Arrow’s theorem: continuation of the demonstration

Continuing the demonstration in the article at this link, we demonstrate that the voter we called Mister X, besides being definitely decisive for a given choice A>B, it is so for any alternative between the candidates.

In the first part of the demonstration, we saw that Mister X, which we could now call “the dictator“, is decisive for A over B.
Let’s see what happens if we consider the alternative A and Q where Q is any candidate whatsoever.

X L
A B
B Q
Q A

 

We already know that X is decisive for A over B and globally must be A>B.
For the unanimity axiom must be B>Q because it is preferred by both X and L.
For the transitivity property, A>B>Q implies A>Q.
Following the chain A wins over Q despite L disagree, so it turns out that X is decisive in getting A winning over Q.

Now let’s consider the dictator’s opinion by considering C and any alternative that is not A.

X L
C Q
A C
Q A

 

We’ve established above that A>Q.
Because of the unanimity axiom C>A (they all agree on this, both X and L).
For the transitivity property C>A>Q it implies that C>A.
Again this reflects the will of the dictator, not the one of L thus X is decisive in the comparison between Q and C.

Finally consider the case where C and A are in the presence of another Q candidate.

X L
C Q
Q A
A C

 

We know from above that C>Q.
Because of the unanimity axiom Q>A.
Being C>Q>A, the transitivity property implies that C>A.
Once again C>A reflects the will of the dictator against that of everyone else, so X is decisive also in this case.