Potentially all pairwise rankings of all possible alternatives

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Potentially All Pairwise Rankings of All Possible Alternatives (PAPRIKA)

What PAPRIKA is
- PAPRIKA is a method for deciding between options when each option is described by several criteria (like education, experience, and references) that each have different levels (like poor or good).
- The method uses people’s preferences expressed as pairwise comparisons (which of two options is better) to derive weights for the criteria. These weights then produce an overall ranking of all possible alternatives.

How PAPRIKA works, in short
- You define a value model: a set of criteria, each with a few performance levels. Each level is assigned a point value that reflects how important that criterion and level are.
- Every possible combined profile (an alternative) gets a total score by adding up its points across all criteria.
- Most pairs of alternatives are “dominated” (one is better in every criterion). Those pairs don’t need judgment. The remaining undominated pairs are the ones that need a human judgment to decide which is preferable.
- PAPRIKA asks only a small, manageable set of undominated pairs. After each explicit ranking, the method uses a rule called transitivity to deduce many other rankings implicitly, reducing the number of questions the decision-maker must answer.
- The process goes from comparing pairs that differ in just two criteria, then to three criteria, and so on, stopping whenever the decision-maker wants.

A simple example: three criteria, two levels each
- Criteria: a, b, c
- Levels: 1 (lower) and 2 (higher) for each criterion
- This yields eight possible alternatives, labeled by the triple of levels (abc): 222, 221, 212, 122, 211, 121, 112, 111.
- The “undominated” pairs (the ones that need ranking) for this simple model are six:
1) b2 + c1 vs b1 + c2
2) a2 + c1 vs a1 + c2
3) a2 + b1 vs a1 + b2
4) a2 + b2 + c1 vs a1 + b1 + c2
5) a2 + b1 + c2 vs a1 + b2 + c1
6) a1 + b2 + c2 vs a2 + b1 + c1
(Here, a2 means Education is high/“2,” a1 is low/“1,” and similarly for b and c.)

How the rankings build the overall order
- If you compare three of these six undominated pairs (in the example, pair 1, pair 2, and pair 5), and you keep the natural ordering constraints a2 > a1, b2 > b1, c2 > c1, you can deduce the rest by transitivity. After ranking these three pairs, you end up with all six undominated pairs ranked.
- From the explicit rankings, you solve a simple linear problem to assign concrete point values to a1, a2, b1, b2, c1, c2. Those values define how important each criterion and level is.
- In the job-candidate example, a likely result is that Experience (b) is the most important, References (c) the next, and Education (a) the least. With these weights, the eight alternatives are ranked from best to worst, e.g., 222 is top, followed by 122, then 221, 212, 121, 112, 211, and 111.

Why PAPRIKA is efficient
- The number of possible pairwise rankings grows very fast as you add criteria and levels (millions or billions of possible pairs in larger models).
- PAPRIKA dramatically cuts the burden by:
- Only asking about undominated pairs.
- Using transitivity to deduce many other rankings from a few explicit ones.
- Stopping early if the decision-maker is satisfied, since many real-world decisions don’t require ranking every possible undominated pair.

Larger value models
- Real-world problems often use many criteria with several levels, which would create enormous numbers of undominated pairs. PAPRIKA uses efficient computational methods to identify the unique undominated pairs and implicitly ranked pairs.
- This approach is implemented in decision‑making software tools (for example, packages like 1000minds and MeenyMo), which handle the heavy lifting behind the scenes.

Where PAPRIKA is used
- PAPRIKA can be used for:
- Multi-criteria decision making (MCDM) to rank or choose among options.
- Conjoint analysis to estimate how much different product attributes matter to people (part-worth utilities).
- It has found applications in health technology prioritization, marketing research, environmental planning, urban planning, and more—where decisions involve weighing many criteria.

In sum
- PAPRIKA is a practical, efficient way to derive a clear, overall ranking of many possible alternatives by asking people to compare only a small, strategically chosen set of undominated pairs. It uses the math of additive value models and transitivity to infer lots of related judgments, reducing the workload while producing consistent, interpretable weights and rankings.