Classification of matches played in the last rounds of sports competitions is a well-established tool for evaluating tournament designs. Both deterministic and probabilistic approaches are available for this purpose. Our paper offers the first comparison of them by analysing the most prominent example of four-team round-robin competitions, the group stage of the FIFA World Cup. We show that both methods are highly relevant in practice: all (four) deterministic and (six) probabilistic match types occurred in the 2014 and 2018 FIFA World Cups, respectively. The probabilistic model, which accounts for the relative benefits of attacking and defending, provides deeper insights; for instance, the competitive matches from the deterministic approach can be of any of the six probabilistic types. Finally, the probabilistic framework is used to quantify and decompose the impact of the main reforms introduced for the 2026 FIFA World Cup: the expansion to 48 teams, as well as the modified qualification and tie-breaking rules.

The energy sector is a cornerstone of national strategic autonomy, yet its increasing financialization has transformed ownership structures into complex networked configurations. This paper investigates the distribution of economic power in the Italian energy sector by introducing two sector-level extensions of the Network Power framework: the Aggregate Network Power Index (A-NPI) and the Aggregate Network Power Flow (A-NPF). Unlike traditional macro-level measures, these indices aggregate firm-level control and influence into a systemic framework that accounts for the relative economic weight of each operator. Applying this framework to the Italian case reveals a "Governance Paradox": while the State retains formal majority ownership, the sector's deepening reliance on global capital markets and the pervasive presence of common ownership by transnational institutional investors have progressively hollowed out public strategic direction. The results show that capital centralization enables global financial actors to internalize sectoral competition, fostering a regime of tacit strategic convergence in the management of critical infrastructure. This configuration challenges European strategic autonomy, raising questions about the adequacy of traditional Foreign Direct Investment (FDI) screening and antitrust tools in addressing the systemic influence exerted through networked ownership structures.

Many selection problems are multilayered: agents first decide whether to participate and then sort among ordered or unordered categories. This paper shows that the sorting layer changes the geometry of identification. Unlike binary selection, in which selection bias can be summarized by a scalar control function, ordered and multinomial sorting generally produce multi-index control functions whose dimension determines the continuous covariate variation needed for identification. I establish matched non-identification and point-identification results for both architectures, showing how nonlinearity in the selection structure can substitute for excluded variables. I also show how additional structural restrictions reduce the control-function dimension and make estimation practical. I propose root-n-consistent two-step sieve plug-in estimators and apply the framework to gender wage gaps among Korean college graduates. Accounting for sorting reshapes the entry-level gap along the firm-size margin, where the corrected female coefficient turns positive for large-firm employment.

Generative artificial intelligence (GenAI) is the first automation wave to reach high-cognitive tasks at scale, yet its effects on intra-urban inequality remain largely unknown. Using 5 million job postings from Beijing (2018--2024), we construct a neighborhood-level GenAI Exposure Index by aggregating task-level assessments from five leading large language models. We examine the spatial, structural and causal mechanisms of this shock. We find that GenAI exposure is highly concentrated in the city's core districts, deepening the intra-urban AI divide. Since 2023, high-exposure neighborhoods have experienced wage stagnation even as they continue to attract high-skilled workers -- a "high-skill trap." This wage penalty is driven by task de-skilling and intensified labor-market crowding. A difference-in-differences design centered on ChatGPT's release supports a causal interpretation. These findings challenge the prevailing theory of skill-biased technological change and provide a basis for inclusive AI governance in global technology hubs.

The estimation of causal effects using quasiexperiments often relies on the use of unusual or serendipitous sources of exogenous variation. When the goal is estimating the same causal effects across many different settings, the same unusual exogenous variation often does not exist in all settings, and the only available form of identification is selection-on-observables, which relies on a conditional indepdendence assumption. Partial identification is especially valuable in this context, as it allows conditional independence to not hold perfectly. This paper proposes a method that sharpens the jointly identified set of causal effects across many settings by making use of unobserved relationships between omitted variable biases across settings.

We study whether adoption of an AI coding assistant causally expands the technological frontier of individual software developers. We exploit the staggered rollout of Claude Code across GitHub between May 2025 and January 2026 in a panel of 5,838 developers observed monthly over 28 months, with treatment defined by the developer's first Claude-co-authored commit and not-yet-treated developers as controls. Using the doubly robust Callaway and Sant'Anna (2021) estimator, we find positive and significant effects on monthly commits (+41), repositories contributed to (+1.5), distinct programming languages used (+0.83), Shannon language entropy (+0.14), newly-used languages (+0.31), and cumulative lifetime languages (+0.51). The cumulative-languages effect grows with time since adoption, matching a Bayesian-learning model in which AI provides free signals about unfamiliar technologies and lowers the switching barrier. Results are robust to two stricter activity filters. The estimates document a sharp, persistent shift in developer behavior coincident with AI adoption; identification limits prevent a strict causal claim and we outline an agenda for cleaner tests.

This paper studies the effectiveness and incidence of the renewable energy Investment and Production Tax Credits. I leverage new geographical variation in these credits, introduced by the Inflation Reduction Act, to test whether renewable energy credits had real economic impacts. Communities with greater tax credits accrued 33% more renewable energy capital and produced 31% more renewable energy compared to similar counties. This suggests elasticities of 1.62 and 6.11 for the Investment and Production Tax Credits respectively. I augment these results using an understudied dataset on planned investment to disentangle preplanned from additional projects. Accounting for inframarginal investment significantly reduces the Investment Tax Credit's investment elasticity to 0.6. After characterizing the supply side responses to these renewable tax credits, I document a new political feedback loop between increased incentives and support for renewable energy policies. Areas with greater tax incentives experienced jumps in support for renewable energy policies, contrary to the Not In My Backyard narrative. Heterogeneity in political responses suggests that the Investment and Production Tax Credits garnered support through two channels: 1) labor market spillovers, with construction wages increasing by 7% in areas with greater tax incentives, and 2) public goods spillovers, with parents across party lines increasing support for renewable energy by 13%.

In this paper, we study the public procurement market through the lens of game theory by modeling it as a strategic game with discontinuous and non-quasiconcave payoffs. We first show that the game admits no Nash equilibrium in pure strategies. We then analyze the two-player case and derive two explicit mixed-strategy equilibria for the symmetric game and for the weighted $(p,1-p)$ formulation. Finally, we establish the existence of a symmetric Nash equilibrium in the general $N$-player case by applying an extended version of Dasgupta and Maskin result, which allows us to extend equilibrium existence to the mixed-strategy setting despite payoff discontinuities.

We study nonparametric distance-based (isotropic) local polynomial methods for estimating the boundary average treatment effect curve, a causal functional that captures treatment effect heterogeneity in boundary discontinuity designs. We establish identification, estimation, and inference results both pointwise and uniformly along the treatment assignment boundary. We show that the geometric regularity of the boundary, a one-dimensional manifold, plays a central role in determining feasible convergence rates and valid inference procedures. Our theoretical contributions are threefold. First, we derive uniform lower and upper bounds on the convergence rate of the misspecification bias of isotropic local polynomial estimators. Second, we obtain uniform distributional approximations that justify boundary-robust inference. Third, we establish minimax lower bounds for a broad class of nonparametric isotropic regression estimators. These results yield practical guidance for empirical implementation, including new bandwidth selection rules that adapt to local irregularities of the treatment-assignment boundary. We illustrate the proposed methods using simulation evidence and an empirical application, and provide companion general-purpose software.

How does competition in markets for information affect the creation and division of surplus? We study this question in a search environment in which an agent searches sequentially for a high-quality good and learns about the quality of sampled goods by repeatedly purchasing signals from profit-maximizing information brokers. Brokers design and price signals but can commit only to spot contracts. We characterize the equilibrium payoff set as a function of the market structure -- the number of competing brokers. When search costs are low, market structure affects neither surplus generation nor its division. When costs are high, however, competition benefits the agent but reduces total surplus relative to monopoly. Methodologically, we extend repeated-games theory to stopping problems such as sequential search.

Parties in spatial competition rarely choose platforms that reverse their ideological order. Mutual leapfrogging is the strongest form of reversal: each party locates beyond the other party's ideal point. In voting models without abstention single-peakedness rules out such reversals. We show that this conclusion does not survive endogenous abstention. There is a spatial voting model in which voter and party preferences are single-peaked, yet mutual leapfrogging occurs in pure-strategy equilibrium. The equilibrium survives because some deviations change which voters participate. We prove that such equilibria are impossible under a sufficient ordinal condition: parties agree on how to rank leftward and rightward deviations from their ideal points. The condition is general enough to cover symmetric single-peaked utilities and common translated utility shapes.

Strategic communication often relies on anchors observed by the sender but not by the receiver. An analyst may report against a proprietary valuation model, an auditor against an internal score, a manager against an accounting estimate, or an institution against its own standard. I study a sender-receiver game in which reports are costly to move away from such privately observed anchors. Anchor heterogeneity changes the geometry of communication. Rather than relying on partitions, privately anchored reporting generates continuous variation in messages because different senders find different reports costly to make. This mechanism can improve information transmission, but it can also pull reports toward noisy private anchors. I show that (i) small positive reporting costs can make communication approach full revelation, even though zero costs return the model to cheap talk, (ii) uninformative anchors can transmit information through strategic distortions. Anchored reports and cheap-talk messages can coexist as endogenous hard and soft information, but cheap-talk alone is preferred by all parties under sufficiently low misalignment, explaining why organizations may rely exclusively on informal channels.

Two-sided matching platforms rely on preferences from both sides, yet participants can evaluate only a small fraction of potential partners. In practice, they use low-cost pre-match screening, e.g., interviews, profile views, or trial tasks, to form noisy impressions before committing to applications and offers. We study bandit learning in matching markets with interviews, modeling these interactions as queried \emph{hints}~\citep{DBLP:conf/innovations/BhaskaraGIKM23} that reveal partial preference information to both sides while constraining subsequent applications. Our framework also allows firm-side uncertainty: firms, like agents, learn their preferences and may make early hiring mistakes. To address this, we introduce strategic deferral, a firm-side action that permits temporary vacancy, corrects premature commitments, and enables decentralized learning under coarse anonymous feedback. We design algorithms for centralized and decentralized markets and show that a constant number of interviews per round suffices for horizon-independent regret, improving over the $O(\log T)$ guarantees known without interviews. Our bounds are near-optimal: the centralized guarantee is within a factor $m$ of an information-theoretic lower bound, while decentralized algorithms match it up to polynomial factors in structured markets and remain horizon-independent in general markets.

Public delay can be informative when the existence, custodian, and review dates of hard evidence are observed. I study a disclosure protocol in which a sealed record is docketed, held by a public custodian, and revealed only at terminal disclosure. At each review, retention is not silence: it rules out the states in which the holder would have relayed or disclosed. This censoring event yields an exact Bayesian filter. Under interval strategies, the public posterior is summarized by finitely many support endpoints. A compact reputation benchmark verifies Markov perfect Bayesian equilibria with such strategies and gives finite time resolution on compact unresolved slices. With two certified routes for the same record, retention on one route changes what remains feasible on the other before it acts. Common record censoring creates network value that pairwise formation can miss.

We study core concepts in the Shapley-Scarf housing market model under unrestricted weak preferences. Among standard concepts, the strong core may be empty while the nonempty weak core may contain Pareto inefficient allocations. We propose the rectified strong core by adding a single condition: an unaffected agent may join a blocking coalition only if every object he views as indifferent to his current assignment is owned by the coalition. The rectified strong core is always nonempty and Pareto efficient, lies between the weak and strong cores, and coincides with the strong core whenever the latter is nonempty. This condition is the most permissive admissibility requirement for unaffected blocking agents that guarantees nonemptiness, and extends the behavioral foundation of the strong core to weak preferences.

We employ the consistency principle to evaluate core concepts in an indivisible object allocation model with intricate co-ownership. In this environment, the strong core is consistent but may be empty, the weak core is nonempty but neither consistent nor Pareto efficient, and the exclusion core introduced by Balbuzanov and Kotowski (2019), although nonempty and Pareto efficient, fails consistency. We introduce the concept of refined exclusion core that preserves the exclusion-rights approach while restoring consistency. The refined exclusion core eliminates unreasonable allocations admitted by the exclusion core and delivers sharper predictions than existing alternatives.

This paper shows that diagnostic expectations (DE) and rational expectations (RE) are not observationally equivalent in dynamic stochastic general equilibrium (DSGE) models. Using the frequency-domain framework of Qu and Tkachenko (2012, 2017), I show that no RE parameterization yields the DE-implied autocovariance structure of the macroeconomic observables considered in either small- or medium-scale DSGE models, even after structural frictions and shock processes are reparameterized. Incorporating DE preserves overall identification but weakens the identification of shock variances. In the medium-scale model, among the frictions, wage rigidity emerges as most important for generating the benchmark DE model dynamics.

Emerging economies, particularly the MINT countries (Mexico, Indonesia, Nigeria, and Türkiye), are gaining influence in global stock markets, although they remain susceptible to the economic conditions of developed countries like the G7 (Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States). This interconnectedness and sensitivity of financial markets make understanding these relationships crucial for investors and policymakers to predict stock price movements accurately. To this end, we examined the main stock market indices of G7 and MINT countries from 2012 to 2024, using a recent graph neural network (GNN) algorithm called multivariate time series forecasting with graph neural network (MTGNN). This method allows for considering complex spatio-temporal connections in multivariate time series. In the implementations, MTGNN revealed that the US and Canada are the most influential G7 countries regarding stock indices in the forecasting process, and Indonesia and Türkiye are the most influential MINT countries. Additionally, our results showed that MTGNN outperformed traditional methods in forecasting the prices of stock market indices for MINT and G7 countries. Consequently, the study offers valuable insights into economic blocks' markets and presents a compelling empirical approach to analyzing global stock market dynamics using MTGNN.

In panel predictive regressions with persistent covariates, coexistence of the Nickell bias and the Stambaugh bias imposes challenges for estimation and hypothesis testing. This paper introduces an innovative estimator, the Double IVX (DIVX), inspired by the IVX technique in time series. DIVX effectively removes this composite Nickell-Stambaugh bias and reinstates standard inferential procedures based on the t-statistic. This new procedure achieves unified inference across a wide range of modes of persistence in panel predictive regressions when the cross-sectional dimension and the time dimension are comparably large. Such desirable properties were unattainable by existing methods, including the popular within-group estimator. Extensive Monte Carlo simulations demonstrate the robustness of DIVX under a variety of settings. We apply DIVX to panel data of financial markets in developed economies to examine the predictability of stock returns.

Large Language Models (LLMs) are reshaping knowledge work, yet their impact on voluntary, self-guided open innovation forums (contributors choose tasks without managerial direction) may differ fundamentally from effects observed in organizational settings. We study this question in open-source software development, where individuals' contributions collectively drive innovation at a community level. Unlike product innovation, where typologies for classifying innovation are well established, knowledge work in open-source settings calls for a distinction grounded in the cognitive demand a task places on the contributor. Burgeoning literature distinguishes substantive contributions, which require creative problem formulation to introduce new functionality, from incremental contributions, which draw on comprehension of existing code to maintain and refine it. We exploit a natural experiment around GitHub Copilot's launch in October 2021, where Copilot supported languages like Python while not supporting R for business reasons, creating an exogenous partition between otherwise comparable ecosystems. Using three complementary identification strategies and two classification approaches, we find that Copilot availability increases open-source contributions by 28 to 40 percent. The increase in incremental contributions is significantly larger than the increase in substantive contributions across all specifications. This disparity is more pronounced in projects with higher activity levels and widens following a model upgrade: LLMs function more effectively when existing context helps define the problem and constrain solutions, tilting collaborative innovation toward exploitation of established codebases rather than exploration of new functionality. This paper provides a rare instance of causal field evidence on LLM effects, given the speed at which GenAI has exploded across the knowledge economy.

This paper establishes non-asymptotic convergence of the cutoffs in Random serial dictatorship in an environment with many students, many schools, and arbitrary student preferences. Convergence is shown to hold when the number of schools, $m$, and the number of students, $n$, satisfy the relation $m \ln m \ll n$, and we provide an example showing that this result is sharp. We differ significantly from prior work in the mechanism design literature in our use of analytic tools from randomized algorithms and discrete probability, which allow us to show concentration of the RSD lottery probabilities and cutoffs even against adversarial student preferences.

The global push for electric vehicles (EVs) has sharply increased demand for critical minerals such as cobalt and lithium, creating a tension between rapid industrial growth and long-term sustainability. Extraction is concentrated in a few regions -- notably the Democratic Republic of Congo (DRC), Chile, and Argentina -- where it has produced serious socio-environmental harms, including ecosystem degradation, labour exploitation, and the displacement of Indigenous communities. In the DRC, cobalt mining is frequently linked to child labour and hazardous working conditions; in Chile, lithium extraction intensifies water scarcity and threatens local agriculture and biodiversity. Policy instruments such as the U.S. Inflation Reduction Act (IRA) seek to promote ethical sourcing, but an extraction-driven model continues to deepen global inequalities. This chapter examines the contested temporalities of the transition, in which the short-term economic incentives of extraction conflict with longer-term environmental and social goals. It argues for a place-based framework built on community-centred governance, sustainable mining practices, and circular-economy strategies, including recycling and material substitution, to align resource security with equity and ensure that the shift to EVs does not reproduce the injustices it aims to address.

A user faces a list returned by a search system, ordered by a noisy proxy for relevance, and decides sequentially whether to pay a fixed cost to inspect another item or stop with the best she has uncovered. She does not enter the page knowing how good its items are, so each inspection both produces a candidate item and refines her belief about the page's underlying quality. We show the optimal policy is a standout rule: the user stops as soon as her best find exceeds her posterior mean of an average item on the page by a depth-dependent threshold. The induced dynamics collapse to a one-dimensional Markov chain, which yields the full distribution of inspection depth through a closed-form recursion. The model uncovers three hidden mechanisms (trust, commit, and cut-losses) on why users stop and yields a rich set of testable implications. Moreover, the Bayesian-rational view delivers a novel learning-to-rank likelihood: an observed depth censors the latent relevance path into a polyhedron of survival inequalities, whose Gaussian probability is a differentiable function of any feature-based relevance prediction.

Emergency departments (EDs) often use a shared-queue setup in which physicians self-assign cases from a pool of triaged patients. We conduct a multi-method study to examine this self-assignment behavior and its effects on system performance. Using data from five EDs spanning 1.4 million patient visits, we show that batching, i.e., self-assigning multiple patients at once, is common and associated with longer stays for batched patients, even after controlling for clinical acuity, physician fixed effects, and ED congestion. We then develop a continuous-time queueing model that characterizes the optimal self-assignment policy under individual and group throughput incentives. We use the model predictions to test experimentally with 203 healthcare workers and 73 ED physicians whether batching is a rational response to incentives or a deeper behavioral tendency that persists independent of incentives. Indeed, batching is pervasive across both samples, with 94% of healthcare workers and 73% of physicians choosing to batch even when it reduces their own payoffs -- a behavior that we term the personal productivity bias. Together, these results suggest that compensation redesign alone is unlikely to eliminate batching, and suggest changes to the assignment interface in the electronic health record system as a more promising remedy.

In algorithmic markets, predictive models become part of the data-generating process they aim to forecast. Once their outputs are converted into trades, allocations, execution schedules, or risk controls, they change the future data on which they are evaluated. I introduce algometrics, a framework for time series whose evolution depends on the predictive algorithms forecasting them. The framework distinguishes historical risk, measured under passive forecasting, from deployment risk, measured when forecasts drive actions. I prove three results. First, deployment risk is not identifiable from passive historical data alone: even in a one-step linear feedback model, infinitely many algorithm-mediated environments induce the same historical law while implying different deployment risks for the same forecaster. Second, historical model rankings can invert under crowding, so a predictor with lower passive error can have higher deployment error once similar algorithms are adopted. Third, randomized or instrumented actions identify short-horizon linear feedback, and I derive a finite-sample bound for deployment-risk estimation. These results suggest that time-series benchmarks in algorithmic markets should report feedback sensitivity alongside predictive accuracy.

M365 Copilot is used every week by millions of people across more than a million companies around the world as part of their workflows. Uniquely positioned in the AI landscape given its near-exclusive use for work purposes, M365 Copilot can offer a clear picture of how people use AI for work and where that usage may expand next. This paper characterizes that usage through direct classification of user interactions with M365 Copilot Chat. Based on an anonymized and privacy-preserving analysis of a sample of approximately 5.5 million sessions, we combine a learned classification of user intent with a classification of O*NET work activities done with M365 Copilot Chat. We find that M365 Copilot is emerging as an everyday assistant for knowledge work: writing dominates, but users also rely on it for information retrieval, analysis, decision making and strategizing, and evaluating and diagnosing programs and systems, among others. Information seeking tasks remain common, but time trends suggest a relative shift away from ``chat as search'' and toward content and communication-related work. Comparisons across occupational groupings and to work done in the labor market further show that usage is broad but uneven, where the relative share of work done with M365 Copilot Chat cuts across jobs in some cases and is occupation-specific in others. Areas of relative underrepresentation in the labor market suggest the next frontier for enterprise AI adoption.

LLM tool registries function as unregulated advertising platforms: providers write free-text descriptions that agents use for selection, yet no measurement infrastructure -- no viewability standard, quality score, or outcome audit -- exists to make this market accountable. We provide the first systematic framework, combining 17,700+ trials across five LLMs and ten domains with a constructive registry design prescription. Legal puffery alone (subjective superlatives, benefit framing) captures 100% of the optimization effect; fabricated claims add zero incremental bias -- rendering FTC enforcement of deceptive advertising rules ineffective against the active mechanism. Disclosure fails structurally: system-prompt warnings produce zero measurable effect for four of five models, and behavioral ceilings leave no headroom for label-based correction. Superlatives are the dominant single feature (SBC = +0.35). Registry-layer description normalization achieves first-best welfare model-independently. We propose separating selection-facing descriptions (structured, registry-controlled) from marketing-facing descriptions (provider-authored, shown post-selection), and introduce the Agent Attention Quality Score to distinguish capability from copywriting.

We investigate the emergence of synchronization in a network of coupled chaotic macroeconomic systems. Each node represents an economy characterized by three key variables savings, gross domestic product (GDP), and foreign capital inflows. These economies interact or are connected through a fitness-based probability that depends on the potential GDP of each node. This formulation allows both structural heterogeneity, arising from uneven network connectivity, and dynamical heterogeneity, due to differences in local parameters, to be explored within a unified framework. Using both numerical simulations and a mean-field approximation, by varying the coupling strength and the degree of heterogeneity of both network topology and dynamical behavior of the nodes, we analyze synchronization transitions. Our results show that the mean-field approach accurately captures the collective dynamics in homogeneous and fully connected networks even with heterogeneity within the intrinsic dynamic of the nodes but fails when strong heterogeneity in the structure of the network is introduced. In heterogeneous networks, the system exhibits partial synchronization and on--off intermittency, where coherent phases of global synchronization alternate with abrupt desynchronization bursts. The distribution of laminar phase durations follows a power-law scaling, consistent with theoretical predictions for intermittent synchronization. From an economic perspective, these results suggest that global business cycle synchronization is inherently fragile: strong integration can promote temporary coordination among economies, but structural and dynamical disparities inevitably lead to intermittent breakdowns of collective behavior.

We study inference for linear quantile regression with two-way clustered data. Using a separately exchangeable array framework and a projection decomposition of the quantile score, we characterize regime-dependent convergence rates and establish a self-normalized Gaussian approximation. We propose a two-way cluster-robust sandwich variance estimator with a kernel-based density ``bread'' and a projection-matched ``meat'', and prove consistency and validity of inference in Gaussian regimes. We also show an impossibility result for uniform inference in a non-Gaussian interaction regime.

Min-max optimization arises in many domains such as game theory, adversarial machine learning, etc. For these problems, gradient-based methods are well understood and enjoy strong guarantees. However, in the absence of convexity or concavity, existing approaches study convergence to an approximate saddle point or first-order stationary points, which may be arbitrarily far from global optima. In this work, we present an algorithmic framework for computing the global minimax value in convex--non-concave and non-convex--concave min-max optimization. For convex--non-concave min-max problems, we use a reformulation that transforms the problem into a non-concave--convex max-min optimization problem with suitably defined feasible sets and objective function. This reformulation can be viewed as an extension of Sion's minimax theorem to the convex--non-concave setting. We then introduce EXOTIC -- an Exact, Optimistic, Tree-based algorithm for solving the reformulated max-min problem. EXOTIC combines an iterative convex optimization solver for the inner minimization with an optimistic hierarchical tree search for the outer maximization, inspired by StroquOOL~\cite{bartlett2019simple}. Unlike StroquOOL, which assumes stochastic zero-mean noisy evaluations, EXOTIC handles deterministic, biased, and budget-dependent evaluation errors arising from finite-time solutions of the inner convex subproblems. We establish an upper bound on its optimality gap. The same framework also applies to non-convex--concave min-max optimization. Empirically, EXOTIC outperforms gradient-based methods on popular benchmarks from the literature. Finally, we demonstrate the utility of EXOTIC by computing security strategies in multi-player games with three or more players -- a computationally challenging task that, to our knowledge, no prior method solves exactly.