This paper studies how long-run earthquake risk shapes national identity, separating a distributive margin (national membership as a rule for allocating scarce resources) from an expressive margin (pride, willingness to fight, and affective attachment). Linking World Values Survey respondents (1981-2022; 63 countries, 494 subnational regions) to subnational seismic-risk geography, I find that people living closer to high-risk zones express stronger national in-group orientation: more pride, more willingness to fight, and more priority for nationals when jobs are scarce. Family attachment and out-group hostility do not rise, while religiosity increases in parallel. The expressive margin is conditional: the pride response is pronounced where state-religion alignment and a cohesive religious field lend the symbolic infrastructure to cast disaster as a shared national ordeal, and indistinguishable from zero where they do not. A complementary design exploiting earthquakes between adjacent survey waves finds no average short-run response, yet the response it does detect concentrates among older, place-attached residents who cannot leave -- consistent with attitudes tracking a chronic, inescapable risk rather than single events. Together, the results point to a demand-side origin of national attachment: where a covariate shock would overwhelm local and family insurance, people turn to larger communities of protection and meaning -- the nation and religion -- a logic I formalize in a simple social-interaction model.

Despite near-universal electrification in many countries, electricity supply shortages continue to shape household energy use. This paper examines how households adapt to chronic grid failure in high-electrification, high-dependence contexts, using Lebanon as a case study. Drawing on original survey data from 1,000 households, we analyze both supply-side coping mechanisms such as diesel generators and solar photovoltaic (PV)-battery systems, and demand-side adaptations, including load shifting and demand suppression. The results reveal a landscape of household responses, where socioeconomic status plays a central role in determining access to backup solutions and the extent of met demand. While diesel generators remain widespread, a transition toward PV-battery systems is observed, especially among financially capable households. However, decentralized self-generation is associated with inefficiencies, including substantial levels of curtailed solar generation. On the demand side, households exhibit reductions in electricity use, leading to distinct consumption profiles depending on the type of backup system employed. These findings highlight the importance of distinguishing between met and unmet demand when assessing energy needs under unreliable supply. The paper contributes to the literature by providing a quantitative characterization of the interaction between self-generation and demand adaptation in a supply-constrained high-electrification context. It also offers empirical demand profiles that incorporate suppressed consumption, addressing a key gap in electricity system planning. From a policy perspective, the results underscore the need to account for unmet demand, address inequities in access to coping technologies, and reduce inefficiencies in decentralized systems.

Prediction markets are usually evaluated after their contracts exist, by asking how well prices forecast outcomes. We study the prior institutional margin of market formation, asking which uncertainties become tradable contracts at all. Using an audited dataset of 6,047 Africa-topic and Latin America-topic contracts listed on Polymarket and Kalshi, we construct a coded measure of settlement legibility, the degree to which an uncertainty can be worded, sourced, and credibly resolved by third parties, and validate it on 451 units under a frozen codebook, where independent double scoring reaches ordinal reliabilities of 0.92 and 0.96 on the primary dimensions and blind human benchmarks reach 0.97 and 0.92. Using this measure, we find that formation is selective in ways that public importance does not explain, with African inventory concentrated overwhelmingly in football while salient civic events produce little or no inventory, and Latin American inventory deeper but dominated by Venezuela, where attention to prospective United States military action sustains the largest civic cluster in the data. Legibility orders the inventory steeply, with sports and elections near the top of the scale and conflict at the bottom. In a formation test against an externally assembled frame of 131 civic events, legibility predicts listing in the expected direction but falls short of pre-specified acceptance criteria, while among listed contracts the relation between legibility and trading value is negative, as a model of selective listing implies and as we predicted before estimation. Prediction-market inventories therefore measure what platforms can settle as much as what traders believe, and reading them as maps of public interest conflates the two.

We introduce Martingale Doppelgänger-Eval, a public shadow-market benchmark for auditing whether vision-language models (VLMs) use candlestick evidence rather than extrapolate past trends. The central difficulty is identification: on real market histories, chart evidence and trend are strongly coupled, so an observational score cannot determine whether a fluent technical-analysis narrative is grounded in local visual evidence. We prove this limitation formally: no evaluation functional computed from observational chart--label data can distinguish a grounded responder from a trend-shortcut responder under strong coupling, whereas matched evidence interventions separate the same responders at an exponential rate and trend--label swaps provide an independent shortcut stress test. The benchmark therefore evaluates frozen VLMs on rendered OHLCV charts under four controlled mechanisms: a martingale-null market, injected-alpha counterfactual pairs, trend-confounder swaps, and regime shifts. A structural behavioral model identifies null-market bias, trend sensitivity, evidence sensitivity, prompt/renderer fragility, and evidence faithfulness; the accompanying statistical toolkit provides minimum detectable effects, block-aware sequential testing for metered APIs, and an overlap-weighted artifact check. Across frozen commercial and open VLMs, the identified regression assigns large positive coefficients to past trend but evidence coefficients that are zero or opposite to the rule-implied sign. Matched-pair analyses show that models either ignore injected candlestick semantics or move opposite to the rule-implied direction conditional on responding. The benchmark isolates a failure mode that standard observational chart benchmarks cannot detect and gives a reusable audit template for time-series imagery with controllable label mechanisms.

Private Property is one of the central institutions of civilized society. We first consider its social, legal, and economic aspects. We then follow the Lockean tradition by focusing on a specific procedural definition: Homesteading is the acquisitive act of first using an object that is initially unowned. The ontology of concrete objects and the nature of their uses determine how objects may be acquired. In this article, we address the open-texture problem in the definition of property, then provide the necessary conditions for an object to be property in the Lockean Scheme.

Climate policy in global network industries is implemented across fragmented jurisdictions, yet firms respond through integrated operational networks. We develop a two-stage game-theoretic framework to analyze how firm-level responses interact with alternative governance structures. Regulators first choose emissions charges. Firms subsequently compete through pricing, service capacity and capital deployment decisions. The analytical results demonstrate that uniform global regulation maximizes welfare in symmetric markets. However, in sufficiently asymmetric markets, a uniform global charge is dominated by decentralized regimes. Multiple regulatory instruments better accommodate region-specific market externalities. We apply this framework to a calibrated case study of North American, Western European and transatlantic aviation markets. The numerical results establish that a globally coordinated regulator setting region-specific charges achieves the highest aggregate welfare. These aggregate gains nonetheless mask substantial distributional disparities across jurisdictions. Effective climate governance in network industries therefore requires more than determining an efficient emissions charge. Policy instruments ought to accommodate regional heterogeneity and transfer mechanisms will be necessary to ensure efficient, politically stable cooperation.

Which regional exposure conclusions are identified when public data do not observe buyer-seller links across states? We study this question by treating the missing intermediate-input spatial kernel as an unknown coupling constrained by regional activity margins, support restrictions, and auxiliary shipment moments. For linear exposure statistics, the sharp identified set is computed by transportation linear programs. Applying the method to U.S. state-sector data, we find that shipment data are inconsistent with the spatial diffuseness implied by proportional regionalization in key goods sectors. However, they do not identify a unique regional production network or a precise ranking of state exposure to local shocks. Bilateral shipment restrictions tighten the bounds, but much of the remaining uncertainty comes from large service and mixed sectors that are weakly covered by goods-movement data. The results show which exposure conclusions are supported by public data and which are imposed by maintained regionalization assumptions.

Agentic artificial intelligence systems introduce a new class of model risk. Unlike traditional predictive models, autonomous agents continuously acquire information, form beliefs regarding latent states of the environment, generate forecasts, select actions, and adapt their behavior over time. Existing validation methodologies focus primarily on predictive accuracy and therefore provide limited insight into the quality of the underlying decision process. This paper proposes a model validation framework for agentic AI based on Partially Observable Markov Decision Processes (POMDPs). The framework decomposes autonomous decision making into information, beliefs, forecasts, actions, and utility, allowing each component to be validated independently. Large language models (LLMs) are formalized as approximate Bayesian filtering operators, and a model-risk taxonomy is developed encompassing state-space, filtering, forecast, policy, utility-specification, and parameter risks. The model risk validation methodology is demonstrated through a portfolio-management case study in which an agent infers latent market regimes from market and macroeconomic information, generates belief-conditioned forecasts, and constructs portfolios using a Black--Litterman framework. Empirical validation combines performance analysis, belief calibration diagnostics, coverage tests, ablation studies, and parameter-sensitivity analysis. The results indicate that latent-state inference contributes independently to decision quality and that the principal conclusions remain robust across a broad range of parameter values. The principal contribution of the paper is a practical framework for extending established model risk management concepts to autonomous AI systems and providing a rigorous foundation for their validation, governance, and monitoring.

Modern option-learning systems operate in two coordinates: price space, where markets quote and no-arbitrage constraints are most naturally enforced, and implied volatility (IV) space, where volatility surfaces are smoothed, regularized, and evaluated. The bottleneck is interface, not approximation: Jäckel's seminal "Let's Be Rational" (LBR) solver already inverts the Black-Scholes price to machine precision efficiently. What is missing is a differentiable layer that preserves LBR in the forward pass and avoids backpropagating through its branch logic. Such a layer must also confront the unavoidable singularity of the inverse map in the low-vega regime, where the sensitivity 1/vega diverges as vega -> 0. We close this gap with PIVOT, the Price-Implied-Volatility Objective Translator. PIVOT keeps the LBR forward pass intact and supplies the backward pass by implicit differentiation through the smooth Black-Scholes/Black-76 price map, with an explicit gating contract: invalid domains return NaN, well-conditioned rows receive the exact 1/vega gradient, and low-vega rows are attenuated rather than silently regularized. On a single H100, a fused Triton kernel reaches 1.79e9 IV/s at machine precision (9.3e-14 max relative error vs. the reference C solver); end-to-end label generation sustains 48.9M/s on synthetic chains and 16.6M/s on SPX OptionMetrics. In a HyperIV-style one-day reproduction on SPX, PIVOT-augmented objectives Pareto-dominate the baselines, reducing held-out price MAE by up to 43.4% and the strongest three-seed gated objective improving price MAE by 38.8% and IV MAE by 21.3% jointly; cross-asset results on RUT, VIX, and NDX show directional price-MAE gains of 40.1%, 24.2%, and 16.7%, while an ungated IV-roundtrip control collapses to a degenerate near-zero surface, confirming the gate as a correctness contract rather than a tuning knob.

Large language models (LLMs) are increasingly deployed as autonomous agents that make consumption decisions on behalf of users. This shift raises fundamental questions for consumer theory, which has traditionally modeled humans as the primary decision-makers. In this paper, we introduce LLM Consumer Behavior Theory, a new field of study concerned with analyzing consumer behavior in agentic markets. Drawing on classical and behavioral economics alongside recent advances in Natural Language Processing, we formalize how human preferences are reflected and acted upon by LLM-based agents, and how agent-level decisions aggregate into market demand. We unify previously fragmented literature on LLM decision-making, human behavior simulation, and preference elicitation under a common economic lens, highlighting where assumptions, such as rationality and heterogeneity, may fail in agentic markets. Rather than providing empirical validation, this paper outlines the scope of LLM consumer behavior and identifies open research questions related to alignment, preference representation, and market dynamics.

Simulation based solvers for optimal stopping problems must discretize the stopping decision. Under classical dynamic programming, a coarse exercise grid with only a few stopping opportunities can materially undervalue the optimal expected reward, whereas on a very fine grid, approximation errors accumulate through the backward recursion. To remove this limitation, we develop a new reinforcement-learning inspired algorithm that enables us to learn the exercise rule at arbitrarily fine time resolution. Our CARLOS (Continuous-time Adaptive Reinforcement Learning for Optimal Stopping) algorithm utilizes an aggregate deep neural network (ADNN) to learn a joint space-time decision boundary. Starting from a coarse time grid, we progressively increase the frequency of stopping opportunities, while in parallel training the ADNN to refine its timing-value estimates. We moreover design an adaptive sampling strategy that gradually concentrates training effort near the stopping boundary. Benchmarked results show that CARLOS delivers higher prices than existing Bermudan solvers, approaching the American upper bound, and achieves high computational efficiency relative to non-RL comparators.

This paper extends classical conformal frameworks for constructing prediction intervals with global marginal coverage $1-α$ to intervals that provide explicitly calibrated guarantees for the upper and lower tails separately. Focusing on split conformal prediction, we first construct lower and upper one-sided conformal intervals that achieve marginal validity, and then derive the induced two-sided interval by intersection. Theoretical results prove both tail-specific and global marginal coverage of the induced two-sided interval. Results are presented first for the exchangeable setting, where coverage has finite-sample guarantees, and then for non-exchangeable data, where guarantees are asymptotic. Simulation studies show that the proposed approach achieves improved directional calibration relative to classical two-sided intervals, especially relevant in skewed data. Finally, the benefit of the proposed framework is showcased in a financial application, where one aims for return maximization while seeking strict control on the left tail.

New York City implemented the nation's first cordon-based congestion pricing program in January 2025, providing an opportunity to evaluate how system-wide urban mobility responds to large-scale pricing interventions. Because such policies generate spillovers across modes and locations, credible control groups are difficult to construct. We address this challenge using time series foundation models to generate probabilistic counterfactual demand forecasts with calibrated uncertainty. Applying this framework to bus, subway, and aggregate trip volume data, we find that post-policy bus and subway ridership increased significantly relative to expected no-policy demand, while overall travel demand decreased modestly. The effects are spatially heterogeneous: while reductions in overall travel demand are concentrated within the Congestion Relief Zone, transit gains extend beyond Manhattan's core. Socio-demographic analyses further reveal uneven adaptation across neighborhoods, highlighting spatial equity implications. Our framework provides a scalable approach for the uncertainty-aware evaluation of system-wide urban interventions when clean control groups are unavailable.

According to the recent Wealth Thermalization Hypothesis (WTH) the wealth inequality in the world is described by the Rayleigh-Jeans (RJ) thermal distribution of interacting agents in a society with social stratification. In this concept, the wealth layers of society are associated with energy levels from a nonlinear dynamical system conserving two integrals of motion being total energy and probability norm. This leads to RJ condensation and the formation of a huge poverty phase of low wealth and a tiny oligarchic phase that captures a main part of total society wealth. This RJ phenomenon has similarities with self cleaning in multimode optical fibers and constraint driven condensation in various physical systems. We analyze real Lorenz and Pareto curves for wealth of households in countries and the world, Gross Domestic Product of countries, market capitalization of companies at stock exchange of Hong Kong, Shanghai, London, bitcoin transactions, world trade between countries and show that the WTH theory gives a good description of these curves. On the basis of this comparison we argue that the RJ thermal distribution provides a universal description of wealth inequality in the world.

Short-dated index options make scheduled macro-announcement risk visible in market prices, but visibility does not imply identification: a flexible no-event surface fitted to event-spanning quotes can absorb event premia, while a jump calibrated without event-spanning quotes is unidentified. To separate the continuous surface from the scheduled jump, we model Federal Open Market Committee (FOMC) decisions, Consumer Price Index (CPI) releases, and nonfarm payroll (NFP) reports as deterministic-time jumps in risk-neutral option pricing and propose a non-spanning identification protocol. Non-spanning expiries identify the no-event volatility surface, event-spanning training quotes calibrate the scheduled jump, and held-out event-spanning quotes are used only for pricing evaluation. On PM-settled S\&P 500 index (SPX) options from May 2022 to August 2025, Gaussian and two-component mixture jumps improve held-out event-spanning pricing, with the clearest gains in robust median pricing errors and in event-volatility option combinations (straddles and strangles) rather than directional risk reversals. A contaminated-surface stress test confirms the identification concern: allowing event-spanning training quotes into the no-event surface fit produces strong held-out performance by absorbing event premia rather than identifying scheduled jump risk. An amortized mixture density network (MDN) benchmark shows limited cross-event transfer: pure leave-one-event-out amortization reduces implied-volatility errors but not mean dollar or mean spread-normalized pricing errors, while the scale-calibrated variant restores Gaussian-level performance yet remains below event-specific mixture calibration. Scheduled-jump identification is strongest for CPI and FOMC and weaker for NFP.

We develop a dynamic stochastic model of markets with an externality and multiple trading frictions, and cap-and-trade as the leading application. Slow participation, limited intermediation, and heterogeneous information interact in equilibrium: agents choose costly market access, access determines residual compliance demand, intermediary constraints translate residual demand into a surrender-month premium, and the premium feeds back into access incentives. These interactions shape how effectively the market corrects the externality. We characterize access choices in closed form, prove that the equilibrium premium is unique, and show that endogenous access dampens the response to each friction in isolation, while the interaction of multiple frictions is non-additive and can amplify the price response. We quantify the model using 2.7 million EU ETS registry transactions and compliance records from 2005-2021. About 40% of operators do not trade annually, purchases concentrate in April when returns are systematically high, and operator flow predicts future returns.

This paper presents a comprehensive study on the use of ensemble Reinforcement Learning (RL) models in financial trading strategies, leveraging classifier models to enhance performance. By combining RL algorithms such as A2C, PPO, and SAC with traditional classifiers like Support Vector Machines (SVM), Decision Trees, and Logistic Regression, we investigate how different classifier groups can be integrated to improve risk-return trade-offs. The study evaluates the effectiveness of various ensemble methods, comparing them with individual RL models across key financial metrics, including Cumulative Returns, Sharpe Ratios (SR), Calmar Ratios, and Maximum Drawdown (MDD). Our original experimental results demonstrate that ensemble methods often outperform base models in terms of risk-adjusted returns, providing better management of drawdowns and overall stability. However, both the original analysis and the additional reproduction reported in this version show that ensemble performance is sensitive to the choice of variance threshold \(τ\), classifier group, RL-agent pair, and market universe. The reproduction evidence strengthens the conclusion that classifier-assisted ensemble selection can improve robustness, while also clarifying that the advantage is conditional rather than automatic across all datasets. This study emphasizes the value of combining RL with classifiers for adaptive decision-making, with implications for financial trading, robotics, and other dynamic environments.

Renewable deployment and rising demand from electrification and large digital loads are transforming electricity markets. However, how these developments reshape electricity price dynamics remains poorly understood, leaving system planners, capacity investors, and market participants reliant on assumptions from a thermal-dominated era that may no longer hold. We use causal discovery to study the evolution of wholesale electricity prices in Texas, which is undergoing rapid transformation. Our findings overturn the view of Texas as a gas-price-driven market, demonstrating that wind generation has become the dominant causal driver of day-ahead prices, with effects more than three times greater than those of natural gas. Yet wind's price-suppressing effect is weakening during peak periods, and wind growth redistributes congestion costs to distant load centres. Furthermore, rising load in South and West Texas alters system prices and regional differentials. Uncovering the evolving spatiotemporal nature of causal drivers, our analysis reveals that the pace, geographic siting, and relative scale of new generation and large loads will be decisive for future electricity price risks, infrastructure needs, and investments.

In airport operations, optimally using dedicated personnel for baggage handling tasks plays a crucial role in the design of resource-efficient processes. Teams of workers with different qualifications must be formed, and loading or unloading tasks must be assigned to them. Each task has a time window within which it can be started and should be finished. Violating these temporal restrictions incurs severe financial penalties for the operator. In practice, various components of this process are subject to uncertainties. We consider the aforementioned problem under the assumption of time-dependent stochastic travel times across the apron. We present two binary program formulations to model the problem at hand and propose a novel solution approach that we call Branch-Price-Cut-and-Switch, in which we dynamically switch between two master problem formulations. Furthermore, we use an exact separation method to identify violated rank-1 Chvátal-Gomory cuts and utilize an efficient branching rule relying on task finish times. We test the algorithm on instances generated based on real-world data from a major European hub airport with a planning horizon of up to two hours, 30 flights per hour, and three available task execution modes to choose from. Our results indicate that our algorithm is able to significantly outperform existing solution approaches. Moreover, an explicit consideration of stochastic travel times allows for solutions that utilize the available workforce more efficiently, while simultaneously guaranteeing a stable service level for the baggage handling operator.

We perform a behavioral experiment of karma, a class of mechanisms for repeated resource allocation with attractive fairness and efficiency properties, in theory. Individuals in these mechanisms bid non-tradable credits that flow from resource consumers to yielders, like karma. Human subjects recruited on Amazon MTurk are repeatedly and randomly paired to bid karma according to time-varying and stochastic individual preferences or urgency to acquire resources. Treatments varied in the dynamic urgency process (frequent moderate urgency versus sporadic high urgency) and the richness of the bidding scheme (binary versus full range). Results are benchmarked against random allocation, and karma achieves a (almost) Pareto improvement over random, despite the MTurk subjects deviating significantly from the theoretically optimal Nash bidding policy. Maximum improvement is attained by subjects that deviate from Nash by up to one karma bid unit on average, and positive improvement is attained with average deviations of up to 3-4 bid units. These findings hold across all treatments, among which no significant differences are found, with the exception of the sporadic high urgency process with binary bidding treatment being (weakly) favorable over others. These results offer behaviorally robust lower bounds for the expected performance of karma in human populations. They also provide guidance for future testing and implementation of karma mechanisms in the real world.

In many insurance contexts, dependence between risks of a portfolio may arise from their frequencies. We investigate a dependent risk model in which we assume the vector of count variables to be a tree-structured Markov random field with Poisson marginals. The tree structure translates into a wide variety of dependence schemes. We study the global risk of the portfolio and the risk allocation to all its constituents. We provide asymptotic results for portfolios defined on infinitely growing trees. To illustrate its flexibility and computational scalability to higher dimensions, we calibrate the risk model on real-world extreme rainfall data and perform a risk analysis.

This paper introduces a newly constructed individual-level dataset of prewar Japanese elites using the ``Who's Who'' directories published in 1903--1939. Covering approximately the top 0.1\% of the population, the dataset contains rich information on social group, education, occupation, and family structure. By reconstructing intergenerational links and family networks, we provide descriptive evidence on elite formation and persistence across geography, social groups, and education during transitions from a feudal system to a modern system. We also use family records to document elite marriage patterns and family-based mobility, showing stable age assortative matching, widening husband--wife age gaps, and associations between marriage-age structure, adoption, and elite persistence. The dataset provides a foundational empirical resource for studying intergenerational and intergroup mobility, and institutional development during Japan's transition to a modern society.

Cryptocurrency portfolio management requires the fusion of heterogeneous multi-modal signals, including structured price and on-chain time series, unstructured news text, and technical indicators, under high-volatility and real-time constraints. While deep learning approaches show predictive capability, their opacity limits practical adoption, and single large language model (LLM) agents struggle to process the breadth of modality-specific inputs needed for robust decision-making. We propose a multi-agent system (MAS) framework in which three modality-specialised agents, a Crypto Agent for market dynamics, a News Agent for weekly news sentiment, and a Trading Agent for signal fusion and portfolio execution, decompose the task across three communication architectures: hierarchical, collaborative, and debate. We evaluate four capability configurations: zero-shot, chain-of-thought (CoT), retrieval-augmented generation (RAG), and skill-augmented. In a 52-week backtest over calendar year 2025 across the top 15 L1 blockchain native cryptocurrencies by market capitalisation as of January 2025, the best configuration, Hierarchical (Skill), achieves a cumulative return of 133.52% and a Sharpe ratio of 1.502, outperforming single-agent variants, passive benchmarks, and deep learning baselines. An ablation study identifies the Crypto Agent as the most critical component, with its removal reducing cumulative return by 42.57 percentage points. A cross-model comparison further shows that MAS outperforms the single-agent baseline under GPT-4o, GPT-5, and Claude Sonnet 4.5, suggesting that the benefit of multi-agent coordination is model-agnostic. Unlike black-box deep learning models, every portfolio decision is traceable to explicit agent reasoning, offering an interpretable and effective approach to multi-modal cryptocurrency portfolio management.

A key determinant of monetary-fiscal interactions in Heterogeneous Agent New Keynesian (HANK) models is the liquidity value of public debt and its effect on interest rate dynamics. Yet, while household heterogeneity shapes this channel, it doesn't pin it down. In both analytically tractable and quantitative 2-asset HANK models, asset market assumptions unrelated to standard micro moments give rise to disparate implications of fiscal policy for inflation, as well as model determinacy and fiscal self-financing. To address this issue, I propose a simple model extension and discipline it with macro-level evidence on the relationship between public debt and treasury returns. This moderates the inflationary impact of public debt dynamics but does not render it negligible. After large fiscal shocks, it can still generate persistently elevated ``last mile'' inflation.

Model risk in credit portfolio models is a serious issue for banks but has so far not been tackled comprehensively. We will demonstrate how to deal with uncertainty in all model parameters in an all-embracing, yet easy-to-implement way.

We study Stackelberg Equilibria (Bowley optima) in a monopolistic centralized sequential-move insurance market, with a profit-maximizing insurer who sets premia using a distortion premium principle, and a single policyholder who seeks to minimize a distortion risk measure. We show that equilibria are characterized as follows: In equilibrium, the optimal indemnity function exhibits a layer-type structure, providing full insurance over any loss layer on which the policyholder is more pessimistic than the insurer's pricing functional about tail losses; and no insurance coverage over loss layers on which the policyholder is less pessimistic than the insurer's pricing functional about tail losses. In equilibrium, the optimal pricing distortion function is determined by the policyholder's degree of risk aversion, whereby prices never exceed the policyholder's marginal willingness to insure tail losses. Moreover, we show that both the insurance coverage and the insurer's expected profit increase with the policyholder's degree of risk aversion. Additionally, and echoing recent work in the literature, we show that equilibrium contracts are Pareto efficient, but they do not induce a welfare gain to the policyholder. Conversely, any Pareto-optimal contract that leaves no welfare gain to the policyholder can be obtained as an equilibrium contract. Finally, we consider a few examples of interest that recover some existing results in the literature as special cases of our analysis.