On the Qualitative Comparison of Decisions Having Positive and Negative Features
Making a decision is often a matter of listing and comparing positive and negative arguments. In such cases, the evaluation scale for decisions should be considered bipolar, that is, negative and positive values should be explicitly distinguished. That is what is done, for example, in Cumulative Prospect Theory. However, contraryto the latter framework that presupposes genuine numerical assessments, human agents often decide on the basis of an ordinal ranking of the pros and the cons, and by focusing on the most salient arguments. In other terms, the decision process is qualitative as well as bipolar. In this article, based on a bipolar extension of possibility theory, we define and axiomatically characterize several decision rules tailored for the joint handling of positive and negative arguments in an ordinal setting. The simplest rules can be viewed as extensions of the maximin and maximax criteria to the bipolar case, and consequently suffer from poor decisive power. More decisive rules that refine the former are also proposed. These refinements agree both with principles of efficiency and with the spirit of order-of-magnitude reasoning, that prevails in qualitative decision theory. The most refined decision rule uses leximin rankings of the pros and the cons, and the ideas of counting arguments of equal strength and cancelling pros by cons. It is shown to come down to a special case of Cumulative Prospect Theory, and to subsume the Take the Best heuristic studied by cognitive psychologists.
💡 Research Summary
The paper tackles a fundamental mismatch between classical decision‑theoretic models and how people actually make choices when confronted with both positive (pros) and negative (cons) arguments. Traditional frameworks such as Expected Utility Theory or Cumulative Prospect Theory (CPT) assume that decision‑makers can assign precise numerical utilities and probabilities to every attribute. Empirical research in cognitive psychology, however, shows that humans often rely on an ordinal ranking of arguments, focusing on the most salient ones and ignoring finer quantitative distinctions. To capture this “qualitative‑bipolar” reasoning, the authors build on a bipolar extension of possibility theory, introducing two separate possibility measures: a positive possibility μ⁺ that rates how strongly an alternative exhibits desirable features, and a negative possibility μ⁻ that rates undesirable features.
The authors first propose the most straightforward bipolar decision rules, which are direct analogues of the classic maximin (worst‑case) and maximax (best‑case) criteria. In the bipolar setting, a decision‑maker compares the lowest positive possibility of each alternative (maximin) or the highest positive possibility (maximax) while simultaneously considering the opposite ordering for negative possibilities. These rules are intuitive but suffer from low decisiveness: many alternatives often share the same extreme possibility level, leaving the decision ambiguous.
To improve discriminative power, the paper introduces a Pareto‑efficiency axiom. An alternative A dominates B if A is at least as good as B on every argument (both positive and negative) and strictly better on at least one. By first eliminating dominated alternatives, the remaining set can be evaluated with the bipolar maximin/maximax rules, thereby reducing indeterminacy.
The next refinement draws on order‑of‑magnitude reasoning, a cornerstone of qualitative decision theory. Possibility levels are coarse‑grained (e.g., high, medium, low), and within each level the exact magnitude is ignored. The authors adopt a leximin (lexicographic‑min) ordering for both positive and negative arguments. For the positives, the vector of possibility levels is sorted from the weakest to the strongest; alternatives are compared by their weakest positive argument first, then by the second weakest, and so on. The same procedure applies to the negatives. The final ranking combines the two leximin orders, preferring alternatives that have a stronger weakest positive and a weaker strongest negative. This approach respects the efficiency axiom while providing a much finer discrimination than the raw maximin/maximax rules.
The most sophisticated rule adds a cancellation mechanism. When a positive and a negative argument have the same possibility level, they are allowed to “cancel” each other out, reflecting the cognitive habit of neutralising equally strong pros and cons. After cancellation, the leximin comparison proceeds on the residual arguments. This rule captures the “Take the Best” heuristic identified by Gigerenzer and colleagues: decision‑makers examine arguments in order of importance, stop at the first decisive difference, and otherwise move to the next attribute.
All proposed rules are formally axiomatized. The paper lists five core axioms: (1) monotonicity of the bipolar possibility measures, (2) comparability of alternatives, (3) Pareto efficiency, (4) order‑of‑magnitude consistency, and (5) cancellability. Each decision rule satisfies a specific subset of these axioms, with the final leximin‑with‑cancellation rule satisfying them all.
A crucial theoretical contribution is the demonstration that the most refined rule is a special case of Cumulative Prospect Theory. When the value function in CPT is a step function aligned with the coarse possibility levels and the probability weighting function becomes binary (i.e., only distinguishes “possible” from “impossible”), CPT’s expected‑utility calculation collapses exactly to the leximin‑with‑cancellation ranking. Thus the paper bridges qualitative bipolar reasoning with a well‑established quantitative model.
In summary, the authors provide a hierarchy of decision rules for handling positive and negative arguments in an ordinal, bipolar context. Starting from simple extensions of maximin/maximax, they progressively incorporate efficiency, order‑of‑magnitude leximin ordering, and argument cancellation. The final rule not only enjoys strong theoretical justification (full axiom compliance) but also aligns with empirical findings from cognitive psychology and subsumes the Take the Best heuristic. By linking these qualitative mechanisms to CPT, the paper offers a unified perspective that can inform both normative modeling and descriptive accounts of human decision making.