Papers

Can simple learning rules keep their regret bounded in self-play? Recent work achieves constant regret bounds through modified regularization and higher-order prediction. Yet for Optimistic Hedge, arguably the most canonical method in games, the best known individual regret bound remains logarithmic. In this work, we prove that plain Optimistic Hedge with a constant step size can attain $O_{n,d}(1)$ individual regret in general-sum games with $n$ players and $d=(d_1,\ldots,d_n)$ actions, under expected loss-vector feedback. As a corollary, its time-averaged play enjoys an $O_{n,d}(1/T)$ coarse correlated equilibrium (CCE) gap. Our analysis represents Optimistic Hedge as a real-analytic recurrence on a compact space, which yields an exact finite-order difference relation that eliminates horizon dependence. Our proof hinges on nonconstructive Noetherianity argument of Frisch (1967), so the $(n,d)$-dependence remains implicit.

A (coarse) correlated equilibrium (CE) is information-value-free (IVF) if a player can match the payoff obtained from recommendations by committing to a fixed action. Motivated by the problem of regulating algorithmic collusion, this refinement was introduced by Hartline, Wang, and Zhang [EC'26], who showed that it can be computed in polynomial time in explicitly represented normal-form game. In this paper, we examine the complexity of IVF(C)CEs in succinct games, which model more realistic strategic interactions that feature either many players or exponentially many pure strategies. We first show that computing an information-value-free CE is PPAD-complete in many-player polymatrix games or two-player Bayesian games, even when the approximation is a constant. We also prove an unconditional exponential query lower bound. Our results establish that IVFCEs are intractable, even in the centralized model, and rule out the existence of any efficient learning dynamics. This significantly strengthens the impossibility result of Hartline, Wang, and Zhang, which concerns a particular class of learning algorithms, and furnishes strong computational critiques of recent regulation on algorithmic collusion. To sidestep these hardness results, we examine the complexity of information-value-free CCE. Certain no-regret algorithms—such as regret matching or FTRL—provide a fully polynomial-time approximation scheme (FPTAS) for this problem. The complexity when the approximation is exponentially small turns out to be nuanced. On the one hand, leveraging no-regret dynamics, we establish membership in $\text{CLS} = \text{PPAD} \cap \text{PLS}$. On the other hand, we show that it is at least as hard as the P-matrix linear complementarity problem, and hence as hard as simple stochastic games. This shows that even IVFCCEs are unlikely to admit a polynomial-time algorithm barring a major breakthrough.

We study single-agent combinatorial contracts under linear payments. Under a reward share $\alpha\in[0,1]$, an agent chooses a subset $S$ of $n$ hidden actions, generating reward $f(S)$ at cost $c(S)$, to maximize $\alpha f(S)-c(S)$, while the principal receives $(1-\alpha)f(S)$. For nonnegative additive rewards and monotone supermodular costs, Dütting et al. (SODA 2026) proved an exponential supply-query lower bound for exact optimization and left open whether a representation-independent approximation is possible. We resolve this question in a stronger form, removing all structural assumptions on the cost function: for nonnegative additive rewards, arbitrary normalized nonnegative set costs, and every $\varepsilon\in(0,1)$, we give a deterministic $(1-\varepsilon)$-approximation using $O(n\log(n+1)/\varepsilon)$ supply queries, with query complexity independent of numerical bit lengths and breakpoint separation. More generally, the same guarantee and asymptotic query complexity hold for normalized monotone subadditive rewards and arbitrary normalized nonnegative costs under exact best-response and value-query access. The key idea is a cost-independent, reward-side scale certificate. The algorithm tries all singleton rewards as candidate anchors; one of them brackets the optimal retained share within a factor $n^2$. Geometric search and monotonicity of the induced response reward then yield the approximation without enumerating best-response breakpoints.

Assortment optimization asks a seller which of $n$ products to display to each customer. Under the multinomial-logit (MNL) model, a revenue-maximizing policy may concentrate most purchases on only a few products. Balanced market share (BMS) limits this disparity by requiring every product to receive either zero aggregate sales or at least an $\alpha$-fraction of the largest product's sales. We study randomized BMS under a per-assortment cardinality constraint: every displayed assortment contains at most $K$ products, although the seller may rotate a larger catalog across customers. This couples per-realization feasibility with aggregate balance over an exponential policy space. We resolve the model on all three algorithmic fronts. First, reweighting assortments by their MNL denominators turns the cardinality restriction into a single uniform-matroid rank inequality, yielding an exact compact sales-space formulation in which every feasible sales vector can be implemented by a policy supported on $O(n)$ assortments. Second, deciding whether there exists a feasible policy attaining a given revenue target is NP-complete even for $\alpha=1$ and $K=2$, whereas optimization is polynomial-time solvable once the set of products with positive sales is prescribed. Third, despite this hardness, BMS confines all positive sales to one multiplicative band, and a geometric scale search combined with a standard multiple-choice-knapsack dynamic program yields an FPTAS. For every $\varepsilon\in(0,1)$, it achieves at least a $(1-\varepsilon)$ fraction of the optimal expected revenue, preserves BMS and per-assortment cardinality exactly, and outputs a rational policy supported on $O(n)$ assortments.

Fair division of indivisible bads seeks allocations that guarantee every agent a bundle whose cost is no larger than a meaningful fairness benchmark. The canonical minimax share has widely been used; unfortunately, it is not a simultaneous guarantee. Hill (Ann. Probab., 1987) initiated a complementary approach in which the share depends only on the number of agents and the largest possible single-item value. Li et al. (ACM Trans. Econ. Comput., 2024) gave the exact Hill formula and proved that its monotone closure is a simultaneous guarantee. The closure treats the largest-item cost only as an upper bound and can be strictly larger than the share conditioned on the actual largest item. They asked whether this smaller exact share is itself simultaneously guaranteed for three or more agents. We resolve this open question affirmatively. For any number of agents and arbitrary heterogeneous largest-item costs, there is one allocation that satisfies every agent's exact Hill's share. We further provide a polynomial-time algorithm computes such an allocation. Our algorithm combines an ordered moving knife with a tail-domination invariant, and a one-sided trimmed subset-sum routine for the two-agent endpoint without computing an exact minimax partition.

We study online allocation with convex costs (\OACC), a generalization of limited-supply models in which additional resources can be produced dynamically at convex cost. Prior work largely focuses on constructing a single algorithm that achieves strong competitive guarantees. In contrast, our main contribution is a structural characterization of the optimal design space of reserve functions (i.e., normalized pricing rules) for \OACC, yielding a family of optimal online algorithms. Using a principled dynamical-systems approach, we show that optimal online algorithms arise as solutions of a nonlinear eigenvalue problem: the optimal competitive ratio corresponds to the dominant eigenvalue, and the associated eigenfunction determines the optimal reserve function. This perspective establishes the existence of infinitely many optimal designs attaining the best possible competitive ratio, thereby unifying and generalizing a broad class of prior algorithms. Beyond unification, our characterization uncovers new structural phenomena, including the ability to flexibly mix dynamic and static pricing without loss of optimality and the construction of universally competitive algorithms under unknown or stochastic costs. We further extend the analysis to more structured settings such as supply-oblivious arrivals and hard supply constraints, obtaining sharper guarantees that improve upon previous bounds.

Strategic classification studies how a decision maker should choose a classifier when the agents being classified can adjust their features in response to it. In existing models, the classifier is the only instrument available to the decision maker, and therefore a feature that is predictive but easy to fake can only be down-weighted or discarded. However, in many settings the decision maker can also verify: lenders verify income, admissions offices check documents, and tax authorities audit returns. In this paper, we propose a model of strategic classification in which the firm jointly designs a linear classifier and an audit profile, which assigns to each fakeable feature a probability of detection and a penalty when caught. We show that under linear costs, the classifier affects the audit problem only through the distribution of gaming rents it induces, so that the joint design problem decomposes into the choice of a score rule and an audit allocation problem. We use this decomposition to characterize the optimal audit allocation, to identify when the allocation problem is tractable and when it is NP-hard (namely, when agents can game through overlapping features under an inspection cap), and to bound the regret of a firm that has to learn the rents by auditing. We further show that audit intensity is a quantity to be tuned rather than maximized: welfare is single-peaked in it, and a firm and a social planner disagree on the mix of detection and penalty that delivers a given level of deterrence. Notably, two populations with identical costs and causal structure can game in one domain and improve in the other, a difference that a cost-only model cannot account for. Together, our findings highlight that whether a feature is “gameable” depends on the institution's verification policy as much as on the feature itself.

Dashboards, feeds, and leaderboards determine whom agents believe they are playing against. We study how such information architectures shape strategic behavior, and how a designer who controls them can use visibility as a lever. We propose a behavioral model in which a transparency parameter governs how selectively agents are exposed to opponents of nearby reasoning depth, and embed it in a Cognitive Hierarchy environment, which provides a tractable language for tying agents' beliefs about the sophistication of others to what they see. We show that because the exposure operator is an exponential tilt of the type distribution, the effect of transparency on aggregate behavior has a closed form: the derivative of the aggregate action with respect to $p$ is proportional to the population-average variance of perceived sophistication. This identity signs the effect in games of strategic complements (opacity raises aggregate effort) and in coordination environments (opacity lowers the mean action), and an ordinal version holds for any response rule that is monotone in the belief. Notably, under a Poisson prior, opacity acts as a sophistication multiplier: every reasoner plays as if the population were $\mathrm{Poisson}(\tau/p)$. We use this structure to characterize the disagreement-versus-level tradeoff faced by a coordination-minded designer, to identify the reasoning depths that respond most to visibility, and to show that opacity imposes a “sophistication tax” under which the deepest reasoners can lose to moderate ones. Simulations in which agents see only a noisy leaderboard of published actions reproduce both the reversal and the tax. Together, our findings highlight that who an agent sees, and not only how deeply it reasons, shapes strategic behavior, and point to visibility as a design lever.

Models of opinion dynamics on networks, provide a framework to study how a network of agents aggregates dispersed opinions into a consensus. However, existing models assume that agents truthfully report their opinions, and do not account for environments in which a platform aggregates agents' reports across multiple networks, and agents can strategically report different opinions on each. In this paper, we propose a model of strategic opinion manipulation on multiplex networks, in which a platform merges L network layers with attention weights, and each agent, holding a private opinion, chooses (potentially different) reports on each layer, at a layer-specific misreporting cost, so as to pull the merged consensus toward their own opinion. We show that this game has a unique Nash equilibrium in closed form, that the resulting consensus is the truthful consensus under a centrality tilted toward a manipulability index of each agent, and that the resulting distortion is the (centrality-weighted) covariance of agents' manipulability and opinions. We further show that agents' reports on every layer are more extreme than their opinions, so that the polarization observed by the platform overestimates the true polarization. Notably, we highlight that merging layers is a double-edged sword: as manipulability depends on the square of an agent's influence on each layer, spreading attention across layers dilutes manipulation, while heterogeneous misreporting costs across layers and shifts in agents' centralities can amplify it. Finally, we characterize the platform's optimal choice of attention weights, in closed form when the layers share a stationary distribution, and through an exact gradient and a marginal test otherwise. Together, our findings shed light on when aggregating opinions across networks is (not) robust to strategic manipulation, and point out potential interventions to alleviate it.

We introduce Multiplicatively Optimistic Regret Matching (MORM), an uncoupled learning rule for finite general-sum games. Under simultaneous full-information self-play, every player achieves external regret $O(\sqrt n\log d)$ uniformly over all horizons, using only one-step optimism. The analysis combines a potential-based regret-matching argument with multiplicative stability and Hellinger control of strategy movement. A learning-rate safeguard additionally gives $O(\sqrt{T\log d})$ regret in the face of adversarial utilities.

In recent years, quantum game theory has gained significant attention as a framework for studying decision-making in multi-agent systems using quantum principles. However, computing equilibrium strategies is challenging because the dimension of the joint Hilbert space grows as the product of the players' local dimensions. In this paper, we consider an extended Gutoski-Watrous (EGW) game in which each player's quantum strategy is represented by a local density matrix. We derive tensor-contraction expressions for the payoff functions and their gradients, thereby avoiding the explicit construction of the full joint density matrix and its computationally expensive multiplication by the payoff operators. Building on the resulting effective Hamiltonians, we propose the Matrix Exponential Fixed-Point Iteration with Annealing (MEFPIA) algorithm to search for equilibrium points in EGW games. We compare MEFPIA with the Matrix Multiplicative Weights Update (MMWU) algorithm in terms of convergence. For the tested instances and parameter settings, both algorithms approach the same strategy profiles and payoffs, while MEFPIA achieves lower relative error in fewer iterations. These results indicate that MEFPIA is a promising numerical method for equilibrium search in multi-agent quantum games. Our findings provide important insights into the quantum game theory's potential for addressing complex decision-making processes, as well as opening up new paths for future research and exploration in multi-agent quantum systems.

We initiate the study of $\alpha$-fair prophet inequalities. This interpolates between utilitarian welfare $(\alpha=0)$, Nash welfare $(\alpha=1)$, and Rawlsian max-min fairness $(\alpha\to\infty)$. Given the non-linearity of the objective, it matters when the expectation is applied. For instance, for the Rawlsian objective, it matters whether we aim to maximize $\min \mathbb{E}[u_i]$ or $\mathbb{E}[\min u_i]$. We refer to the former as the ex-ante model, and the latter as the ex-post model. For ex-ante fairness, full distributional knowledge yields a tight competitive ratio of exactly $1/2$ for every $\alpha\ge 0$. Under sample access, $O(n\log n)$ samples per distribution suffice for a constant competitive ratio when $\alpha\in(0,1]$. In contrast, for every $\alpha>1$, no finite number of samples improves upon the trivial $1/n$ guarantee. Thus, unlike in the utilitarian setting, full-information and sample-access prophet inequalities become fundamentally separated. For ex-post fairness, under full information, we obtain a uniform constant ratio for all $\alpha\in(0,1)$, while for every $\alpha>1$ the competitive ratio collapses to $1/n$. In the sample-access model, one sample per distribution suffices for each fixed $\alpha<1$, but no sample budget depending only on $n$ yields a uniform constant guarantee as $\alpha\to 1$. Beyond these phase transitions for $\alpha$-fairness, our results open the door to a broader theory of prophet inequalities for non-linear welfare objectives.

Cross-platform mobile frameworks promise code reuse, shorter delivery cycles, and lower implementation effort, but their trade-offs relative to native development remain difficult to assess objectively. Many comparisons rely on simplified applications, inconsistent feature sets, or a narrow set of metrics. This paper compares five implementations of the same plant-management application: native iOS, native Android, Flutter, React Native, and Kotlin Multiplatform. The shared approaches target both Android and iOS, yielding eight executable variants. Guided by ISO/IEC 25010, the study examines time behavior, implementation footprint, source-code organization, and observable rendering responsiveness. All implementations share the same domain, backend services, functional requirements, and benchmark contract. The supplied dataset contains 2,000 completed runs per variant and covers authenticated and cached retrieval, image transfer and decoding, list rendering and scrolling, local synchronization, and media upload. Backend preparation and framework-specific UI drivers are analyzed separately from the primary client workflow. Native records the lowest non-UI client subtotal on both operating systems, with Kotlin Multiplatform the closest cross-platform implementation, while operation and UI-wrapper rankings vary by task. The shared approaches contain less authored mobile source than the native applications combined, and the source inventory shows different patterns of file size, organization, and dependency use. Rather than identifying a universally superior technology, the study shows that each approach's advantages and costs depend on the quality attribute, workload, platform, and measurement boundary.

In strategic interactions, deterrence is an outcome where one party incentivizes the other party to not participate. This concept can be applied to Colonel Blotto games between two generals where troops can be deterred from following orders by the actions of the opposing general. In this setting, deterrence depends not only on how generals allocate troops, but also on the information available to troops when they decide whether to follow orders or defect. In this work, we study a two-battlefield Colonel Blotto model of deterrence under multiple information structures, ranging from decisions made with only aggregate information to decisions made after local battlefield conditions are observed. We show how different information structures alter the structure of the induced game, affect the existence of equilibria, and substantially change expected utilities. Our findings also prove that the ratio of troops available to each general combined with a capture probability threshold is enough to inform generals about what information they want their troops to have access to.

This article introduces mutual evaluation of a replicable task worker and a critic that incentivizes truthful reporting, both modeled as strategic agents. The critic chooses a finite-valued rule that induces an evaluation score on joint report laws. Their common payoff is analyzed through regret relative to the unrestricted critic envelope. The critic rule is distinct from the evaluation score. This class enables a peer-free information elicitation mechanism using conditionally independent replications of a worker on the same task. This replication-loop mechanism implements a type-agreement payoff using same-task replications and new-task samples. In contrast to the peer-prediction and scoring-rule literature, implementations are shown that produce unbiased Pearson and Shannon information scores without requiring peers, a ground-truth reference, or likelihood-ratio estimation. A valid binary critic also can be represented by shared finite type annotations of worker returns. One runtime restriction is that the number of required replicas is random and can depend on the critic rule. Other timing effects, such as commitment and reoptimization, yield distinct incentives, connecting the framework to variational peer prediction. This mechanism class illustrates why strategic considerations matter for both critic and worker agents.

We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $\Omega(q/\log q)$ lower bound on the competitive ratio. Together with the known $O(q)$ upper bounds, this resolves, up to a logarithmic factor, the optimal dependence on $q$, an open question posed by Correa, Cristi, Fielbaum, Pollner, and Weinberg (IPCO 2022) and Saxena, Velusamy, and Weinberg (ITCS 2023). Our construction also yields an $\Omega(d/\log d)$ lower bound for $d$-single-minded auctions, where buyers request fixed bundles of at most $d$ unit-capacity items. Our construction and analysis build on the "big-decisions-first" framework of Rubinstein and Singla (STOC 2026).

PJR$^+$ is a polynomial-time verifiable proportionality axiom for approval-based committee elections, but its known polynomial-time verification procedure relies on general submodular-function minimisation. We show that its objective is a maximum-closure problem and give a direct mincut formulation of the problem on a bipartite graph. Using an almost-linear-time maximum-flow algorithm, this yields an $\mathcal{O}(m(nk)^{1+o(1)})$-time verifier, where $n$, $m$, and $k$ are the numbers of voters, candidates, and committee members, respectively. The dependence of this bound on each parameter separately is almost linear: it is linear in $m$, and almost linear in $n$ and $k$. The verifier also returns an explicit group witnessing a violation and admits a slower but immediately implementable variant based on the preflow–push mincut algorithm. Finally, for the parameterised axiom $\alpha$-PJR$^+$, where $\alpha$ is used as a multiplier in the group size, we demonstrate how to compute the largest value of $\alpha$ for which a committee still fails the axiom using this mincut formulation.

This paper studies resource-limited external influence in time-varying opinion networks when a controller must choose both a small set of agents and a scalar intervention at each update. We use the affine free response, which includes DeGroot and Friedkin–Johnsen dynamics, followed by a direct sparse action. Eliminating the scalar action reduces the one-step problem to a cardinality-constrained support objective whose global optimum is obtained from the largest or smallest residual components. Hence exact sparse one-step-ahead optimal control requires one sort and \(O(|\mathcal T|\log|\mathcal T|)\) operations, where \(\mathcal T\) is the support set. For periodic affine dynamics, the resulting state-dependent low-rank feedback admits cycle-based ultimate bounds and exact tracking of model-invariant targets. The DeGroot and Friedkin–Johnsen specializations expose the role of target invariance and the affine mismatch term. With several competing external players, each sparse best response retains the same sorting structure and the complete binary–continuous stage game is an exact potential game, so a pure-strategy equilibrium exists at every frozen state. Same-state welfare benchmarks separate loss due to scalar competition from the additional loss due to strategic support selection. Numerical examples validate exact support selection and illustrate switching-order effects, DeGroot/Friedkin–Johnsen tracking, resource tradeoffs, and competitive implementations.

Agricultural soils are a major untapped carbon sink. Carbon farming is emerging as a promising practice for tapping this potential. Smallholder farmers, who dominate agriculture across South Asia and sub-Saharan Africa, are key to scaling climate mitigation via carbon farming. It is ironic that real-world carbon programs largely fail to reach them. We study this important gap through the lens of contract design. An aggregator offers a single pooled contract to a heterogeneous population of smallholder farmers who have private adoption costs (adverse selection) and exert unobserved effort (moral hazard), with agronomic outcomes evolving over multiple seasons. We formulate this evolving contracting problem as a POMDP and use reinforcement learning to learn a dynamic profit-maximising contract. We analyse the performance of the aggregator under various conditions. We find that a profit-maximising aggregator does not merely inherit the exclusion of smallholders, it amplifies it. On large farms the aggregator realises 87.7% of achievable adoption, against only 8.2% on smallholdings. Per-hectare Measurement, Reporting and Verification (MRV) costs fall as farm size rises, and the aggregator's pooling contract compounds this gradient rather than offsetting it. A counterfactual that makes MRV costs purely area-proportional eliminates this disparity. Our results and simulation can guide contract and policy design that opens carbon income to smallholders while enabling agricultural soils to contribute to climate mitigation at scale.

Exploration in centralized serial-dictatorship matching bandits must use complete matchings, so learning one player-arm pair can impose regret on others. We study this externality under a known common priority order and Gaussian rewards with unit variance. We show that the matching-level Graves-Lai constraints reduce to finitely many pairwise exploration quotas and, at top-choice-separated instances, yield a polynomial-size marginal linear program. At these instances, the exact attainable set of expected logarithmic regret coefficients is $G(\theta)\mathcal{X}(\theta)$, where $\mathcal{X}$ is the feasible matching-allocation set and $G$ maps allocations to player regret. The usual upper-closed Graves-Lai region can be strictly larger despite having the same Pareto-minimal boundary. We further show that identical exploration quotas can induce very different regret through their scheduling. Finally, we construct estimate-solve-track policies, uniformly good on the full row-strict class, that attain every fixed positively weighted optimum without assuming optimizer uniqueness. Every Pareto-minimal point is pointwise attainable, possibly through an instance-calibrated target.

Regularized self-play – the family behind DeepNash's Stratego play – drives a two-player zero-sum policy to a Nash equilibrium by best-responding to a slowly moving, entropy-regularized reference policy $\rho$. When the game has a polytope of value-equivalent equilibria, the regularizer silently breaks the tie: with a uniform reference it selects the maximum-entropy member, the I-projection of $\rho$ onto the Nash set. Can the reference be used to choose the equilibrium on purpose? On five exactly solvable games plus a 2-D polytope, with exact best responses and equivalence tests over independent seeds, anchoring the reference at a target member and refining steers self-play to that member with mean coordinate error 0.007 at median exploitability $5\times10^{-5}$, TOST-equivalent to the request within $\pm0.05$; the anchoring persists through refinement and follows the reference, not the initialization. Selection follows the reach-weighted I-projection (slope 0.969 [0.950, 0.987]). We report with equal emphasis where the story breaks: fixed off-manifold references cost 0.08-0.25 exploitability; stiff or flat families require a smaller mirror step, set by a pre-registered rule; boundary targets undershoot; curvature predicts where boundary saturation bites (rank correlation 0.90, p=0.037) while interior precision is curvature-independent. Table and MLP steering maps are equivalent within $\pm0.03$ at every target (30 seeds); matched control arms show attention's robust signature is excess seed variance, any systematic shift bounded at 0.018 and not significant. Against a best response the selection-robustness trade-off is degenerate: steering matters only against fixed, non-equilibrium opponents. The recipe – anchor the reference at the desired member and refine – reinterprets the KL anchor of RLHF-style RL as a selection knob, not only a stability leash.

We study a Stackelberg model where two players compete in a sequential contest, and the player that successfully invests the most effort wins. Our formulation considers a probability that each player fails to invest their chosen level of effort, as captured by a failure function that increases with the amount of effort. The players each have their own cost of failure, as well as a maximum level of effort (capturing different abilities of the players). We prove three key results. First, we provide the optimal strategy for Player 1 (the follower), to maximize their utility given an observed effort exerted by Player 2 (the leader). Next, we characterize the optimal strategy to maximize the expected utility of Player 2, anticipating the optimal response by Player 1. Finally, we analyze the expected utilities of each player when both play their optimal strategy, and provide key insights into how heterogeneous player abilities and failure costs affect the outcomes of the game.

Can simple no-regret dynamics attain smaller regret in self-play than against arbitrary adversaries? In $n$-player general-sum games, Daskalakis et al. 2021 proved an $O(n\log d_i\log^4 T)$ individual regret bound for Optimistic Hedge, which improves upon the classical $O(\sqrt T)$ adversarial regret bound. In this work, we show that Optimistic Hedge with a constant step size can further achieve $O(\sqrt n\log d_i\log T)$ individual external regret under expected loss-vector feedback. The time-averaged play consequently enjoys a coarse correlated equilibrium gap $O(\sqrt n\log d\log T/T)$, where $d=\max_i d_i$. The improvement comes from a larger admissible step size $\eta=\Theta(1/(\sqrt n\log T))$. Our analysis proves factorial bounds on high-order differences of probability-weighted pairwise loss gaps, then applies finite-difference interpolation in a fixed Euclidean norm. These estimates sharpen the analysis of Daskalakis et al. 2021 and yield a logarithmic regret bound.

The Kemeny rule aggregates multiple strict rankings into a single strict ranking that minimizes the sum of its distances from the input rankings. The resulting optimization problem, called the Kemeny problem (KP), is a special case of the linear ordering problem (LOP). The Kemeny rule satisfies several desirable properties in social choice theory, including the extended Condorcet criterion (XCC). Ando et al. strengthened this result by introducing the strong Condorcet criterion (SCC) and showing that it holds for every optimal solution to an arbitrary LOP instance. Yoo and Escobedo extended the Kemeny rule to rankings with ties and showed that the resulting rule satisfies the non-strict extended Condorcet criterion (NXCC). This criterion gives a condition under which one candidate must be ranked strictly above another in every optimal solution. In this paper, we introduce the non-strict strong Condorcet criterion (NSCC), a counterpart of the SCC for rankings with ties, and show that it holds for every optimal solution to an arbitrary instance of the linear ordering problem with ties (LOPT). We also establish a complementary structural property that gives conditions under which two candidates must be tied in every optimal solution to an arbitrary LOPT instance.

Solvency games are a gambling problem on infinite-state Markov decision processes in which the state $n \in \mathbb{N}$ represents an investor's fortune. In every round, the investor chooses an action from a finite action set, and every action yields a distribution over integer-valued gains in an interval $\{-\ell,\ldots,m\}$. The risk-averse investor wants to minimise the probability of eventual ruin (reaching a fortune $\le 0$). It was shown in [Berger et al.] that memoryless deterministic optimal strategies exist, but they are not eventually constant in general. Even in the special case of gains in $\{-2,\ldots,1\}$, the optimal strategy may need to make use of two different actions at arbitrarily high fortunes. We show that optimal strategies in solvency games need not be ultimately periodic in general (thus disproving a 2012 conjecture of Kučera). Already in the case of gains in $\{-3,\ldots,1\}$, it is possible for the optimal strategy to be unique but aperiodic. For gains in $\{-2,\ldots,1\}$, there always exists an ultimately periodic optimal strategy whose tail is constant or alternates between two actions. Finally, we show that the optimal strategy is computable if it is unique. Moreover, (some) optimal strategy can always be computed in the case of gains in $\{-\ell,\ldots,1\}$ for any $\ell \in \mathbb{N}$. Computability in the general case however remains open.

Game-theoretic analyses of cyber defence often compute equilibria of games whose payoffs exist only as the output of a simulator. Iterative equilibrium-finding methods grow a set of attacker and defender policies and need the payoff of every attacker–defender pair, so they are bottlenecked by payoff estimation: each payoff costs many simulator runs. We introduce Regret-Weighted Payoff Sampling (RWPS), which spends a fixed simulation budget on the payoffs the equilibrium actually depends on and predicts the rest with a model trained on every payoff measured so far. Standard error bounds for estimated games are driven by the worst-estimated payoff, so they cannot credit an estimator that is inaccurate only where accuracy does not matter. We prove a bound that weights payoff errors by the opponent's equilibrium strategy, a certificate that can be computed from simulated payoffs alone, and a condition under which errors in the predicted payoffs cannot change either player's regret. On three synthetic general-sum games, one of them a Colonel Blotto game of military resource allocation, the new bounds are four to six times tighter than the standard one, and RWPS finds less exploitable equilibria than minimum-regret-first search, information-gain search and progressive sampling at the same budget. On two cyber-defence simulators, CyGym and a new game whose hosts are LLM agents exposed to prompt injection, it gives the least exploitable equilibria at the smallest budgets.

We study deterministic strategyproof aggregation in finite-dimensional Euclidean spaces. For every odd number $n\ge3$ of agents and every finite dimension, we prove that the coordinate-wise median minimizes the worst-case approximation ratio for total Euclidean distance among all continuous, anonymous, deterministic strategyproof mechanisms. The same optimality result holds for every even $n\ge4$ when each coordinate uses a fixed choice of the lower or upper middle rank. The proof combines a rigidity theorem with a normalization that does not increase the approximation ratio: any hypothetical mechanism outperforming the median has a normalized representative that is a fixed coordinate-wise order-statistic rule in a single orthonormal frame. A reflection argument then shows that no such rule improves on the median.

We investigate the asymptotic behavior of max-min fair allocations for indivisible goods under i.i.d. random utilities. For $N$ agents and $K$ goods with utilities ${\mU_{i,j}}$ drawn independently from a common distribution $F$, we derive asymptotic characterizations of the max-min value in the balanced case $K=N$ (and in $K=LN$ extensions) via distributional quantiles. We then study the efficiency impact of max-min fairness by comparing the resulting total welfare with the optimal sum welfare. For distributions with sufficiently light tails, we prove that the relative efficiency loss converges to zero as the market grows, implying that max-min fairness incurs negligible welfare loss in large random instances for a broad class of distributions.

We study continuous-time fictitious play in 2-bidder, symmetric first-price auctions with independently distributed discrete values and a discrete bid set. We first exhibit a minimal instance — two bidders, two values, three positive bids — on which fictitious play with the standard uniform-split tie-breaking rule does not converge to the symmetric Bayes–Nash equilibrium: the equilibrium is unstable and the dynamics converge to a stable limit cycle far from the Nash equilibrium. We then show that a small modification of the tie-breaking rule — awarding a payoff of zero to every bidder in case of a tie — restores convergence: fictitious play converges to a Nash equilibrium of the modified game. This limit is an $\epsilon$-equilibrium of the original auction in a broad range of settings.

We study state-adversarial Markov decision processes (SA-MDPs) as games of observation-space attacks: at each step, an agent selects an action from a received observation while an adversary$\unicode{x2014}$who knows the true state the agent is in$\unicode{x2014}$chooses a perturbed observation within a state-dependent proximity set. While existing work focuses on Markovian policies, we develop a solution concept and computational approach for SA-MDPs under history dependence. History dependence can materially change equilibrium outcomes and can force both the agent and the adversary to adapt their strategies. First, we prove the non-existence of universal (agnostic of the initial state distribution) history-dependent equilibrium policies. Our main result presents the first algorithmic route to computing $\epsilon$-approximations of initial-state dependent equilibria. We do so by reducing SA-MDPs to a strategically equivalent constrained zero-sum one-sided partially observable stochastic game. We test our algorithm on small analytically verifiable games and show that it scales to larger, more realistic benchmarks, including Atari Freeway rollouts with a 12-period ahead horizon.