Minimizing Volatility: Optimal Adjustment with Evolving Feasibility Constraints

2026-02-17

AI Summary

This study investigates how a decision-maker selects actions to minimize adjustment costs or action variance under stochastic dynamic feasibility constraints. By integrating dynamic optimization, stochastic process analysis, and convex optimization theory, the authors derive two classes of optimal policies: under strictly convex adjustment costs, the optimal rule minimizes the distance to a target action, which—when constraints follow a random walk—coincides with the previous period’s action; under a variance-minimization objective, the optimal policy follows a constant-target rule. The proposed approach substantially outperforms the commonly used midpoint heuristic, reducing quadratic variation and variance by over 50% in natural settings. Furthermore, it offers a unified explanation for reference price phenomena observed in contexts such as stock auctions, over-the-counter trading, supply chain capacity planning, and political agenda setting.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Heuristic Search
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Minimizing volatility and adjustment costs is of central importance in many economic environments, yet it is often complicated by evolving feasibility constraints. We study a decision maker who repeatedly selects an action from a stochastically evolving interval of feasible actions in order to minimize either average adjustment costs or variance. We show that for strictly convex adjustment costs (such as quadratic variation), the optimal decision rule is a reference rule in which the decision maker minimizes the distance to a target action. In general, the optimal target depends both on the previous action and the expectation of future constraints; but for the special case where the constraints follow a random walk, the optimal mechanism is to simply target the previous action. If the decision maker minimizes variance, the optimal policy is also a reference rule, but the target is a constant, which is not necessarily equal to the long-term average action. Compared to mid-point heuristics, these optimal rules may substantially reduce quadratic variation and variance, in natural environments by $50\%$ or more. Applied to stock market auctions, our results provide an explanation for the wide-spread use of reference price rules. We also apply our results to bilateral trade in over-the-counter markets, capacity planning in supply chains, and positioning in political agenda setting.

Problem

• volatility minimization
• adjustment costs
• feasibility constraints
• decision making
• stochastic constraints

Innovation

• reference rule
• volatility minimization
• evolving feasibility constraints
• adjustment costs
• optimal targeting