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Extending choice assessments to choice functions: An algorithm for computing the natural extension

Arne Decadt, Alexander Erreygers, Jasper De Bock

TL;DR

This work extends decision-making beyond single optimal choices to the framework of choice functions, introducing the natural extension as the unique, most conservative coherent extension compatible with a given assessment. It develops a practical algorithmic pipeline that reduces consistency checks and extension calculations to tractable linear feasibility problems via IsFeasible, and it introduces systematic generator simplifications (conjunctive and disjunctive) that preserve the induced set of coherent orders. The authors demonstrate, through extensive experiments, that these simplifications significantly improve scalability, enabling consistent assessments of realistic size and imprecision, and show that a fully simplified approach often outperforms more conservative strategies for large problems. The resulting methodology provides a principled, implementable path to inferring new choices under uncertainty, with potential applications in multicriteria decision making, risk assessment, and collaborative or group decision processes.

Abstract

We study how to infer new choices from prior choices using the framework of choice functions, a unifying mathematical framework for decision-making based on sets of preference orders. In particular, we define the natural (most conservative) extension of a given choice assessment to a coherent choice function -- whenever possible -- and use this natural extension to make new choices. We provide a practical algorithm for computing this natural extension and various ways to improve scalability. Finally, we test these algorithms for different types of choice assessments.

Extending choice assessments to choice functions: An algorithm for computing the natural extension

TL;DR

This work extends decision-making beyond single optimal choices to the framework of choice functions, introducing the natural extension as the unique, most conservative coherent extension compatible with a given assessment. It develops a practical algorithmic pipeline that reduces consistency checks and extension calculations to tractable linear feasibility problems via IsFeasible, and it introduces systematic generator simplifications (conjunctive and disjunctive) that preserve the induced set of coherent orders. The authors demonstrate, through extensive experiments, that these simplifications significantly improve scalability, enabling consistent assessments of realistic size and imprecision, and show that a fully simplified approach often outperforms more conservative strategies for large problems. The resulting methodology provides a principled, implementable path to inferring new choices under uncertainty, with potential applications in multicriteria decision making, risk assessment, and collaborative or group decision processes.

Abstract

We study how to infer new choices from prior choices using the framework of choice functions, a unifying mathematical framework for decision-making based on sets of preference orders. In particular, we define the natural (most conservative) extension of a given choice assessment to a coherent choice function -- whenever possible -- and use this natural extension to make new choices. We provide a practical algorithm for computing this natural extension and various ways to improve scalability. Finally, we test these algorithms for different types of choice assessments.
Paper Structure (32 sections, 38 theorems, 62 equations, 9 figures, 1 table, 8 algorithms)

This paper contains 32 sections, 38 theorems, 62 equations, 9 figures, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Choice functions
  3. Examples
  4. Lower expectations
  5. Decision-making based on lower expectations
  6. Consistency and the natural extension of a choice assessment
  7. Introducing consistency and natural extension
  8. Alternative characterisations of the set of preference orders of the assessment
  9. Consistency and natural extension for generators
  10. Consistency and natural extension for generator sets
  11. Connecting back to generators
  12. Practical methods for finite generator sets
  13. Checking if an option belongs to the natural extension for finite G
  14. Consistency and natural extension in practice
  15. An algorithmic approach
  16. ...and 17 more sections

Key Result

Lemma 2.3

For any preference order $\prec$, $G_{\prec}$ is a coherent set of desirable options. Moreover, the map $\prec\mapsto G_{\prec}$ is a bijection between the set of preference orders and the set of coherent sets of desirable options, with inverse $G\mapsto \prec_G$.

Figures (9)

  • Figure 1: Comparison of the number of option sets in the disjunctive generator of an assessment $\mathcal{A}_{1:\ell}$ of size $\ell$ (horizontal axis), each pair $(V,W)\in \mathcal{A}_{1:\ell}$ containing between two and eight options in total, using the first method (), the second method () or the third method (). The results are the average of 7 individual experiments and plot with logarithmic vertical axes.
  • Figure 2: Comparison of the number of option sets in the disjunctive generator of an assessment $\mathcal{A}_{1:\ell}$ of size $\ell$ (horizontal axis), each pair $(V,W)\in \mathcal{A}_{1:\ell}$ containing between two and eight options in total, and constructed using $C_{\text{lin}}$ (), $C_{\text{max}}$ (), $C_{\text{adm}}$ () or $C_{\text{imp}}$ (). The results are the average of 7 individual experiments and plot with logarithmic vertical axes.
  • Figure 3: Comparison of the number of option sets in the simplified disjunctive generator $\mathcal{G}_{\square}$ of an assessment $\mathcal{A}_{1:\ell}$ of size $\ell$ (horizontal axis), each pair $(V,W)\in \mathcal{A}_{1:\ell}$ containing between two and eight options in total, and constructed using $C_{\text{lin}}$ (\ref{['groenkotje']}), $C_{\text{max}}$ (\ref{['roodkotje']}), $C_{\text{adm}}$ (\ref{['blauwkotje']}) and $C_{\text{imp}}$ (\ref{['paarskotje']}). The results are the average of 7 individual experiments and shown in a non-logarithmic plot.
  • Figure 4: Total size of the conjunctive generator of an assessment with size 10 as a function of $\epsilon$, the amount of contamination of an expectation with the vacuous model, with and without simplifications. The first method, without simplifications, is indicated by . The second method, with simplifications, is indicated by . (The third method has the same conjunctive generator as the second.) The results are the average of 100 individual experiments.
  • Figure 5: Total size of the disjunctive generator of an assessment with size 10 as a function of $\epsilon$, the amount of contamination of an expectation with the vacuous model, using three different methods. The first method is indicated by , the second method is indicated by and the third by . The results are the average of 100 individual experiments and plot with a logarithmic vertical axis.
  • ...and 4 more figures

Theorems & Definitions (74)

  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 3.2
  • Definition 3.4
  • Proposition 3.6
  • Lemma 3.7
  • proof
  • ...and 64 more