Proof: The group decision in the Condorcet case is indifference, so that B = C. Under the axiom of universality we can look at various preference profiles, of which Condorcet’s example is only one. Now regard the adjusted profile such that the preferences on B and C remain the same, but the preference on A drops to the lowest position. The new profile thus is {A < B < C, A < C < B, A < B < C}. Since the preferences on B and C have not changed, the APDM outcome on B and C should be the same. Majority voting now however results into B < C which differs from B = C. Contradiction. Thus there is a counterexample to the axioms. So the axioms are inconsistent. Q.E.D.
The merit of this short proof is that it clearly shows the awkwardness of the APDM. In the case of Condorcet’s example the conclusion B = C is a sound decision, and in the case of the adjusted example the conclusion B < C is sound too. That preferences outside of the pair B and C have changed is vital to the group decision, since the shift helps a change from clear indifference to clear preference. The preferences on other topics are quite relevant, and not ‘irrelevant’. APDM excludes vital information about the preferences - to be precise: it destroys information that exists - and it should come as no surprise that paradoxes and inconsistencies arise. The APDM is incongruent with the notion of group decision making. Perhaps an individual can exclude information about other topics, but a group cannot. (Or a brain that works as a group cannot.) It is a surprise that APDM has not been killed right in 1951.
A note on the name of APDM
Arrow (1951, 1963) introduced an axiom “Independence of Irrelevant Alternatives” (AIIA) that has caused much misunderstanding. That axiom here has been baptised the “Axiom of Pairwise Decision Making” (APDM). Thus the axiom remains the same, only the name is different. The new name is much clearer about what the axiom really means in normal English.
Since the name “IIA” is so entrenched in the literature, this change of name requires some explanation. The explanation is along the lines:
· There is the distinction between voting and deciding.
· Items that cause cycles cannot be called ‘irrelevant’ for decision making.
· The criterion to separate the relevant items from the irrelevant ones is rather the budget and is not necessarily found in pairwise voting for all items.
Arrow's axioms on using the whole commodity domain and universal preferences introduce the possibility that we might also be obligated to consider farfetched items. Arrow introduced the APDM to limit this effect again, since it allows that a decision on our current issues can be taken independently from other farfetched possibilities. It is reasonable that people neglect farfetched possibilities. Thus Arrow on one hand opens the door wide for such farfetched possibilities, and on the other hand introduces a strict condition that kills the relevance of this. The whole looks reasonable, since people in fact neglect farfetched possibilities.
Yet, the whole does not conform with the practical situations in Parliaments, where the problem is defined for existing voters and where the issues on table are given by the budget set.