From my doctoral course
This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).
The problem
In most organisational decisions one person does not decide: several experts or stakeholders each see the attributes and alternatives differently. The question of group decision making is: how do we turn several people’s opinions into one ranking or choice without one person stealing the others’ opinion?
Two approaches in the course
- Group agreement: if the experts agree on the attributes and alternatives, the data are unified first and the decision is made afterwards.
- Individual evaluation: if there is no agreement, each expert evaluates separately and then the results are combined. (The same applies if the attributes are shared but the weights differ.)
Ordinal combination methods
The judgements are ranks (ordinal) and we want to build a group ranking. Three well-known techniques: Borda, Bernard, and Cook and Seiford. The simplest is Borda.
Borda count: with alternatives, first rank gets points, second , … last zero; the sum of points decides. (Jean-Charles de Borda, 1770.) Cook and Seiford’s method picks the group rank that minimises the total distance from the individuals’ ranks.
A warning: Condorcet’s paradox
Three experts, four alternatives:
| Expert | Ranking |
|---|---|
| 1 | A > B > C > D |
| 2 | B > C > A > D |
| 3 | C > A > B > D |
- By Borda count: A , B , C , D 0; a three-way tie.
- Pairwise comparison: A over B (two votes to one), B over C (two to one), C over A (two to one): a vicious cycle! The group majority has an intransitive preference, while each individual is transitive. This is Condorcet’s paradox (1785).
Arrow’s impossibility theorem
Kenneth Arrow (1951) proved that for more than two alternatives there is no ranked aggregation method that satisfies several reasonable conditions at once: unrestricted domain (any order allowed), unanimity (if all prefer A to B, the group says the same), independence of irrelevant alternatives, and no dictator. That is, a “perfectly fair method of combining votes” does not exist. I think the key lesson is that choosing the method is itself a political decision.
Combining numerical judgements
When the judgements are numbers:
- Arithmetic or geometric mean (for pairwise-comparison ratios the geometric one is right, because it preserves reciprocity; see AHP: the analytic hierarchy process).
- Delphi: several rounds of anonymous polling until the experts converge.
My suggestion
For important decisions: 1) agree on the attributes first, 2) take evaluations independently (to avoid the anchoring effect), 3) run the combination with two methods, 4) when the results differ, discuss, rather than calling one “right”.
Related: Introduction: what is multi-attribute decision making? and Decisions under uncertainty: fuzzy numbers.