threshold policy design

This paper studies optimal dynamic pricing for time-sensitive goods (e.g., expiring vouchers) in a setting with strategic buyers possessing private valuations and stochastic arrival times. The central challenge lies in the strategic tension between the seller’s urgency to liquidate inventory before expiration and buyers’ incentive to delay purchase in anticipation of price reductions. To address this, we propose a Value-Based Threshold (VBT) policy that decouples the buyer’s two-dimensional private type—valuation and arrival time—and rigorously establish existence and constructive characterization of the Bayesian Nash equilibrium. Leveraging stochastic processes, dynamic game theory, and ordinary differential equations, we develop an analytically tractable equilibrium framework. Numerical analysis reveals: (i) linear discounting is near-optimal in thick markets, whereas fixed pricing dominates in thin markets; (ii) high buyer time-sensitivity favors linear markdowns, while patient sellers benefit from a quasi-auction mechanism—charging premium prices early and steeply slashing prices near expiration. Our results fundamentally characterize how market thickness and seller patience jointly determine optimal pricing structure.

Technical analysis lacks a rigorous mathematical foundation for identifying support and resistance levels, leading to ad hoc and subjective trading rules. Method: We propose a path-dependent three-state stock price model and formulate timing decisions for buying and selling as a coupled optimal stopping problem. Employing probabilistic methods, optimal stopping theory, and multi-state Markovian dynamics, we analyze various price processes under heterogeneous risk-aversion preferences. Contribution/Results: We establish, for the first time, the C¹-continuity of the value function with respect to the scale function and develop a unified framework for jointly solving the dual free boundaries corresponding to support and resistance levels. Our approach yields closed-form or semi-analytical optimal trading strategies. The results significantly enhance the mathematical rigor, theoretical interpretability, and computational tractability of technical trading rules.

This paper addresses how dynamic information design influences agents’ stopping times and action choices in optimal stopping problems, particularly when the principal lacks intertemporal commitment power to induce dynamically consistent stopping behavior. Method: We develop a unified framework that jointly models dynamic persuasion and optimal stopping, grounded in game theory, Bayesian updating, dynamic programming, and stopping-time theory. Contribution/Results: We prove that, for any agent preference structure, there exists an optimal information structure that guarantees dynamic consistency without commitment. This structure endogenously determines the optimal stopping time, state-contingent actions, and the path of information revelation. Our framework provides both a rigorous theoretical foundation and a computationally tractable design paradigm for information manipulation in sequential decision-making contexts—including algorithmic recommendation systems and regulatory interventions.

This study investigates the performance trade-offs of data-driven approaches in finite-horizon dynamic pricing, with a focus on complex settings involving high-dimensional multi-product offerings, heterogeneous demand structures, and intertemporal revenue constraints. By systematically comparing Fitted Dynamic Programming (Fitted DP) against several reinforcement learning algorithms—integrating demand estimation, trajectory sampling, and expectation-based optimization—the work comprehensively evaluates their relative strengths in terms of revenue generation, stability, constraint satisfaction, and computational scalability. The findings reveal that Fitted DP exhibits superior scalability in structured, complex environments, whereas reinforcement learning demonstrates greater flexibility and adaptability. These insights provide both theoretical grounding and empirical evidence to inform method selection for real-world dynamic pricing systems.

This paper addresses identification and estimation in continuous-time dynamic discrete-choice games, focusing on the previously overlooked challenges of endogeneity and heterogeneity in decision arrival rates (i.e., timing of actions). Under the realistic constraint of fixed-interval discrete observations, we are the first to model the arrival rate as an estimable parameter and allow it to vary across agents. Within a Markov perfect equilibrium framework, we derive sufficient conditions for nonparametric identification of the underlying continuous-time primitives—including policy functions, the discount factor, and the distribution of heterogeneous arrival rates—using only discrete-time data. Monte Carlo simulations and empirical application to Rust’s (1987) bus engine replacement data demonstrate the method’s estimation accuracy, robustness, and computational feasibility across sampling frequencies. Results show that neglecting arrival-rate heterogeneity systematically biases behavioral inference, underscoring the model’s significant contribution to structural econometrics and empirical industrial organization.

This study investigates how decision-makers dynamically choose what to explore and when to stop before taking action. By reformulating the dynamic exploration–stopping problem as a static optimization with an information budget constraint, the authors introduce an information shadow price and establish that the optimal policy exhibits a concavified structure of stopping gains net of this shadow price. They precisely characterize, for the first time, the set of attainable joint distributions and uncover the critical role of time preference curvature in shaping exploration: convex preferences induce Poisson-like exploration, whereas concave preferences restrict stopping to specific time windows. The framework also elucidates the emergence of pure exploration phases and is unifiedly applied to real options, speed–accuracy trade-offs, and continuous-time exploration contests, clearly delineating the structural impact of time preferences and information costs on exploration behavior.

This study addresses the pervasive issue of dynamic inconsistency in classical statistical decision rules based on ex ante criteria—such as minimax regret—which often become irrational to follow once data are observed. The paper provides the first systematic characterization of this problem, develops a formal framework for analyzing dynamic consistency, and axiomatically introduces two novel optimality criteria. By integrating decision theory, axiomatic reasoning, and minimax regret analysis, the proposed criteria effectively prevent post-data deviations across a range of empirical settings, thereby ensuring that decision rules remain coherent before and after information updating. This approach rectifies the behavioral discrepancies inherent in existing methodologies and enhances their practical applicability.

This study addresses optimal control in restart-type partially observable Markov decision processes (Restart POMDPs), where the system state is observable only at restart epochs. By introducing a sufficient statistic—comprising the last observed state and the time elapsed since the most recent restart—the problem is transformed into a fully observable MDP. Under both discounted and average cost criteria, the authors establish that when the one-stage cost satisfies a deterioration condition and the transition kernel exhibits monotonicity, the optimal policy admits a time-threshold structure, with thresholds non-increasing in the observed state. The analysis leverages dimensionality reduction via sufficient statistics, stochastic monotonicity, geometric ergodicity, vanishing discount techniques, and uniform boundedness of the relative value function. These results provide a theoretical foundation and a structured policy representation for efficiently solving Restart POMDPs.