Papers

This survey introduces contract design to computer scientists: using outcome-based payments to incentivize effort. We discuss applications that motivate a computational approach to contract design, ranging from online labor markets, to incentive-aware algorithmic classification, to emerging ecosystems of AI agents. We present the classic principal-agent model and introduce basic concepts. We show how optimal contracts are computable via linear programming, and why they can be opaque and brittle. Motivated by this we turn to simpler contract formats, specifically linear contracts. We discuss approximation guarantees, and robust (max-min) optimality of such contracts. We then move beyond the single-agent setting to combinatorial contract settings with multiple agents, mapping the algorithmic landscape of tractability and approximation. Finally, we discuss partial information settings through the lens of learning algorithms, and outline open directions. The goal is a concise, accessible entry point that connects economic foundations with algorithmic tools and highlights opportunities for new theory and applications.

We study the online fair division of indivisible goods. In the standard model, where assignments are irrevocable, no algorithm can guarantee any positive fraction of the maximin share (MMS). Reassigning earlier goods can restore fairness, but it may take value away from agents who already hold them. We ask how much fairness can be maintained if every reassignment must leave each affected agent at least as well off. We consider two forms of such upgrade recourse. Under item-level upgrades, each good an agent holds must be kept or replaced by a distinct good she values at least as much; under bundle-level upgrades, only her total value must not decrease. For item-level upgrades, we show that the optimal MMS factor is essentially $1/H_n \approx 1/\ln n$, where $H_n$ is the $n$-th harmonic number, and that fixed picking sequences nearly attain it. Since these sequences use only the agents' rankings of individual goods, knowing the actual values cannot improve the optimal factor for up to 33 agents, and improves it by at most a vanishing margin as $n$ grows. Bundle-level upgrades allow substantially stronger guarantees. By permitting several-for-one exchanges, they turn this vanishing factor into a constant: we achieve exact MMS for two agents and more than half of MMS for any number of agents, in both cases together with relaxations of envy-freeness. Finally, we show that predictions about future goods let online algorithms inherit any offline MMS guarantee, losing only an additive term bounded by the prediction error. Under item-level upgrades, however, insisting on exact MMS when predictions are correct leaves no positive guarantee when they are wrong.

We prove that, for every fixed rational $c>0$, computing a $2^{-n^c}$-approximate Nash equilibrium in an $n$-player two-strategy anonymous game is $\mathsf{PPAD}$-complete. This holds even when one strategy always pays $1/2$ and, for all but two players, the other is constant outside at most five opponent counts. The reduction uses coefficient estimates to simulate circuit constraints approximately. Exact computation requires eliminating the simulation error: with three strategies, a separate construction uses polynomial roots to implement circuit gates exactly. This yields $\mathsf{FIXP}$-completeness of exact equilibrium computation under separable-linear reductions and $\mathsf{FIXP}_a$-completeness of approximation in distance to an exact equilibrium.

Digital Twin (DT) combined with reinforcement learning (RL) can support adaptive resource allocation and security-aware decision-making by comparing expected and observed network behavior. This paper presents a simulation-based DT reference and a belief-guided Q-learning (BGQL) for joint jammer identification and beam-power control in a site-specific 28-GHz network. For each action, the same channel simulator is evaluated under an attack-free and an attacked configuration. Receiver-level residuals are combined with spatial jamming footprints and accumulated to form a normalized suspicion vector. BGQL incorporates this vector into an aggregated state, a belief-concentration reward, and exploratory beam-selection rule. The BGQL scheme is compared with SARSA, Greedy, and Random under matched attacker sets. Across 30 paired Monte Carlo runs in four configurations, with the number of active jammers supplied to every method, BGQL achieves mean aggregate full-band link-rate scores of approximately 437.4–564.2~Mbit/s and rate-to-device-power scores of 60.37–76.79~Mbit/J. Its detection accuracy is 100% in one-jammer configurations and approximately 78%–81% in two-jammer configurations. BGQL requires approximately 80–130~s per run, compared with 5–14~s for SARSA, reflecting evaluation of 625 beam pairs during exploratory steps. These findings provide a matched-model simulation proof of concept for DT-referenced belief-guided learning for security-aware beam-power control in 6G networks.

We study the voter model with opinion graph, a generalization of the standard (pull) voter model that adds a notion of bounded confidence. In this process, we consider a finite set $A$ of $n$ agents. Each agent $u \in A$ has an opinion $x_u \in O$, where we model opinions as vertices in an undirected opinion graph $G = (O,E)$ with $|O|=q$. In each step, an agent $u$ interacts with another agent $v \in A$. In contrast to classical voting, $u$ adopts the opinion $x_v$ if and only if $x_v$ and $x_u$ are neighbors in $G$. The process is stable once all remaining opinions are an independent set in $G$. We analyze two variants of the process: the ordinary EdgeVOG process, which samples a pair of interacting agents uniformly at random from all possible pairs, and the productive EdgeVOG process, which samples only pairs with neighboring opinions in $G$. We show the following: - For every opinion graph and every initial configuration, ordinary EdgeVOG stabilizes within $O(n^2\log n)$ steps in expectation and with high probability. The expectation bound is asymptotically tight. For the productive EdgeVOG process, the expected stabilization time is exactly the initial number of compatible agent pairs, while $O(n^2\log n)$ steps suffice with high probability. - An opinion $i$ with initial support $s_i$ survives with probability at least $p_i:=s_i/(s_i+\sum_{j\in N(i)}s_j)$. Further, let $\mu = \sum_{i \in O} p_i$, then we show the probability that less than $(1-\delta)\mu$ opinions survive is at most $\exp(-\delta^2\mu/(2(\Delta^2+1)))$ for every $0\leq\delta\leq1$. -If $G$ has maximum degree $\Delta$ and every opinion initially has support at most $\hat{s}$, then no opinion ever acquires more than $\tilde{O}(\min\{n,\Delta^6\hat{s}\})$ agents with high probability. This yields near-linear stabilization bounds for productive EdgeVOG for a large class of configurations.

Thompson sampling (TS) achieves logarithmic and asymptotically optimal regret in classical stochastic bandits, but its behavior in multi-agent learning remains less understood. We study TS in an unknown finite two-player zero-sum matrix game with Gaussian bandit feedback. Under centralized self-play, TS samples a single payoff matrix from the posterior and plays a Nash equilibrium of the sampled game. For every fixed zero-sum game with a unique Nash equilibrium, we show that the expected cumulative duality gap is $O(\sqrt T)$. The rate improves to $O(\log T)$ for pure equilibria. We then identify the exact coefficient of the $\sqrt T$ term, which depends only on the equilibrium strategies and the noise level. We also establish a minimax lower bound matching the square-root dependence on $T$ and an instance-dependent lower bound matching the logarithmic dependence on $T$ for a class of games with a unique pure equilibrium. Experiments support the predicted rates and coefficient.

Computing equilibria in high-dimensional continuous games requires learning and representing rich, multi-modal strategy distributions, something conventional neural policies struggle to achieve. Diffusion models offer a natural solution, but standard diffusion training requires direct samples from a fixed target distribution, which are generally unavailable under iterative game-solving updates. We propose a diffusion-based framework for equilibrium computation that enables diffusion-based variants of Online Mirror Descent (OMD) and Fictitious Play (FP). Our approach derives a reweighted score-matching objective that trains diffusion policies without requiring direct samples from each updated policy. Our main method, DiffNash, combines the fast convergence of OMD with the sample efficiency of FP by reusing past actions through a replay buffer. To correct the resulting distribution shift from sample reuse, we derive an unbiased estimator of the difference between current and past policies' Evidence Lower Bounds, whose exponential provides a tractable proxy for the corresponding importance weight without evaluating diffusion likelihoods. We evaluate our approach on forest protection games and continuous Colonel Blotto games, reducing the Nash gap over the strongest baseline by 35.6% and 24.7%, respectively, with minimal hyperparameter tuning.

In multi-group learning, a single predictor must perform nearly as well as the best hypothesis in a class on every group in a rich, possibly infinite and overlapping, collection of subpopulations. The known algorithms with the optimal sample complexity keep track of every pair of a group and a hypothesis, which is impossible for the large classes of interest, and the known oracle-based algorithms have a worse rate. We ask whether the optimal rate can be attained by a learner that accesses the classes only through an empirical risk minimization (ERM) oracle. We give an algorithm that uses $\tilde{O}((d_G+d_H)/\varepsilon^2)$ samples and polynomially many calls to a weighted ERM oracle over pairs of a group and a hypothesis, where $d_G$ and $d_H$ are the VC dimensions of the group and hypothesis classes and $\varepsilon$ bounds the excess risk on every group, weighted by its probability. We also show a matching lower bound of $\Omega((d_G+d_H)/\varepsilon^2)$ samples for every learner even with unlimited computation, so oracle efficiency costs no additional samples. We also consider a natural special case in which every input belongs to at most $k$ groups and these groups can be listed. In this case, an ERM oracle for the hypothesis class alone suffices, with $\tilde{O}((d_H+\log k)/\varepsilon^2)$ samples in the agnostic setting and $\tilde{O}((d_H+\log k)/\varepsilon)$ samples when every group has a hypothesis that is correct on it.

We study stationary discrete-time mean-field games with finite state and action spaces under the average-reward criterion, with an entropy-regularized objective and population-dependent dynamics and rewards. Under a minorization condition, the soft Q-operator is a span-seminorm contraction for the full transition kernel, so relative value iteration does not require the minorizing measure. When the Lipschitz constant of the kernel is below the minorization mass and the temperature exceeds an explicit threshold, the equilibrium fixed-point operator is a contraction: an equilibrium exists, its population distribution is unique, and a fixed-point iteration converges to it geometrically. We then learn the equilibrium model-free from a simulator, combining soft Q-learning at a fixed population distribution with an empirical transition kernel. We prove a finite-sample bound for soft Q-learning in the span seminorm, with temperature-independent constants, and derive convergence rates for the learned population distribution and policy. A congestion-aware routing game illustrates the results.

We study the problem of fair allocation when agents have dichotomous submodular preferences over groups of, rather than individual, items. These preferences naturally extend existing models, and have a natural interpretation in settings such as course allocation. We show that this seemingly minor extension results in computational intractability barriers: computing a utilitarian or Nash-optimal allocation with item complements is NP-hard. However, when agents' complementary bundles have size at most $k$, there exist poly-time algorithms that compute $\frac1k$-$\texttt{maxUSW}$ allocations and approximate Nash-welfare-maximizing allocations. While PO+EF-1 allocations may not exist in the complementary preference setting, it is possible to compute $\frac1{k+1}$-approximately $\texttt{maxUSW}$ EF-X allocations in polynomial time; furthermore, these allocations offer each agent $i$ a utility of at least $\frac{\texttt{MMS}_i - k}{k+1 + \frac1n}$, where $\texttt{MMS}_i$ is agent $i$'s maximin share guarantee. While the standard dichotomous binary submodular setting admits simple sequential techniques for computing fair and efficient allocations, our results for the complements setting require more complex tools from combinatorial optimization.

We investigate the problem of allocating indivisible chores among agents with arbitrary positive weights and additive nonnegative cost functions. An allocation is said to be $\alpha$-weighted-maximin-share (WMMS) fair if the cost incurred by each agent is at most $\alpha$ times her weighted maximin share. We establish that every instance admits a $4.124234528$-WMMS allocation. Moreover, for instances in which all weights and costs are rational numbers represented in binary, we give an algorithm that computes a $4.12424$-WMMS allocation in time polynomial in the number of agents, the number of chores, and the total binary encoding length of the input. Complementing these upper bounds, we construct a family of instances with strictly positive, identically ordered costs for which the optimal WMMS approximation factors converge to $e$. Together, our results substantially narrow the previously best-known gap of $[2,12]$ established by Li et al.~\cite{LWX26} to $[e,4.12424]$.

We study the model of participatory budgeting, in which a subset of candidates is selected subject to a budget constraint. Voters express their preferences by assigning a numerical utility to each candidate. Our focus is on formal analysis of proportionality, a central notion of fairness in multi-winner elections. Intuitively, proportionality means that each group of voters should influence the election outcome roughly in proportion to its size. Recent "plus" axioms of proportionality are particularly appealing notions: they are verifiable in polynomial time, and they provide strong representation guarantees without requiring groups of voters to have cohesive preferences. However, their existing formulations are restricted to approval-based preferences. We generalise extended justified representation plus (EJR+), and Core plus (Core+) to general additive utilities. Our approach builds on a payment-based interpretation of proportionality: candidates' costs are shared among their supporters, with the aim of balancing voters' total payments.

We study games on finite graphs with labelled or weighted edges in which plays end as soon as the first loop is formed. The resulting play has a lasso shape: a path (head) followed by a cycle (loop). We consider two semantics for determining the winner: the loop semantics, where only the loop matters, and the lasso semantics, where the entire play is taken into account. We show that adopting the lasso semantics may have a dramatic impact on both the complexity of the game and the memory required to play optimally. Mean-payoff games with lasso semantics are no longer positionally determined, and the threshold problem – deciding whether the value of the game is at least 0 – is PSPACE-complete. In contrast, lasso parity games are equivalent to weak parity games, remain positionally determined and are solvable in polynomial time. Furthermore, we show that if a positionally determined winning objective is closed under cyclic permutation and both the objective and its complement are closed under concatenation, then it is equivalent to the parity objective. Finally, we consider mod-k games, where the winner depends on the length of the path modulo k >= 2. We show that these games, in both semantics, are PSPACE-complete. Alongside computational complexity, we analyze the memory required to play optimally. We establish that for any game positionally determined in loop semantics, memory of size 2^n is sufficient in lasso semantics. We complement this by showing exponential memory lower bounds for lasso mean-payoff and mod-k games.

One of the most well-known connections between logic and games is Pauly's representation theorem: the induced powers of individuals and coalitions in concurrent game frames correspond, in a precise sense, to a certain class of neighborhood frames. The precise sense here is what is called alpha effectivity (or alpha power): the power of a coalition is characterized by the sets of states in which it can ensure that the outcome lies by taking some joint action. This definition is inherently monotonic, and, as pointed out by van Benthem, Bezhanishvili, and Enqvist (2019), that fact can obscure relevant information about the power structure in the game: we do not know whether two sets that a coalition has the power to enforce correspond to the same joint action or to different joint actions. An alternative is to characterize the power of a coalition by its actual power (called basic power by van Benthem, Bezhanishvili, and Enqvist (2019)): the collection of sets of states, each of which corresponds to one joint action by the coalition and all possible joint actions by the other agents. It has recently been argued by Li and Ju (2025, 2026) that standard concurrent game frames rely on three assumptions that may, in some cases, be too strong: seriality, independence of agents, and determinism. This yields a total of eight different classes of general concurrent game frames. In this paper, assuming two agents, we prove that, in terms of actual powers, the eight classes of general concurrent game frames are represented by eight corresponding classes of neighborhood frames. This generalizes a result by van Benthem, Bezhanishvili, and Enqvist (2019).

We study the fair allocation of $m$ indivisible chores among $n$ agents in the agent-arrival model against an oblivious adversary. The chore set and the number of agents are known in advance. When agent $i$ arrives, her nonnegative additive cost function $c_i$ is revealed, and the algorithm must immediately and irrevocably assign her a subset of the unallocated chores. We focus on the notion of maximin share (MMS) and measure the fairness of the realized allocation $X$ by the ex-post MMS approximation ratio $\rho(X)$. We measure an algorithm's performance by its expected ratio, and also the guarantees that hold with high probability. For binary additive costs, we achieve an expected ex-post MMS ratio of $3/2+o(1)$, and show that every randomized online algorithm has an expected ratio of at least $3/2-o(1)$. For every $n$, the expected ratio is at most $1.755$. A variant of our algorithm guarantees an approximation ratio of $2$ with high probability, which is optimal. % For general additive costs, we present a weighted-rounding algorithm that achieves an expected ratio of $2+o(1)$, which also holds with high probability. The expected ratio is at most $2.594$ for every $n$. We also show that similar algorithms extend to additive goods, achieving a constant expected ex-post MMS ratio and guaranteeing every agent an $x$-fraction of her MMS with probability at least $1-85x^2$ for every $x\le1/15$.

We study the simultaneous achievement of share-based and envy-based fairness in the allocation of indivisible goods with nonnegative additive valuations. The AnyPrice share (APS) is a strengthening of the maximin share (MMS). A polynomial-time algorithm is known to guarantee a $2n/(3n-1)$ approximation of APS, but without an accompanying envy-based guarantee. Separately, allocations simultaneously satisfying minimum EFX share (MXS) and envy-freeness up to one less-preferred good (EFL), and subsequently the stronger requirement of residual maximin share (RMMS) together with EFL, have been shown to exist. However, the known constructions do not provide pseudopolynomial-time guarantees. We develop algorithms that combine these share-based and envy-based fairness requirements. First, for every $\varepsilon \in (0,1)$, we give an algorithm that computes an allocation satisfying exact EFL, $(1-\varepsilon)$-MXS, and $2n/(3n-1)$-APS. Its running time is polynomial in the binary input length and $1/\varepsilon$. Thus, we match the previously established polynomial-time APS approximation while simultaneously guaranteeing exact EFL and an arbitrarily close approximation to MXS. Second, we provide a pseudopolynomial-time algorithm that simultaneously guarantees exact MXS, exact EFL, and $2n/(3n-1)$-APS. Third, for every fixed number of agents, we obtain a pseudopolynomial-time algorithm that simultaneously achieves exact RMMS and exact EFL. Finally, we establish that computing a numerical threshold between MXS and RMMS is weakly NP-hard, even for additive valuations. This identifies a computational barrier to extending our polynomial-time approach to exact MXS through independently computed residual-valid share thresholds.

EIP-4844 gave Ethereum rollups a separate fee dimension for data, governed by a multiplicative update rule that targets three “blobs” per block. Whether this target is reached depends on how many independent, self-interested rollups choose to post data, a question that closed-form models address only for a single agent under exogenous prices or in a static equilibrium. We propose a calibrated multi-agent reinforcement learning environment for the blob fee market in which the blob base fee is endogenous, set by the EIP-4844 rule on the aggregated decisions of heterogeneous rollups fit to post-Dencun Ethereum data. After confirming that the environment reproduces the known single-agent analytical optimum, we show that independent learners, with no convergence guarantee, still settle near the target without coordination ($3.32\pm0.01$ blobs per block over five seeds). Over a fiftyfold demand range ($10\times$ to $500\times$), supply stays at the target from $10\times$ to $200\times$ and rises above it at higher demand, where the fee can no longer hold it. Posting frequency increases with rollup size, an emergent analog of the size threshold predicted by prior theory. A costless layer-1 calldata outside option pushes the fee down by roughly an order of magnitude while aggregate supply stays at the target, so the fee rule acts as a negative-feedback controller on decentralized supply.

Learning agents, whether humans or algorithms, adapt to one another rather than to a fixed environment. Selfish learning need not produce a good outcome for the group, especially when learners face a social dilemma. The iterated prisoner's dilemma (IPD) is the classical example, but it is usually studied in populations of agents who imitate successful strategies. Here, we instead study selfish gradient ascent learners on the two-dimensional space of reactive strategies. A reactive strategy is described by two continuous traits: reciprocity, the chance of cooperating after the co-player cooperates, and forgiveness, the chance of cooperating after the co-player defects. Each learner updates its strategy in the direction that most rapidly improves its payoff against all other learners, by an amount governed by its learning rate. We show that heterogeneity in learning rates can qualitatively change the long-term outcome: a population with uniform learning rates converges to always-defect (ALLD), whereas a population with variable learning rates approaches generous-tit-for-tat (GTFT), which produces nearly optimal collective payoffs. In simulations, every population passes through a transient state of minimal forgiveness, in which the distribution of reciprocity determines the outcome. Variable learning rates promote cooperation because slow learners preserve high reciprocity until the population reaches this state. We prove sufficient conditions on the distribution of reciprocity for convergence to ALLD or to tit-for-tat (TFT). Variable learning rates also allow cooperation to recover from perturbation, which is provably impossible under uniform learning. These results suggest that purely selfish and myopic learning can produce persistent cooperation provided the population has intrinsic variability in learning rates.

Language models are frequently aligned to human preferences via imperfect evaluators whose errors can be exploited during optimization. We train policies to search for comparisons on which an evaluator may reverse the underlying preference, providing candidates for additional supervision. Since human preferences are unavailable during policy training, we use disagreement among an ensemble of preference models as a proxy for prediction error. We formulate this search as a zero-sum Stackelberg game between two opposing policies, a Leader which tries to minimize disagreement, and a Follower which maximizes it. For a general class of finite contextual Stackelberg games, we establish equilibrium existence and, under regularization, uniqueness and a closed-form characterization. We introduce Stackelberg Mirror Descent-Ascent (SMIDA), a two-timescale algorithm with linear last-iterate convergence in the tabular setting, and develop a neural policy-gradient analogue. Experiments with a synthetic oracle support disagreement as an informative proxy for prediction error. On held out data the trained policies outperform reference policies and increase the fraction of comparisons that exceed a disagreement threshold and reverse the oracle's preference.

We study the relationship between fair division with $\textit{subsidies}$ and $\textit{best-of-both-worlds}$ (BoBW) fairness for indivisible items. We prove a general reduction of BoBW existence to envy-freeability. Specifically, for any scale-free ex-post property $P$ (e.g., EF1, EFX, PO) and any scale-free valuation class $\mathcal{V}$ (e.g, additive, submodular, subadditive, monotone), if every instance in $\mathcal{V}$ admits a $P$-allocation that can be made envy-free using subsidies, then every such instance admits an ex-ante envy-free lottery supported on $P$-allocations. The proof uses linear programming duality. As applications, the reduction gives alternative proofs of BoBW existence results for additive goods, chores, and mixtures of goods and chores. The converse of our general reduction holds for two agents but fails for three: We provide a four-good counterexample with monotone submodular valuations where an ex-ante envy-free and ex-post EF1 lottery exists, but no allocation is simultaneously EF1 and envy-freeable. For lexicographic valuations, we prove simultaneous EFX, Pareto optimality, and envy-freeability (and hence BoBW guarantees for EFX+PO) for three agents, and show that even EFX and envy-freeability can be incompatible for four agents.

Conversational assistants are becoming the place where people ask what to buy, and sponsored content is moving into their answers. Several advertisers then compete for one sponsored slot, and all of them read the same conversation, so their willingness to pay usually moves together. Existing auctions for LLM advertising treat these values as independent. We therefore present AffMyerson, an auction that captures this correlation and turns it into revenue. We model the advertisers' values given the conversation as a multivariate normal and turn the economic premise of positive dependence (affiliation) into a sign condition on the precision matrix. We design a neural parameterization whose every output satisfies it. We prove that every prior in this family yields a truthful, ex-post individually rational auction, so estimation error can cost revenue but never truthfulness. We derive the conditional virtual values in closed form. By the design of the prior, pricing an auction needs no matrix inversion and takes O(n^2) time, fast enough to run at query time. We also work out the log-normal family used in empirical ad-auction work. There we prove that the auction is truthful for every prior and revenue-optimal when the prior is correct, and that the virtual value is regular everywhere whenever the conditional spread is below a loose threshold, which our fitted priors stay well under. With LLM-agent bidders, AffMyerson earns 6.2% more than a second-price auction with a tuned reserve, 6.1% more than Ronen's lookahead auction with the same prior, and 9.2% more than Myerson's auction run as if values were independent. This gain is still 4.5% on real keyword search queries where willingness to pay is hard to read. The correlation comes from common demand. Agents that judge the conversation bid together and agents told to calculate do not. The gain from conditioning moves with the correlation.

We propose a new framework for banner ad auctions, leveraging diffusion models to allow for multiple bidders to co-win a single banner ad opportunity, and in particular influence the ad shown on the banner depending on the strength of their bids, enabling a far richer outcome space compared to standard banner ad auctions, where there is a single winner who shows their ad. In our framework, the outcomes are images (what ad to show) and each bidder expresses their preferences over outcomes by declaring their value per unit of alignment between the image chosen by the mechanism and a textual description of their desired image. Our proposed mechanism combines the advertisers' bids and textual descriptions into a score that drives the reverse process of a diffusion model. This is used to sample a set of candidate images from which the welfare maximizing one is eventually chosen. Extensive experimentation evaluates our proposed mechanism around the following axes: (1) efficiency improvements of our allocation rule relative to the single-winner allocation rule; (2) monotonicity of our allocation rule; and (3) approximate incentive compatibility when combining our allocation rule with Myerson lemma-based or Clarke rule-based payments. Improvements along all these axes are shown with the number of Monte Carlo samples. Meanwhile, these benefits are attained while preserving the image quality.

A predictor is multicalibrated with respect to a class of subpopulations (formally, of functions on the inputs, called auditors) if, for every subpopulation in the class and every value the predictor outputs, the average outcome among the members of that subpopulation who receive that value is close to the value itself. For finite classes, about $\varepsilon^{-3}$ samples are necessary and sufficient to learn a predictor with multicalibration error $\varepsilon$, but the algorithms that attain this rate enumerate the class, which is impossible for the large classes of interest. We ask how many samples suffice when the learner accesses the class only through an empirical risk minimization (ERM) oracle. We give an algorithm that uses $\tilde{O}(d/\varepsilon^{3})$ samples and polynomially many ERM calls, where $d$ is the VC dimension of the class. Its dependence on $\varepsilon$ is optimal for all learners. When $d$ is at least about $1/\varepsilon$, every learner that can only access the class through polynomially many ERM calls needs $\tilde{\Omega}(d/\varepsilon^{3})$ samples. Learners with unlimited computation need only $\tilde{O}(\varepsilon^{-3}+d\varepsilon^{-2})$ samples, so the extra factor $1/\varepsilon$ on the dimension term is the price of oracle efficiency. For swap multicalibration, which allows a different function from the class for each predicted value, $\tilde{\Theta}(d/\varepsilon^{3})$ samples are necessary and sufficient. For estimating the error of a given predictor, the ordinary notion is the harder one: the swap error can be estimated with an ERM oracle, whereas the ordinary error is NP-hard to approximate even for classes with linear-time ERM. Consequently, there are classes for which it is computationally easier to learn a multicalibrated predictor than to compute the multicalibration error of a given predictor.

The Mallows model is a well-established construct in preference modeling, describing random rankings through their Kendall-Tau distance from a reference permutation. Its compact parameterization and interpretable treatment of preference uncertainty have made it influential across statistics, machine learning, probability, and revenue management. In the latter domain, recent literature has studied the embedding of Mallows-driven choice into assortment optimization. Early approximation schemes in this context require stylized structural simplifications or restrictive parametric assumptions. In the absence of these levers, a highly involved quasi-polynomial-time approximation scheme has subsequently been proposed, leaving wide open the question of whether near-optimal assortments can be computed in true polynomial time. The main contribution of this focused technical paper resides in closing these gaps by uncovering a striking independent-utility structure within Mallows choices, and turning this representation into a polynomial-time approximation scheme. Our central structural characterization explicitly constructs a single collection of mutually independent utilities, reproducing Mallows choice probabilities simultaneously for all offered sets. This exact choice equivalence connects pending Mallows-related challenges to the recent approximation framework of Bowers et al. (FOCS 2025) for assortment optimization under independent random utilities. We make this connection algorithmically feasible through a finite-support reduction that produces distributions of polynomial size while controlling revenue loss uniformly over all assortments. Fused together, these ingredients lead to an $O(n^{\tilde{O}(1/\epsilon^2)})$-time approximation scheme for assortment optimization under the Mallows model, while allowing arbitrary product prices, concentration parameters, and modal rankings.

Turn-based deterministic games on graphs are two-player zero-sum games played on finite directed graphs, where the vertex set is partitioned between the Max and the adversarial Min Player. The respective player chooses outgoing edges from their vertices. We consider these games with the classical discounted-sum and mean-payoff objectives. The goal of the Max Player is to maximize the objective against the adversarial Min Player. These games lie in NP $\cap$ coNP and are among the rare combinatorial problems that belong to this complexity class, yet the existence of a polynomial-time algorithm is a major open question. All known deterministic algorithms require exponential time in the worst case, while the classical strategy iteration (policy iteration) algorithm is remarkably efficient in practice, which motivates the study of the smoothed complexity of these games. Previous results established a smoothed polynomial-time bound only for the restricted class of ergodic games and presented only a high-probability guarantee. Our main result is a new analysis technique for a variant of the strategy iteration algorithm, and we show these algorithms achieve smoothed polynomial complexity for general game graphs with both high-probability and expected runtime guarantees. In particular, our result removes the ergodicity restriction of prior work and strengthens the high-probability guarantee to the stronger expected-runtime guarantee, thereby resolving the open question on the smoothed complexity of these games.

Pandora's problem is a fundamental model of sequential search in which opening each box incurs a cost. We study the variant with deadlines, where each box has a publicly known deadline for both opening and selection. We prove that a simple threshold strategy achieves a $1/2$-approximation to the optimal expected utility, improving the previous guarantee arbitrarily close to $1/4$ due to Amanatidis et al. (2025).

We study a dynamic lending game with limited but reusable supply. Borrower arrivals and loan durations are stochastic. Demand is allocated cheapest-first, subject to available supply, so a lender's quote affects both its current payoff and its ability to serve future borrowers. Motivated by industry practice, we study this problem through the lens of information disclosure: Lenders observe either only their own available supply or also their rivals' aggregate available supply. We ask how providing this information to all, some or none of the lenders affects payoffs as demand varies relative to supply, and how these effects change when a fixed total supply is divided among different numbers of lenders. Interestingly, in the case of asymmetric information disclosure, total payoffs can increase, while total lending remains the same. We identify sufficient mathematical regimes in which this phenomenon occurs. Who receives supply information matters differently as demand rises. For the case of two lenders, informing only the larger lender yields higher total lender payoff at lower demand, while informing only the smaller lender can yield the highest total lender payoff at higher demand. For a platform deciding which lenders can observe rival supply, the results suggest that disclosure can change the fees paid by borrowers without changing how much is lent. Wider access to supply information need not lower those fees: its effect depends on who receives the information and the balance between demand and supply.

We study multiwinner voting with $n$ voters and $m$ candidates, where the goal is to select a committee of size $k$. Voters have unit-sum cardinal utilities but report only coarse information induced by $t$ utility thresholds. In particular, each voter partitions the candidates into approval sets, with the utilities of all candidates in the same set lying between the same pair of consecutive thresholds. We model the utility of each voter for a committee as a normalized monotone symmetric norm, capturing, among others, additive welfare, best-representative utility, top-$r$ welfare, $\ell_p$ welfare, and normalized non-increasing ordered weighted averages. We design a deterministic rule with distortion at most $2(2m)^{1/t}$ and a randomized rule with distortion at most $2\min\{m^{1/(t+1)},m/k\}$. These guarantees hold for every committee size $k$ and every norm in our framework. Consequently, a single threshold yields distortion $O(m)$ for deterministic rules and $O(\sqrt{m})$ for randomized rules, while $O(\log m)$ thresholds suffice for constant distortion. We complement these results with norm-dependent lower bounds that asymptotically match our upper bounds for constant committee sizes, including the single-winner case.

We revisit autobidding equilibria in GSP sponsored shopping auctions, recently proposed and studied in Dütting et al. On the existence side, we show that autobidding equilibria under both budget constraints (ABE-B) and return-on-investment constraints (ABE-R) always exist and, moreover, always admit equilibria with polynomial-size support over the allocation space. On the computational side, we show that computing an exact equilibrium is in FIXP, and that computing an approximate equilibrium is in PPAD. We also prove that computing an approximate equilibrium is PPAD-hard for both ABE-B and ABE-R. Together, these results provide a fairly comprehensive picture of the computational complexity of autobidding equilibria in GSP sponsored shopping auctions.

We study concurrent stochastic games in which a team of agents plays against an adversarial environment. If the agents could access a shared source of randomness, then these games reduce to two-player games. Despite being common in the literature on coalition games, this assumption does not accurately model real-world systems. In this paper, we consider the setting where the agents only have access to individual sources of randomness, and must satisfy a given reachability or safety objective. The limit-sure problem asks whether the team can ensure that the objective is satisfied with probability at least $1-\lambda$ for every $\lambda>0$. For reachability objectives, we show that this problem is $\mathsf{NP}$-complete. For safety objectives, we show that limit-sure winning coincides with almost-sure winning, which is known to lie in $\mathsf{P}$. As a corollary of these results, we conclude that model checking of the qualitative fragment of the logic Individually Randomised ATL – introduced in the recent work of Brice et al. – is $\mathsf{P}^{\mathsf{NP}}$-complete.