1 Introduction

This study examines situations in which the evaluation of a finite set of alternatives is required, involving the evaluations of these alternatives by a number of individuals. Examples illustrating this situation include a faculty considering several candidates for new members, or a company evaluating various project proposals. In such organizations, decisions are often influenced by the evaluations submitted by their members.

Then, some individuals should not or cannot evaluate specific alternatives for several reasons such as those as follows. First, they may lack sufficient information or knowledge to evaluate these alternatives. For example, evaluating potential faculty members may require expertise in their specific field. Second, an individual is known to be biased in favor of their family members, colleagues, themselves, their own ideas, and the projects they were previously involved in. As a result, they may not be well-suited to provide objective evaluations of these alternatives. Third, planners may encounter difficulties in obtaining the complete evaluations from all individuals. For example, when new alternatives are added after some individuals have already submitted their evaluations, there may be no opportunity to gather their evaluations for the newly added ones. Consequently, the evaluations of these individuals may not be incorporated into the collective evaluation of the alternatives.

2 Model and technical results

We consider a model that is originally introduced by Ohbo et al. (2005). LetA be the set of finite alternatives. LetVbe the set of finite individuals (voters) who evaluate alternatives. We assume|A| ≥3 and|V| ≥3.

ForA' ⊆A,R⊆A'×A' be a binary relation onA'. If (a, b)∈R, then we write aRb. LetP(R) andI(R) be the asymmetric part and the symmetric part of a binary relation onA' denoted byR; that is...

This implies that if a binary relation is asymmetric and acyclic, then there is a linear order extension of the binary relation.

However, ifRis asymmetric and acyclic forA', then there is a linear order extension ofRforA'.

First, we derive the necessary and sufficient condition for the evaluability profile, ensuring the existence of a collective evaluation function that satisfies transitive valuedness and the Pareto criterion.

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