Field expansion/Group Contraction fallacies in Arrow's theorem

Arrow's theorem is mathematically equivalent to Brouwer fixed-point or Sperner lemma or HEX theorem. Sadly, actual Social Choice does not feature any mathematical sets, connected spaces or relation algebra. However, it can feature 'Muth Rational Expectations'- i.e. a coordination game or collective action problem is solved by applying the correct economic theory which may have a mathematical description- i.e. a canonical form. But, even in that case, the 'intensions' used to define sets would only have fixed extensions if the correct theory was known a priori with certainty. An incorrect theory would yield absurdities very quickly- e.g. proving that a cat is only a cat if it is a dog or a Dictator is only a Dictator if nobody knows he is a Dictator because he isn't a dictator at all. 

Arrow's Mathematical theory of Social Choice is not 'Muth rational'. It is foolish and yields paranoid results. The thing is like using Pythagoras's theorem to prove that getting involved in a love triangles increase the size of your dick. 

Arrow's theorem says Democracy is actually a Dictatorship and if you buy your kids ice-cream even though you wanted to buy them a healthy dessert, then you will inevitably end up buying them guns.

In this post I briefly describe two fallacious lemmas- 'field expansion' and 'group contraction'- which arise because 'weakly decisive' is actually a misnomer. It is a stronger condition than 'decisive'.  

Wikipedia gives the following account of Arrow's theorem- 

Let A be a set of outcomes,

We don't know outcomes. We may know who is standing for election or what proposition is being put to a particular plebiscite but we don't know what the outcome would be of a particular person or proposition getting more votes. Thus A is not a set. Its elements are not well-defined.

On the other hand if there was a 'correct economic theory' which everybody would have a strong incentive to adopt, then that theory has a mathematical representation in which there is no intensional fallacy because extensions are unique and well-defined. 

N a number of voters or decision criteria.

We don't know who will vote or what decision criteria will have salience. Also, though voters are distinct, we can't be sure that some may not vote at another's direction. 

We denote the set of all total orderings of A by L(A).

A set is defined as a collection of well-defined distinct elements. In a total order any two elements are comparable. But the total ordering of any voter or decision criteria could depend on the total order of some other voter or decision criteria and vice versa. This means that elements of the set may be impredicative in a manner which makes them uncomputable- i.e. not well-defined and distinct. Thus, suppose I want what you want but you want what I want. Take it a step further and have me want what you want me to want not because it is what you want but because I should want that for reasons of my own. Clearly there is an infinite regress here. There may be several arbitrary solutions which are good enough but there is no unique solution. Thus, some 'total-orderings' aren't well defined or distinct and thus there can be no 'set' of the sort that is assumed.  On the other hand, if everybody has the same Muth rational economic theory (or deviations from it are normally distributed) then a mathematical representation of that theory would yield a set of total orderings. But that is an artefact of the theory. Actual Social Choice would be unproblematic. It would be like the way the vast majority of us are able to walk down the street without bumping into each other. It would be something we take for granted.

An ordinal (ranked) social welfare function is a function which aggregates voters' preferences into a single preference order on A.

A function f from a set X to a set Y is an assignment of one element of Y to each element of X.

But we have just seen that L(A) is not a set. It may be, under very restricted conditions- e.g. nobody giving a fuck about what anybody else wants and there is perfect information, no externalities etc. But where there is voting, this generally isn't the case. 

The other problem is that Social Welfare isn't just about outcomes for Society, it's also about how people feel about how Society got there. This means that voters may be indifferent between two different outcomes, provided they are arrived at differently, and thus several elements of X may map to the same element of Y. Indeed, they may all do so. It is perfectly possible that if we had an infinity of time to reflect on what everybody ought to prefer on the basis of a proper regard for other preferences, then there may be some 'bliss-point'- a Social Arrangement which everybody considers the best, the fairest, the most just, the most humane etc. But, equally, if we had an infinite amount of time to reflect, we might decide there is no such thing as life or society or language or choice. Indeed, the Universe itself is an illusion. 

An N-tuple (R1, …, RN) ∈ L(A)N of voters' preferences is called a preference profile.

Speaking generally there will be no fucking N-tuple because of impredicativity- i.e. I want to choose what the smart people are choosing. They don't want to waste their time choosing something thickos like me won't get. There's a 'Keynesian Beauty contest' aspect to this. Anyway, the very fact that there is some social process of deciding things itself yields utility or disutility or alters preferences.  

We assume two conditions:Pareto efficiency 

That's mad. Nobody knows at what point no future mutually beneficially trade can be made or how and when concurrency or hold out problems will be resolved. Assuming Pareto efficiency means assuming you know there is nobody in the world who wouldn't swap something you want for something you have but don't value. Indeed, it is a more extreme assumption because it assumes you can always find a way to make the trade without 'hold out' in the bargaining.  

If alternative a is ranked strictly higher than b for all orderings R1 , …, RN, then a is ranked strictly higher than b by F(R1, R2, …, RN). This axiom is not needed to prove the result,[13] but is used in both proofs below.

This is silly. As individuals we may all want a convicted felon to become President. But, it is a probably a good thing that we can't get the outcome we want. Democracy can be great but Democracy under the Rule of Law is likely to be even better. 

Non-dictatorship There is no individual i whose strict preferences always prevail.

There might be a guy who always votes for the winner. Equally, there might be a wise sage whose advise the majority always listens to.  Neither is a Dictator. 

That is, there is no i ∈ {1, …, N} such that for all (R1, …, RN) ∈ L(A)N and all a and b, when a is ranked strictly higher than b by Ri then a is ranked strictly higher than b by F(R1, R2, …, RN).

A Dictator is a guy who kills those who oppose him. He does not hold elections. Non-Dictatorship already holds if elections decide matters. A wise sage whom people listen to is not a Dictator.  

Then, this rule must violate independence of irrelevant alternatives: Independence of irrelevant alternatives 

Which is violated in any social process where there is Tardean mimetics or information asymmetry or externalities or 'Knightian Uncertainty' etc.  

For two preference profiles (R1, …, RN) and (S1, …, SN) such that for all individuals i, alternatives a and b have the same order in Ri as in Si, alternatives a and b have the same order in F(R1, …, RN) as in F(S1, …, SN).

We may all prefer an illegal option but are secretly relieved it is off the menu.  

Formal proof

You can't prove anything using mathematics about things which aint mathematical objects.  

Definition:

• A subset of voters is a coalition.

The word coalition suggests an alliance constituted by some horse-trading or a mutual enemy. This word is being misused just as the word 'Dictator' was misused. If it is known that actual coalitions exist, then preference orderings change in a strategic manner. So not only is there not a set in the general case, still, in the special case where there is a set, the existence of a subset means that there is no longer a set. In other words, if coalitions exist, you have counter-coalitions and a struggle for 'Agenda Control' and McKelvey Chaos.

• A coalition is decisive over an ordered pair ${\displaystyle (�,�)}$ if, when everyone in the coalition ranks ${\displaystyle �{\succ }_{�}�}$, society overall will always rank ${\displaystyle �\succ �}$.

Coalitions don't work that way. There is horse-trading within them and there can always be further horse-trading or persuasion. A decisive coalition may give up something they can implement over a given (x,y) for a compromise over some other (a,b)

 

• A coalition is decisive if and only if it is decisive over all ordered pairs.

In which case it could also force everyone to be a part of itself. Don't forget that who votes for what is itself part of Social Choice.

Our goal is to prove that the decisive coalition contains only one voter, who controls the outcome—in other words, a dictator.

Because it assumes preferences aren't strategic. The problem here is that even actual Dictators face constraints of various sorts. What they prefer is different from what they actually choose. Thus, what these cretins are actually doing is proving that nobody at all wants any outcome of Social Choice though, no doubt, by coincidence there might be some people who always vote for the winner.

The following proof is a simplification taken from Amartya Sen^[17] and Ariel Rubinstein.^[18] The simplified proof uses an additional concept:

• A coalition is weakly decisive over ${\displaystyle (�,�)}$ if and only if when every voter ${\displaystyle �}$ in the coalition ranks ${\displaystyle �{\succ }_{�}�}$, and every voter ${\displaystyle �}$ outside the coalition ranks ${\displaystyle �{\succ }_{�}�}$, then ${\displaystyle �\succ �}$.

If their actions are not coordinated, then there is no coalition.

Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes.

Field expansion lemma — if a coalition ${\displaystyle �}$ is weakly decisive over ${\displaystyle (�,�)}$ for some ${\displaystyle �\ne �}$, then it is decisive.

i.e. if the kids vote for ice-cream and you give in, your kids can also get you to buy them guns. The fact is kids are only decisive over ice-cream because they aren't weakly decisive. The fact is you like ice-cream and only mentioned the healthy desert option because you wanted to be able to boast about this to their Mum. 

The sleight of hand here involves pretending that 'weakly decisive' is weaker than 'decisive' whereas the opposite is the case. Thus, you are decisive when you refuse to buy guns for kids. But you aren't 'weakly decisive' because some of the kids don't really want guns. 

This lemma is false. Suppose you vote against whatever your enemy votes for. You are part of G only because it is weakly decisive- i.e. excludes your enemy. Here G can only be weakly decisive it is not decisive. 

Proof

Let ${\displaystyle �}$ be an outcome distinct from ${\displaystyle �,�}$.

Z is buying guns for your kids. X is your taking them for ice-cream. Y is your taking them for a healthy dessert. You give in to their 'voting' and buy them ice-cream. You now have to buy them guns. 

Claim: ${\displaystyle �}$ is decisive over ${\displaystyle (�,�)}$.

The kids get you to buy them ice-cream because they decide that asking for guns might be counter-productive

Let everyone in ${\displaystyle �}$ vote ${\displaystyle �}$ over ${\displaystyle �}$. By IIA,

it is irrelevant that buying kids guns is illegal 

 changing the votes on ${\displaystyle �}$ does not matter for ${\displaystyle �,�}$. So change the votes such that ${\displaystyle �{\succ }_{�}�{\succ }_{�}�}$ in ${\displaystyle �}$ 

This is illicit. Kids genuinely would prefer guns to a 'healthy dessert'. 

and ${\displaystyle �{\succ }_{�}�}$ and ${\displaystyle �}$ outside of ${\displaystyle �}$.

You would rather give your kids a healthy dessert than either ice-cream or guns. 

By Pareto, ${\displaystyle �\succ �}$. 

No. It may be the case that giving the kids guns is the one thing which saves the whole family from a predator. We don't know what is or isn't Pareto efficient. 

By coalition weak-decisiveness over ${\displaystyle (�,�)}$, ${\displaystyle �\succ �}$. Thus ${\displaystyle �\succ �}$. 

This assumes transitivity. But preferences aren't transitive. They aren't even anti-reflexive. We don't necessarily prefer our own preferences which, in any case, change as the knowledge base changes. What is being committed here is an 'intensional fallacy' of great stupidity. 

Similarly, ${\displaystyle �}$ is decisive over ${\displaystyle (�,�)}$.

It isn't. The kids aint gonna get guns however much they prefer them to a healthy dessert. 

By iterating the above two claims (note that decisiveness implies weak-decisiveness), we find that ${\displaystyle �}$ is decisive over all ordered pairs in ${\displaystyle \{�,�,�\}}$. Then iterating that, we find that ${\displaystyle �}$ is decisive over all ordered pairs in ${\displaystyle �}$.

This is like saying, if you give the kids ice-cream, you will have to buy them guns. It is stupid shit. 

Group contraction lemma — If a coalition is decisive, and has size ${\displaystyle \ge 2}$, then it has a proper subset that is also decisive.

One of the kids is the ring-leader. But for that ring-leader, the other kids would have wanted the healthy desert. You have created a monster who will compel you to buy guns for kiddies all over the world. 

The fallacy in this 'lemma' is that a coalition may be decisive only because it is weakly decisive. The moment it stops being weakly decisive- e.g. your enemy changes his vote- it stops being decisive because it has lost a sub-set it previously had. 

Proof

Let ${\displaystyle �}$ be a coalition with size ${\displaystyle \ge 2}$. 

Like the kids who 'vote' for ice-cream rather than a healthy dessert

Partition the coalition into nonempty subsets ${\displaystyle {�}_1,{�}_2}$.

Fix distinct ${\displaystyle �,�,�}$. Design the following voting pattern (notice that it is the cyclic voting pattern which causes the Condorcet paradox):

(Items other than ${\displaystyle �,�,�}$ are not relevant.)

Since ${\displaystyle �}$ is decisive, we have ${\displaystyle �\succ �}$. So at least one is true:  or . If , then  is weakly decisive over .

by arbitrary stipulation regarding 'voters outside G.' Some of them may have preferred y over x but z over y. 

 If , then  is weakly decisive over .

by arbitrary stipulation

 Now apply the field expansion lemma.

You can't. It is fallacious. 

By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma,

which is fallacious

 there is a size-one decisive coalition—a dictator.

And yet no such thing exists. 

Another approach is to identify a pivotal voter whose ballot swings the societal outcome. We then prove that this voter is a partial dictator (in a specific technical sense, described below). Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator.

This is a simple 'intensional' or 'masked man' fallacy. The 'extension' of the intension 'pivotal voter' is unknown. No voter in a big enough election is actually pivotal. Moreover, if known to be pivotal that voter would not be pivotal. This is like the notion of hidden dictators. But why stop there? We may speak of the cat which on any specific day utters the miaow which permits the multiverse to continue to exist. One can misuse math to show that such a cat exists. But it doesn't really. In any case, identifying that cat and causing it to stop saying miaow won't destroy the multiverse because some other cat is now pivotal.