We develop asymptotic theory for principal component analysis (PCA) of a high-dimensional factor model in which the working dimension $R$ is fixed and only required to satisfy $R \ge r$, where $r$ is the true number of factors. Building on anisotropic local laws from random matrix theory, we show that the ``extra'' empirical eigencomponents beyond the $r$-th are asymptotically noise-governed, incoherent, and nearly orthogonal to the factor loadings. We introduce two rotations, an expanded $r\times R$ map $H'$ and a compressed $R\times r$ map $H^{+}$, and establish consistency of the estimated factors under both. As an application, we analyze a factor-augmented regression for treatment-effect inference and prove $\sqrt{T}$-asymptotic normality for every fixed $R \ge r$. These results provide a theoretical underpinning for the common empirical practice of adopting a conservative upper bound on the number of factors, and shift the analytical burden from consistent dimension selection to the milder requirement of bounding $r$ from above.

Content that drives engagement need not be the same content that drives willingness to pay. We study how polarizing content affects engagement (time on site) and commitment (subscriptions and retention) on a major news platform. We measure article-level polarization with deep-learning classifiers and large language models tailored to a multiparty system, and identify causal effects with two complementary instrumental variables: a Bartik instrument exploiting supply-side editorial variation, and an election instrument exploiting demand-side political salience. We find that supply-driven increases in polarizing content raise engagement but not subscriptions. During the high-salience election window, the same content reduces subscriptions and accelerates churn, with affective polarization driving the sharpest divergence. On the mechanism, we find evidence inconsistent with confirmation bias: three pre-determined ideology proxies do not moderate the engagement or subscription effects. By contrast, on ideological dimensions where the publisher covers both sides, exogenous shifts in the publisher's supply of content opposite readers' baseline ideology raise their consumption of that content, consistent with balanced consumption. These results document an asymmetric engagement-commitment trade-off for digital publishers: polarizing content reliably captures attention but does not convert to subscriptions, and actively damages commitment when political salience is elevated

The social ranking is a recently proposed framework for evaluating the power of individuals according to the performance ranking of their coalitions. Although its origin can be traced to the classical power indices in simple games, social ranking approaches carry out this evaluation within the ordinal framework of social choice theory. This article introduces the Borda rule into social ranking. Specifically, we focus on two essential properties of the Borda rule--consistency and closeness to unanimity--and investigate the social ranking solutions (SRSs) satisfying these properties. Among several possible definitions of the Borda rule as an SRS, we characterize one of such solutions by (a weak version of) consistency, closeness to unanimity (under the linear and symmetric domain), neutrality (i.e., names of the individuals do not matter), and independence of perfunctory participation (i.e., adding a perfunctory coalition into the worst class of the coalitional ranking does not affect the social ranking). We therefore propose a new Borda-type SRS for evaluating the competence of individuals in coalitional contexts.

We introduce a class of cooperative games induced by weighted directed graphs. Specifically, the coalitional value combines an internal interaction term given by the induced subgraph game with an external component based on minimal incoming edges from outside the coalition. The resulting game has a convenient representation in terms of unanimity games. This representation enables closed-form polynomial-time formulas for the Shapley and Banzhaf values. We further establish that the game has a nonempty core and is totally balanced. The class of such games therefore provides an analytically and computationally tractable example of structured network- induced cooperative games in which stability-based allocations and fairness-based solution concepts do not coincide.

We examine marital sorting using novel data from a marriage-matching platform that records both a dating-to-marriage pipeline and pre-marital attributes, including preferences for children and for the division of housework and childcare. Unlike census or post-marital surveys, characteristics are collected before matching, and objectively measurable attributes are verified using official documents, providing an ideal setting to study matching and sorting free from post-marital adjustment. Using a multidimensional matching framework across twelve attributes, we find assortative matching along all dimensions. Age is the most salient trait, while preferences for children are second--exceeding education--an economically important margin invisible in standard data. A low-dimensional factor representation shows that fertility preferences constitute a distinct sorting dimension. Exploiting the platform's dating-to-marriage pipeline, we show that sorting on fertility preferences emerges at later serious-relationship stages. A theoretical analysis suggests that the magnitude of sorting along this margin is consistent with the veto model of fertility decisions.

This paper develops a theoretical model of platform competition where user-generated content (UGC) quality arises endogenously from the composition of the user base. Users differ in their relative preferences for content quality and network size, and platforms compete by choosing advertising intensity, which affects user utility through perceived quality. We characterize equilibrium platform choice, identifying conditions under which equilibria are stable. The model captures how platforms' strategic decisions shape user allocation and market outcomes, including coexistence and dominance scenarios. We consider two types of equilibria in advertising levels: Nash equilibria and Stackelberg equilibria, and discuss the industry and policy implications of our results.

How should well-being be prioritised in society, and what trade-offs are people willing to make between fairness and personal well-being? We investigate these questions using a stated preference experiment with a nationally representative UK sample (n = 300), in which participants evaluated life satisfaction outcomes for both themselves and others under conditions of uncertainty. Individual-level utility functions were estimated using an Expected Utility Maximisation (EUM) framework and tested for sensitivity to the overweighting of small probabilities, as characterised by Cumulative Prospect Theory (CPT). A majority of participants displayed concave (risk-averse) utility curves and showed stronger aversion to inequality in societal life satisfaction outcomes than to personal risk. These preferences were unrelated to political alignment, suggesting a shared normative stance on fairness in well-being that cuts across ideological boundaries. The results challenge use of average life satisfaction as a policy metric, and support the development of nonlinear utility-based alternatives that more accurately reflect collective human values. Implications for public policy, well-being measurement, and the design of value-aligned AI systems are discussed.

We study optimal contract design for large populations of heterogeneous agents whose actions generate network spillovers represented by an interaction function. In a linear-quadratic framework, we solve the finite-agent problem and its continuum limit, obtaining explicit optimal contracts and equilibrium efforts. We show that the continuum contract can be evaluated on a large finite sample of agents to obtain admissible contracts that achieve the finite-agent principal's value up to an error of order 1/N. This provides a scalable approximation for settings with many interacting agents. We also prove stability with respect to perturbations of the interaction function and provide comparative statics and numerical examples showing how network position affects effort, incentives, and the principal's value. The results identify how optimal incentives should be targeted toward agents whose actions generate larger spillovers.

This paper examines how digital transformation reshapes employment structures within Chinese listed firms, focusing on occupational functions and task intensity. Drawing on recruitment data classified under ISCO-08 and the Chinese Standard Occupational Classification 2022, we categorize jobs into five functional groups: management, professional, technical, auxiliary, and manual. Using a task-based framework, we construct routine, abstract, and manual task intensity indices through keyword analysis of job descriptions. We find that digitalization is associated with increased hiring in managerial, professional, and technical roles, and reduced demand for auxiliary and manual labor. At the task level, abstract task demand rises, while routine and manual tasks decline. Moderation analyses link these shifts to improvements in managerial efficiency and executive compensation. Our findings highlight how emerging technologies, including large language models (LLMs), are reshaping skill demands and labor dynamics in Chinas corporate sector.

We develop a Bayesian state-space model for analyzing the dynamic evolution of income distributions using grouped income data. The model combines the generalized beta distribution of the second kind (GB2) with latent time-varying parameters to capture changes in the entire income distribution over time. Using Japanese household income data, we examine how demographic factors, particularly population aging and declining household size, affect inequality dynamics. The results show that demographic changes have heterogeneous effects across different parts of the income distribution and contribute substantially to the evolution of inequality. Counterfactual analyses indicate that aging and household size changes affect the lower and upper tails of the distribution differently. Because the proposed framework requires only grouped income data, it can be applied to countries where micro-level income data are unavailable.

We present a novel parameterized 4-qubit Eisert-Wilkens-Lewenstein (EWL) quantum game circuit for recommender systems in quadruple helix innovation ecosystems (academia, industry, government, and civil society). The local strategy operators $U_{i} = R_y(θ_{i})$ for each helix actor are directly tuned by normalized dominance weights extracted from real participant funding data (\texit{ecContribution}) in the European Commission CORDIS Horizon Europe database (project COVend, ID 101045956). The circuit employs a multi-qubit EWL entangler followed by parameterized local rotations, inverse entangler, and full measurement, achieving only 22 gates and circuit depth 11 while scaling as $O(n)$ for $n$-round helix communications. Measurement probabilities after the quantum game serve as recommender scores for disruptive versus sustaining innovation trends. These scores are subsequently mapped into the diagonal Dirac potential of a Dirac-Solow-Swan Hamiltonian, enabling time-evolution simulation of capital accumulation and bifurcation dynamics under disruptive innovation. Numerical experiments on real CORDIS quadruple-helix collaboration networks demonstrate the circuit's NISQ compatibility and its ability to forecast disruptive capital trajectories with high fidelity. The proposed framework bridges quantum game theory, parameterized quantum circuits, and relativistic economic growth models, offering a computationally efficient tool for innovation policy and strategic decision-making in complex socio-economic ecosystems. Complexity analysis and reproducibility are provided through open Qiskit implementations.

Kusumegi et al. (2025) study whether researchers' preprint output rises after adopting large language models (LLMs), dating adoption as the first month in which at least one submitted abstract exceeds an LLM-detection threshold. We show that this treatment-timing rule is mechanically related to output. The probability that at least one paper is flagged in a month is increasing in the number of papers submitted in that month, so detected-adoption months are disproportionately high-output months. An event study centered on first detection can therefore display positive post-event dynamics even when the flagging rule contains no information about true LLM adoption, because the omitted pre-treatment period is selected from months with no prior detection. We demonstrate this in a simulation: with i.i.d. productivity and no causal effect, first-detection timing generates a spurious positive post-treatment path. We also replicate the stacked event study of Kusumegi et al. (2025) and show that three placebo exercises (random paper-level assignment, neutral keyword flags, and a pre-ChatGPT observation window) each produce a similarly positive post-treatment pattern.

We study whether and how innovation intensity attracts foreign portfolio investment (FPI) using a panel of 60 countries from 1996 to 2021. Using an instrumental variable strategy based on regional shift-share and global push instruments, we estimate the causal response of debt and equity inflows to innovation intensity in the host country. We find that innovation increases FPI, with larger effects for equity than debt inflow. Moreover, the effect of innovation on equity inflow increases with technological development and institutional quality, whereas the effect on debt inflow is positive and significant only at high levels of these factors. We also find that countries with a higher risk-taking environment attract more FPI and that equity inflow responses are immediate and persistent, whereas debt inflow responses are modest and dampen over time.

The increasing penetration of renewable energy necessitates improved power system flexibility, driving the deployment of independent energy storage operators (ESOs). Existing research extensively investigates capacity sizing for price-taker storage systems or the operational coordination of aggregated distributed resources, lacking the joint optimization of capacity planning and multi-market bidding for a price-maker ESO with hybrid energy storage system (HESS) that preserves the technological heterogeneity of the integrated components. We propose a bi-level optimization framework to jointly optimize profit-oriented decisions on capacity and multi-market operation. The upper-level problem determines the optimal capacities of two heterogeneous storage systems while coordinating their bidding across day-ahead joint energy-reserve and real-time balancing markets. The lower-level problems represent market clearing of the system operator (SO). The model is reformulated into a mixed-integer linear program and solved with a Benders' decomposition algorithm. Results demonstrate that the ESO can allocate capacity between energy arbitrage and reserve provision strategically. The system with the high power-to-capacity ratio is used to capture arbitrage profits while the system with low power-to-capacity ratio is used to specialize in reserve markets. There can be internal power transfer between storage systems if there exist grid access constraints. The framework provides differentiated bidding strategies and market participation flexibility for HESS to enhance overall profitability.

Residential batteries increasingly serve two roles: they can earn money by arbitraging wholesale prices and providing grid services, and they provide backup power during outages. This dual use creates a basic tradeoff between earning market value and preserving outage readiness. Coordination across many batteries can help, but a provider cannot treat the fleet as a single virtual battery when each household is promised its own backup protection. We compare standalone control, in which each home is dispatched independently, with pooling, in which homes are coordinated while each battery retains its own state of charge and household-specific backup requirement. Both regimes are implemented as model predictive control problems with 15-minute decision intervals and evaluated using household telemetry together with ERCOT market inputs. The empirical design focuses on the 543 homes in our sample that can support at least one backup product in standalone operation and studies backup caps ranging from 2 to 24 hours. Lower caps relax backup obligations, while the 24-hour cap coincides with assigning each home its own longest feasible backup tier. Pooling remains beneficial in this service-constrained setting, but its value declines smoothly as backup obligations tighten. Standalone firm margin ranges from \$11.06 per home per week at the 2-hour cap to \$10.79 at the 24-hour cap, while pooling benefit falls from \$1.49 to \$1.27 per home per week. Relative to standalone firm margin, pooling is worth about 13.5% at the 2-hour cap and about 11.8% at the 24-hour cap. Coordination therefore still helps after preserving household-level backup guarantees, but its value declines as backup obligations tighten.

This paper develops a model to evaluate the viability of blockchain markets as the sole venue for price formation. Blockchains clear at discrete intervals called block time, and transactions are executed sequentially according to priority fees paid by traders who compete for queue position. We show that these features undermine the viability of markets. Paid-priority ordering induces endogenous selection, where only traders with sufficiently high valuations participate. The participation cutoff rises with competition, which intensifies with lower information costs or higher liquidity demand. This hinders price discovery and biases prices. It also impairs liquidity: the cutoff concentrates trading among aggressive traders and increases adverse selection that liquidity suppliers absorb in a single clearing round. Although longer block times enhance consensus security, they amplify these effects and can cause markets to shut down.

This paper combines traditional panel econometrics with random forest machine learning to revisit the relationship between exchange rate regimes and economic growth for 27 transition economies over 1991-2019. Exploiting the Couharde-Grekou (2024) probabilistic synthesis classification, the random forest approach non-parametrically confirms and sharpens what fixed-effects and system GMM estimation establish parametrically intermediate exchange rate regimes consistently underperform fixed arrangements, with growth penalties ranging from -1.0 to -10.4 percentage points, while floating regimes show negative but largely insignificant differentials. Beyond regime effects, the machine learning analysis reveals that the intermediate regime penalty is sharpest precisely where institutions are weakest - non-parametric validation that institutional capacity, not regime label alone, determines whether exchange rate anchoring pays off. The regime-growth relationship is further concentrated in the pre-2003 stabilization era and is absent among EU member economies, suggesting the growth dividend from exchange rate anchoring eroded as institutional convergence advanced. Together, these findings demonstrate how machine learning variable importance metrics can corroborate and enrich causal inference from panel methods, while supporting the view that exchange rate anchoring carried a meaningful credibility dividend during the formative phase of transition.

Health artificial intelligence (AI) adoption presents a paradox: point-solution tools diffuse readily through clinical populations, yet system-change AI, which carries the greatest potential for pathway-level transformation, consistently stalls at partial adoption. An evolutionary game theoretic model is developed to explain this pattern. Doctors choose among three strategies: genuine adoption, partial adoption, and rejection, where genuine adoption is required for systemic benefits to materialise above a population threshold. The system is shown to be generically bistable, with a stable partial adoption equilibrium coexisting alongside full genuine adoption. The basin of attraction of the partial adoption trap is enlarged by three compounding failure modes: a threshold coordination failure arising from the non-appropriable nature of systemic benefits; a trust failure arising from the organisation's inability to credibly commit to sharing productivity gains; and a cultural failure arising from negative coordination norms among doctors. These failure modes are shown to be most severe precisely for the technologies with the greatest systemic value: the Value-Adoption Paradox. A cost ratchet dynamic implies that failed adoption attempts permanently lower barriers even when embedding fails, but this benefit is offset when trust erosion is rapid. Conditions are derived under which sustained but imperfect adoption pressure is welfare-improving, and the policy architecture required to escape the trap (targeting trust, sequencing, and team-level adoption) is characterised. Standard health system digital transformation policy, which typically addresses only the threshold failure through individual incentives, is predicted to systematically produce the partial adoption trap.

Advances in quantum computing introduce long-term security challenges for widely deployed public-key cryptographic systems used across blockchain platforms and decentralized applications. Although post-quantum cryptography (PQC) standards are emerging, understanding quantum risk remains fragmented across research, engineering, governance, and investment communities. This demo presents Quantum Futures Interactive, a live interdisciplinary demonstration platform combining educational visualization, participatory interaction, and cryptographic artifact generation to illustrate the transition from classical to quantum-resilient blockchain systems. Participants engage in a structured interaction flow including quantum threat education, sentiment capture, technology prioritization, infrastructure tradeoff exploration, and generation of post-quantum cryptographic outputs. The system integrates distributed trust concepts, sustainability-aware infrastructure considerations, and responsible innovation within an interactive decision framework. The demonstration supports interdisciplinary dialogue on blockchain resilience while aligning with United Nations Sustainable Development Goals (SDGs).

Triple differences (DDD) is a workhorse quasi-experimental design in applied economics. But, under staggered adoption, its conventional three-way fixed-effects (3WFE) implementation inherits the interpretation issues now well understood in the difference-in-differences literature. I introduce stacked DDD. I extend the stacked difference-in-differences approach to the DDD setting by creating self-contained stacks, each consisting of four cells over an event window: treated and clean comparison cohorts, each with treatment-eligible and treatment-ineligible units. Appending these stacks yields a unified dataset for estimating treatment effects. I prove that, at each post-treatment event-time, a linear regression with fully saturated fixed-effects applied to the stacked dataset identifies a strictly positive, cell-size-weighted average of stack-level conditional average treatment effects, with stack weights proportional to stack-level cell sizes. Building on this characterization, I outline alternative weighting schemes that recover causal estimands with clear interpretations. Stacked DDD complements recent GMM and imputation-based frameworks by trading efficiency for regression-based transparency, pairwise (rather than global) parallel changes-in-trends, and direct control over both the comparison group for each treated unit and the aggregation weights. I provide two empirical illustrations where stacked DDD yields substantially different quantitative conclusions compared to existing procedures.

Central banks cannot observe market reactions to their communications before release. We propose a framework in which Large Language Models simulate 30 heterogeneous traders interpreting European Central Bank press conference transcripts, yielding a measure of cross-sectional disagreement among synthetic agents. Across 293 Governing Council events from 1998 to 2026, this measure correlates at approximately 0.5 with realized Overnight Index Swap volatility, outperforming standard text-based alternatives in explaining market reactions. LLM-implied disagreement adds information beyond volatility clustering and remains robust in out-of-sample validation on genuinely unseen conferences from January 2025 onwards. We further show that providing historical examples of pre and post-conference volatility improves the calibration of model responses. The framework offers a practical tool for assessing, prior to release, how central bank communication is likely to be interpreted by financial markets

The synthetic control method (SCM) is widely used for constructing the counterfactual of a treated unit based on data from control units in a donor pool. Allowing the donor pool contains more control units than time periods, we propose a novel machine learning algorithm, named SCM-relaxation, for counterfactual prediction. Our relaxation approach minimizes an information-theoretic measure of the weights subject to a set of relaxed linear inequality constraints in addition to the simplex constraint. When the donor pool exhibits a group structure, SCM-relaxation approximates the equal weights within each group to diversify the prediction risk. Asymptotically, the proposed estimator achieves oracle performance in terms of out-of-sample prediction accuracy. We demonstrate our method by Monte Carlo simulations and by an empirical application that assesses the economic impact of Brexit on the United Kingdom's real GDP.

Local projections (LPs) are widely used for impulse response analysis, but Bayesian methods face challenges due to the absence of a likelihood function. Existing approaches rely on pseudo-likelihoods, which often result in poorly calibrated posteriors. We propose a quasi-Bayesian method based on the Laplace-type estimator, where a quasi-likelihood is constructed using a generalized method of moments criterion. This approach avoids strict distributional assumptions, ensures well-calibrated inferences, and supports simultaneous credible bands. Additionally, it can be naturally extended to the instrumental variable method. We validate our approach through Monte Carlo simulations.

In this paper, we reveal that the signal representation of information introduced by Gentzkow and Kamenica (2017) can be applied profitably to dynamic decision problems. We use this to characterize when one dynamic information structure is more valuable to an agent than another, irrespective of what other dynamic sources of information the agent may possess. Notably, this robust dominance is equivalent to an intuitive dynamic version of Brooks, Frankel, and Kamenica (2022)'s reveal-or-refine condition.

We derive the asymptotic theory of Bai (2009)'s interactive fixed effects estimator for unbalanced panels in which the source of attrition is conditionally random. For inference, we propose a method of alternating projections algorithm based on straightforward scalar expressions to compute the residualized variables required for bias correction and covariance matrix estimation. Simulation experiments confirm that our asymptotic results provide reliable finite-sample approximations. We also reassess Acemoglu et al. (2019). Allowing for a more general form of unobserved heterogeneity, we confirm significant effects of democratization on economic growth.

We decompose the U.S. consumption inequality distributional changes during the COVID-19 phase. Analyzing the Consumption Expenditure Interview Survey data, we decompose observed changes in consumption inequality into components attributable to several individual variables. Using a distribution regression method, we construct counterfactual distributions under the scenario in which the consumption structure or any specific variable would have remained the same between the two years before and after the onset of the COVID-19 pandemic. We find that changes in the conditional distribution of consumption explain most of the observed decline in consumption inequality among male-headed households between 2018 and 2022. The rise in asset holdings has significantly increased the consumption inequality in all measures. Moreover, the changes in a set of household characteristics have significantly reduced the consumption inequality. Our analyses focus on the well-measured consumption components that are robust to the measurement errors in consumption data.

Automation affects the labour content of work differently across different contexts. Yet, most existing exposure measures assign fixed scores to tasks or occupations, limiting comparisons of automation exposure across countries. We develop a task-based and country-specific approach to classify automation exposure across the world to disentangle labor-substituting from labor-augmenting automation, the relevant technology channel, and the material role of AI. Our measure spans 124 countries, generating an atlas of 2.33 million task-country labels for economies covering 99% of world population and GDP. We present five descriptive results. First, exposure is highly uneven, ranging from 3.3% of tasks in South Sudan to 61.6% in China, and rises strongly with income, although substantial variation remains within income groups. Second, across countries, exposed tasks are skewed towards substitution rather than augmentation, but low-income countries are disproportionately exposed to substitution, whereas middle-income countries are more heterogeneous. Third, less technologically advanced forms of automation account for more than half of exposed tasks in low-income countries but about one quarter in high-income countries; while other more complex channels generally rise with income levels. Fourth, AI tends to be less prevalent in simpler channels of automation, but also more prevalent in labour-substituting margins in lower income settings and to augment labour in higher income settings. Fifth, we find that females seem to be disproportionately more exposed to labour-substituting automation than males. Our methodology provides a basis for comparing automation exposure across development stages, linking it with cross-country data and allowing us to treat exposure levels, labour margins, technological channels and AI involvement as separate dimensions.

I propose Nonparametric Bayesian Policy Learning (NBPL) as a framework for uncertainty-aware treatment choice. I consider a decision-maker (DM) seeking to select an expected welfare-maximizing treatment rule using observable characteristics. A key observation is that, for a given welfare criterion and policy class, uncertainty about welfare-relevant objects is entirely induced by uncertainty about a reduced-form distribution. I assume the DM places a nonparametric Dirichlet process prior on this reduced-form parameter and uses the resulting posterior to conduct inference on optimal treatment assignments, optimal welfare, and comparisons across policy classes. The NBPL framework is flexible, and its implementation via the Bayesian bootstrap is highly tractable. I establish two main theoretical properties of NBPL. First, posterior welfare regret under NBPL converges at the minimax-optimal rate. Second, posterior model comparison across policy classes is pointwise consistent. I illustrate NBPL in two empirical applications: the bednet subsidy experiment of Bhattacharya and Dupas (2012) and the JTPA experiment studied by Kitagawa and Tetenov (2018).

We study inference using trimmed least squares (TLS) and trimmed least absolute deviations (TLAD) estimators of \citet{honore_trimmed_1992} in censored two-period panel-data models with fixed effects. We show that the published asymptotic variance formulas rely on additional regularity conditions that are not fully stated in the original analysis. For TLS, the published Hessian formula requires that the regressor-difference index vanish only when the regressor difference itself is zero, a restriction not explicitly stated in the original paper and violated, for instance, with a zero parameter vector. We derive the correct Hessian, establish asymptotic normality without imposing this restriction, and obtain a consistent plug-in variance estimator. We also show that the Hessian estimator proposed in \citet{honore_trimmed_1992} {\em is} actually consistent for the {\em correct} TLS asymptotic variance. For TLAD, we show that the published variance formula omits a conditional-probability term and that asymptotic normality requires additional continuity conditions. Under these conditions, we derive the corrected asymptotic variance and provide a tuning-parameter-free bootstrap variance estimator.

We extend the concept of meta-Nash equilibrium, introduced by Eshaghi Gordji and Bagha [2026] for complete-information games, to environments with incomplete information. We define a meta-Bayesian Nash equilibrium as a profile of type-dependent mixed meta-strategies together with an environmental move such that no player type can profitably deviate and the environment cannot improve its expected payoff. For each transformed game, meta-payoffs are determined by the unique Bayesian Nash equilibrium of that game. Using Kakutani's fixed point theorem, we establish existence under finiteness assumptions on type spaces, meta-actions, and transformations, together with the assumption that each transformed game admits a unique Bayesian Nash equilibrium. Several illustrative examples, including adaptive subsidy competition, cybersecurity protocol selection, and platform rule formation, demonstrate that private information at the meta-level plays an essential role in endogenous game transformation. The framework contains both classical Bayesian games and complete-information meta-games as special cases.