Maskin rescuing Arrow

Last year, Maskin published a paper arguing 

 that Arrow’s (1951) independence of irrelevant alternatives condition (IIA) is unjustifiably stringent.

It is necessary otherwise preferences could be impredicative and thus any determination of them would be arbitrary or non-unique. In other words, any voting rule applied would be either wholly arbitrary of else indeterminate. Like Arrow-Debreu general equilibrium, the outcomes are 'anything goes'. 

Although, in elections, it has the desirable effect of ruling out spoilers and vote-splitting (Candidate A spoils the election for B if B beats C when all voters rank A low, but C beats B when some voters rank A high - - because A splits off support from B),

Why would this be a desirable property of a voting system? We want to be able to spoil the chances of dangerous but charismatic nutters if any such should arise. 

it is stronger than necessary for this purpose. Worse, it makes a voting rule insensitive to voters’ preference intensities.

So what? Voters can express their preference intensities by running amok or threatening to do so.  

Accordingly, we propose a modified version of IIA to address these problems.

The bigger problem is that there are no desirable properties for a voting system save that it already exists and things haven't already totally turned to shit in the sense that administrations change from time to time without large scale violence. 

Rather than obtaining an impossibility result, we show that a voting rule satisfies modified IIA, Arrow’s other conditions, May’s (1952) axioms for majority rule, and a mild consistency condition if and only if it is the Borda count (Borda 1781), i.e., rank-order voting. 

Sadly, this isn't true.  

In his monograph Social Choice and Individual Values (Arrow 1951), Kenneth Arrow introduced the concept of a social welfare function (SWF) – a mapping from profiles of individuals’ preferences to social preferences.

But 'individuals' preferences' are intensional and epistemic. Thus, they have no non-arbitrary or unique representation. There is no actual mapping here of a mathematical type. We may speak of Sociology or Social Psychology or Culture and affiliation as determining electoral outcomes but, it will also be the case, those outcomes won't matter very much. Public policy will be largely determined by economic and geopolitical considerations which are independent of preferences. 

No doubt, if politics didn't matter at all and voting in general elections was like voting for sexiest Soap Opera star then Borda might come into its own. But nobody would greatly care.  

The centerpiece of his analysis was the celebrated  Impossibility Theorem, which establishes that, with three or more social alternatives, there exists no SWF

No SWF ever exists save by arbitrary stipulation. Nor does any individual's preference function. Both are essentially epistemic and indeterminate. Why? The menu conveys information and thus changes the knowledge base and thus the 'extension' of any given 'intension' in welfare econ. 

There is a further problem of impredicativity relating to implementation of SWFs. Our preferences may change if we think certain Social Welfare configurations won't be implemented because preferences may change if they are. Now, it may be possible to arrive at an arbitrary 'equilibrium' but it would be non unique. Indeed, actual politics is about passing laws which everybody knows won't be fully implemented. Indeed- as with Prohibition- the thing may be so badly implemented that it is wholly counter-productive.

satisfying four attractive conditions: unrestricted domain (U), the Pareto Principle (P), non-dictatorship (ND), and independence of irrelevant alternatives (IIA). Condition U requires merely that a social welfare function be defined for all possible profiles of individual preferences (since ruling out preferences in advance could be difficult).

So my support for Trump might be conditional on the proof that P=NP while that for Biden on the proof that this isn't so. Preferences are epistemic. But the time class of calculating the general equilibrium under different economic policies would be exponential. Thus Preferences are approximate and ad hoc. The Law of large numbers means this doesn't matter too much if we are speaking of millions of voters. But, otherwise, there could be deterministic chaos. Arrow restricts the SWF to deterministic functions. 

P is the reasonable requirement that if all individuals (strictly) prefer alternative x to y, then x should be (strictly) preferred to y socially as well.

Robert Aumann points out that the Sanhedrin had a rule against unanimity.  We move closer to the Pareto frontier when the information set expands and/or Transaction costs fall such that more arbitrage occurs. If everybody always prefers x to y, that may be a reason to try y. However, the real objection to 'P' is that we need Society to do certain things even if no voter wants those things done- e.g. killing invaders. More importantly, we may want constitutional checks and balances such that unanimity on a particular issue is not action guiding. There is a 'cooling off' period.

ND is the weak assumption that there should not exist a single individual (a “dictator”) whose strict preference always determines social preference.

What if some guy always happens to vote for the winning side? Why should he be described as a Dictator when he is nothing of the sort?  

These first three conditions are so undemanding that virtually any SWF studied in theory or used in practice satisfies them all.

None do. Unrestricted domain is impossible. Pareto means there can be no voting system because there could be unanimity not to implement it- which is a choice voters must have under U- but this also means there will be no voting system because it is always possible that those who dispute the outcome vigorously enough can put society on a slippery slope to that very unanimity. 

For example, consider plurality rule (or “first-past-the post”), in which x is preferred to y socially if the number of individuals ranking x first is bigger than the number ranking y first.

Plurality rule satisfies U

No. The majority of voters may want a guy who won't or can't stand. Equally, they may prefer candidate X but only if he represented Party Y and vice versa. No voting rule is operating on an unrestricted domain of voter preferences. Moreover voting is going to have a strategic component precisely because of restricted domain. We may also speak of 'substitution' and 'income' effects which are menu dependent. In other words, a given U has no determinate relationship with any restriction on Domain. 

because it is well-defined regardless of individuals’ preferences.

No. It is intensional- i.e. epistemic. The extension is arbitrary or by stipulation. People will say 'X only won because people were against Y and didn't get a chance to vote for Z' or things of that sort. 

It satisfies P because if all individuals strictly prefer x to y, then x must be ranked first by more individuals than y. 

Not if x can't or won't stand or is disqualified or dies.  

Finally, it satisfies ND because if everyone else ranks x first, then even if the last individual strictly prefers y to x, y will not be ranked above x socially.

Everyone writes in for Obama except me because I know Obama can't have a third term. My vote puts Biden in the White House. But this isn't the real problem with ND. There is bound to be some guy in Peoria who has always voted for the winning candidate since the time of Truman. Is he a Dictator?  

Turning to the mathematical meat of the paper, we find

The Arrow conditions for a SWF F are:

Unrestricted Domain (U): The SWF must determine social preferences for all possible preferences that individuals might have.

I prefer Biden to Trump iff there is a proof that P not equal to NP.  How is this evaluated by the SWF? The fact is our preferences are conditional. There is a 'characteristic space' and our support for a candidate is conditional on something in that space. How is the SWF to evaluate this? As a matter of fact, political scientists do 'factorize' the platforms of different candidates along socio-economic and ideological lines and try to work out whether this can give them a majority given what is known about the constituency. But candidates themselves use this type of analysis so you might have a Hotelling type convergence to what would be supported by 'the median voter'. 

Formally, for all [0,1], i i ∈ ℜ consists of all strict orderings of X.

There are no such orderings because X is unknown and some elements of it are ambiguous, impredicative or uncertain.  

Pareto Property (P): If all individuals (strictly) prefer x to y, then x must be strictly socially preferred. 

All individuals may prefer to have good Social Choice by magic or the intervention of providence without the bother and expense of holding an election. The thing is a 'second best' solution to minimize social conflict. Just as a harmonious society with optimal mechanisms wouldn't have much need of Law Courts, so too would it dispense with Elections and Parliaments.  

Nondictatorship (ND): There exists no individual who always gets his way in the sense that if he prefers x to y, then x must be socially preferred to y, regardless of others’ preferences.

Yet such an individual always exists. I and I alone get to decide whether I will scratch my arse when it feels itchy to me. Also, actual Dictators don't get to choose any outcome they like. This is because their bodyguards may shoot them. More generally, polities have to choose to do the things which enable them to survive as polities. Otherwise they get conquered or implode into anarchy. 

 Independence of Irrelevant Alternatives (IIA): Social preferences between x and y should depend only on individuals’ preferences between x and y, and not on their preferences concerning some third alternative

We don't have any such preferences. True, I prefer eating a burger to eating stewed eel but if the third alternative is that my family is butchered in front of my eyes, I choke down the stewed eel. All our preferences are like that. They depend on a ceteris paribus condition. Yet, new knowledge constantly causes us to discover that ceteris is not paribus. Smoking may look cool but it fucks up your lungs. 

Because we have argued that IIA is too strong, we are interested in the following relaxation: Modified IIA: If, given two profiles and two alternatives, each individual (i) ranks the two alternatives the same way in both profiles and (ii) ranks the same number of other alternatives between the two alternatives in both profiles, then the social preference between x and y should be the same for both profiles

Two people agree that someone should do the washing up. But the one assigned the task may change his mind. The mistake Maskin is making is to think that alternatives can be fully specified. They can't. Everything is connected. You may say 'you can vote for either Biden or Trump' but what we actually do is vote for an imagined outcome that we like. If things don't pan out the way we think they would we renege. Now there be some enforcement mechanism. But where did it come from? Was it itself created by voting? What about the creators of that creation? Were they empowered by voting? At some point there was some arbitrary, non-voting based, intervention. In that case voting rules are historical and hysteresis based. They are not ergodic or mathematical.

May (1952) characterizes majority rule axiomatically in the case X = 2 .

In which case there is a third alternative which is not voting at all.  

We will consider natural extensions of his axioms to three or more alternatives:

in which case there are an infinity of alternatives e.g. not voting but saying 1 instead, not voting but saying 2 instead, not voting but saying 3 instead... 

Anonymity (A): If we permute a preference profile so that individual j gets i’s preferences, k gets j’s preferences, etc., then the social ranking remains the same.

Very true. You will definitely take the trouble to vote if you are told your attendance at the booth will enable the preferences of your lazy Trump-loving neighbour to be registered.  

 Neutrality (N): Suppose that we permute the alternatives so that x becomes y, y becomes z, etc., and we change individuals’ preferences in the corresponding way.

Suppose word of this gets out. There is a hue and cry. Trumpistas burn your house down because you are obviously part of the conspiracy which stole the fucking election! 

Positive Responsiveness (PR) 28: If we change individuals’ preferences so that alternative x moves up relative to y in some individuals’ rankings and doesn’t move down relative to y in anyone’s ranking, then, first, x moves up socially relative to y; second, it does so continuously.

If such a mechanism were implemented for some x, it will become a 'wedge issue' and a site of strategic preference falsification.

 Ranking Consistency (RC): If, given a profile of individual preferences, each of a set of disjoint subpopulations has the same strict social ranking, then the (unique) top-ranked alternative for that ranking is also the (unique) top-ranked alternative for the union of those subpopulations.

If these genuinely are disjoint sets then it must be possible to unambiguously assign individuals to each. But, if that were the case, why not let representative agents of the disjoint sets engage in a bargaining game? Corporatism is the solution, not Democratic voting. 

 We are not aware of a SWF actually used in practice that fails to satisfy RC.

No SWF is actually used in practice. I suppose Maskin means 'theoretical SWFs' but they are all vitiated by the intensional fallacy.  

Indeed, RC holds for almost any standard SWF studied in the literature.

No SWF can be implemented because everybody wants to have an above average IQ. Stuff we all agree on is not feasible.  

As mentioned in Section 1C, RC is satisfied by all scoring rules – including plurality rule and the Borda count – because these rules satisfy the much stronger requirement of consistency.

Which is met by the assumption that there is a well-ordering on the set of finite rooted trees. However the 'reverse mathematics' type results this gives rise to would themselves need a 'Divine Axiom' for consistency. However Unrestricted Domain requires completeness. Also, the SWF is restricted to deterministic computability. Thus either the SWF can only be computed 'at the end of mathematical time' or else it is incomplete, inconsistent or arbitrary. 

The Borda count does have obvious advantages but in 'rich domains' (i.e. where there is no clear 'best' or 'worst') first past the post may be less, not more 'strategy proof'. Still, because of the intensional fallacy, this entire line of inquiry is vitiated.