# Optimal Counterfactual Explanations for Scorecard modelling

arXiv: https://arxiv.org/abs/2104.08619 · HTML: https://mlsanity.com/paper/2104.08619/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 16 references that are in the map.
- First posted on arXiv 17 Apr 2021.

## Builds on (most cited first, 8 of 8)

- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Multi-Objective Counterfactual Explanations](https://mlsanity.com/paper/2004.11165/) (2020)
- [Efficient Search for Diverse Coherent Explanations](https://mlsanity.com/paper/1901.04909/) (2019)
- [Optimal binning: mathematical programming formulation](https://mlsanity.com/paper/2001.08025/) (2020)
- [Multivariate outlier detection based on a robust Mahalanobis distance with shrinkage estimators](https://mlsanity.com/paper/1904.02596/) (2019)

## Built on by (most cited first, 6 of 6)

- [Achieving Diversity in Counterfactual Explanations: a Review and Discussion](https://mlsanity.com/paper/2305.05840/) (2023)
- [Generating collective counterfactual explanations in score-based classification via mathematical optimization](https://mlsanity.com/paper/2310.12822/) (2023)
- [Counterfactual Explanations Using Optimization With Constraint Learning](https://mlsanity.com/paper/2209.10997/) (2022)
- [Designing User-Centric Behavioral Interventions to Prevent Dysglycemia with Novel Counterfactual Explanations](https://mlsanity.com/paper/2310.01684/) (2023)
- [GlyTwin: Digital Twin for Glucose Control in Type 1 Diabetes Through Optimal Behavioral Modifications Using Patient-Centric Counterfactuals](https://mlsanity.com/paper/2504.09846/) (2025)
- [Counterfactual Explainable AI (XAI) Method for Deep Learning-Based Multivariate Time Series Classification](https://mlsanity.com/paper/2511.13237/) (2025)

## Abstract (arXiv)

Counterfactual explanations is one of the post-hoc methods used to provide explainability to machine learning models that have been attracting attention in recent years. Most examples in the literature, address the problem of generating post-hoc explanations for black-box machine learning models after the rejection of a loan application. In contrast, in this work, we investigate mathematical programming formulations for scorecard models, a type of interpretable model predominant within the banking industry for lending. The proposed mixed-integer programming formulations combine objective functions to ensure close, realistic and sparse counterfactuals using multi-objective optimization techniques for a binary, probability or continuous outcome. Moreover, we extend these formulations to generate multiple optimal counterfactuals simultaneously while guaranteeing diversity. Experiments on two real-world datasets confirm that the presented approach can generate optimal diverse counterfactuals addressing desired properties with assumable CPU times for practice use.