Markovian Search with Ex-Ante Constraints: Theory and Applications to Socially Aware Algorithmic Hiring

TL;DR

We address incorporating ex-ante constraints into stateful sequential search with costly inspection, unifying Pandora's box and joint Markov scheduling (JMS) under affine and convex constraints for socially aware algorithmic hiring. The core result is that optimal constrained policies retain index-based structure via dual-adjusted indices and randomized tie-breaking, with polynomial-time computation; for multiple affine constraints, we leverage an exact Carathéodory reduction, and for convex constraints we provide a primal-dual FPTAS (G-RDIP) that yields near-optimal near-feasible policies. A value-specific constraint extension and a broader JMS generalization extend these ideas, showing that index-based adjustments still govern optimal policies, albeit with more nuanced tie-breaking and optimization in the dual space. Numerical experiments reveal that socially aware ex-ante constraints induce only small short-term utilitarian losses while potentially improving long-term outcomes, suggesting practical viability for demographic parity and quotas in hiring pipelines. Overall, the framework offers actionable, economically interpretable dual-adjusted policies and efficient algorithms to balance efficiency with fairness in sequential candidate search and selection.

Abstract

We develop an algorithmic framework to incorporate "ex-ante" constraints on outcomes (that hold only on average) into stateful sequential search with costly inspection. Our framework encompasses the classical Weitzman's Pandora's box [Weitzman, 1979] as well as its extensions to joint Markovian scheduling [Dumitriu et al., 2003; Gittins, 1979], modeling richer processes such as multistage search with multiple layers of inspection. Ex-ante constraints in search are particularly motivated by social considerations in algorithmic hiring, where they adjust outcome distributions to promote equity and access. Building on the optimality of index-based policies in the unconstrained problems, we show that optimal policies under a single ex-ante constraint (e.g., demographic parity) retain an index-based structure but require (i) dual-based adjustments of the indices and (ii) randomization between two such adjustments via a "tie-breaking rule," both easy to compute and economically interpretable. We then extend our results to handle multiple affine constraints by reduction to a variant of the exact Carathéodory problem and providing a polynomial-time algorithm to construct an optimal randomized dual-adjusted index-based policy that satisfies all constraints simultaneously. For general affine and convex constraints, we develop a primal-dual algorithm that randomizes over a polynomial number of dual-based adjustments, yielding a near-feasible, near-optimal policy. All these results rely 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. Finally, through a numerical study, we investigate the implications of various socially aware ex-ante constraints on the utilitarian loss (price of fairness), and examine whether they achieve their intended socially desirable outcomes.

Figures (45)

• Figure 1: The Lagrange dual function $\boldsymbol{\mathcal{G}_{\textsc{cons}}}$ as a function of $\boldsymbol{\lambda}$
• Figure 2: Candidates as Markov reward processes in JMS: The numbers on the states represent rewards, and those on the edges are transition probabilities; $\CIRCLE$ and $\blacklozenge$ denote non-terminal and terminal states, respectively; (a) Pandora's box problemweitzman1979optimal: After inspection, the value is realized from $\boldsymbol{\{V_i\}_{i=1}^4}$, and then the box can be selected; (b) Multi-stage search with rejection (\ref{['ex:reject']}): The candidate passes the phone interview with probability $\textbf{P}\left[\boldsymbol{\texttt{pass}}\right]$. If successful, there is an onsite interview, after which the her value is realized from $\boldsymbol{\{V_i\}_{i=1}^4}$. Lastly, if an offer is made, the candidate accepts it with probability $\textbf{P}\left[\boldsymbol{\texttt{accept}}\right]$.
• Figure 3: (a), (b), and (c): Sample histograms of the generated values $\boldsymbol{\{{v}_i\}_{i\in[1:60]}}$ for the groups $\boldsymbol{\mathcal{Y}}$ (cyan) and $\boldsymbol{\mathcal{X}}$ (black); (d) Distribution of skill scores for all U.S. workers in 1999 bls2017wage.
• Figure 4: Comparing the short-term outcomes of unconstrained and constrained optimal policies.
• Figure 5: Comparing the long-term outcomes of unconstrained and constrained optimal policies.

Theorems & Definitions (35)

• Remark 2: Equality vs. Inequality Constraint
• Proposition 1: Properties of $\boldsymbol{\mathcal{G}_{\textsc{cons}}}$
• Definition 1: Extreme Tie-Breaking Rules
• Proposition 2: Slack Signs for Extreme Rules
• Remark 3
• Theorem 1: Optimal Policy for Constrained Problem
• Example 1: Multi-stage Search with Rejection
• Theorem 2: Approximate Ex-ante Feasibility and Optimality
• Lemma EC.1: Checking for a Binding Constraint
• Proposition EC.1