)$, again improving over reductions to parity games. Notably, our approach yields memory-optimal strategies, in contrast to those obtained via reductions to parity games. Finally, we show how universal trees can be used to bound the recursion tree of the Zielonka-McNaughton algorithm for Emerson-Lei games. This leads to a symbolic algorithm that replaces the factor $n^c$ in the time complexity of existing symbolic approaches with $|U(n,c)|$.

Shared autonomy requires principled mechanisms for allocating and transferring control between a human and an autonomous agent. Existing approaches often rely on blending control inputs or heuristic switching rules, which lack theoretical guarantees and fail to account for the dynamics of authority transfer. This paper develops a cooperative game-theoretic framework for authority switching in shared autonomy. We formulate the control switching problem as an identical-interest dynamic game in which authority transitions are embedded into the system dynamics, yielding optimal switching policies rather than ad hoc rules. We establish the existence and characterization of team-optimal policies in pure strategies under stochastic human override, accounting for asymmetric authority where humans retain override capability. For linear-quadratic systems, we derive closed-form recursions for the optimal switching policies and value functions, enabling efficient computation independent of the continuous state. We validate the framework on scalar and multi-dimensional linear systems, demonstrating how optimal switching adapts to varying system dynamics, cost structures, and override probabilities. The results reveal fundamental trade-offs between human adaptability and autonomous efficiency, illustrating the practical benefits of grounding shared autonomy in cooperative game theory.

Shared autonomy requires principled mechanisms for allocating and transferring control between a human and an autonomous agent. Existing approaches often rely on blending control inputs or heuristic switching rules, which lack theoretical guarantees and fail to account for the dynamics of authority transfer. This paper develops a cooperative game-theoretic framework for authority switching in shared autonomy. We formulate the control switching problem as an identical-interest dynamic game in which authority transitions are embedded into the system dynamics, yielding optimal switching policies rather than ad hoc rules. We establish the existence and characterization of team-optimal policies in pure strategies under stochastic human override, accounting for asymmetric authority where humans retain override capability. For linear-quadratic systems, we derive closed-form recursions for the optimal switching policies and value functions, enabling efficient computation independent of the continuous state. We validate the framework on scalar and multi-dimensional linear systems, demonstrating how optimal switching adapts to varying system dynamics, cost structures, and override probabilities. The results reveal fundamental trade-offs between human adaptability and autonomous efficiency, illustrating the practical benefits of grounding shared autonomy in cooperative game theory.

We study the group-fair distortion of metric facility assignment problems, where a set of agents, partitioned into unknown groups, must be assigned to a collection of facilities, possibly subject to capacity or other feasibility constraints. Given an assignment, each agent incurs a cost that depends on both its distance to its assigned facility and, via an affinity factor, the average distance of the other members in its group to their assigned facilities. We consider full-information algorithms, which have complete knowledge of the metric space, and ordinal-information algorithms, which know the distances between facilities and only the rankings of the agents over facilities (sorted by increasing distance). We establish worst-case distortion upper bounds in terms of the Max-of-Sum and Sum-of-Max social objectives, which combine the classic utilitarian and egalitarian social cost measures. We also derive informational lower bounds for one-sided matching and clustering, two fundamental and well-studied problems captured by our model, that match our upper bounds exactly for Max-of-Sum and asymptotically for Sum-of-Max.

In over-the-counter corporate bond markets, dealers compete for client trades by quoting bid and ask prices. Tighter quotes attract more business, but also informed customers more likely to trade ahead of adverse price moves, leaving the dealer holding the risk. As dealers increasingly use machine learning to set quotes, they retrain these models on the trades their own quotes attract, creating a feedback loop in which each model reshapes the market that generates its next training data. The question is therefore not only whether a quoting model performs well, but whether the market it creates stays stable as the model learns from it. Existing performative prediction theory gives a sharp stability condition, yet expresses it through abstract properties of the learning objective a trading desk cannot measure before deployment. We introduce REFLEX, a framework that replaces those unobservable quantities with three measurable features of dealer behavior: how strongly trading volume responds to tighter quotes, how sharply the dealer's objective bends around its optimum, and how quickly informed flow increases as spreads narrow. REFLEX combines these into a single retraining modulus, a pre-deployment stability margin estimated from a desk's own quote and execution history that predicts whether repeated retraining will converge or amplify itself. In simulation, predicted and measured stability agree within 8%, and competing dealers increase instability by 1.74x with two and 3.16x with three, as predicted. Where ordinary retraining becomes unstable at modulus 1.21, a structurally anchored correction converges as blind retraining collapses. Calibrated over 36 years of public market data, stability headroom falls roughly 4.4x for investment grade and 4.3x for high yield from calm to crisis regimes. Ultimately, REFLEX turns an abstract convergence theorem into a market-level safety margin.

Electric vehicles (EVs) represent a critical intelligent terminal within the Internet of Things (IoT). Despite the year-on-year growth in EV penetration, the highway driving experience still requires improvement. Accurate prediction of EV highway charging behavior is crucial to addressing this issue. This paper introduces a novel bounded rationality framework to analyze highway charging decisions. Specifically, we utilize prospect theory to capture the tendency of drivers to reserve more electricity than theoretically necessary. We then propose a Bayesian game in which EV drivers, unaware of others' decisions, aim to minimize costs, including range anxiety, charging fees, and queuing time. To gain insights into the game, we prove the existence and uniqueness of the Bayesian Nash Equilibrium in two practical scenarios. Our numerical experiments, based on real-life data, demonstrate that drivers' risk aversion tendency significantly influence EV charging decisions, charging demand, queuing lengths at charging stations, and the departure rate on the highway network. Furthermore, our strategy reduces cumulative EV cost and CSs' charging costs compared to other benchmarks.

Proportionality (PROP) is one of the simplest fairness criteria for allocating items among agents with additive preferences. With indivisible chores, however, PROP is not always satisfiable. We study proportionality up to any item (PROPX), which requires every agent to satisfy proportionality after any chore is removed from her bundle. Under strictly positive costs, we settle the weighted compatibility question negatively: weighted PROPX and Pareto optimality are incompatible already for two agents and four chores. Moreover, for every $n\geq3$, we give an $n$-agent, $(n+1)$-chore counterexample whose shares can be arbitrarily close to equal. These counterexamples are item-minimal: under strictly positive costs, weighted PROPX and Pareto optimality are always compatible when the number of chores is at most the number of agents, and they are compatible for two agents with at most three chores. Our impossibility result contrasts with the compatibility theorem of Mahara (2026) for weighted envy-freeness up to one item (EF1) and Pareto optimality .

We study fair division of indivisible goods when agents' valuations are accessed only through ordinal comparisons between bundles, with arbitrary tie-breaking. In this model, even deciding whether a given allocation is envy-free up to one good (EF1) can be impossible. This suggests explicit fairness certificates as a natural algorithmic object. Our main contribution is a certificate-preserving scaling framework, which recursively contracts goods, solves a smaller instance, and expands the solution while repairing an explicit envy-eliminating witness certificate. For arbitrary identical monotone valuations, this yields a certified EF1 allocation for $n$ agents and $m$ goods using $O(n \log n \log(m/n))$ comparison queries, within an $O(\log n)$ factor of the $\Omega(n \log (m/n))$ communication lower bound. For identical additive valuations, we additionally obtain a $1/2$-MMS guarantee within the same query complexity. For non-identical additive valuations, exploiting our EF1+$1/2$-MMS algorithm, we accelerate the existing matching-based PROP1+$1/2$-MMS framework, improving the query complexity from $O(n^4\log m)$ to $O(n^3\log m)$. Finally, we study the structure of such certificates through $k$-witness EF1, a hierarchy between EF1 and EFX.

We consider strategic facility location in Euclidean space $\mathbb R^d$, where a mechanism selects a single facility based on the reported locations of $n$ agents and seeks to minimize the maximum distance from any agent to the facility. The optimal approximation ratio of deterministic strategyproof mechanisms is $2$. Whether randomization can yield a universal constant improvement over this factor has remained a major open question. We show that for $d \geq 2$, every strategyproof-in-expectation mechanism has approximation ratio at least \[ \alpha(\mathcal M) \ge 2 - e^{-\Theta(\sqrt d)} - O\left(n^{-2/(d-1)}\right). \] Therefore, no strategyproof-in-expectation mechanism can guarantee a $(2-\varepsilon)$-approximation uniformly over all $n$ and $d$, for any universal constant $\varepsilon>0$.

Deciding whether a Sokoban puzzle is solvable is PSPACE-complete (Culberson, 1997): solutions can be exponentially long and there is no short certificate to check. Solvability is also a fragile property, since even a single misplaced wall can silently render an entire puzzle unsolvable. In this work, we show that a transformer-based discrete diffusion model trained purely on tile completion, with no access to solvers, rewards, or solvability labels, achieves a solvability rate of 77.4%, with 94.5% of the remaining failures rendered solvable by removing a single wall. In other words, a global, search-heavy property follows from a local training objective: trained only to fill in masked cells, the model inherits solvability it was never trained on. An autoregressive model factorizes as $p(c_k \mid c_1 \dots c_{k-1})$, meaning a fixed order, always conditioned on a prefix. Masked diffusion does not: it hides a random subset of cells and learns $p(c_k \mid \text{any subset})$, so at generation time it can reveal cells in any order, each one conditioned on everything already placed, wherever it sits on the board. A puzzle's difficulty comes from exactly this kind of non-local interaction, a decision in one part of the grid constraining what will work somewhere else entirely. A generator that is not locked into a single fixed order is therefore a better structural match for the problem than one that is. The training pipeline is adapted from MD4 (Shi et al., 2024) and the dataset is DeepMind's Boxoban (Guez et al., 2019). The trained model and instructions for generating puzzles are publicly available.

Generative search engines (GSEs) answer user queries directly from crawled web content. The capture of value from the corpus without a visit returned to the source (we call this capture extraction) diverts the traffic that finances content production. In response, publishers may restrict crawler access to their websites. In this paper, we model the crawlable corpus as a common-pool resource: the crawlable commons. It is described by three quantities: volume, average quality, and lifetime. Under two types of responses of publishers we prove that extraction degrades all three at once: publishers opt out, renewal loses its funding, and content becomes more perishable. After a given erosion threshold, the corpus goes extinct. A myopic GSE can cross this threshold, a long-run oriented GSE stays below it. We extend our model to several competing engines and prove, under a concavity condition on the steady-state value of the commons, that the symmetric equilibrium extraction rate is nondecreasing in their number and converges to the threshold. Adding users who strictly prefer direct answers, the assumption most favorable to extraction, we prove that the socially optimal extraction rate lies strictly below the erosion threshold, and no higher than the single engine's sustainable optimum. Finally, we discuss seven survival mechanisms.

This paper presents CoupVisor, a decision-support system for the hidden-information card game Coup. It addresses two questions: what a player should do on each turn, and when a player should challenge an opponent's claim. The system is built around a single description of game events, which is shared across manual play, replay of recorded games, simulation, belief tracking, advisor recommendations, and learning-based policies. CoupVisor estimates the chance that a claim is truthful by combining how likely each role is with how many cards the claimant still holds, which corrects a case where the very first claim of a game was flagged as suspicious despite no evidence. We compare a rule-following advisor and several learned and heuristic players across many simulated games and different opponent styles. Our main finding is that the choice of reward, whether it rewards short-term gains or ultimately winning the game, decides which learning approach performs best, and that a win-oriented reward produces a policy that outperforms all baselines.

The maximin share (MMS) is a central fairness benchmark for allocating indivisible goods and chores. We study additive valuations in the personalized bivalued setting, where each agent assigns one of two agent-specific values to every item. Whether exact MMS allocations always exist in this setting has remained a major open question, as highlighted by Ebadian, Peters, and Shah and by Garg, Huang, and Segal-Halevi. We answer this question affirmatively: we prove that exact MMS allocations always exist for both goods and chores and can be computed in polynomial time. Our proof combines a quota-based reformulation with an envelope relaxation, a sparse extreme-point construction, and flow-based rounding that controls the total rounding loss.

In Shapley-Scarf housing markets, Ma (1994) shows that top trading cycles (TTC) is the unique mechanism satisfying individual rationality (IR), Pareto efficiency (PE), and strategy-proofness. We ask what other mechanisms become possible when strategy-proofness is replaced by a weaker condition called non-obvious manipulability (NOM), introduced by Troyan and Morrill (2020). We first show that this weaker condition does not help on its own: every IR and PE mechanism is already NOM. We therefore introduce a new condition: NOM with groups, under which each agent knows the preferences of the other members of her group, but not those of agents outside the group. This condition reduces to strategy-proofness when all agents belong to one group, and to standard NOM when every group is a singleton. Also, we introduce a participation condition called group rationality (GR), which requires that no group do worse than it would by trading only among its own members. We then define a class of mechanisms called TTC with super-groups, whose members satisfy GR, PE, and NOM with groups. The class includes mechanisms that differ from standard TTC, including mechanisms that are not strategy-proof. Furthermore, we show that every mechanism that satisfies GR, PE, and NOM with groups has the same best- and worst-case outcomes as every TTC with super-groups mechanism.

We study approval-based multi-winner elections with Justified Representation (JR) when voters strategically report their ballots. We prove that there does not exist a strategy-proof mechanism that outputs JR committees, even when the mechanism can be randomized and only Ex-Ante strategyproofness is required. The impossibility already holds with three voters, four candidates, and target committee size three, even when the mechanism may select fewer than three candidates. It applies to every common strictly increasing cardinality-based utility function and scales to every target size $k\geq3$, every number of voters divisible by $k$, and every candidate-set size at least $k+1$. Motivated by our negative results, we then ask for weaker strategy-proof guarantees under voters' partial information. We use the notion of Risk-Avoiding Truthfulness and its RAT-Degree proposed by Hartman, Segal-Halevi, and Tao (EC'25), the number of other voters whose information a manipulator must know before a safe and profitable deviation is possible. Standard proportional rules, such as Greedy Approval Voting, Proportional Approval Voting, and the Method of Equal Shares, have poor performances under the RAT-Degree metric (with low RAT-Degrees). Finally, for $n$ voters, we construct a JR mechanism with RAT-Degree at least $n-5$, without a polynomial-time bound. We then use the same idea together with algebraic ballot encodings to obtain a polynomial-time JR mechanism with RAT-Degree at least $n-7$. When voters also have immutable private identifiers and both the identifiers and ballots of unknown voters are unobserved, we obtain a polynomial-time JR mechanism with RAT-Degree at least $n-1$ for $2\le k<m$. This degree is exactly $n-1$ when $k\ge3$ and $k$ divides $n$.