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Optimal Matching of Random Parts

Thomas Alois Weber

This paper examines the minimization of the cost for an expected random production output, given an assembly of finished goods from two random inputs, matched in two categories. We describe the optimal input portfolio, first using the standard normal approximation of the binomial classification distributions, and second using a tight concave envelope instead of the exact output objective. The latter approach yields closed-form expressions for the factor demands and total costs which are linear in the expected output and which approximate the solution to the original minimum-cost matching problem for sufficiently large production batches. A key structural insight is that depending on the ratio of input prices, one of the inputs should be considered as “critical component” while the other assumes the role of a “buffer component.” As long as the cost ratio does not reach a critical threshold, which is proportional to the ratio of the grade-attainment likelihoods, the relative composition of the optimal input portfolio remains largely invariant. A numerical study confirms the practicality of the envelope approach, both as a seed for a numerical solution of the exact optimality conditions and as an approximate solution in closed-form.

2022

Relatively robust decisions

Thomas Alois Weber

It is natural for humans to judge the outcome of a decision under uncertainty as a percentage of an ex-post optimal performance. We propose a robust decision-making framework based on a relative performance index. It is shown that if the decision maker's preferences satisfy quasisupermodularity, single-crossing, and a nondecreasing log-differences property, the worst-case relative performance index can be represented as the lower envelope of two extremal performance ratios. The latter is used to characterize the agent's optimal robust decision, which has implications both computationally and for obtaining closed-form solutions. We illustrate our results in an application which compares the performance of relative robustness to solutions that optimize worst-case payoffs, maximum absolute regret, and expected payoffs under a Laplacian prior.

SPRINGER2022

Optimal recovery of unsecured debt via interpretable reinforcement learning

Huanxi Liu, Michael Mark, Thomas Alois Weber

This paper addresses the issue of interpretability and auditability of reinforcement-learning agents employed in the recovery of unsecured consumer debt. To this end, we develop a deterministic policy-gradient method that allows for a natural integration of domain expertise into the learning procedure so as to encourage learning of consistent, and thus interpretable, policies. Domain knowledge can often be expressed in terms of policy monotonicity and/or convexity with respect to relevant state inputs. We augment the standard actor–critic policy approximator using a monotonically regularized loss function which integrates domain expertise into the learning. Our formulation overcomes the challenge of learning interpretable policies by constraining the search to policies satisfying structural-consistency properties. The resulting state-feedback control laws can be readily understood and implemented by human decision makers. This new domain-knowledge enhanced learning approach is applied to the problem of optimal debt recovery which features a controlled Hawkes process and an asynchronous action–feedback relationship.

2022