Does anybody know a MDCM system that is immune to Rank Reversal, that is:
* Change in a ranking when an alternative, worse than others is introduced in the problem?
* Comparing alternatives or projects?
* Deleting alternatives or projects?
* Replacing alternatives or projects?
* Breaking down the problem in pair of alternatives and comparing the ranking obtained for them with the ranking when the same alteratives are consisdered in the complet set?, that is transitivity
Rank reversal occurs because the collection of rankings, called a profile, contains what Don Saari calls Condorcet n-tuples. For example, a Condorcet triple is A ≻ B ≻ C, B ≻ C ≻ A, and C ≻ A ≻ B where ≻ means “is preferred to.” Any positional voting scheme aggregating this set of rankings will yield a tie. If one of the candidates is removed or another is added, however, the tie is broken. If a profile contains Condorcet n-tuples, it can result in inconsistencies of this type. An excellent reference is Donald G. Saari, Disposing Dictators, Demystifying Voting Paradoxes: Social Choice Analysis, 2008, Cambridge University Press: New York.
Note that any attempt to avoid rank reversal generally involves discarding critical information provided by the rankings, for example, looking only at the relative position of two alternatives and ignoring any that are in between. In other words, when comparing A and B, failing to differentiate between A ≻ B ≻ C ≻ D and A ≻ C ≻ D ≻ B. This can result in an overall or aggregate ranking that is nontransitive or irrational! Saari has examples.
This issue is related to a property called Independence of Irrelevant Alternatives (IIA). Saari showed that this property negates another property, transitivity of outcome, and substitutes one called Intensity form of Independence of Irrelevant Alternatives (IIIA) that considers not only the relative position of two candidates but also some measure of the strength of preference. In the case of the Borda Count, the number of alternatives between the two being considered is the measure of strength of preference.
I refer you to a paper I coauthored entitled “Deflecting Arrow by Aggregating Rankings of Multiple Correlated Criteria According to Borda.” It appears in the Journal of Multi-Criteria Decision Analysis http://onlinelibrary.wiley.com/doi/10.1002/mcda.1568/full.
Article Deflecting Arrow by Aggregating Rankings of Multiple Correla...
Dear Khrisnendu
Thank you for your answer, Naturally, if you add an alternative that is better than all the alternatives in a ranking, it will go to the first place. But the order of the balance of the alternatives in the ranking shouled not change. In this case there is not RR.
For instance, if the original ranking is C>A>D>E and if you add alternative G that is better tan any of the existent in the ranking, it should be G>C>A>D>E
Then there is not RR. However, if the addticon of G chaqnges the original ranking as it happens some times, as for instance G>E>D>A>C, then you have RR.
From my point of view the problem is:
How do you know that an added alternatvie is better or worse tan the others in the ranking?
By comparing its contribution to benefit or to cost?
This is not enough , because you need to analyze how the whole vector of the added alternative interacts with all the vectors of the exsiting alternatives.
This can only be done by Linear Programming which bases its rational in comparing opportunity costs of the added alternative with those of the existing.
This is the reason why a problem solved with LP, with only one objective function, reaches an optimal solution that cannot be improved because it is Pareto effcient
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