What is the base case for a Social Choice function?
Robinson Crusoe. An individual can be a Society if he chooses to think of himself as composing a Society- in other words, Crusoe sees himself as part of something bigger than himself. But, precisely because a Society's preferences are different from a Class or Clan or a Family's preferences, Crusoe's preferences as an individual must be different from his Society's preferences. Also, any goats or trees or possible future Crusoes, or possible personalities of Crusoe, have different preferences from their own Social preferences. In other words, even in the base case, Social Choice
1) is not a function of individual preferences which are themselves so impredicative as to have no unique or natural (non arbitrary) representation. Indeed, neither individual preferences, nor their Social preferences, nor Social preferences are 'intensions' with a knowable, well defined 'extension'. The intensional fallacy arises when we stipulate otherwise. To do so is to act arbitrarily.
2) There is no 'categoricity' or unique model and there is no 'naturality'. No sets or functions (i.e. sets of ordered pairs) or graphs of functions exist save by arbitrary stipulation. One may as well speak of unicorns or mermaids and how they interact with Crusoe. Forget about imprimitivity, there is no primitivity because there can be singletons which only exist if they don't exist. Nothing here is 'decomposable'. Nothing of interest to economists in this field has any mathematical representation whatsoever save by arbitrary stipulation.
Under what circumstances, in the base case, can there be a functional relationship between the individual's preferences and the Society's preferences? The answer is, this would be the case if for any and every Choice there was 'common knowledge' of all objective determinants and there was an inerrant, ubiquitous, and costless mechanism to ensure actual Choice was based wholly on those objective determinants. But if this were the case, any sentient being would also magically know the 'Muth rational' or canonical answer to any subjective or 'inter-subjective' question. Thus waking up in the morning, I'd know I should walk into McDonalds where the staff would already know what food to give me. No words or money need change hands. I'd then walk down a strange road and enter the house of a stranger and go fix his toilet and leave. The houseowner would already know that I would do this and would have left the door open for me. Again, no words or money need change hand. As I walk towards the house of my next client, an Alien space-ship lands. They show me the Mochizuki proof of the abc theorem. Though I am a mere plumber and received no education of any kind, I silently indicate the place where the proof goes wrong (assuming it is wrong) before keeping my next appointment.
For 'revealed preference' to be the same as 'preference', it is this extreme type of common knowledge which must be fulfilled. But, to our present knowledge it can't be. There is no non-arbitrary way of solving concurrency problems or, by the ugly duckling theorem, removing bias or 'uncorrelated asymmetries' such that 'naturality' and categoricity prevails.
Consider Crusoe. The fact is, he sometimes ignores his own feelings or needs to do what is best for his Society- at least, this is how he would himself see it- at other times it is obvious that he is doing something reckless which, no doubt, is Crusoe's own free choice, but which we feel could be disastrous for the 'Social' Robinson. We can imagine David Niven playing Crusoe, taking the trouble to dress for dinner. The moment he neglects this custom of the fastidious English gentleman, we know he is likely to come to a sticky end.
If Crusoe chooses to admit another- e.g. Man Friday- we could say there is a 'dokimasiai' type eligibility criterion. This is a 'Social Choice Rule'. We may equally say that Man Friday has a set of criteria determining whether he stays, flees, or tries to eat Crusoe. There could be a lot of negotiation and then a contract or constitution established between Crusoe and Friday. But, this would be an 'incomplete' contract or a defeasible constitution. Once again there is no functional relationship between individual preferences and Social Choice.
Can there ever be a functional relationship between individual preferences and Social Choices? No. Don't be silly. It has never happened and it will never happen. This does not mean some people can't make a little money doing market research of various sorts or that political scientists can't study how Legislatures and Judiciaries actually work. In doing so, some assumptions may be made about what individuals, or 'representative agents' would consider to be in their interest. But, these imputed preferences may have no relation to actual views or behavior.
Social Choices are likely to be driven by Tardean 'mimetic' effects- i.e. trying to imitate what other, supposedly better, Societies are doing. But this is equally true of individual behavior. This means there is massive 'impredicativity' and thus the problem of computing the 'equilibrium' is in any exponential time class- i.e. even if it could be done, the Universe will have ended before it is done.
True, we have preferences and can make reasoned choices- except we change our minds when we see other, smarter, people are doing the opposite. It is safer to go with the crowd. This is rational because of the Condorcet Jury theorem and is also the reason Condorcet's voting paradox does not matter in the slightest. However, Condorcet and many of his fellow liberals were killed because the Parisian crowd went too far in one direction before going too far in another direction. There may be both danger and safety in numbers. We need to damp down 'cobweb effects' such that there are wide swings of the political pendulum. Hence the Anglo-Saxons went in for 'checks and balances' rather than trusting to a plebiscitary type of despotism. It appears the Muth Rational solution- i.e. the one everybody would pick if they had the right information and the right economic theory- for Social Choice is to limit Social Choice in a protocol bound manner. This does mean there will be hysteresis. But it also means that ergodicity too can prevail provided there is competition between polities or sub-units of a polity.
Apart from mimetic effects, what other features will characterize Social or individual choice?
1) Society will choose to do things which no individual wants for the same reason that individuals do things aren't beneficial to themselves but which they may regret not doing. This means 'Pareto' or 'Unanimity' is out of the window even in a Society composed of only one. Knightian Uncertainty- i.e. the fact that some future states of the world are unknown- means that there are no unique expected utility functions. All you can have is arbitrary stipulations.
2) Coordination mechanisms (e.g. voting) will be associated with 'discoordination' mechanisms such that there is hedging, arbitrage, transferable utility, and other 'income effects'. This means that ex poste outcomes change qualitatively as a result of the process whereby they are 'achieved' ex ante. Thus a guy who does what needs to be done to get re-elected President may be either much more powerful or much less powerful than previously if this is known to be the case. It's as though, the can of tomatoes you buy can suddenly shrink or expand because of 'irrelevant alternatives' which certainly did not feature in the 'mechanism' by which the outcome was realized. It is a different matter that under Knightian Uncertainty- i.e. where possible states of the world are not known (indeed, we don't really know that much about past states of the world either )- decision making will be highly non-linear in the manner of other 'co-evolved processes'. But this also means it will be relatively insulated from the fitness landscape even though, in the final analysis, changes in the fitness landscape are the 'determinants' of any evolving entity.
3) There is bound to be a disjunction between Policy implementation and the Principles which that Policy wishes to uphold. Policy will display economia- a suave, discretionary, accommodative type of management which is context dependent. Principles display akreibia- i.e. rigidity & universality. Aristotle says it is a mistake to seek for greater precision (akreibia) than the subject-matter will allow. Arrow's theorem and the vast literature it has given rise to, is precisely such a mistake.
In Law, the difference between akreibia and economia gives rise to a defensive de jure/ de facti disjunction. Judges decide the former on the basis of a rigid deontic logic. Juries decide the latter on the basis of an empirical 'reasonable man' test. In the Anglo Saxon tradition- which after World War 2 also prevailed on the Continent- the Judiciary can at times seem more powerful than the Executive though, no doubt, the Legislature can permit the latter to prevail.
As a result, the type of Social Choice mechanism we have actually chosen has spent a lot of time is restricting Social Choice by creating Hohfeldian incidents- e.g. immunities for individuals to act dictatorially- but this is a justiciable- i.e. defeasible- matter.
Dasgupta & Maskin write-
We shall assume that the objectives of a society are embodied in a certain social choice rule. A social choice rule (SCR) selects a set of feasible social states for each possible configuration of individual preferences and other characteristics.
Why not assume there is Command of God (CoG) which determines which State of Society will replace, at time t +1, what ever it is that obtains at time t? At least, that way you avoid the problem of impredicativity and infinite regress. As it is, the SCR must also be governing the way it is itself applied and the way the way it is itself applied etc.
It is a different matter to say, in rough and ready fashion, Societies change over time because people start wanting different things or start behaving differently or there are exogenous factors. But it simply isn't true that there is a 'Social Choice Rule' unless decisions really are made by a Giant Computer which decides every detail of everybody's life because that's the way everybody wants things to operate.
One interprets the choice set as the set of welfare optima.
So, they assume perfect information, no Knightian Uncertainty etc. But, in that case, why should there be a Society? Why should there be Language? Why is any coordinating device required? The Arrow Debreu price vector encodes all information including who will 'defect' or be shit at their job. On the basis of this information, which is available for free to everybody, Society would consist of 'windowless monads' synchronized in 'pre-established harmony'. This is Liebniz's Occasionalist Universe.
For example, given an Arrow social welfare function which embodies individual preferences in a social ordering,
this can only happen if individual preferences are completely independent of each other and there are no preferences for anything specifically 'Social'. But in that case, why would there be a Society? Alternatively, a bunch of people living far from each other but whose actions are coordinated by markets would be called a Society. But their kids and Moms and neighbors would be excluded from it. True, if they are aware of each other, they may form a 'community' or 'Society' but this is merely a manner of speaking.
then a natural social choice rule is derived by maximizing this social ordering over the feasible set.
this assumes that the 'maximizing' won't alter the feasible set. Yet, this could only be the case if the situation was knowingly and deliberately made sub-optimal in the previous time period. In other words, the Rule is only applied when it hasn't been applied. But Rules of that sort aint Rules at all. They are crazy shit you make up as you go along. There is no 'naturality' here. We don't know any individual or social ordering or what is or isn't the feasible set. This entire subject is about locating the unicorn of the mermaid who can maximize our sexual pleasure as a Society. Crusoe may have been interested in such things. Sadly, he had to settle for goats. Debasing such critturs is the base case for Social Choice Theory.
Alternatively, the Pareto rule
which can never be implemented because we don't know if someone else will pay more for something we have than we ourselves value it. But we also don't know whether we could get more value from it in some novel way.
is the social choice rule which selects all Pareto efficient states,
which are unknowable. I think it a good idea to take a job in a Bank. Unknown to me, this will cause my death at the hands of a Bank robber. The transaction wasn't actually Pareto efficient.
given individual preferences and the feasible set. These are two particular social choice rules which have received much attention, but our discussion will cover social choice rules in general. If the relevant characteristics of individual agents, such as preferences, happen to be publicly known,
even I don't know what my preferences ought to be. I shouldn't have bought that hotdog instead of a burger. I now have diarrhea.
then the social choice rule can be implemented trivially because the choice set itself is known. The problem of incentive compatibility arises precisely because these characteristics are not known by the planner a priori.
In which case, neither is the feasible set. In any case, incentive incompatibility only arises when at least some individuals produce goods and services in response to expected incentives or penalties. That's why, during a war, they may not matter at all. The penalty for not doing all you can for the War effort is defeat and enslavement.
The planner may attempt to learn characteristics directly by asking agents to reveal them. In general, however, if the agents realize how the information they reveal is to be used, they will have an incentive to misrepresent. Then the task of the planner in implementing the social choice rule is more difficult.
Planners can say they are implementing anything they like. Who the fuck will implement their plan once the money to do so runs out? Plans don't matter. Budgets do.
Obviously, he must use a planning mechanism of some kind, whose outcomes are possible social states. We shall assume that, when he devises the mechanism, the planner knows what social states are feasible,
in which case he knows the relevant preferences for incentive compatibility to obtain. But this means he is an omniscient god. By acting as an oracle or allocating resources in the private sector, he can gain a reputation for sagacity or else become as rich as fuck. Then, as in a Newcombe problem, his utterances will be heeded. People will do as he says even if he isn't a 'planner' and holds no official post.
so that he can ensure that the final outcome is feasible. The planner, however, relies on signals from the individual agents to help him implement the social choice rule.
Those signals are that individual agents will pay him money or kick his head in unless it turns out that the Social Choice rule results in their unjust enrichment.
It is assumed that each individual agent sends his own signal.
However such signals are not univocal. They are 'multiply realizable'. There is no functional relationship though there may be correlation and Granger causality. Noise has to be separated from Signal. But this is an arbitrary process.
The planner's mechanism is then a rule which specifies a social state for each list of signals sent by the individual agents.
Possible states of the world are not known. We don't even know all the details of our current state of the world.
It is assumed that each agent knows the precise form of the mechanism the planner is using. Then each agent realizes that he is involved in a game, because the outcome of the mechanism depends on the signals which he and all the other agents send to the planner. More precisely, this is a " game form ", in which there is a fixed set of strategies, consisting of signals to the planner, and in which the outcomes of these strategies are known to all "players ".
This isn't a 'game form'. It is stupid arbitrary shite.
This is a " game form " rather than a "game ",however, because the players' preferences over outcomes have not been specified.
The players won't play this stupid game. I can say 'there are two possible states of the world. One where you give me a beejay. In the other you give me a beejay and do the washing up.' The outcome of this game form is that I get kicked in the crotch.
It is then assumed that the players in this game form, who are the individuals in the society, reach some kind of equilibrium which depends on their true characteristics-in particular, their preferences.
Which nobody knows. I thought picking the hotdog over the burger was a good idea. It so wasn't.
The mechanism generates a particular social state given these equilibrium signals. Presumably, one wants this social state to be in the social choice set given the agents' true characteristics-i.e. to be something the planner might have chosen had he known these characteristics right from the start. The basic problem of the planner, then, is to devise a game form which always has at least one equilibrium, and whose possible outcomes in equilibrium all belong to the appropriate social choice set for the individuals' true characteristics.
Yet life and the economy are disequilibrium phenomena. Ex post ought not to equate with ex ante otherwise no 'discovery' is being done.
A mechanism (or game form) with this property is said to implement the social choice rule.
Neither is there a rule, nor is there a mechanism. On the other hand, there is a flying unicorn which implements the rainbow farting rule.
This task of implementing social choice rules is what has come to be called the problem of " incentive compatibility ", a term due to Hurwicz (1972) (and hinted at in Hurwicz (1960, p. 28)).
It is useful enough but what isn't 'incentive compatible' is Social Choice theory because it is a waste of fucking time. Auction theory is fine. Incomplete contract theory may be okay. Anything which ties in with Hohfeldian incidents can pay for itself in 'Law & Econ'. But this shite is shite because it is induction based on an incompossible base case.
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