Terence Tao & the imbecility of Arrow's theorem

The 'masked man' or 'intensional fallacy' arises when we treat something which can be named as if it is already known. We can give a name- e.g. 'optimum'- to the best possible arrangement. But this does not mean we know what that arrangement is. We may say, to our current knowledge it is x, but if more is discovered about y, we know it won't be x. In other words, the thing that is named changes as the knowledge base changes.

If we speak of 'the set of optima for a polity over time', we are not speaking of a set whose elements can be specified. The 'extension' is unknown though the 'intension' appears clear. Thus the set is not well defined. In fact, it isn't a set at all. No mathematical operation can be performed with it. 

Terence Tao, with his usual lucidity, sees that Arrow's 'paradox' arises from assuming 'decisive coalition' (which he terms 'quorum') is well defined. Yet this can't be the case if we assume (as he does) 

1)  it is rational for voters to have a total ordering over candidates 

More precisely, rationality means choosing the best total order from all possible total orders. However, a total ordering of candidates can only robustly obtain if voters don't have a total ordering of possible social states. This is because, in politics, 'uncorrelated asymmetries' can change in the blink of an eye. We may want a particular candidate, for some personal reason, to implement a particular social state that she does not currently propose to do. Tactical voting is possible in this case. The politician's advisers will explain to her that she has to change her platform if she wants to get elected. It frequently happens that a politician's 'face fits' with a platform which is the opposite of what they thought it strategic to support. 

A different point is that comparability is only required for one pair of candidates for any given voter. The requirement for a total ordering is onerous. When we make a choice, we are only concerned with the next best alternative foregone. Why should this be considered 'irrational'? Information is costly to acquire. There is little benefit in getting to know that the candidate of the Monster Raving Loony Party is less mad than that of the Giant Spaghetti Monster party. 

Notice that if political parties exist, then candidates may not be comparable in one respect- i.e. which party they belong to- that is an uncorrelated asymmetry. Thus, if a candidate changes his party affiliation, or a voter changes her party affiliation, then the total ordering chosen also changes. But, surely, 'party affiliation' is not independent of the preferences of other voters? This in turn raises the possibility that whatever total ordering obtains or is imputed to a voter, it is not really based on the total ordering she would choose on the candidate space. Indeed, her only choice may be to stick with one party or defect from it. This could be extended to other traits- e.g. gender, race, etc. In India, people are said to vote their caste not cast their vote. Ackerlof has explained why this may be rational under information asymmetry.

A different problem has to do with what rationality means. Is it procedural? In that case the total ordering chosen by a rational voter must be 'constructible'. If it is constructible is it 'lawlike' or 'lawless'? If it is 'lawlike' then surely it is based on the attributes of the candidates or their expected actions or some such criteria. But, if this is the case, choice isn't really about candidates but about something associated with the candidates. This means candidates can do things to change the total ordering. But, in that case, the total ordering is a work in progress. It seems a choice is made about when to stop deciding and make your pick. But this means that rationality is a criteria about when to decide as much as it is about what decision to make or else it isn't procedural at all. Of course, one could simply say- well, the total ordering procedure could be 'lawless' for all we care. But, by 'the axiom of open data', this means we can't know we are proceeding along the 'rational' choice sequence which, by the 'axiom of density', must exist in a pre-specifiable way. In other words our intuition of rationality, in this context, is empty. We are simply deeming the outcome to be rational no matter what it is.

The same problem arises if we say rationality isn't procedural but is substantive with respect to the 'end of time'. But if this rationality is displayed by a 'creating subject'- i.e. has a representation as a choice sequence- then anti-symmetry is violated and thus no total order obtains because a mathematical object has changed over time. Of course, we could say we are not concerned with whether rationality is procedural or substantive. We are only concerned with showing that there are problems with aggregating rational choices. However we have no rational means of proving anything with respect to this assertion. All that we have shown is that when we treat 'Tarskian primitives' as 'well defined', then we have committed a stupid blunder. 

It may be rational for voters to rank some candidates but it is not rational to have a total ordering of candidates just so something else we need to make our argument could be called 'well defined'. Why? Because we have no means to prove that assertion independently. All we have done is begged the question with a vengeance. Thus we are no better off. 

2) such ranking is independent of that of other other voters. 

When a voter offers herself as a candidate- this affects other voters. Similarly, if a candidate withdraws, and endorses another candidate, this may have a big affect on voters. Surely, the preferences of the voter-who-can-be-a-candidate, to be or not be a candidate, influences the preferences of others?

Suppose a voter decides to run as a 'strategic candidate'- i.e. to split the vote or to convey some information to other voters. Such a voter may not want to win but also may not want to lose her deposit or suffer a big loss of face. Then she will vote for herself as may some other people despite the fact that they don't really want her

 Tao's notion that we should have a total ordering over candidates is reasonable if it costs enough money to be a candidate and fear of losing your deposit deters nutters and publicity hounds. But is it reasonable to say that the outcome of an aggregation procedure should yield a total ordering? We may be interested in knowing the 'runner up' in an election. But do we really want to know how each candidate would fare with respect to every other? I suppose, if candidates are ranked purely on 'objective' grounds- e.g. technical knowledge- then you could have an exam which ranks all candidates (except those who get the same mark). But this isn't a voting system. It is an examination system. 

Tao stipulates for 'determinism'- presumably this rules out non-deterministic or 'NP' solutions- e.g. consulting an oracle or expert with superior intuition. Is that rational? Anything biological must involve coevolved processes on an uncertain fitness landscape such that 'solutions' are NP though 'verification' is P.  Furthermore, for robustness, no biological or 'bio-political' mechanism should be deterministic because it could then be 'hacked' by a predator or parasite. In any case, it would be sub-optimal because of known concurrency, complexity and computability problems. That's why what actually obtains is the 'Consensus' that no deterministic preference or judgment aggregation procedure should be implemented. Instead we have a complicated system of checks and balances and 'logrolling' and rent-contestation and Coalition formation and deformation and so on and so forth. 

Tao, with admirable economy, shows the tension between 'voter rationality' and 'voter freewill'- for Arrow's paradox to work voters have to have something irrational- viz a total ordering- and must get it by irrational means (i.e. independently of others, though society is all about communication and mutual adjustment)

Is there a 'perfect voting system'? Sure, if the thing makes any difference at all, there is always a way, by backward induction, to get something everybody involved would agree was the perfect voting system at that time and place- though this might involve 'transferable utility'- i.e. pay-offs. Moreover, after the fact- i.e. once the human race had died out- there would always be a 'harmonious construction' permitting 'univalent foundations' for a sufficiently rich mathematical language encoding a perfect, type theoretic, voting rule. It would feature 'decomposability' because after Humanity has disappeared, there would be a 'slingshot' function with this property. True it may be in a computational class exponential to the time frame of this Universe. Still, there is an existence proof for it. However, this would be no way to show it wasn't mere correlation, not causation nor that the thing was unique. Indeed, such solutions may be non-denumerably infinite. 

A simple example of the problems in a voting system occurs when there is only one voter and one candidate. Basically, you start to feel a bit of a fool for electing yourself President of the Cosmos, more particularly when, quite suddenly, you lose the election to NOTA. 

If there is a post of 'Chairperson', presumably there is a Committee of some sort tasked with some particular matter. If these guys can't get along and achieve anything worthwhile by working together, the Committee should be dissolved. A voting rule can't change this outcome. It may be better for this Committee to be dissolved immediately. Graciella Chichilnisky's work shows that where Preference or Endowment diversity is too little or too great there may be no point in having any type of coordinating mechanism. However, a coercive mechanism of a dictatorial kind may prevail. Killing or chasing away people is one way to solve problems of Social Choice. 

It is perfectly possible that ordinary people like me could have a judgment re. the validity of the Mochizuki proof of the abc conjecture that is just as good as Terence Tao's. But would any mathematician really want the Condorcet Jury theorem to prevail in determining whether or not proofs are valid? 

The fact is, voting rules fail the 'base case'- i.e where there is only one voter and one option. Why? The thing is foolish because the option must be 'don't bother with voting rules'. For any n peeps who are doing something sensible, they will tell the n+1th person to fuck off if their inclusion prevents that sensible thing being done. What is that sensible thing? It is to say 'voting rules' have no magical properties. True, Arrow was paid by RAND during the Cold War but, so what?, so was Isaac Asimov who invented 'psychohistory'. 

Tao illuminates the notion of a 'decisive coalition'.

Tao is using 'quorum' for what is termed 'weakly decisive coalition' in the literature. However, it is not well defined. There may be some who only vote with the quorum if some one outside it votes the other way. Terence thinks 'independence' and 'impartiality' gets rid of this problem. But this is not the case. Some unknown mechanism may produce the same result. You would have to introduce an infinite number of axioms to use the notion of 'quorum' using 'field expansion' and 'group contraction' to get Arrow's insane result. 

Another way to look at it, is to say if 'quorum' is well defined then 'determinism' is violated because

1) either members know they are part of the quorum- in which case they know they can change the output, at least stochastically, by defecting- in which case the assumption re. voter total ordering is violated.

or 

2) they don't know they are part of the quorum- in which case 'voter free will' is violated- they can't really chose the best order given their preferences. Why? The voters have been disallowed information which would have changed their total ordering. 

The truth is, if 'quorum' is well defined then something silly- viz that voters have access to the Muth rational ordering they ought to have- is 'asserted'. But it can't be proved- so 'quorum' isn't well defined at all.  Thus, showing that a contradiction arises is trivial. We don't know the 'correct economic theory' which, if common knowledge, would yield Muth Rational Expectations based pure strategies in small sample spaces- forget about those with strategic behavior. Maybe we could, if we knew how to unfailingly distinguish pseudo-random from random strings. But in that case, move over Evolution!, we can very quickly fulfil the Biblical injunction 'ye are as Gods', not by being scrupulous Jury members, but by actually becoming Gods and creating really nice Universes. 

Tao illuminates the stupidity of imagining 'coalitions' without 'transferable utility'- i.e corrupt deals in smoke filled rooms- can spontaneously arise out of a stupid type of mathematics. Either a 'preorder' is associated with a directed graph or it does not exist. But, if there is a directed graph, then all 'preorders' must, to be rational, at the very least, have 'maximal' properties w.r.t Kantorovich-Monge type problems. 

This is not a lemma. It is an unproven assertion of a silly type. It means- if at such and such time these guys decide things and then at such and such other time those guys decide things then it must be the case that the guys who are members of both decisive sets at different times are actually SECRETLY IN CONTROL! I heard about this guy in Peoria who has always voted for the winning candidate in every election. He must be the head of the Lizard People who control everything!

The fact is we all may at some times in our lives be on the winning side of an election. Some of us may even always be on the winning side. But this doesn't mean anything very sinister at all. 

Of course, if we add in crazy assumptions- which basically amount to giving us super-powers- then we can find very strange and sinister results. But this involves pretending something is 'well-defined' when we haven't actually given a proof of that assertion.