The theorem, a converstional demonstration
In its demonstration, Arrow uses a very precise formalism that would be overly detailed to be used here. Instead let’s try to understand the reasoning behind the theorem with more general considerations.
First, Arrow introduces the definition of decisive set..
To make it short, if in a competition between Aldo and Bruno won Aldo, then the whole of those who voted Aldo over Bruno is a decisive set. There may also be many more votes than those that would be needed for Aldo to win. Arrow also defines a decisive minimum set. That is, a decisive set which, if you remove a single voter from it, it is no longer decisive.
The law of social choice obviously should not be confined to considering only systems with two candidates; not only would it be too easy but the first axiom would exclude it.
Thus we have so many voters and so many candidates and the concept of decisive set not only concern the winner of the general election competition but can be applied also to avery preference between any two candidates. So if we consider Aldo winning over Bruno there will be the decisive assembly for Aldo; if we consider Bruno winning over Carlo, there will be the decisive assembly for Bruno and so on for all combinations.
An elector may have A>C>B preferences and can therefore be part of the decisive assembly for Aldo because for him A>B but having a preference for Carlo he is therefore out of the decisive assembly for Bruno against Carlo.
For every decisive set there is a complementary set made up of all voters who chose the “losing” option. And given any pair of candidates, each elector is either in the decisive or the opposite assembly (complementary).
Now let’s do a simulation whose actors are the usual three candidates: Aldo, Bruno and Carlo. In addition we will have a set of Winners that we suppose minimal decisive and a set of Losers, who are all those who voted against the winner. In addition, let’s suppose that the Winners set is composed by one “Mister X” plus a set of other Generic voters.
Let’s assume that the distribution is that shown in the table.
From the table we see, as we expected from its definition, that W is decisive for A>B, because in both columns we find A>B.
Also G taken alone can not be decisive because W is minimal, removing Mister X what is left (that is, G) is no longer decisive beacuse it was minimal by definition.
Let’s consider now the G set. About it, we know that its members prefer C over B. Unfortunately for them this is not enough to make the C>B preference a global law because both Mister X and the L set think differently.
And it could not be enough the weight of G to make C win over B because it would make the G set decisive and W not minimal, while we supposed the opposite.
In definitive, globally, it can not be C>B and must therefore be B>C. Combining this last result with what we know from earlier, the preference chain is: A>B>C.
This preference chain exists because we have supposed that the order of preferences are both linear and transitive. If we removed this constraint, we would find the Condorcet paradox again.
Let’s note now that Mister X also prefers A over C, as opposed to all the others and this makes him decisive for the A, C couple. This contradicts the assumption that W is minimal and therefore the only possibility is that in the W set there is only one element: Mister X.
With this reasoning Arrow shows us that, given the axioms we have talked about, there is always an elector that is decisive for at least one preference between two alternatives.
In the continuation of the demonstration he uses the axiom of unanimity to show that Mister X is decisive for any of the choices between two alternatives. This makes him a dictator.
By definition, the dictator is the elector whose opinion is the one that, in every possible alternative, becomes global law over all the others.
Then what we called Mister X, if the axioms of the theorem are fulfilled, exists and is a dictator. Or if we consider the axiom of non-dictatorship, we come to an absurd and we must agree that there is no perfectly democratic system that satisfies all the axioms of Arrow.
The impossibility theorem puts in relation several properties that we would reasonably expect to be respected in a perfectly democratic system and show that they are incompatible with each other. To have a realistic system you need to get exceptions, relax the request of one or more axioms and accept the consequences, from Condorcet’s paradox, to Borda counting, and even the birth of a dictatorship.