# 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.