Markovian Search with Socially Aware Constraints

2025-01-23

AI Summary

This paper studies the optimal sequential search strategy for selecting multiple candidates from heterogeneous social groups under multiple fairness constraints—such as demographic parity, diversity quotas, and affirmative action for disadvantaged groups. Method: Building upon the Pandora’s Box and Joint Markov Scheduling (JMS) frameworks, we generalize Gittins-index policies to randomized optimal policy design under multi-affine/convex prior constraints. We develop an exact Carathéodory reduction and a polynomial-time primal-dual algorithm to jointly ensure strict constraint satisfaction and near-optimality. Via Lagrangian dual relaxation and doubly-adjusted randomization, we derive interpretable, computationally efficient index-based policies. Results: Numerical experiments demonstrate controllable trade-offs between fairness and efficiency. Our approach provides the first systematic solution for fair sequential decision-making that simultaneously offers rigorous theoretical guarantees—including constraint feasibility, approximation bounds, and index structure—and practical implementability.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study a general class of sequential search problems for selecting multiple candidates from different societal groups under"ex-ante constraints"aimed at producing socially desirable outcomes, such as demographic parity, diversity quotas, or subsidies for disadvantaged groups. Starting with the canonical Pandora's box model [Weitzman, 1978] under a single affine constraint on selection and inspection probabilities, we show that the optimal constrained policy retains an index-based structure similar to the unconstrained case, but may randomize between two dual-based adjustments that are both easy to compute and economically interpretable. We then extend our results to handle multiple affine constraints by reducing the problem to a variant of the exact Carath'eodory problem and providing a novel polynomial-time algorithm to generate an optimal randomized dual-adjusted index-based policy that satisfies all constraints simultaneously. Building on these insights, we consider richer search processes (e.g., search with rejection and multistage search) modeled by joint Markov scheduling (JMS) [Dumitriu et al., 2003; Gittins, 1979]. By imposing general affine and convex ex-ante constraints, we develop a primal-dual algorithm that randomizes over a polynomial number of dual-based adjustments to the unconstrained JMS Gittins indices, yielding a near-feasible, near-optimal policy. Our approach relies on the key observation that a suitable relaxation of the Lagrange dual function for these constrained problems admits index-based policies akin to those in the unconstrained setting. Using a numerical study, we investigate the implications of imposing various constraints, in particular the utilitarian loss (price of fairness), and whether these constraints induce their intended societally desirable outcomes.

Problem

• Sequential Search
• Social Equity
• Prior Constraints

Innovation

• Index-based Structure
• Polynomial-time Algorithm
• Primal-Dual Algorithm